Correspondence of John Wallis (1616-1703). Vol. 2, 1660 - September 1668

The Correspondence of John Wallis, Volume II PHILIP BEELEY CHRISTOPH J. SCRIBA, Editors OXFORD UNIVERSITY PRESS John...

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The Correspondence of John Wallis, Volume II

PHILIP BEELEY CHRISTOPH J. SCRIBA, Editors

OXFORD UNIVERSITY PRESS

John Wallis c.1668 engraving by William Faithorne (16167-91)

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The Correspondence of John Wallis Volume II (1660-September 1668) Editors PHILIP BEELEY CHRISTOPH J. SCRIBA With the Assistance of Uwe Mayer

OXPORD UNIVERSITY PRESS

OXPORD UNIVERSITY PRESS

Great Clarendon Street Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. Tt furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Honk Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toroanto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungray Italy Japan South Korea Poland Portugal Singapore Switzerland Thailand Turkey Ukraine Vietnam Published in the United States by Oxford University Press Inc., New York © Oxford University Pi-ess, 2005 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0-19-856601-8 1 3 5 7 9 1 0 8 6 4 2 Typeset by the Editors Printed in Great Britain on acid-free paper by Bukkes Ltd., King's Hynn, Norfolk

The Correspondence of John Wallis Volume II (1660 - September 1668)

PHILIP BEELEY CHRISTOPH J. SCRIBA Editors With the Assistance of Uwe Mayer

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For Inge Scriba

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PREFACE

The second volume of the Correspondence of John Wallis (1616-1703) covers a decisive period of political and scientific reorganization in England following the eleven years of the Interregnum. It begins just a matter of weeks before the triumphal return of Charles II from the Netherlands and his accession to the throne on 29 May 1660 and sees the emergence of the Royal Society, soon to become the leading organization of its kind in Europe, from two main scientific circles which had flourished in London and Oxford during the revolutionary years. Wallis, who had been appointed Savilian professor of geometry at the University of Oxford largely in recognition of his services to parliament as a decipherer during the Civil Wars was in this period establishing himself as one of the central figures in the growth of modern science in England. Fittingly, his name stands alongside those of Robert Boyle, Kenelm Digby, Robert Moray, John Wilkins, and Christopher Wren as one of the original members of council, appointed on 15 July 1662, when the first charter 'for the incorporation of the society under the title of the Royal Society' was passed. The years encompassed by the present volume witness the Savilian professor consolidate his reputation both in London and in Oxford. At the Royal Society he was a close associate of its secretary Henry Oldenburg, with whom he corresponded regularly and in whose journal, the The Philosophical Transactions of the Royal Society, he published articles on a wide range of themes from the fields of mathematics and the physical sciences. Within the walls of the University of Oxford he continued to fulfil his professorial duties and at the same time succeeded in acquiring considerable authority and influence through the office of keeper of the archives, to which he had been elected under somewhat questionable circumstances in 1657/8. Although not a trained lawyer, Wallis displayed considerable legal acumen and an unswayable commitment to defending the University's rights in the face of numerous attempts to infringe these on the part of city authorities. The volume contains a total of 232 existent or identified letters exchanged between Wallis and over 30 correspondents at home and abroad Vll

between February 1660 and September 1668. More than sixty of these letters have never appeared in print before, while many of the rest are only to be found in publications which are rare or not widely accessible. Eighteen of the letters printed are part of the former Macclesfield Collection, now in the possession of Cambridge University Library. Although these letters have been previously available in S. J. Rigaud's Correspondence of Scientific Men (1841), they are published here completely in the form of a critical edition for the first time. As in the first volume, additional letters and material have been included where these form an essential part of exchanges between Wallis and other scholars or where they are able to throw light on correspondence for which no other documentary evidence has survived. In this way it has been aimed to provide as complete a picture as possible of the literary activity of the Savilian professor during the nine years covered. Work on the edition has been made possible through funding from the Deutsche Forschungsgemeinschaft. The editors should like to express their sincere thanks to this organization for its continued generosity. An enterprise such as the Wallis correspondence project can only succeed through close international cooperation of scholars working in the field. The editors are therefore once more glad to record their debt to numerous friends and colleagues who through advice or assistance have contributed significantly to realizing the volume in the present form. The editors are once again grateful to Robert Hatch for providing information on manuscript sources in France and the United States. David Cram and Jackie Stedall are to be thanked not only for sharing their profound knowledge of Wallis and his contemporaries but also for their willingness to provide help in almost every conceivable way. Further, the editors should like to express their gratitude to Jordan Avramov for useful discussions on questions concerning Henry Oldenburg and Robert Moray. They are grateful also to Jason Rampelt for kindly allowing them to read parts of his as yet unfinished Cambridge dissertation on the life and work of Wallis. Special thanks go to Mordechai Feingold who read the manuscript at an early stage of completion and who as always provided many helpful suggestions on commentaries and general presentation. On the technical side the editors have once again had the good fortune to be able to fall back on the sound advice of Menso Folkerts and his colleagues in Munich. They are particularly grateful to Albert Krayer for giving vital assistance on questions concerning the ET^X system used for editing. vin

The editors have continued to enjoy the generous support of the Institute for History of Science, Mathematics, and Technology at the University of Hamburg. They should like to express their deep gratitude in particular to Karin Reich in her capacity as director of the Institute for ensuring that the correspondence project continues to nourish. The University of Hamburg has generously provided rooms to accommodate the project, and the Department of Mathematics has made sure that those employed on the project receive all the necessary technical assistance. Without this help much of the work on the edition would not have been possible. A personal note of thanks goes to Siegmund Probst who has continued to share generously in his knowledge of seventeenth-century mathematics. The editors are also grateful for the dedication shown by Ines Harrie and Oliver Leistert while working on the project in the course of their studies. The editors' greatest debt goes to Uwe Mayer who has taken on a substantial part of the workload over the last years. His editorial efforts have been enormous. Without his energy and commitment the second volume could not have been realized in the short length of time it has taken. Staff at numerous libraries and archives have provided often invaluable help in preparing the present volume. The editors should like to express their particular gratitude to Adam Perkins of Cambridge University Library, Rupert Baker of the Library of the Royal Society, Frances Harris of the British Library, William Hodges of Duke Humfrey, Bodleian Library, and Peter Rau and staff of the Staatsbibliothek Hamburg. Once again, the editors are especially indebted to Simon Bailey, the Keeper of the Archives of the University of Oxford, and to Alice B. Millea, the Assistant Keeper. The editors are grateful to the following persons and institutions for granting permission to publish copyright material held in their possession: The Syndics of Cambridge University Library; the Librarian of the Bibliotheek der Rijksuniversiteit, Leiden; the British Library Board; the Librarian of the Bodleian Library, Oxford; the Keeper of the Archives, University of Oxford; the Koninklijke Bibliotheek, The Hague; the Bibliotheque Nationale de France; the Royal Society; the National Archives, Kew; and the Governors and Guardians of Marsh's Library, Dublin. Quotations from the Devonshire Manuscripts, Chatsworth House, are published by permission of the Chatsworth Settlement Trustees. IX

Finally, the editors should like to thank editorial staff at Oxford University Press for their continued help in realizing the edition, their care in proof reading, and their patience in the light of often considerable delays in the submission of material. Philip Beeley Christoph J. Scriba Miinster (Westf.) and Hamburg, June 2004

x

CONTENTS

Introduction Restoration politics and scientific practice Wallis, Oldenburg, and the Royal Society Correspondence with Boyle Mathematical correspondence Correspondence with Collins Wallis, Gregory, and Huygens Cryptanalysis University affairs Personal affairs Theological affairs Editorial principles and abbreviations

xix XX

xxii XXV

xxvi xxix XXX

xxxi xxxi xxxiii xxxiii xxxv

Correspondence 1. HUYGENS to C ARC AVI, [16] /26 February [1659] / 1660 . . . . 2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660 3. HUYGENS to WALLIS, [21]/31 March [1659]/1660 4. CARCAVI to WALLIS, June? 1660 5. CARCAVI to HUYGENS, [15]/25 June 1660 6. HUYGENS to WALLIS, [5]/15 July 1660 7. WALLIS to DIGBY, 24 August/[3 September] 1660 8. WALLIS to HUYGENS, 31 August/[10 September] 1660 . . . . 9. FELL to YATE, 5/[15] February [1660/1 ?] 10. WALLIS to DILLINGHAM, 16/[26] March 1660/1 11. WALLIS: Humble Petition to Charles II, [March? 1661] . . . 12. DILLINGHAM to WALLIS, 1/[11] April 1661 13. COOPER to WALLIS, 23 April/ [3 May] 1661 14. [HOBBES]: La duplication du cube, [June? 1661] 15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661 16. FRENICLE to WALLIS, [October/November ? 1661] XI

1 1 7 11 13 13 20 22 27 30 32 35 36 37 38 41 45

Contents 17. WALLIS to FRENICLE, [November/December ? 1661] . . . . 18. FRENICLE to WALLIS, [[10]/20 December 1661] 19. WALLIS to BOYLE, 30 December 1661/[9 January 1662] . . 20. BOYLE to WALLIS, 4 or 5/[14 or 15] January 1661/[1662] . 21. WALLIS to BOYLE, 20 February/[2 March] 1661/2 22. BOYLE to WALLIS, 26 February/[8 March] 1661/[1662] . . . 23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) 24. [WALLIS:] Note on letter to Boyle, 11/[21] July 1670 . . . . 25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) 26. BOYLE to WALLIS, 5/[15] April 1662 27. WALLIS to MORAY, 7/[17] April 1662 28. MORAY to WALLIS, April/May 1662 29. WALLIS to MORAY, 6/[16] May 1662 30. MORAY? ET AL. to WALLIS, 8/[18] May 1662 31. WALLIS to TITUS, 12/[22] June 1662 32. WHARTON to WALLIS, 30 September/[10 October] 1662 . . 33. BROWNE to CHYLINSKI, 3/[13] October 1662 34. CHYLINSKI to BROWNE, 4/[14] October 1662 35. CHYLINSKI to WALLIS, 4/[14] October 1662 36. WALLIS to DILLINGHAM, 21/[31] October 1662 37. DILLINGHAM to WALLIS, 27 October/[6 November] 1662 . . 38. HOBBES to WALLIS, end of 1662 39. WALLIS to HEVELIUS, 9/[19] February 1662/3 40. WALLIS to BOYLE, 27 March/[6 April] 1663 41. WALLIS to HEVELIUS, 30 March/[9 April] 1663 42. BOYLE to WALLIS, August/September 1663 43. WALLIS to BOYLE, 10/[20] September 1663 44. WALLIS to BOYLE, 10/[20] September 1663, enclosure (i) . . 45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) . 46. WALLIS for the ROYAL SOCIETY, 24 September/[4 October] 1663 47. HEVELIUS to WALLIS, [25 December 1663]/4 January 1664 48. WALLIS to BROUNCKER?, ? March 1664 (i) 49. WALLIS to BROUNCKER?, ? March 1664 (ii) 50. WALLIS to HEVELIUS, 5/[15] April 1664 51. WALLIS to OLDENBURG, 6/[16] April 1664 52. WALLIS to OLDENBURG, 7/[17] April 1664 53. OLDENBURG to WALLIS, 21 April/[l May] 1664 54. WALLIS to OLDENBURG, 30 April/[10 May] 1664 Xll

45 46 47 50 50 51 51 60 63 68 69 71 71 72 73 73 74 75 76 77 77 79 79 80 82 86 87 88 91

99 100 102 102 103 106 110 Ill Ill

Contents 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92.

OLDENBURG to WALLIS, 4/[14] May 1664 WALLIS to OLDENBURG, 7/[17] May 1664 WALLIS to OLDENBURG, 14/[24] May 1664 WALLIS to OLDENBURG?, 16/[26] May 1664 WALLIS to OLDENBURG, 25 May/[4 June] 1664 WALLIS to BLANDFORD, 13/[23] June 1664 JENKINS to WALLIS, 15/[25] June 1664 WALLIS to BLANDFORD, 15/[25] ? June 1664 BLANDFORD to WALLIS, 16/[26] June 1664 WALLIS to BLANDFORD, 18/[28] June 1664 WALLIS to BLANDFORD, 20/[30] June 1664 JENKINS to WALLIS, 21 June/[l July] 1664 BLANDFORD to WALLIS, 21 June/[l July] 1664 WALLIS to BLANDFORD, 23 June/[3 July] 1664 BLANDFORD to WALLIS, 27 June/[7 July] 1664 WALLIS to BLANDFORD, 28 June/[8 July] 1664 JENKINS to WALLIS, early-mid July 1664 WALLIS to BLANDFORD, 18/[28] July 1664 HEVELIUS to WALLIS, c.31 August/[10 September] 1664 . . CONSTANTUN HuYGENS for WALLIS, 7/[17] September 1664 WALLIS to OLDENBURG, 21 September/]! October] 1664 . . OLDENBURG to WALLIS, 29 September/[9 October] 1664 . . CONSTANTUN HUYGENS for WALLIS, 6/[16] October 1664 . OLDENBURG to WALLIS, 13/[23] October 1664 OLDENBURG to WALLIS, 21/[31] October 1664 WALLIS to OLDENBURG, 29 October/[8 November] 1664 . . WALLIS to CHYLINSKI, c.!7/[27] November 1664 CHYLINSKI to WALLIS, 19/[29] November 1664 WALLIS to BURSCOUGH, ? November 1664 BURSCOUGH to WALLIS, 24 November/[4 December] 1664 . OLDENBURG to WALLIS, mid-December 1664 WALLIS to OLDENBURG?, 24 December 1664/[3 January 1665] WALLIS to OLDENBURG, 21/[31] January 1664/5 H. L. to WALLIS, 10/[20] April 1665 BOYLE to WALLIS, early 1665 WALLIS to BOYLE, 24 April/[4 May] 1665 WALLIS to BOYLE, 29 April/[9 May] 1665 WALLIS to OLDENBURG?, 8/[18] May 1665 Xlll

114 114 118 137 137 139 142 144 144 145 146 147 148 149 154 155 157 157 158 159 160 162 163 164 164 164 166 166 168 168 170 171 174 179 182 182 182 184

Contents 93. MATTHEW WREN to WALLIS, 30 May/[9 June] 1665 . . . . 185 94. WALLIS to MATTHEW WREN, 4/[14] June 1665 186 95. OLDENBURG to WALLIS, 28 September/[8 October] 1665 . . 186 96. OLDENBURG to WALLIS, 26 December 1665/[5 January 1666] 187 97. OLDENBURG to WALLIS, 30 December 1665/[9 January 1666] 187 98. COLLINS to WALLIS, 2/[12] January 1665/[1666] 188 99. WALLIS to COLLINS, January/February 1665/6 191 100. COLLINS to WALLIS, 28 February/[10 March] [1665/6] . . 191 101. OLDENBURG to WALLIS, early 1666 195 102. WALLIS to OLDENBURG, 3/[13] April 1666 195 103. WALLIS to OLDENBURG, 23 April/[3 May] 1666 197 104. WALLIS to BOYLE, 25 April/[5 May] 1666 200 105. OLDENBURG to WALLIS, 5/[15] May 1666 223 106. WALLIS to OLDENBURG, 7/[17] May 1666 223 107. WALLIS to OLDENBURG, 12/[22] May 1666 226 108. OLDENBURG to WALLIS, 17/[27] May 1666 231 109. WALLIS to OLDENBURG, 19/[29] May 1666 232 110. WALLIS to OLDENBURG, 24 May/[3 June] 1666 235 111. OLDENBURG to WALLIS, 29 May/[8 June] 1666 238 112. OLDENBURG to WALLIS, 31 May/[10 June] 1666 240 113. WALLIS to OLDENBURG, 2/[12] June 1666 240 114. WALLIS to OLDENBURG, 8/[18] June 1666 246 115. OLDENBURG: Objections against Dr Wallis's Hypothesis of Tides, [June? 1666] 250 116. WALLIS to OLDENBURG, 18/[28] July 1666 251 117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 . . . . 263 118. OLDENBURG to WALLIS, 26 July/[5 August] 1666 270 119. OLDENBURG to WALLIS, 31 July/[10 August] 1666 . . . . 270 120. WALLIS to OLDENBURG, 2/[12] August 1666 271 121. COLLINS to WALLIS, 2/[12] August [1666] 275 122. OLDENBURG to WALLIS, 4/[14] August 1666 278 123. WALLIS to COLLINS, 7/[17] August 1666 278 124. WALLIS to OLDENBURG, 11/[21] August 1666 281 125. OLDENBURG to WALLIS, mid-August 1666 283 126. WALLIS to OLDENBURG, 18/[28] August 1666 284 127. WALLIS to [OLDENBURG?], 23 October/[2 November] 1666 287 128. FELL to WALLIS, October/November 1666 (i) 288 129. FELL to WALLIS, October/November 1666 (ii) 288 130. FELL to WALLIS, 18/[28] November [1666] 288 XIV

Contents 131. 132. 133. 134. 135. 136. 137. 138. 139.

FELL to WALLIS, 5/[15] December [1666 ?] OLDENBURG to WALLIS, c.8/[18] January 1666/7 OLDENBURG to WALLIS, 15/[25] January 1666/7 WALLIS to OLDENBURG, 19/[29] January 1666/7 OLDENBURG to WALLIS, 24 January/[3 February] 1666/7 . WALLIS to OLDENBURG, 31 January/[10 February] 1666/7 COLLINS to WALLIS, [January? 1666/7] WALLIS to COLLINS, January? 1666/7 COLLINS to WALLIS, 2/[12] February 1666/7

290 291 291 291 295 296 298 301 301

140. COLLINS to WALLIS, 2/[12] February 1666/7, enclosure . . 307

141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166.

WALLIS to COLLINS, 5/[15] February 1666/7 COLLINS to WALLIS, [c. 10/[20] February 1666/7] WALLIS to OLDENBURG, 12/[22] February 1666/7 FELL to WALLIS, 19 February/[l March] [1666/7?] . . . . OLDENBURG to WALLIS, 19/[29] March 1666/7 WALLIS to OLDENBURG, 21/[31] March 1666/7 WALLIS to CLENDON, 5/[15] May 1667 OLDENBURG to BOYLE, 17/[27] September 1667 WALLIS to JENKINS, 26 October/[5 November] 1667 . . . . JENKINS to WALLIS, 26 October/[5 November] 1667 . . . . WALLIS to JENKINS, 27 October/[6 November] 1667 . . . . WALLIS to JENKINS, 31 October/[10 November] 1667 . . . JENKINS to WALLIS, 31 October/[10 November] 1667 . . . WALLIS to HOPKINS, 1/[11] November 1667 WALLIS to JENKINS, 2/[12] November 1667 WALLIS to JENKINS, 2/[12] November 1667, enclosure . . . WALLIS: Notes on Martivell's case, [1667 ?] JENKINS to WALLIS, 5/[15] November [1667] WALLIS to JENKINS, 11/[21] November 1667 WALLIS to OLDENBURG, 16/[26] November 1667 JENKINS to WALLIS, 16/[26] November 1667 OLDENBURG to WALLIS, 26 November/[6 December] 1667 WALLIS to OLDENBURG, 30 November/[10 December] 1667 OLDENBURG to WALLIS, 10/[20] December 1667 WALLIS to OLDENBURG, 13/[23] December 1667 OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] 167. JENKINS to [WALLIS?], [January 1667/8?] 168. WALLIS to JENKINS, 21/[31] January 1667/8 XV

309 311 318 321 322 322 325 325 330 332 332 334 337 339 339 346 350 352 353 354 359 360 361 365 368 375 378 379

Contents

169. WALLIS to JENKINS 21/[31] January 1667/8, enclosure . . 380 170. WALLIS to JENKINS, 28 January/[7 February] 1667/8 . . . 382

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure 384 172. BROUNCKER: Solution to Dulaurens's Problem, [December 1667/January 1667/8]

394

173. 174. 175. 176.

395 396 400 400

JENKINS to WALLIS, 30 January/[9 February] 1667/[1668] WALLIS to OLDENBURG, I/[11] February 1667/[1668] . . . OLDENBURG to WALLIS, 4/[14] February 1667/8 WALLIS to OLDENBURG, 8/[18] February 1667/8

177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February

178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199.

1667/8 WALLIS to JENKINS, 11/[21] February 1667/8 OLDENBURG to WALLIS, 11/[21] February 1667/8 JENKINS to WALLIS, 13/[23] February 1667/[1668] WALLIS to COLLINS, early 1668? WALLIS to COLLINS, 15/[25] February 1667/8 WALLIS to LEOTAUD, 17/[27] February 1667/[1668] . . . . OLDENBURG to WALLIS, 25 February/[6 March] 1667/8 . COLLINS to WALLIS, 25 February/[6 March] 1667/8 . . . . WALLIS to COLLINS, 27 February/[8 March] 1667/8 . . . . WALLIS to OLDENBURG, 29 February/[10 March] ? 1667/8 MORAY? to WALLIS, ? February 1667/8 WALLIS to MORAY?, ? February 1667/8 WALLIS to OLDENBURG, 7/[17] March 1667/8 (i) WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) OLDENBURG to WALLIS, 10/[20] March 1667/8 WALLIS to OLDENBURG, 17/[27] March 1667/8 (i) WALLIS to OLDENBURG, 17/[27] March 1667/8 (ii) . . . . OLDENBURG to WALLIS, 26 March/[5 April] 1668 WALLIS to OLDENBURG, 30 March/[9 April] 1668 BROUNCKER to OLDENBURG, March? 1667/8 [WALLIS?] to OLDENBURG, March/April? 1668 PHILIPS to WALLIS, 6/[16] April 1668

402 405 406 408 409 409 412 428 429 429 432 432 432 434 435 440 440 441 446 446 454 461 462

200. CONSTANTUN HuYGENS to WALLIS, [21 June]/I July 1668 467

201. 202. 203. 204.

OLDENBURG to WALLIS, 30 June/[10 July] 1668 WALLIS to COLLINS, early July 1668 WALLIS to OLDENBURG, 2/[12] July 1668 (i) WALLIS to OLDENBURG, 2/[12] July 1668 (ii) xvi

469 469 470 471

Contents 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242.

WALLIS to OLDENBURG, 4/[14] July 1668 OLDENBURG to WALLIS, 4/[14] ? July 1668 WALLIS to OLDENBURG, 6/[16] July 1668 WALLIS to OLDENBURG, 8/[18] July 1668 WALLIS to OLDENBURG, 11/[21] July 1668 COLLINS to BRERETON, 11/[21] July 1668 OLDENBURG to WALLIS, 13/[23] July 1668 WILKINS to WALLIS, 13/[23] July 1668 WALLIS to MORAY, 14/J24] July 1668 COLLINS to WALLIS, 14/[24] July 1668 WALLIS to WILKINS, 16/[26] July 1668 WALLIS to OLDENBURG, 16/[26] July 1668 WALLIS to OLDENBURG, 18/[28] July 1668 OLDENBURG to WALLIS, 18/[28] July 1668 COLLINS to PELL, 18/[28] July 1668 COLLINS to PELL, 18/[28] July 1668, enclosure WALLIS to OLDENBURG, 20/[30] July 1668 WALLIS to COLLINS, 21/[31] July 1668 PELL to BRANCKER, 21/[31] July 1668 BRANCKER to PELL, 25 July/[4 August] 1668 WALLIS to BRAMSTON, 27 July/[6 August] 1668 WALLIS and CHRISTOPHER WREN to the ESSEX COMMISSIONERS, 27 July/[6 August] 1668 WALLIS to BROUNCKER, July/August 1668 OLDENBURG to WALLIS, July/August 1668 WALLIS to OLDENBURG, 3/[13] August 1668 WALLIS to BROUNCKER, 3/[13] August 1668 WALLIS to BROUNCKER, 3/[13] August 1668, enclosure . . BROUNCKER to WALLIS, 3/[13] August 1668 WALLIS to BROUNCKER, 5/[15] August 1668 WALLIS to BROUNCKER, 6/[16] August 1668 WALLIS to BROUNCKER, 8/[18] August 1668 WALLIS to BROUNCKER, 8/[18] August 1668, enclosure . . COLLINS to WALLIS, 15/[25] August 1668 WALLIS to COLLINS, 25 August/[4 September] 1668 . . . . WALLIS to OLDENBURG, 25 August/[4 September] 1668 . OLDENBURG to WALLIS, 28 August/[7 September] 1668 . PELL to COLLINS, 29 August/[8 September] 1668 WALLIS to HUYGENS, 31 August/[10 September] 1668 . . XVll

477 479 479 481 489 491 492 492 494 495 496 497 498 525 525 527 530 531 533 534 536 538 541 542 542 545 545 554 554 558 559 559 561 562 564 565 566 568

Contents 243. 244. 245. 246. 247. 248. 249. 250. 251. 252.

WALLIS to BROUNCKER, [August ? 1668] OLDENBURG to WALLIS, I/[II] September 1668 WALLIS to OLDENBURG, 3/[13] September 1668 PELL to COLLINS, 6/[16] September 1668 WALLIS to COLLINS, 8/[18] September 1668 WALLIS to COLLINS, 10/[20] September 1668 WALLIS to LALOUBERE, c.lO/[20] September 1668 COLLINS to WALLIS, 22 September/[2 October] 1668 . . . WALLIS to COLLINS, 26 September/[6 October] 1668 . . . WALLIS to COLLINS, 26 September/[6 October] 1668, enclosure 253. GREGORY to WALLIS, September/October 1668 254. OLDENBURG to WALLIS, September/early October 1668 .

573 592 592 594 596 601 603 603 604 610 614 615

Biographies of correspondents

617

List of manuscripts

629

Bibliography

631

List of letters

651

Index: persons and subjects

661

XVlll

INTRODUCTION

The restoration of the monarchy in England in May 1660, following almost a year of political instability and uncertainty after Richard Cromwell's (1626-1712) resignation, meant not only the return to some form of constitutional order, but also provided conditions in which the creation of a central scientific organization at last became possible. Although a group of some of the key figures had begun to meet regularly in London in the mid-1640s, and later more formally also in Oxford, it was only now that the efforts of these two centres of activity were able to be merged into one fully fledged institution dedicated to the promotion of mathematical and experimental knowledge. This transformation took place remarkably quickly. While the official beginning of the Royal Society coincides with the granting of the first royal charter in July 1662, its predecessor in all but name began to take shape in London already towards the end of 1660. At regular meetings at Gresham College, which at this time typically took place after the lecture of the astronomy professor Christopher Wren (1632-1723), concrete plans were drawn up by William Brouncker (16207-84), Paul Neile (1613-86), William Petty (1623-87) and others to create a college whose aim would be to promote learning in the mathematical and experimental sciences. Particularly prominent from the outset in these moves to provide an institutional focal point for scientific activity in England and Scotland was Robert Moray (1608-73), who had spent much of the Interregnum in exile and was able to combine his interest in experimental philosophy with influence at the royal court. It was he who promoted the constitution of the society into a corporation. But others, too, played a decisive role in the creation of what was soon to become the most important institution of its kind in Europe. Many of these were men who had pursued their scientific endeavours in the intervening years almost entirely in Oxford and had been active in the philosophical association which had formed there around the pivotal figures of John Wilkins (1614-72) and Robert Boyle (1627-91). Wallis was among the most notable of these together xix

Introduction with Jonathan Goddard (1616-75), Seth Ward (1617-89), and Christopher Wren.

Restoration politics and scientific practice The Restoration was in many respects, at least in the early days, a time of reconciliation; only in this way could the political and religious divides which had driven the country into civil war be overcome. However, goodwill on the royalist side extended usually only to those who had not aligned themselves too openly with the Independents during the revolutionary years. Despite having widely different political leanings, many of the most accomplished mathematicians and experimental philosophers who gathered in the future Royal Society from 1660 onwards had made their careers during the Commonwealth and the Protectorate. Wallis had been appointed Savilian professor of geometry in place of Peter Turner (1586-1652), just a matter of weeks after the proclamation of the republic in 1649. There he had joined Wilkins who had replaced John Pitt (1584?c.1648) as warden of Wadham College, while his colleague, the Savilian professor of astronomy, Seth Ward moved into the place forceably vacated by John Greaves (1602-52). Another of Wallis's long-standing associates, the physician Jonathan Goddard was made warden of Merton College in place of Nathaniel Brent (1573?-1652) in 1651. Fates at the Restoration were largely decided by political sympathies of old. Wallis came out of the Cromwellian period largely unscathed, whereas Goddard and Wilkins, who had in the meantime been appointed master of Trinity College, Cambridge, witnessed a reversal of earlier events and were ejected from their posts. In effect, both men were punished on account of having sided too openly with Cromwell—although Wilkins, with the help of influential friends at court, was soon thereafter able to embark upon a successful career in the church. Undoubtedly, one of the main reasons for Wallis's survival at the University was his moderation in both political and religious affairs. Not only had he vigorously opposed the execution of Charles I in 1649, but also, as he was later careful to point out in his autobiography, he had consistently followed a Presbyterian path throughout the Cromwellian period.1 But this was not all. At least two further factors counted heavily in his favour: his renown as Europe's greatest living decipherer and his proven skill in serving the interests of the University of Oxford as keeper 1

ScaiBA, 'Autobiography of John Wallis', 35, 42-3. XX

Introduction of the archives. In these areas of activity Wallis had succeeded in making himself absolutely indispensable. Both of Wallis's posts at the University, the Savilian professorship in geometry and the office of Gustos archivorum, were confirmed at the Restoration and his efforts in each of them during the period covered by the present volume played a part in extending still further his already considerable scholarly reputation. On the one side he continued to produce important contributions to the mathematical and physical sciences, particularly through the medium of papers or letters read at meetings of the Royal Society and articles published in Henry Oldenburg's (1618-77) Philosophical Transactions. On the other side he was actively engaged by successive vice-chancellors in defending University rights in the face of repeated moves on the part of the city of Oxford to have the University's privileges removed or at least significantly reduced. In his publication practice Wallis made full use of the modern means available for the communication of scientific results. Some of the papers presented at meetings of the Royal Society were later incorporated into his monumental Mechanica sive de motu tractatus geometricus (1670-1) which served to generate new interest in a mathematical approach to physical problems and which had immense impact on the scientific world. His Treatise of Algebra (1685), which like the Mechanica was printed in London, also contains pieces largely written or even completed before the end of 1668. Among these is his Cono-Cuneus, a tract on the shipwright's circular wedge, which he wrote at the request of Robert Moray, in 1662 (No. 27). In many ways Cono-Cuneus reflected the interests and the utilitarian focus of the Royal Society. Not only did it arise out of a whole series of discussions on the nature of various solids, but also the demonstrations in piano which Wallis produced of sections of the double-coned wedge by means of analytical geometry were conceived as being potentially useful for the construction of ships. Other works appeared as publications in their own right. In continuation of the dispute with Thomas Hobbes (1588-1679), which in the long term did justice to neither of the two men, Wallis published Hobbius Heauton-timorumenos (1662) by way of reply to the philosopher's Examinatio et emendatio mathematicae hodiernae (1660) and his more recent Dialogus physicus de natura aeris (1661). In the Dialogus physicus Hobbes had attacked not only the statutes and the scientific practice of the Royal Society, but also Boyle's New Experiments Physico-Mechanicall (1660) and not least for this reason Wallis chose the form of an open letxxi

Introduction ter to Boyle for the publication of his reply (No. 21). As an appendix to the Dialogus physicus Hobbes ambitiously published his solution to one of the three great classical mathematical problems, that of constructing a cube double in volume to a given cube. This solution had already been published by him anonymously in the summer of 1661 as a printed paper purportedly originating from France (No. 14). It was subsequently refuted by Laurence Rooke (1622-62), professor of geometry at Gresham College, as well as by Wallis (No. 15), and Hobbes used the opportunity of Dialogus physicus not only to reprint his solution but also to reply to the objections raised by Rooke and Wallis. Not to be outdone by Wallis's latest attempt to humiliate him, Hobbes replied to Hobbius Heauton-timorumenos by means of an epistolary tract entitled Mr Hobbes considered in his loyalty, religion, reputation, and manners (No. 38). A more substantial publication, which however did not appear separately until 1684, reflects the close cooperation which existed between Wallis and Oldenburg. In a series of papers presented to the Royal Society, the Savilian professor of geometry put forward a new hypothesis to explain apparent irregularities of the tides (Nos. 104, 106, 109, 111, 11314). Galileo (1564-1642) had attributed this phenomenon to the different velocities of different parts of the earth. The new hypothesis consisted in assuming the earth and the moon to be a single body with a common centre of gravity. The variation of the tides from spring to neap was correspondingly to be accounted for by movements in the centre of gravity. In perfecting his hypothesis, Wallis was able to profit from numerous discussions of parts of it at meetings of the Royal Society as well as from observations of the tides which Oldenburg enlisted from some of his correspondents living near to the sea (Nos. 108, 115-16, 134, 136, 146, 179, 191, 199).

Wallis, Oldenburg, and the Royal Society Like Boyle, who remained in Oxford until 1668, Wallis relied on correspondence not only to keep abreast of developments in the scientific world, but also to play an active part in discussions taking place a considerable distance away in London. Oldenburg was the decisive figure in this respect. In return for communicating Wallis's papers to the Royal Society, and for acting as an intermediary in subsequent discussions, Oldenburg supplied the Savilian professor with news from the Republic of Letters and occasionally sent him material for perusal and criticism. He also set up the xxu

Introduction direct correspondence between Wallis and the Liege mathematician Rene Frangois de Sluse (1622-85). Although regrettably only a few of Oldenburg's letters have survived, the correspondence between the two men is by far the most extensive individual collection in the present volume. Any disadvantage which Wallis might have felt by being in Oxford rather than London vanished in the second half of 1665 on account of the outbreak of what became known as the Great Plague. With the exception of Brouncker and Oldenburg, most leading members of the Royal Society normally resident in London sought refuge in the university city where they were soon joined by the king and parliament. By September, there was almost a return to the old state of affairs when a group including Moray, Neile, Petty, and Wallis decided to meet regularly in the rooms of Robert Boyle. In view of increasing disruption to normal life in the capital, Oldenburg also placed the publication of Philosophical Transactions temporarily in the hands of Boyle, Moray, and Wallis in Oxford (No. 96). At the same time, the secretary of the Royal Society expressed the hope that all this scientific activity might have some influence on academic life, suggesting in his letter to Boyle of 5 October 1665 that the Oxonians might come to prefer 'that solidity of knowledge' associated with the Society rather than their usual 'scholasticall contentions'.2 Apart from this, Wallis usually went at least once a year to London and as a matter of course used the opportunity of being there to attend meetings of the Royal Society. Occasionally, his stays in the capital occured en route to and from his home town of Ashford. However, the most common reason for his travelling to London was to deal with legal proceedings involving the University, often being required to stay there for many weeks on end until a satisfactory solution had been reached. Sometimes both were combined. During one visit to Kent in September and October 1666, following the death of his brother Henry, he devoted time to making observations on the tides which he subsequently communicated to the Royal Society (No. 127). On arriving in London on his way back to Oxford he was called upon to look after the University's interests in the latest legal dispute (Nos. 128-30). During his stay in the capital, a large part of which had been devastated by the Great Fire in September, he was able to be present at least at one meeting of the Royal Society and to give members a personal account of the observations he had sent earlier based on his new hypothesis. 2

OLDENBURG-BOYLE 5/[15].X.1665, The Correspondence of Henry Oldenburg, ed. A. R. Hall and M. B. Hall, II, 543-5, 545. XXlll

Introduction But Wallis's being in Oxford was at times advantageous for the Royal Society, too. When Henri Justel (1620-93), who was assisting in the publication of Peter of Blois's (c.H35-c.l204) Opera omnia in Paris, requested Oldenburg's help in obtaining transcripts of the French author's manuscripts contained in English collections, the secretary engaged Wallis to investigate the holdings of libraries in Oxford and have copies made of those which were required (Nos. 102). Wallis in fact transcribed some of the shorter pieces himself (Nos. 103, 110, 134). This was in many ways typical of his sense of scientific cooperation and reflects at the same time his general attitude to the importance of the accurate transmission of texts, which in later years finds its expression in numerous exemplary editions of classical works such as Ptolemy's Harmonics. In precisely this spirit, Wallis conveyed the wishes of the Royal Society to the oriental scholar and Bodley librarian Thomas Hyde (1636-1703) that a translation of Ulug Beg's (1394-1449) catalogue of the fixed stars be made for Johannes Hevelius (1611-87). In order to minimize any possibility of error in the numerical tables, the Savilian professor made a transcript of the corresponding part of the translation and collated this against the Persian originals. He subsequently enclosed this transcript in a letter to the Danzig astronomer (Nos. 50-1). This continued a pattern of cooperation which by then was already well established. A number of years beforehand, Wallis had sent to Hevelius the account of the transit of Venus which the English astronomer, Jeremiah Horrox (16177-41), had written on the basis of observations he had made in November 1639. Hevelius subsequently published Horrox's Venus in Sole visa as an appendix to his own account of the transit of Mercury, Mercurius in Sole visus. Appropriately, he sent copies of this book, printed in Danzig in 1662, both to Wallis and to Seth Ward, who had only recently been succeeded as Savilian professor of astronomy by Christopher Wren (No. 39). Wallis, who had been a contemporary of Horrox at Emmanuel College, was in 1664 assigned the task by the Royal Society of publishing a selection or composition of his papers and in numerous letters to Oldenburg he reports on the progress of this project (Nos. 51, 75-6, 80). The result of these efforts, Horrox's Opera posthuma, was eventually published in Oxford in 1673. Wallis also pursued astronomical investigations of his own. In 1664-5 he collaborated with Christopher Wren and Robert Hooke (1635-1703) in observing a new comet and reported to Oldenburg on the successes and failures of his undertakings made from the Schools' Tower in Oxford XXIV

Introduction (Nos. 86-7). With Robert Moray operating as an intermediary, Adrien Auzout (1622-91) and Christiaan Huygens (1629-95) in Paris worked together with the English side in collecting and comparing observational data. When subsequently a controversy over results broke out between Auzout and Hevelius, who had carried out his investigations from his own observatory, Wallis was among the members of the Society who were called upon to act as impartial judges. Ultimately, it was concluded that Auzout's observations were more reliable than those of Hevelius and Oldenburg tactfully communicated this verdict to his friend in Danzig.3 Just as astronomy, so too was music closely tied in with Wallis's mathematical interests. When Oldenburg reported to him that John Birchensha (fl. 1664-72) had written to the Royal Society on the question of perfecting the art of music, the Savilian professor responded by writing an extensive paper on the theory of music in which he points out defects in ancient and modern accounts (Nos. 56-7). Only after he had sent the paper to London did Wallis find time to consult Marin Mersenne's (15881648) work on harmonics and was thereupon forced to concede that his proposals for improvement had already been anticipated by others (No. 59). Further letters to Oldenburg reflect the interest of members of the Royal Society in gathering news of unusual natural phenomena or innovations in the field of technology. For example, Wallis reports observations he made on the hatching of pigeons (No. 46), on a man killed in a thunderstorm in Oxford (No. 107), and the use of an otacousticon to relieve defects in hearing (No. 245). As in the case of his astronomical or tidal observations these, too, were presented at meetings of the Society in London for discussion. Correspondence with Boyle Wallis's correspondence with Boyle likewise covers a variety of topics and provides evidence of the two men's close professional ties. Thus Wallis gladly obliged the Irish scholar's request to comment on his book entitled Some Considerations Touching the Usefulnesse of Experimental Natural Philosophy (No. 43) and at the same time sought his advice as chemist on possible treatment for an ailment affecting his sister in law, Rebecca Wallis (d. 1677). To this end he sent Boyle a copy of a report on her illness which 3 OLDENBURG-HEVELius 24.1/[3.II]. 1665/6, The Correspondence of Henry Oldenburg, ed. A. R. Hall and M. B. Hall, III, 29-30.

XXV

Introduction was almost certainly written by her brother, the physician John Nowell (d. 1701) (No. 44). Wallis also cooperated with Boyle in supporting the Lithuanian scholar and translator Samuel Chylihski (c. 1634-68), who with their help spent a number of years in Oxford before moving on to London (Nos. 25, 33-5, 82). Without doubt the most remarkable topic dealt with in the letters exchanged by Wallis and Boyle concerns their mutual interest in language. In January 1662, Wallis undertook the task of teaching Daniel Whalley, a young man who had been born deaf-mute, to speak distinctly. Just two months later he wrote an extensive letter to Boyle, explaining his motives for taking on the task and giving details of the approach he was using in order to achieve his goal (Nos. 19, 23-4, 29). Shortly afterwards he presented Whalley at a meeting of the Royal Society in order to demonstrate his initial results. Such was his success that he was later also employed by Anne Wharton (d. 1692) to teach her deaf-mute son, Alexander Popham (No. 32). This particular case, however, ultimately brought Wallis into dispute with another member of the Royal Society, William Holder (1616-98). When, in 1670, Wallis gave the Society his account of the cure of Popham's handicap, he neglected to mention Holder's name, despite the fact he had been responsible for the young man's earlier language tuition. In his long letter to Boyle on language training, Wallis considers the possibility of developing and introducing a universal character which would signify things independently of words and sounds. The seventeenth century witnessed numerous projects of this nature to overcome linguistic divides. At the same time, an artifical language based on characters or signs promised a means to improving the acquisition of knowledge. Wallis likens universal character to the role of numbers and symbols in algebra. Although he did not doubt that a philosophical language such as that conceived by John Wilkins could be developed, he felt its universal implementation was unlikely (No. 207). Nevertheless, when Wilkins wrote him a note in his real character (No. 212), Wallis replied in kind (No. 215).

Mathematical correspondence A substantial part of the correspondence in the present volume concerns mathematical themes. At the beginning of 1660, Pierre de Fermat's (1607/8-65) challenges on number theory and Blaise Pascal's (1623-62) XXVI

Introduction prize questions on the cycloid continued to influence discussions. In a letter to Pierre de Carcavi (c. 1600-84), Huygens announces the arrival of Wallis's newly published Tractatus duo in which he provides an account of the history of the cycloid quite different from that of Pascal. In many ways anticipating later academic squabbles on questions of priority, Huygens suggests that the Savilian professor in his latest book had sought at all costs to maintain the honour of his nation (No. 2). Carcavi for his part gives Huygens a clear impression of the prevailing attitude towards Wallis in the circles around Pascal and Roberval (1602-75) at this time (No. 4). Likewise, when Kenelm Digby (1603-65) sent the Savilian professor a copy of Fermat's De linearum curvarum cum lineis rectis comparatione in the summer of the same year, Wallis replied within two days, arguing that the rectification of the semicubic parabola contained therein had already been carried out by the English mathematician William Neile (1637-70) in 1657 (No. 7). By this time Huygens had also made the same claim for his fellow countryman Hendrik van Heuraet (1633-60?) (No. 6). More than once in the following years Anglo-Dutch rivalry in the field of mathematics came to the surface over this particular question of priority. One of the theorems on which Fermat's challenges in 1657/8 had been based was concerned with the indeterminate quadratic equation nx2 + 1 = y 2 , generally known as Pell's equation. Wallis returned to this topic in a paper which he enclosed in a letter sent to Brouncker in August 1668, where he also investigates the French mathematician's so-called last theorem for n = 3, which holds that a rational cube cannot be divided into two further rational cubes, that is, that a;3 + y3 = z3 has no nonzero integer solutions for a;, y, and z (Nos. 230, 231). He later reworked the two parts of this paper with the probable intention of publishing the piece in the Philosophical Transactions (No. 243). Wallis's demonstration of Fermat's other negative theorem, which holds that there is no rightangled triangle in numbers whose area is a square appears likewise to have been intended for publication (No. 236). In the reworked letter for Brouncker, Wallis also addresses once more some of the problems proposed by Bernard Frenicle de Bessy (1605-75) in the context of the discussion with him and Fermat over questions of number theory in the 1650s, including that of finding a number n such that 6n + 1 is a cube. Since the 1650s there had been at least one more exchange between the two men, rendered possible by their long-standing intermediary Kenelm Digby. Towards the end of 1661, Frenicle proposed a new problem on triangles to Wallis. The Savilian professor obliged in subXXVll

Introduction mitting a solution to which Frenicle subsequently wrote a short comment (Nos. 16-18). In the seventeenth century the practice of publishing reviews of the latest books in journals was becoming an increasingly important part of scientific discourse. Wallis was no exception in this respect. Typically, Oldenburg would ask for his comments on a new publication in the field of mathematics or experimental philosophy and Wallis would duly submit his views for possible inclusion in the Philosophical Transactions. Such was the case with Nicolaus Mercator's (1620-87) Logarithmotechnia (1668), in which the German-born mathematician demonstrated his method for finding the sums of logarithms and published his well-known series for the area of the hyperbola based on this. Already by this time there was fierce rivalry in the Republic of Letters on most questions concerning quadratures. After the claim had been made in French circles that Mercator had based his quadrature of the hyperbola on the efforts of Laloubere (160064), Wallis was quick to point out that it owed even more to his own Arithmetica infinitorum (1656) (No. 247). Another work on which Wallis was called upon by Oldenburg to provide an assessment was the Specimina mathematica (1667) of Frangois Dulaurens (d. c.1675). A largely unknown figure at the time, Dulaurens managed to create a stir already in advance of sending copies of his book to London by setting English mathematicians a problem on the properties of the circle (No. 164). After the problem had been read at a meeting of the Royal Society, John Collins (1625-83), who often advised Oldenburg on mathematical topics, Brouncker, and Wallis were asked to consider it and to produce a solution. All three did so, but not without beforehand finding fault with the way in which it had been expressed (Nos. 165, 172, 174, 1767). When the book finally arrived, Wallis was dismayed to find that in it Dulaurens ascribed to him the prize question which had been posed in the 1650s by a certain 'Jean de Montfert'. Apart from attacking this ascription as being without foundation, Wallis claimed that much of the first part of Dulauren's book had been taken from Oughtred (1575-1660) and himself, while much of the second part was openly based on Schooten and Viete (Nos. 182, 186, 196, 204). Initially, Oldenburg published only the first half of Wallis's review of Specimina mathematica. But after Dulaurens had replied to the Savilian professor's account by publishing his Responsio ad epistolam Wallisii, Oldenburg lifted the diplomatic restriction he had imposed on himself and printed the rest. XXVlll

Introduction Correspondence with Collins Occasionally, Collins too asked Wallis for his views on new publications. Alongside Honore Fabri's (1607-88) Synopsis optica (1667), of which Wallis wrote only a summary of contents (Nos. 186, 190), the most prominent example is undoubtedly that of Vincent Leotaud's (1595-1672) Cyclomathia (1663). With this work the French Jesuit mathematician entered a long-standing dispute over the nature of the angle of contact which had been re-ignited some seven years earlier when Wallis published his De angulo contactus (1656). There, he provided arguments in support of the view originally put forward by Jacques Peletier (Peletarius) (151782) against Christoph Clavius (1537-1612) that the angle of contact is not only less than any possible right-lined angle, but also cannot even be classified as a true angle at all. After receiving from Collins a copy of the Cydomathia (No. 182), in which Leotaud maintains against Wallis tha the angle of contact does have a magnitude, the Savilian professor wrote an extensive reply to him on the topic, showing what he considered to be the irreconcilability of Clavius's concept of angulus contactus with Euclid's definition of a plane angle (No. 183). Oldenburg sent Wallis's letter to France for conveyance to Leotaud in February/March 1668, but as fa as can be established no reply was ever written. Wallis's correspondence with Collins covers almost as equally large and varied a number of themes as that with Oldenburg. Many of the letters contain details of recent scientific publications on the Continent, particularly in the field of mathematics. Sometimes Collins presented Wallis with copies of books he had obtained from abroad in repayment for favours he had received (No. 98). Not a few of his letters betray his in terest in sea charts and navigation originating from the time spent at sea earlier in his life (Nos. 99, 100, 137). On one occasion he gives th Savilian professor a short account of some of his own publications (No. 100). In addition, Collins kept Wallis informed on progress in the printing of Thomas Brancker's (1633-76) English translation of Rahn's (1622-76) Teutsche Algebra, which had been significantly reworked and expanded by John Pell (1611-85). Moreover, he ensured that the printer, Moses Pitt (fl. 1654-96), sent Wallis those parts of the book which had already been printed for proofreading and correction (No. 137). In the course of doing this Wallis was able to produce a list of errors in Brancker's table of incomposite numbers, which he subsequently conveyed to Pell (Nos. 220, 222-4). Later on, Collins similarly supervised the printing of Wallis's xxix

Introduction Mechanica, the task of which was also taken on by Pitt (Nos. 247-8, 251). When Wallis drew up plans for reprinting William Oughtred's Clavis mathematicae early in 1667, Collins assisted by presenting the proposal to the same London printer (No. 139). However, Pitt was evidently not to be convinced of the viability of the project and the book was instead reprinted by Lichfield in Oxford later the same year. Although Collins's assistance led to no success with the Clavis project in London, it is nevertheless indicative of a spirit of cooperation which pervades his correspondence with fellow mathematicians. What is particularly remarkable in this case is that he too was clearly doubtful of there being a need for a new edition of Oughtred's book, citing contemporary authors critical of its form and content (No. 142). Wallis, by contrast, had an unswayable respect for Oughtred right up to the end of his life and not seldomly accused other authors such as Dulaurens of having appropriated parts of his work for their own (Nos. 141 and 196). Wallis, Gregory, and Huygens Disputes based on accusations of plagiarism or concerning questions of priority increasingly dominated scientific discourse from the mid-1650s onwards. As one of the leading mathematicians of his day, Wallis was often—indeed, more often than most—involved in such academic quarrels. But sometimes, as with Auzout and Hevelius, the Savilian professor was called upon to adjudicate in controversies involving others. The same task fell upon him in a dispute between James Gregory (1638-75) and Christiaan Huygens which came about towards the end of the period covered by the present volume. As before, tact was required, as the adversaries had close ties with the Royal Society—Huygens had been a fellow since 1663 and Gregory was elected in 1668. The Scottish mathematician had sent Huygens a copy of his Vera circuli et hyperbolae quadratura (1667), in which he set out his discovery of the infinitely converging series for the area of the circle and of the hyperbola, soon after its publication in September 1667. Huygens subsequently wrote a review of the book for the Journal des Sgavans. While recognizing its importance, Huygens pointed out a number of faults and suggested that some of the propositions, including the derivation of logarithms from hyperbolic areas, had already appeared in his own work. Moreover, he attacked the book's central proposition, according to which the circle cannot be squared 'analytically'. By this Gregory meant that TT could not be obtained by a finite sequence of XXX

Introduction the five basic operations: addition, subtraction, multiplication, division, and root extraction. Wallis's initial reaction to Gregory's work, based only on a superficial reading, was one of approval (No. 186). As the dispute emerged and on the basis of a more detailed study he modified his opinion, but nevertheless he remained even-handed towards the two opponents. Thus although he expressed surprise at the ferocity of Huygen's attack, he sided with the Dutch mathematician on a number of points, noting, for example, that Gregory had failed to prove conclusively that there is no 'analytical' means of squaring the circle other than that considered by him (No. 229). As to the derivation of logarithms from hyperbolic areas, Wallis finds that he is able to correct both authors: this had in his view already been set out by Gregoire de Saint-Vincent (1584-1667) in his Opus geometricum (1647) and later by himself in letters published in his Commercium epistolicum (1658) (Nos. 229, 251). Cryptanalysis In comparison to later years of his life, there was apparently little call upon Wallis's deciphering skills during the period covered by the present volume. The only case evident in the correspondence appears to have arisen from fears on the part of the lord chancellor, Edward Hyde (1609-74), that the repressive legislation against dissenters introduced by parliament in the course of re-establishing the authority of the Church of England might lead to a conspiracy against the ruling powers. After two letters written in cipher had been intercepted at Banbury, the lord chancellor, who clearly valued Wallis's skills, commanded his secretary, Matthew Wren (162972), to convey them to the Savilian professor to be deciphered (No. 93). Just a few days later, Wallis was able to send back the results of his labours (No. 94). The encoded letters were, as it turned out, perfectly harmless.

University affairs By contrast, enormous demands were made on Wallis in his capacity as Keeper of the Archives. Legal disputes involving the University abounded, partly as a result of efforts to restore discipline after years of moral decline (No. 9), but mainly because its historical rights and privileges were increasingly called into question by the civic authorities. Relations between the city of Oxford and the University became particularly strained XXXI

Introduction towards the end of 1660, when the mayor refused to take the oath obliging him 'to observe and keep all manner of lawfull liberties and customes of the University of Oxford'.4 Among the privileges which the city subsequently disregarded were the University's rights to licence alehouses and victuallers as well as printers and apothecaries, to govern the market, and to take possession of felons' goods. In the disputes which followed, Wallis was required to assist in preparing the University's case at court by investigating legal precedents and by studying the exact wording of diverse royal charters. On occasion he also sought the collaboration of a colleague in the University of Cambridge, Theophilus Dillingham (1613-78), who had been his contemporary at Emmanuel College (Nos. 10-12, 36-7). Two disputes between 1660 and 1668 were especially time consuming for Wallis. In 1664 an attorney called William Thackwell was accused of libel by a manciple at Balliol College and then imprisoned by the vice-chancellor on refusing to put in a security following his arrest. Since Thackwell was able to provide a writ of privilege and a Habeas corpus from the Court of Common Pleas, the focus of attention switched to London, where it was necessary for Wallis to spend a number of weeks attending the interests of the University. In the end, he was only able to achieve moderate success, despite meetings with the lord chancellor and the lord chief justice (Nos. 60-72). The second major dispute also concerned an attorney. In 1667 Fish Lyne was apprehended in an alehouse by University proctors conducting their night walk. He subsequently ignored a summons to attend the vicechancellor's court and later pleaded himself free from the University's jurisdiction at the Court of Common Pleas. In a series of letters sent to Llewelyn Jenkins (1623-85) in London, Wallis provided material from the Archives in Oxford with which the principal of Jesus College sought to defend the rights and privileges of the University (Nos. 149-59, 161, 167-71, 173, 178). However, the University was not always on the defensive in this way. As one of Wallis's letters to Jenkins makes clear, the city was at least on one occasion concerned about the effects continuing legal disputes might have and sought to bring an end to its most recent conflict with the University. Quite simply, townsmen feared they might lose the University's trade and custom to outsiders (No. 170). l

The Life and Times of Anthony Wood, ed. A. Clark, I, 370-2. xxxu

Introduction

Personal affairs Only rarely does Wallis's surviving correspondence allow us insight into the private side of his life. This makes the few instances there are exceedingly valuable. Among the more personal letters is one from John Burscough (c.l629-c.!707), formerly a fellow of Brasenose College and since 1662 rector of Stoke by Guildford. Having recently married, Burscough evidently arranged a loan from Wallis in order to overcome financial difficulties he was currently experiencing (No. 84). Help of a rather different kind from Wallis is sought in a letter from a mother whose daughter had been lodging in the household of the Oxford printer Ann Lichfield. On learning of her daughter's unsuitable behaviour in that environment, she asks Wallis and his wife to take her into their care and to prevent her from having access to potentially corrupting influences in the future (No. 88). From a number of letters we discover that Wallis's house in Oxford was broken into and plundered by thieves at the beginning of 1667. Even his books were not spared. As he reports to Oldenburg, he found his copy of Hevelius's Descriptio cometae (1666) torn apart after the thieves had departed (Nos. 134, 137, 143). Finally, two pieces of correspondence document in different ways the uncertainties of the time. The stipend of the Savilian professors was raised entirely from the rents of the properties donated by Sir Henry Savile (1549-1622) to the University of Oxford. The imposition of taxes of whatever nature could therefore—and often did—have a direct influence on the income received by Wallis and his counterpart, the professor of astronomy, who from 1661 onwards was Christopher Wren. In a joint letter to the Essex commissioners of monthly assessments, the two men apply for redress for moneys already deducted from the rent of their tenant in Rettendon, and thereby cite a proviso in parliamentary and other acts, according to which the stipends of readers at both universities are exempt from such charges (No. 225-6).

Theological affairs One of Wallis's ways of responding to financial and professional uncertainty of this kind was to take on additional responsibilities. Having succeeded in supplementing his Savilian professorship by the office of Gustos archivorum in 1658, he sought in 1661 to gain an additional stipend, by petitioning the king for the post of prebendary at Christ Church, OxXXXlll

Introduction ford (No. 11). Many dignities, canonries, and prebends had fallen vacant by 1660 and cathedral chapters needed to be restored. This attempt at gaining a prebendaryship reflects at the same time his re-emergence as a prominent theologian at the beginning of the Restoration. Having spent the years following the Westminster Assembly largely in the background, Wallis was one of a dozen or so Presbyterian divines, including Richard Baxter (1615-91) and Edmund Calamy (1600-66), who were appointed chaplains-in-ordinary to the king in the course of 1660 in recognition of the important part leading Presbyterians had played in bringing about the return of the monarchy.5 Moreover, he was a key participant in negotiations which took place well into 1661, aimed at overcoming the significant constitutional and liturgical differences which existed between moderate Presbyterians on the one side and Anglican episcopalians on the other side.6 Although these talks got off to a promising start in the presence of the king at Worcester House, the residence of the lord chancellor, they finally ended in deadlock and acrimony at the Savoy Conference. This failure eventually led to the decision by the Cavalier parliament to take religious settlement into its own hands. Against the background of the resulting Act of Uniformity (1662), the following years were marked by rumours of Presbyterian plots, such as was probably thought to be behind the encoded letters the Savilian professor was asked to decipher. The success of the Royal Society in the 1660s, to which Wallis as one of the leading scientists of his day made a decisive contribution, came about despite the absence of the religious unity he and others, particulary on the Presbyterian side, had sought to achieve.

5 6

R. BAXTER, Reliquiae Baxterianae, ed. M. Sylvester, II, 229. Ibid., 230, 276-7; E. CARDWELL, A History of Conferences, third ed., Oxford 1849,

257. XXXIV

EDITORIAL PRINCIPLES AND ABBREVIATIONS

All letters are preceded by an account (Transmission) of the various manuscript and printed forms in which they have been handed down. In the case of those letters whose text has not survived, the reasons for assuming that they did exist at some time are given. The Transmission section also puts each letter in context, records, when known, how it was conveyed to the addressee, and supplies additional information such as postmarks, details of notes appended to manuscripts, enclosures, and so on. Manuscript and printed sources are denoted according to the following scheme: W w C c E

original manuscript in Wallis's hand copy of Wallis manuscript in scribal (or identified) hand original manuscript in correspondent's hand copy of correspondents manuscript in scribal (or identified) hand contemporary edition

Where there is more than one source in a particular category, these are numbered successively W1, W2, . . . , w1, w2, . . . , and so on. All letters contained in the volume are dated according to both the old style or Julian calendar employed in England until 1752 and the new style or Gregorian calendar widely used on the Continent, with the form not given in a particular letter placed in square brackets. In the period covered by the present volume the difference between the two calendars was ten days. To accommodate the English year, which began on Annunciation or Lady Day (25 March) and which permitted a new style date such as 16 February 1663 to be expressed in a number of ways old style— 6 February 1662, 6 February 1662/3 or even 6 February 1663—the most common form (1662/3) has been used in brackets where a correspondent on the Continent employing new style has not supplied the date old style XXXV

Editorial principles and abbreviations himself. (For reasons of legibility, only the Gregorian calendar has been used in creating the Index of Letters.) Where the place at which a letter was written can only be surmised, this also is set in square brackets. The spelling, capitalization, and punctuation of manuscript and printed sources has been retained throughout. Contractions have been silently expanded, except where they are still in common use today, and thorn has been altered to 'th'. The use of i/j and u/v has been modernized. All symbols, the ampersand, and use of superscripts to denote pounds, shillings, and pence have likewise been kept. Full stops have been diplomatically added at the end of sentences where their absence could lead to misunderstandings. As a general rule, sentences begin with a capital letter even when a miniscule is used in the handed-down text. All underlining in manuscripts is reproduced as italics. The reproduction of italics from printed sources has been treated diplomatically. In mathematical passages, letters used to indicate points or places in figures, and likewise all algebraic formulae, have been italicized where the writer or printer has not already done this.

Editorial signs (text) { }

[paper torn]

11 add. alt. corr. | text del. ed. ins. suppl.

uncertain reading illegible words (the number of dashes indicates the number of illegible words to a maximum of three) words omitted Editor's remarks (N.B. upright square brackets contained in text or in variant readings of the critical apparatus are always either employed by the author himself or represent a contemporary addition, as indicated) new paragraph within a variant reading added contemporary alteration to text by someone other than the author corrected word or words deleted editor inserted supplied

The critical apparatus shows the development of text through its various stages. Each successive stage replaces the preceding one. Thus XXXVI

Editorial principles and abbreviations stage (1) is superseded by stage (2} and this in turn by stage (3). Further subdivisions are indicated by letters: (a) is replaced by (6) and then by (c), (aa) is replaced by (66), (aaa) by (666), and so on. As in the case of the critical apparatus, but placed above this, marginal annotations to texts are referenced by means of line numbers. Editorial comments (footnotes) are indicated by numerical superscripts. Astronomical and mathematical symbols T a)b(c H 6 $ S>

Aries, vernal equinox division Gemini Mars Mercury Moon

a.b:: c.d I I,D t? D O A

xxxvii

proportion rectangle Saturn, Saturday square Sun triangle

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CORRESPONDENCE

1. CHRISTIAN HUYGENS to PIERRE DE CARCAVI [16J/26 February [1659J/1660 Transmission:

C Heavily revised and corrected draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 722, 3 pp, (our source).—printed: HUYGENS, (Euvres completes III, 26-8. c Copy of draft: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 3611, f. 151r-152v. Reply to: CARCAVi-HiJYGENS [3]/13.IX.1659 (HUYGENS, (Euvres completes II, 534-6). Answered by: CARCAVI-HUYGENS [25.II]/6.III.1659/60 (HUYGENS, (Euvres completes III, 38-9).

Carcavy. 26 feb. 1660.

Monsieur. Depuis que j'ay receu vostre derniere7 qui a este au mois de decembre, j'ay escrit8 a Mr. Wallis mon correspondant en Angleterre, sur le suject des Livres de Mr. d'Etonville9, et en attendant tousjours sa response 4 [In left margin in Huygens's hand:] perpet. mob. Pascalin. horologe de Mr de Boismorand.

4 receu (1) vostr breaks off (2) vostre 4 qui (1) fust (2) a este 7

vostre derniere: i.e. CARCAVI-HUYGENS [3]/13.IX.1659; HUYGENS, (Euvres completes II, 534-6. 8 j'ay escrit: i.e. HuYGENS-WALLis XII.1659-II.1659/60. 9 Livres de Mr. d'Etonville: i.e. [PASCAL], Lettres de A. Dettonv^lle, Paris 1658. 1

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pour vous pouvoir satisfaire je suis demeure plus long temps qu'il ne faloit a vous en faire moymesme, de quoy je vous demande pardon. Je ne puis deviner la raison pourquoy il ne m'escrit point depuis tant de mois, scachant bien pourtant que les dits livres avec ceux10 de Mr. Sluse et depuis encore ceux de mon Systeme11 ont este delivrez au libraire12 a Londres a qui j'addresse tous mes pacquets. Peut estre que les siens se sont esgarez. Je luy escriray13 encore une fois, et lorsqu'il me viendra response14 je ne manqueray pas de vous en faire part. Je vous remercie beaucoup des extraits15 qu'il vous a pleu m'envoyer des lettres de Mr. de Fermat. Pource qui est de la demonstration de la spirale et parabole, je vous ay escrit16 que j'y trouvois de la difficulte, et que Monsr. Sluse non plus que moy ne la pouvoit resoudre, c'est a dire

1 satisfaire (1) j'ay t breaks off (2) je 1 qu'il (1) {breaks off (2) ne 2 moymesme, (1) et dont je vous en demande (2) de quoy je vous demande pardon. Je ne (a) scay (b) puis 4 les (1) Livres (2) dits livres avec ceux de Mr. Sluse add] et 6 j'addresse (1) toutes mes lettres (2) tous mes pacquets. (a) Lors (b) Peut 7 et (1) s'il me vient (2) lorsqu'il me viendra 9 remercie (1) des (2) beaucoup 10 de (1) sa (2) la demonstration (a) de Mr. Dettonville (b) de la spirale et parabole, je (oa) ne scay (66) vous 11-3, 2 difficulte, (1) a scavoir pour 1' (2) et que (a) ny Mr. (b) Monsr. Sluse non plus que moy ne la (oa) pouvoit (bb) pouvions resoudre, c'est a dire (aoa) en retenant les paroles (aaaa) de (bbbb) de Mr. dettonville. Car (bbb) que selon nostre jugement il y avoit de la faute en cette demonstration I, comme il y en a en effect add. \ . Mais 10

ceux: i.e. SLUSE, Mesolabum, Liege 1659. mon Systeme: i.e. HUYGENS, Systema saturnium, The Hague 1659. 12 libraire: i.e. Samuel Thompson (d. 1668), London bookseller at the 'White Horse' in St Paul's Churchyard and (from 1664 onwards) at the 'Bishop's Head' in Duck Lane. Cf. WALLIS-HUYGENS 24.XI/[4.XII].1659. 13 Je . . . escriray: Huygens in fact wrote to Wallis at the end of March (HUYGENSWALLIS [21]/31.III.1659/60) and again in July (HuYGENS-WALLis [5]/15.VII.1660). 14 lorsqu'il . . . response: Wallis did not in fact reply to Huygens until August: WALLISHUYGENS 31.VIII/[10.IX].1660. 15 extraits: i.e. excerpts of letters from Fermat to Carcavi concerning the comparison of spiral and parabolic lines, as dealt with in Pascal's Lettres de A. Dettonville. See HUYGENS, (Euvres completes II, 536-8 and 538-40. 16 je ... escrit: i.e. HUYGENS-CARCAVI [25.VIIIJ/4.IX.1659; HUYGENS, (Euvres completes II, 474. Cf. HUYGENS-CARCAVI [12J/22.V.1659; HUYGENS, (Euvres completes II, 411-12. 11

2

1. HUYGENS to CARCAVI, [16]/26 February [1659]/1660 que selon nostre jugement il y avoit de la faute en cette demonstration, comme il y en a en effect. Mais j'ay bien veu d'abbord qu'en la changeant, Ton y pouvoit remedier. Et voycy comme je 1'avois conciie, en gardant de plus pres ce me semble 1'intention de Mr. dettonville que n'a fait 17 Mr. de Fermat. Si elles ne sont pas egales, soit X la difference et soit Z la cinquiesme partie de X, et soient inscrites et circonscrites les figures ainsi que dit 1'autheur. Maintenant puis que la difference entre 1'inscrite en la spirale et 1'inscrite en la parabole est moindre que Z\ et que aussi la difference entre 1'inscrite en la parabole et la circonscrite a la mesnie parabole est moindre que Z; done la diff. entre 1'inscr. en la spirale et la circonscr. a la parabole est moindre que deux Z. Mais la diff. entre la circonscr. a la parabole et la circonscr. a la spirale est aussi moindre que Z: done la difference entre 1'inscr. a la spirale et la circonscr. a la mesme spirale est moindre que 3Z. Et a plus forte raison la diff. entre la spirale mesme et le tour de sa figure inscrite sera moindre que 3Z. Mais la diff. entre 1'inscr. en la spirale, et 1'inscrite en la parabole est moindre que Z, done la diff. entre la spirale et 1'inscrite en la parabole sera moindre que 4:Z. Enfin la diff. entre 1'inscrite en la parabole et la parabole mesme est aussi moindre que Z. done la diff. de la spirale et de la parabole sera moindre que 5Z,

2 qu'en (1) y (2) la changeant, (a) quelque (b) 1'on 3 Et (1) voyla (2) voycy comme je 1'avois conciie, (a) sans (6) sans la reforme entierement comme a fait Mr. de Fermat (c) en 4 ce me semble add. 4 dettonville (1) au lieu qu'il dit, Et soit Z le tiers de X, je (a) mets (6) dis, et soit Z une cinquiesme de X, c'est tout ce (2) que n'a fait Mr. de Fermat. 6 et (1) puis (2) soient 9 que Z; et ( 1 ) dereclief (2) que aussi 12 circonscr. (1) en la spirale (2) a 14 1'inscr. (1) en la et en (2) a 17-4,1 est jderechef del] jmoindre que Z, done la diff. entre la spirale et (1) la (2) 1'inscrite en la parabole sera moindre que 4Z. (a) Mais (6) Enfin la diff. . . . supposition &c. in left margin] 17 fait: i.e. in FERMAT-CARCAVi VIII.1659; HUYGENS, (Euvres completes II, 536-8; FERMAT, (Euvres (1891-1912/22) II, 438-40.

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c'est a dire que .X", centre la supposition &c.| [2] La comparaison des autres sortes de spirales avec les lignes paraboloides que donne18 Mr. de Fermat est veritable mais non pas fort difficile a trouver apres que la premiere est conniie. Et je m'estonne qu'il prend plaisir a inventer des lignes nouvelles, qui n'ont pas autrement de proprietez dignes de consideration. Les propositions touchant les surfaces des Conoides et Sphaeroides comme aussi de la ligne parabolique sont les mesmes que je vous ay cydevant communiquees19, et a plusieurs autres de mes amis. Je croy bien pourtant que Mr. de Fermat n'en avoit veu aucune puis qu'il 1'assure, mais d'autres peutestre seront plus incredules, si en les donnant au public il n'allegue celuy a qui il les ait fait veoir auparavant. La mesure de la superficie du conoide que fait la parabole autour de 1'appliquee la quelle

2 (1) II y (a) aura (6) a (oa) done (oao) cinq (666) ces 6 (bb) premierement ces 4 quantitez dont chacune differe de la suivante (aaaa) d'une moindre (bbbb) que (cccc) de nioins que n'est Z; a scavoir 1'inscrite (aaaaa) a (bbbbb) en la spirale, 1'inscrite en la parabole, la circonscrite a la parabole, la circonscrite a la spirale. (aaaaaa) Done la (bbbbbb) Done la difference entre la premiere et la derniere, a scavoir entre 1'inscrite et la circonscrite a la spirale sera moindre que 3Z. Et a plus forte raison, sera la difference entre la spirale mesme et sa figure inscrite moindre que 3Z. \\ Mais la d breaks off (2) La comparaison (aaaaaaa) que (bbbbbbb) des 2-3 paraboloides (1) est (2) que donne Mr. de Fermat est (a) assez subtile mais pour moy je ne suis pas (b) veritable (oa) mais non p breaks off (bb) mais non add. pas 6 proprietez (1) fort (2) fort remarquables. (3) dignes de (a) remar breaks off (b) consideration. 7 Les (1) propositions (2) inventions (3) propositions (a) qu'il d breaks off (b) touchant 8 comme aussi de la ligne parabolique add. 10 n'en (1) a (2) avoit 11 mais (1) il (2) d'autres 11-5, 2 si (1) ce n'est qu'en (2) en les donnant au public il (a) ne (ao) puisse alleguer 18

donne: i.e. FERMAT-CARCAVI IX. 1659; HUYGENS, (Euvres completes II, 538-49; FERMAT, (Euvres (1891-1912/22) II, 441-4. 19 je . . . communiquees: i.e. HuYGENS-CARCAVi [6]/16.1.1658/9; HUYGENS, (Euvres completes II, 315-17. 4

1. HUYGENS to CARCAVI, [16]/26 February [1659]/1660 il promet en supposant la quadrature de 1'hyperbole sera quelque chose de nouveau si elle est vraye. Vous m'auriez fait grand plaisir si lors que Mr. de Boismorand20 vous parla de son horologe a pendule, vous 1'eussiez demande plus particulierement de quelle fagon ce pendule est applique, et s'il fait un bon effect. II y a de 1'apparence que non, parce qu'il n'auroit pas ainsi laisse se perdre une invention qu'il eust juge utile. C'est une chose estrange que personne devant moy n'ait parle de ces horologes, et qu'a cette heure il s'en de[3] couvre tant d'autres autheurs.| J'espere de vous faire voir bien tost ce que j'y ay adjouste de nouveau, qui est une invention que les Geometres

(66) scache dire (6) n'allegue celuy (aoa) a qui il (bbb) qu'il (ccc) a qui il les ait (aaaa) communique ci devant que les mienes paravent. (bbbb) fait veoir auparavant. La |mesure de la add.\ superficie jpourtant add. and del] du conoide que fait la parabole (aaaaa) estant tournee sur (bbbbb) autour de 1'appliquee (aaaaaa) est quelque qu'il d breaks off (bbbbbb) la quelle il (aaaaaaa) mesure (bbbbbbb) dit mesurer (ccccccc) promet en supposant la quadrature de 1'hyperbole (aaaaaaaa) est (bbbbbbbb) sera quelque chose de nouveau (aaaaaaaaa) et de bien joly a (666666666) la quelle je ne (aaaaaaaaaa) croyois pas (6666666666) scay pas encore mais je doute si elle vraye. (ccccccccc) pourveu qu'elle soit (ddddddddd) si elle est vraye. 3 (1) Vous (2) Je suis marry que vous (—) (3) Lors (4) Vous 5 s'il (1) faisoit (2) fait un bon effect, (a) Ce (6) II 6-9 perdre (1) cette (2) une invention qu'il (a) 1'eust (b) eust juge utile. (ao) II y en a (aoa) plus de (666) 5 ou (aaaa) 6 (bbbb) six qui maintienent d'avoir eu cette (ccc) encore plusieurs qui disent avoir eu cett invention en teste devant moy, et pour ce qui est de Galilee j'ay veu dernierement le dessein du modelle que Le Prince Leopold de Toscane assure avoir trouve apres sa morte. (aaaaa) A tout cela je ne puis respondre, que (bbbbb) Mais tout cela ne fait rien contre moy si ce n'est que l'(—} de sorte que (aaaaaa) cette (bbbbbb) la pensee semble avoir este assez commune, mais mon modelle a jbien add] succede le p(—} et je (aaaaaaa) ne veoy (bbbbbbb) scay bien qu'on ne trouve point, que j'en aye rien emprunte de personne. (66) |C'est une . . . d'autres autheurs. in left margin\ 7 choye corr. ed. 9 autheurs. (1) L'on ne (—) (2) Vous verrez (3) J'espere de vous faire voir bien tost (a) une nouvelle invention (6) ce que j'y ay adjouste (oa) a ces horologes (aaa) pour (666) pour leur derniere perfection |et justesse add. , la quelle [invention add. (66) de nouveau, qui est une invention que 20

Mr. de Boismorand: a magistrate in Anguleme, who was in possession of a pendulum clock made in 1615 or 1616; see CARCAVI-HUYGENS [3J/13.IX.1659 HUYGENS, (Euvres completes II, 534-6, 535.

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estimeront infiniment plus que tout le reste de la fabrique. Celle de la machine de Mr. Pascal21 pour supputer, que 1'on m'a envoyee de Paris, est certainement digne d'admiration, et contient plusieurs belles pensees. Je luy en escriray bientost puis que j'apprens qu'il commence a se porter mieux. Mais toutes nos inventions Monsieur vont estre peu considerables si celle de 1'allemand Johannes Joachimus Becherus22 s'effectue, ou s'il ne nous trompe pas, car il m'a assure, m'ayant este veoir icy, qu'il a construit23 un mouvement perpetuel a Mayence, qui continue d'aller depuis six mois. Et hier il m'en envoya les figures qu'il a fait graver en deux grandes planches a Amsterdam. L'on ne peut pas pourtant comprendre le secret de 1'invention par ces figures, devant que de voir la description qu'il en promet, ayant par tout adjouste de lettres et des nombres. Seulement 1'inscription tient et 1'on le voit a peu pres que 1'une des machines (car il a deux inventions diverses du perpetuum mobile) est purement mechanique et 1'autre physico-mechanique. Pour celle cy la chose ne me paroit pas tout a fait impossible, mais de 1'autre j'advoue qu'elle passe ma croyance. Je suis Mons.

2 (1) J'ay icy (2) Celle de 3 Paris, (1) qui certainement est (2) est certainement digne d'admiration, (a) p breaks off (b) et 4-5 pensees. (1) Je luy en escr breaks off (2) |Je luy ... mieux. add] Mais 5 inventions jmechaniques add. and del] Monsieur (1) doivent ceder (2) vont 6 1'allemand add. 7 m'ayant este veoir icy add. 8 un (1) perpetuum (2) mouvement perpetuel a Mayence, qui (a) va en dessin (b) continue d'aller 11 que (1) d'avoir (2) de voir 13 des (1) chiffres (2) nombres. Seulement (a) 1'invent breaks off (b) 1'inscription tient et 1'on le voit a peu pres 14 (car il a deux inventions diverses |du perpetuum mobile add.\ ) add. 16-18 1'autre (1) je n'en croy rien. (2) j'advoue qu'elle passe ma croyance (a) Je suis (b) Je 21 machine de Mr. Pascal: i.e. Pascal's arithmetical machine; cf. BELLAIR-HUYGENS [24.VI]/4.VII.1659 (HUYGENS, (Euvres completes II, 426-9). 22 Becherus: i.e. Johann Joachim Becher (1635-82), German alchemist and inventor. 23 construit: see SCHOOTEN-HuYGENS [7]/17.1.1659/60; HUYGENS, (Euvres completes III, 10-11.

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2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660 Le livre24 de Mr. de Wit n'est pas encore acheve d'imprimer. Celuy25 de Wallis que vous desirez de veoir ne se trouve pas icy chez les libraires, mais a la premiere occasion qui s'offre je vous envoyeray mon Exemplaire et seray tousjours

2.

CHRISTIAN HUYGENS to PIERRE DE CARCAVI [17J/27 March [1659J/1660 Transmission:

Cl Preliminary draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 735, 1 p.—printed: HUYGENS, CEuvres completes III, 56. C2 Heavily revised and corrected draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 735, 2 pp. (our source).—printed: HUYGENS, (Euvre completes III, 56-7. c Copy of C2: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 3611, f. 147r-148r. Date: given in C1 and c. Reply to: CARCAVl-HuYGENS [25.IIJ/6.III.1659/60 (HUYGENS, (Euvres completes III, 38-9). Answered by: CARCAVl-HuYGENS [15J/25.VI.1660.

Carcavy. Monsieur II en est arrive ce que je me estois imagine du silence de Mrs. les Anglois, car 2 ou trois jours apres que je vous eus envoye ma precedente26 je receus

2 Wallis (1) que v breaks off (2) que 2 icy (1) avec (2) chez 3 Exemplaire (1) . Je suis (2) et seray 7 (1) Deux ou trois jours apres vous avoir escrit ma derniere, je receus nouvelles de Mr. Wallis a s§avoir une lettre (2) II en est arrive (a) cornme je me 1'estois (&) ce que je me estois 7 Anglois, (1) car il y a longtemps que (2) et (3) car 24

livre: i.e. WlTT, Elementa curvarum linearum, ed. F. van Schooten, Amsterdam 1659. 25 Celuy: i.e. WALLIS, Commercium epistolicum, Oxford 1658. 26 ma precedente: i.e. HUYGENS-CARCAVI [16]/26.II.1659/60. 7

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une lettre27 de Mr. Wallis avec un nouveau livre28 qu'il a fait imprimer dont le titre est Joh. Wallisii &c. tractatus duo Prior de Cycloide et corporibus inde genitis, Posterior Epistolaris in qua agitur de Cissoide et corporibus inde genitis. Et de curvarum turn linearum ev'&vvoei^ turn superficierum -K\arva\j1u). Je m'estonne que vous ne 1'ayez pas encore vu lors que vous me fites 1'honneur de m'escrire la derniere fois29, car sans doute il en aura envoye aussi des exemplaires a Paris pour Mr. Pascal et pour vous Mr. qui y estes nomine si souvent. Je ne vous escriray que sommairement ce qu'il contient, parce que vous le verrez vous mesme ou peutestre 1'aurez veu desia. Dans le premier traicte il resout par ses methodes les problemes que Mr. Pascal a proposez et resolus. Et il y adjouste la demonstration de Mr. Wren de la dimension de la Cycloide more veterum, qui est ingenieuse. L'autre traicte est la lettre que j'ay attendue si long temps, au commencement de la quelle il dit avoir receu et distribue les exemplaires du livre30 de Mr. dettonville que je luy avois envoye de vostre part. II donne en suite ses specul. touchant la Cissoide a 1'occasion d'un theoreme que cy devant je luy avois communique pour la dimension de 1'espace infini qui est entre cette courbe et son asymptote. Au reste il y a

(1) Pent estre (2) Je m'estonne que vous ne (a) 1'avez (6) 1'ayez pas 5 encore vu add] lors 8 nomine (1) tant de fois. (2) si souvent. 8 escriray (1) pas le detail de ce qu'il traite dans ce livre (2) que sommairement ce qu'il (a) est contenu dans (6) contient 9 verrez (1) ou ... veu desia vous mesme. (2) vous mesme ... veu desia. (a) Au premier (6) Dans (ao) la premiere partie (66) le premier traicte 11 les jmesmes add. and del.\ problemes 11 proposez et add. 12 dimension (1) des Cycloides (2) de la Cycloide 14 il (1) (avoit) (2) dit avoir 16 part. (1) Pour apres (2) Les speculations touchant la cycloide (a) son breaks off (6) qu'il traite au long sont (3) II (ao) traite en suite d breaks off (bb) donne en suite ses specul. touchant la Cissoide (aaa) auxquelles (bbb) a 17 communique (1) pour (2) concernant (3) pour 18 asymptote. (1) Dans la preface (2) (—) (3) II (4) Au reste il y a (a) (aussi) (b) et 27

lettre: i.e. WALLis-HuYGENS 24.XI/[4.XII].1659. nouveau livre: i.e. WALLIS, Tractatus duo, Oxford 1659. 29 derniere fois: i.e. CARCAVI-HUYGENS [25.II]/6.III.1659/60; HUYGENS, (Euvres completes III, 38-9. 30 livre: i.e. [PASCAL], Lettres de A. Dettonv^lle, Paris 1659. 28

8

2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660 et dans cette lettre et dans la preface du traite de la Cycloide beaucoup de choses centre Mrs. Dettonville et de Roberval et centre leur histoire31 de la Roulette aux quelles peut estre ils ne demeureront pas sans replique. Ce Mr. Wallis tesmoigne certes d'avoir 1'esprit prompt et il y a du plaisir a veoir comme il tasche a toute force de maintenir 1'honneur de sa nation. Je vous rends graces tresh. de 1'extrait32 de la lettre de Mr. de Fermat [2] de qui je trouve cette derniere speculation touchant| les roulettes proportionelles beaucoup plus belle que la precedente33 des spirales. Pour ce qui est de la proposition dont j'estois aucunement en doute, je ne voudrois pas qu'il prit la peine d'en escrire la demonstration separement, pour me la faire veoir, mais plustost Mr. ayant le traite entier et tant d'autres excellens ouvrages de ce grand geometre dont vous pourrez obliger le public lors qu'il vous aura tout envoye, ainsi qu'il a promis. II ne se plaint pas

1 de | la add] Cycloide 3-5 Roulette |aux quelles . . . sans replique add] . (1) (—} (2) Vous le verrez Monsieur et peut estre (a) (ces) (&) les dits Messieurs. II (aa) ne manque point d'esprit (bb) a 'lesprit prompt, ce Mr. Wallis, et dans (aaa) de (bbb) les controverses il tasche (3) Ce Mr. Wallis tesmoigne certes d'avoir 1'esprit prompt, et (aaaa) c'est chose (bbbb) il y a du plaisir a veoir (aaaaa) il (bbbbb) comme il tasche a toute force add. de 6 graces |tresh. add.\ (1) des nouveaux (2) de 6 Fermat (1) qui {—} (2) de 8-13 que (1) celle des spirales la (2) la precedente des spirales. (a) II (b) Ne luy donnons pas la peine de (c) ne luy donnez pas la peine Monsieur d'en (d) Pour ce qui est de la (aa) superficie (66) proposition dont j'estois aucunement en doute je ne voudrois pas qu'il prit la peine d'en escrire la demonstration separement, (aaa) mais insistez tousjours (aaaa) qu breaks off (bbbb) a fin qu'il (bbb) pour me la faire veoir, mais (aaaaa) insistez (bbbbb) ayons plustost quelque (ccccc) plustost Mr. ayant le traite entier |et tant d'autres excellens (aaaaaa) de ce (aaaaaaa) (grand) (bbbbbbb) bel esprit (bbbbbb) ouvrages de ce grand geometre add] dont vous pourrez obliger le public (aaaaaaaa) Mr puis (bbbbbbbb) lors qu'il 13-10,14 promis. (1) \(a) Cette proposition touchant la superficie du conoide 31

histoire: i.e. [PASCAL], Historia trochoidis sive cycloidis, Paris 1658. This work was generally ascribed both to Pascal and to Roberval. 32 extrait: i.e. FERMAT-CARCAVI 11.1659/60; HUYGENS, CEuvres completes III, 39-40; FERMAT, (Euvres (1891-1912/22) II, 445-6. This was quoted by Carcavi in CARCAVIHUYGENS [25.II]/6.III.1659/60. 33 precedente: cf. FERMAT-CARCAVI VIII.1659; HUYGENS, (Euvres completes II, 5368; FERMAT, CEuvres (1891-1912/22) II, 438-40. This was quoted by Carcavi in CARCAVI-HUYGENS [3]/13.IX.1659.

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2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660

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tout a fait sans raison de Monsieur Schooten de ce qu'il n'a pas fait mention de luy en publiant ses lieux plans.34 Car encore qu'il n'ayt jamais veu ce que Mr. de Fermat en avoit escrit comme il m'a assure tousjours, il 1'a bien sceu et pourtant il n'auroit pas deu le dissimuler. Si je n'estois apres a faire la description de mon horologe avec ce que j'y ay adjouste nouvellement je vous en ferois part des a cet heure, mais puis que j'espere de 1'achever bientost et de la faire imprimer je vous prie de ne 1'exiger pas auparavant. Je voy plus que jamais par ce dernier escrit de Wallisius les inconvenients et disputes qui en peuvent naistre lors que des inventions de quelque consequence vont de main en main devant que d'estre publiees. Apres que j'auray acheve cela je vous feray aussi veoir ma methode de mesurer les lignes courbes la quelle je ne croy pas que personne encore s'est imaginee. Le traite35 de Mr. de Wit est imprime et je chercheray quelque occasion pour vous 1'envoyer comme toutes celles par ou je pourray tesmoigner

parabolique (6) Je suis bien loin de craindre (oa) qu'il en la dedans (66) qu'il y puisse avoir en cela quelque chose qui soit (aoa) en (bbb) a mon prejudice, ou outre que nos propositions sont desia connues de plusieurs, (aaaa) je (bbbb) j'y croy (en) (cccc) je scay que Mr. de Fermat ( ) des belles clioses luy mesnie. 11 Si je m'estois apres del.\ || Si je n'estois apres . . . s'est imaginee. || Monsieur de Fermat ne se plaint pas tout a fait sans raison de Monsieur Schooten . . . il n'auroit pas deu le dissimuler. 11 Le traite de Mr. de Wit . . . (2) II ne se plaint pas . . . 1 tout a fait add. 3 assure |tousjours add. , il (1) me semble que la candeur requiert (a) que (b) qu'il (c) que 1'autrement (2) 1'a bien sceu (ao) neanmoins (bb) et pourtant (aaa) ce qu'il n'auroit pas deu (bbb) il n'auroit pas deu le 5 horologe (1) et de (2) avec ce que j'y ay adjouste (a) de (b) nouvellement 8 auparavant. (1) J'ay (2) Je 9 de | Mr. add. and del] Wallisius 9 en (1) peuvent (2) naissent (3) peuvent naistre 12 je (1) scay bien (2) ne 14 est (1) desia sorty (2) imprime 15 1'envoyer (1) et (2) comme (a) toute autre par la quelle (b) toutes celles par ou je pourray |vous del. tesmoigner 34

en publiant . . . plans: i.e. SCHOOTEN, Exercitationum mathematicarum, liber III. Continens Apollonii Pergaei loco, plana restituta, Leiden 1656. 35

Le traite: i.e. WITT, Elementa curvarum linearum, ed. Frans van Schooten, Amsterdam 1659. 10

3. HUYGENS to WALLIS, [21]/31 March [1659]/1660 combien j 'estime I'honneur d'estre Monsieur &c.

3. CHRISTIAN HUYGENS to WALLIS [21J/31 March [1659]/1660 Transmission:

C Preliminary draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 736, 1 p. (our source).—printed: HUYGENS, (Euvres completes III, 58. c Copy of draft: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 3611, f. 149r. Reply to: WALLis-HuYGENS 24.XI/[4.XII].1659. Answered by: WALLIS-HUYGENS 31.VIII/[10.IX].1660. Wallis's letter of November 1659 to which this was a reply only reached Huygens on [10]/20 March 1660.

Wallis. 31 Mart. 60. 20 martio accepi.36

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Dat ick wel gedacht had dat hy niet gesien moet hebben 1'examen &c. dat daer sijn fauten in staen. Sendt het hem. Wrens demonstratie elegans, ingeniosa, sed clarior potuisset esse in prop, bedanck voor d'eer die hy my doet. mittat in Galliam. Heuratius. In Cissoide apparet vis methodi. Horologii mutatio quam dicit nova non est. talia multa hinc in Angliam missa38. Additio mea tota constructione melior. Methodus curvarum. Sa-

2 Monsieur &c. 27.° Mars 1660. c 6 ick (1) g breaks off (2) wel 8 sed (1) obscuri breaks off (2) clarior 9 Galliam. \ ( 1 ) Robervall (2) de Robervalli suspicione {—} quod insinuas { } fatum est ego expertus in horologio {—} hoc me invenisse (ut) et de (a) Saturno (6) Saturni annulo del} Heuratius 36

accepi: i.e. WALLIS-HUYGENS 24.XI/[4.XII].1659. l'examen: i.e. [PASCAL], Recit de 1'examen, Paris 1658. 38 Horologii mutatio . . . missa: on this error on Wallis's part cf. HUYGENS, (Euvres compleies XVII, 27, n. 3. 37

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turni Systema39 me misisse. Balii40 observationes frater41 ejus mihi communicavit42 ad Jun. 17. 1659. Quomodo balteum f? pingat. Exemplum ad Schotenium ipse Broun43 misit44. Nelio45 de telescopiis. 6 diam. non eo modo qui est in systemate46. Dicat et ipse de suis telescopiis. Contigua vitra sunt ocularia. nulla mine diaphragmata. dicuntur optima habere, nominatim Boilius47. Guisonius48 retulerit. Vlack49 spreecken vande exempl. Examinare Wall, de Cycloide.50 De Cissoide more veterum demonstratio51.

2 pingat. |Error in propositione de Conoide del] Exemplum 5 habere, (1) prae (2) nominatim 6 Guisonius (1) dix breaks off (2) retulit. (3) retulerit. 39

Saturni Systema: i.e. HUYGENS, Systema Saturnium, The Hague 1659. Balii: i.e. William Ball (c.1627-90), astronomer and member of Gresham College, later fellow and treasurer of the Royal Society. 41 frater: i.e. Sir Peter Ball (d. 1680), secretary of state to the Queen in the reigns of Charles I and Charles II, DNB. 42 communicavit: i.e. BALL-HuYGENS 17/[27].VI.1659. This apparently detailed description of William Ball's observations of Saturn has not survived. Cf. WALLISHUYGENS 22.XII.1658/[1.I.1659] and HUYGENS, Brevis assertio systematis Saturnn sui, The Hague 1660, 7; HUYGENS, CEuvres completes XV, 446-7. 43 Broun: i.e. Samuel Brown (d. 1665), English bookseller, who established his business in The Hague near to the Anglican church in 1647. Cf. HuYGENS-ScHOOTEN [18]/28.VL1656; HUYGENS, CEuvres completes I, 440. 44 Exemplum . . . misit: i.e. the copy of Wallis's Tractatus duo destined for Schooten. Cf. WALLIS-HUYGENS 24.XI/[4.XII].1659 and HUYGENS-SCHOOTEN [9]/19.III.1660; HUYGENS, CEuvres completes III, 44. 45 Nelio: i.e. Sir Paul Neil(e) (1613-86), courtier and astronomer, founding member of the Royal Society. Cf. WALLIS-HUYGENS 24.XI/[4.XII].1659. 46 in systemate: see HUYGENS, Systema Saturnium, The Hague 1659, 79; HUYGENS, CEuvres completes XV, 345-6. 47 Boilius: i.e. Robert Boyle, q.v. Cf. HuYGENS-MoRAY [14J/24.VI.1661; HUYGENS, CEuvres completes III, 283-4. 48 Guisonius: i.e. Pierre Guisony, medical practitioner in Avignon. 49 Vlack: i.e. Adriaen Vlacq (16007-67). Bookseller, first in London, then in Paris and finally The Hague. From 1626 to 1633 he worked in London on the calculation and edition of a table of logarithms. 50 Wall. de Cycloide: i.e. WALLIS, Tractatus duo, Oxford 1659. 51 De Cissoide . . . demonstratio: Huygens sent Wallis his proof of the quadrature of the cissoid (HUYGENS, CEuvres completes II, 170-3), which he had completed already in April 1658, some time before the appearance of the third part of Mechanica (London 1671), in which it was subsequently printed. In the second edition it was inserted in part two of this work; see WALLIS, Opera mathematica I, 906. 40

12

4. CARCAVI to WALLIS, June? 1660 4.

PIERRE DE CARCAVI to WALLIS June ? 1660 Transmission:

Manuscript missing. Existence and date: Mentioned in CARCAVi-HuYGENS [15J/25.VI.1660. Carcavi apparently sent this letter via Huygens in reply to the letter from Wallis, which reached him at the same time as a copy of his Tractatus duo, Oxford 1659. Reply to: WALLis-CARCAVi 1659?. Enclosure to: HUYGENS-WALLIS [5J/15.VII.1660.

5.

PIERRE DE CARCAVI to CHRISTIAN HUYGENS Paris, [15]/25 June 1660 Transmission:

C Letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 754, 6 pp. (our source).—printed: HENRY, 'Pierre de Carcavy', Bullettino di bibliografia XVII (1884), 317-79, 346-50; HUYGENS, (Euvres completes III, 85-90. Reply to: HuYGENS-CARCAVi [17J/27.III.1659/60. Answered by: HUYGENS-CARCAVI [5]/15.VII. 1660 (HUYGENS, (Euvres completes III, 97-8). Huygens transmitted a copy of this letter to Wallis as enclosure to HUYGENS-WALLIS [5]/15.VIL1660.

de Paris ce 25e. Juin 1660. Monsieur a mon retour de la campagne ou J'ay este oblige de demeurer Environ trois moys pour des affaires qui ne m'ont laisse aucun loisir de me donner 1'honneur de vous escrire J'ay trouve un petit livre52 de Mr. Fermat qu'il vous envoye, il m'a aussy fait tenir pendant ce terns un autre pe52

livre: i.e. [FERMAT], De linearum curvarum cum lineis rectis comparatione dissertatio geometrica, autore M. P. E. A. S., Toulouse 1660; FERMAT, (Euvres (1891-1912/22) I, 211-54; Varia opera mathematica, 89-103.

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tit traicte53 de solutions, problematum geometricorum per curvas simplicissimas et unicuique problematum generi proprie convenientes. que Je vous feray coppier si vous le desirez, II y fait voir plusieurs fautes de Mr. Descartes dans sa geometrie54 dont Mr. Schooten n'a dit mot55. Voicy Encore Trois de ses propositions. 1. Data quadratura hyperboles, datur circulus aequalis superficiei curvae paraboles circa applicatam rotatae.

Sit data parabola AD, cujus axis AE, applicata seu semibasis DE, rectum latus ABC, quaeritur circulus aequalis superficiei curvae solidi quod ex rotatione figurae ADE, circa applicatam DE tanquam immobilem circumductae conficitur. Bisecetur latus rectum AC, in B, et axi AE, ponatur in directum recta EF aequalis rectae AB, seu dimidio recti lateris, et jungatur DF, Exponatur separatim recta IQ aequalis axi AE, cujus dupla sit recta IR, fiat ut FE, sive AB, ad DF ita recta QI, ad rectam QH, et a puncto H, ducatur HG, perpendicularis ad HIR, et fiat HG, aequalis rectae DE, per punctum

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5 Encore (1) deux (2) Trois 53

traicte: i.e. FERMAT, De solutions problematum geometricorum per curvas simplicissimas et unicuique problematum generi proprie convenientes dissertatio tripartita, Manuscript 1657/58; FERMAT, CEuvres (1891-1912/22) I, 118-31; Varia opera mathematica, 110-15. 54 Descartes dans sa geometrie: i.e. DESCARTES, La geometrie, Leiden 1637. 55 Schooten n'a dit mot: i.e. in Geometria a R. Des Cartes, ed. Frans van Schooten, Leiden 1649. 14

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/, tanquam verticem describatur Hyperbole cujus transversum latus sit recta IR, centrum Q et transeat hyperbole per punctum G, et sit IG.\ Describatur item alia hyperbole separatim cujus transversum latus MN, sit aequale quartae parti

recti paraboles lateris, hoc est quartae parti rectae AC, centrum vero sit V, rectum latus OVP, aequale transverse lateri, sit autem hyperbole ita descripta MK, cujus vertex M, axis ML, qui continuetur donee recta ML, sit aequalis axi 15

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5. CARCAVI to HUYGENS, [15]/25 June 1660 paraboles AE, et ducatur perpendicularis seu applicata LK, a rectangulo sub QH, in HG, deducantur duo spatia hyperbolica IGH, MKL, quorum quadraturae supponuntur, et quod supererit aequetur quadrato, Diagonia istius quadrat! erit radius circuli superficiei curvae, cujus dimensionem quaerimus, aequalis. 2. Esto cyclois primaria ANIF, cujus axis AD, semibasis DF, Et ab ea formentur aliae curvae vel extra ipsam vel intra, quarum applicatae sint semper in eadem ratione data, ad applicatas primariae cycloidis. Ex. grat. in curva exteriori AMHG, ducantur applicatae GFD, HIC, MNB, ratio autem GD, ad DF, sit data et sit semper eadem quae HC,

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ad CI, et MB, ad BN. In curva autem interiori AROE, ratio FD, ad DE, sit data, et sit semper eadem quae 1C, ad CO, et NB, ad RB. Dico contingere ut curvae Exteriores qualis est AMHG, sint semper aequales aggregate lineae circularis et lineae rectae. Curvae autem Interiores, qualis est AROE, sint semper aequales parabolis primariis sive archimedeis. Theorematis generalis enuntiationem, quando volueris exhibebo, immo et demonstrationem.56

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10 ad applicatas primariae cycloidis. add. in margin 12 semper (1) et s breaks off (2) eadem 56

1. Data quadratura . . . demonstrationem: i.e. FERMAT-CARCAVI 7.VI.1660; FER-

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[3]

5. CARCAVI to HUYGENS, [15J/25 June 1660 3. Voicy 1'extrait d'une sienne lettre.

Pour me sauver un peu de 1'accusation57 de Mr. de Zulychem, qui dit que mes spirales n'ont pas des proprietez qui soyent autrement considerables vous pourrez si vous voulez luy proposer celle qui suit. Soit le cercle BCDM, duquel le centre A, et le Rayon AB, et soit la spirale BOZA, de laquelle la propriete soit telle, BA, est a AO, comme le quarre de toute la circonferance BCDMB, au quarre de la portion de la mesme circonferance CDMB, cette spirale par mon Theoreme general est esgale a une parabole en laquelle les cubes des appliquees sont en mesme raison que les quarres des portions de 1'axe, laquelle parabole est esgalle a une ligne droitte, J'espere que cette propriete suffira pour me reconcilier avec Mr. de Zulychem. Et puisque je luy cede touts mes droits sur les surfaces courbes des spheroides et conoides, Je souheterois qu'en revenche il m'indiquat s'il scait aucune surface courbe esgale a un quarre par voye purement geometrique et pareille a celle dont je me suis servi en donnant des droittes esgales a des courbes.58 MAT, CEuvres (1891-1912/22) II, 446-8. 57 l'accusation: i.e. in HuYGENS-CARCAVi [16]/26.II.1659/60; HUYGENS, CEuvres completes III, 26-8; 27. Cf. FERMAT-CARCAVI IX. 1659; FERMAT, CEuvres (18911912/22) II, 441-4. The latter was quoted by Carcavi in CARCAVI-HUYGENS [3J/13.IX.1659; HUYGENS, CEuvres completes II, 538-40. 58 3. Pour ... courbes: i.e. FERMAT-CARCAVI 7.VI.1660; FERMAT, CEuvres (1891-

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Voila Monsieur ce que J'ay receu de Mr. de Fermat depuis que Je n'ay eu 1'honneur de vous escrire59, Aprez quoy je vous supplie tres humblement agreer que je vous dise quelque chose de Mr. Walls, et vous en userez avec luy de la sorte qu'il vous plaira, n'ayant pas cru que je deusse respondre autre chose ny a son livre60 ni a une lettre61 qu'il m'escrivit au mesme terns qu'il m'a este rendu, qui fut le Jour avant mon depart de cette ville, que ce qu'il vous plaira de voir par la lettre62 que je luy escris, II est vray que J'ay este surpris de son precede, et que Je n'eusse pas attandu qu'il en diit user| de la sorte Envers Mr. D'Ettonville et par consequent En mon endroit ne luy en ayant Jamais donne aucun sujet, que seroit ce si nous avions fait imprimer non seulement les lettres qu'il m'a escrit, mais encore celles qui sont entre les mains de Mr. de Roberval qui Justifient et ses paralogismes, et son aveuglement, pour ne pas dire davantage, a ne s'en point corriger, II ne faudroit point d'autre responce a toutes ses inpertinences, Et vous verriez Monsieur qu'en ce qui concerne les problemes du dit Seigneur d'Ettonville il n'a pas seulement failly mais encore a advoue qu'il ne pouvoit pas y donner davantage de satisfaction, aprez cela le livre estant imprime il veut qu'on croye qu'il ne luy a servi de rien pour se corriger, et ce qui est le plus outrageux qu'on a pris de luy ou d'autruy ce qu'il n'a Jamais sceu, il faut avoir bien peu de sincerite, pour faire paroistre aux yeux de tout le monde des bagatelles et des bassesses de cette nature, pour moy Je ne scaurois concevoir les raisons qui 1'ont porte a cela, Mr. de Roberval m'a bien dit avoir escrit a un de ses amis quelque chose sur les fautes qui sont tant dans son livre Intitule Elenchus geomet. hobbian.^ que dans son arithmetique des infinis64, mais il n'a rien dit sinon qu'il y avoit telle et telle faute, et II ne 1'a point fait imprimer, Et quand cela seroit qu'est ce qu'il y auroit de commun avec le livre de Mr. Dettonville et la maniere toute genereuse dont il en a use, car il ne s'est pas contente de donner seulement le terns porte dans son deffi, 4 deusse |luy del.\ respondre 11 imprimer (1) ses (2) non 1912/22) II, 448-9. 59 Je . . . escrire: see CARCAVI-HUYGENS [25.IIJ/6.III.1659/60; HUYGENS, (Euvres completes III, 38-9. 60 livre: i.e. WALLIS, Tractatus duo, Oxford 1659. 61 lettre: i.e. WALLIS-CARCAVI 1659?. 62 lettre: i.e. CARCAVI-WALLIS VI?. 1660. 63 Elenchus geomet. hobbian.: i.e. WALLIS, Elenchus geometriae Hobbianae, Oxford 1655. 64 arithmetique des infinis: i.e. WALLIS, Arithmetica infinitorum, Oxford 1656. 18

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5. CARCAVI to HUYGENS, [15J/25 June 1660 mais encore trois moys davantage, durant lesquels Mr. Vallis ny personne [5] autre n'ont fait rien paroistre de ce qui| avoit este demande, aprez quoy il a donne Jusques a ses principes et a ses methodes, Et pour tout cela ce Brave professeur traitte en pedant des personnes de condition, et cherche a leur dire des Injures sur chaque mot qu'il tourne a sa fantesie. II impute a crime d'avoir propose un prix, ad pompam facere visum est. II chicane sur des clauses que nous avons mis, qui ne dependent neanmoins que de nostre volonte, et veut que nous y ayons mis de 1'equivoque, affigat (dit il) quam velit mentem verbis suis, qu'il nous connoit mal! Pourquoy cette longue et inutile apologie de Toricelly repetee en tant d'endroits, que nous pouvons facilement convaincre de faux et de ridicule par les lettres mesme originales de Toricelly que nous avons Entre les mains, Et pouvoit on dire ce qui s'est passe dans la recherche de la ligne dont il estoit question qu'en rapportant fidellement ce qui est icy connu de touts les geometres, Mr. Walis vouloit il qu'on mentit comm'il a fait en tant d'endroits de son livre. Quand II trouvera quelque chose nous ne dirons pas qu'il ne 1'a pas trouve, mais quand nous en aurons veu les demonstrations donnees par un autre, nous dirons librement et en verite qu'il n'en est pas 1'inventeur. Je vous serois trop inportun si Je vous disois tout ce qui ne devroit point estre dans ce livre, Je n'ay fait que le parcourir, et ce que J'y trouve encore de plus beau en achevant de le lire c'est qu'il veut que Mr. D'Ettonville ayt pris de luy ce qu'il a de meilleur, avec quel front me peut il dire cela ayant eu toutes ces demonstrations avant que d'avoir receu aucune nouvelle d'Angleterre. En voyla assez s'il vous plaist et mesme trop, dont| vous me per[6] mettrez de vous dire encore une fois que vous userez comm'il vous plaira, mais je crois que la chose ne vaut la peyne d'en parler davantage, la verite n'ayant point besoin d'autre deffence que d'elle mesme, Je suis de tout mon coeur.

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Monsieur Vostre tres humble et tres obeissant serviteur Decarcavy 9 neanmoins add. 20 librement (1) en (2) et 25 demonstrations avoir que d'avoir corr. ed.

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6. HUYGENS to WALLIS, [5]/15 July 1660 Mr. Boulliaud65 qui ma promis de vous faire tenir cette lettre me presse si fort que je n'ay eu loisir de la relire. Apres avoir Escrit cette lettre J'ay trouve un imprime66 de 1'annee 1640. qui Justine que Mr. de Roberval a pense le premier a la cycloide.

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CHRISTIAAN HUYGENS to WALLIS [5]/15 July 1660 Transmission:

C Preliminary draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 759, 1 p. (our source).—printed: HUYGENS, (Euvres completes III, 96; FERMAT (Euvres (1891-1912/1922) IV, 130. c Copy of draft: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 36 II, f. 157r-157v. W Excerpt from missing letter sent (quoted in WALLIS-OLDENBURG 4/[14].X.1673): LONDON Royal Society Early Letters W2, No. 14, p. 2 of 4 pp. (our source).—printed: Philosophical Transactions No. 98 (17 November 1673), 6148; HUYGENS, CEuvrcs completes VII, 342; OLDENBURG, Correspondence X, 278 (Latin original), 281 (English translation). Answered by: WALLis-HuYGENS 31.VIII/[10.IX].1660. Enclosures: CARCAVI-WALLIS VI?. 1660 and (a copy/part copy of) CARCAVi-HuYGENS [15]/25.VI.1660.

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Wallis. ic send de apologie67 van M. de Carcavy, en den brief68 ut videas quid

7 send (1) den brief (2) de apologie 7 Monsieur |de add] Carcavy 7 brief \ ( 1 ) quorum (2) ut videas .. . reprehendant add. . dat 65

Mr. Boulliaud: i.e. Ismael Boulliau (1605-94). imprime: not identified. 67 apologie: probably CARCAVI-WALLIS VI?.1660. Cf. WALLIS-HUYGENS 31.VIII/ [10.IX].1660. 68 den brief: presumably CARCAVl-HuYGENS [15J/25.VI.1660. Cf. WALLlS-HliYGENS 31.VIII/[10.IX].1660. 66

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6. HUYGENS to WALLIS, [5]/15 July 1660 in tuo libro69 reprehendant. dat ick uyt Brouncker verstaen heb70 dat hij mijn laetste71 heeft ontfangen. Fermat send my een boeck72, forte etiam ad vos, de Curvarum cum rectis comparatione.precipuum in eo vande parabolois die Heuraet73 hier en Nelius74 apud vos rectae aequavit. quorum scripta mirum est ilium non vidisse. veruntamen et alias curvas ex ilia paraboloide enatas subtili admodum ratione rectificare docet. scribe quid novi apud vestrates ubi jam studia si unquam florere incipient, restituta patriae tranquillitate, de qua tibi plurimum gratulor. Exemplaren van 't systema75 alhoewel het laet is, waer daer nae te vernemen. Schoten doot.76 (W)

Fermatii libellum novum simul ad me misit Carcavius, de Curvarum linearum cum rectis comparatione; in quo praecipue agitur de Paraboloide

2 ontfangen. (1) Caxcavy send my een boeck van Fermat, (2) Fermat send my een boeck, 3 comparatione. |sed eundem del. precipuum 3 eo (1) dat (2) vande 4 aequavit. (1) quanquam (2) quorum scripta (a) non oportet (6) mirum est ilium 6 scribe add. 7 studia (1) reflor breaks off (2) magis quam (3) si 8 tibi (1) max breaks off (2) plurimum 69

tuo libro: i.e. WALLIS, Tractatus duo, Oxford 1659. dat . . . heb: This letter from Brouncker to Huygens is missing. n mijn laetste: i.e. HuYGENS-WALLis [21J/31.III.1659/60. 72 een boeck: i.e. [FERMAT], De linearum curvarum cum lineis rectis comparatione dissertatio geometrica, Toulouse 1660. This tract was sent to Wallis by Kenelm Digby. See WALLIS, Algebra, 293; Opera mathematica II, 319. 73 Heuraet: on Heuraet's rectification of the semicubic parabola see HEURAET, Epistola de transmutatione curvarum, linearum, in rectas, in: Geometric, a Renato Des Cartes, ed. Fr. v. Schooten, Amsterdam 1659, 517-20. 74 Nelius: on William Neile's rectification of the semicubic parabola see WALLIS, Tractatus duo, Oxford 1659, 92; Opera mathematica I, 551-2. According to Wallis, Neile achieved this result already in 1657 and could therefore claim priority in this over Heuraet: Tractatus duo, 91; Opera mathematica I, 551. See also his Algebra, 293; Opera mathematica II, 319. 75 systema: i.e. HUYGENS, Systema Saturnium, The Hague 1659. 76 Schoten doot: Frans van Schooten the younger died in Leiden [19J/29.V.1660. 70

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7. WALLIS to DIGBY, 24 August/[3 September] 1660 ilia, quam jam ante apud nos Heuratius, apud vos Gu. Nelius rectae lineae adaequavit.

7. WALLIS to KENELM DIGBY London, 24 August/[3 September] 1660 Transmission:

E1 First edition of missing copy of missing letter sent: WALLIS, Algebra, 294-6 (our source). E2 Second edition: WALLIS, Opera mathematica II, 320-2. Probably in summer 1660, Kenelm Digby sent Wallis a copy of Fermat's tract De linearum curvarum cum lineis rectis comparatione. Within two days of its receipt, Wallis wrote the present letter to Digby in Paris, arguing that the rectification of the semicubic parabola had already been carried out by William Neile in 1657. Cf. WALLIS-OLDENBURG 4/[14].X.1673.

Illustrissimo Nobilissimoque Viro, D. Kenelmo Digby, Equiti Anglo. 5

August! 24. 1660. Londini. Illustrissime Vir,

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Videbam ego nudius tertius Fermatii quod miseras acutum opus77; quo Curvam Paraboloeidem, quam ego Semicubicalem appello, (cujus Ordinatim-applicatae sint in Diametrorum ratione Subtriplicatae-duplicata,) aequalem Rectae ostendit. Quod acute quidem & Geometrice (ut sua solet) peragit. Unum autem est aut alterum, quod monendum duxi. Primo quidem, Eandem ipsam Curvam Rectae aequalem, primus

9 sunt E2 13 Primum E2 77 Fermatii . . . opus: i.e. [FERMAT], De linearum curvarum cum lineis rectis comparatione dissertatio geometrica, autore M. P. E. A. S., Toulouse 1660.

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7. WALLIS to DIGBY, 24 August/[3 September] 1660 (credo) omnium ostenderat78 Gulielmus Nelius, Equitis Pauli films; suamque hujus Demonstrationem jam Anno 1657 divulgaverat: quod & a pluribus apud nos post ilium demonstratum est, & passim notum. Id Ipsum deinde, post annum circiter, ab Heuratio Batavo peractum est,79 quod (nescius, puto, quid apud nos factum fuerat) iteratae suae Cartesiani Operis Editioni subjunxit Schootenius. Eandemque rem in Epistola80, quam Tractatui de Cycloide (Anno praeterito a me edito) subjunxi, fusius prosecutus sum. Quae tamen omnia cum Fermatio, credo, minime innotuerint, non mirum est si ipse se primum invenisse putaverit. Alterum est, Quod, cum (praeter primas illas,) Secundas, Tertias, Quartas, aliasque in infinitum a primis derivatas, in Dissertatione sua memoratas, quas item a primis specie differentes appellat, rectis aequales dederit; non videtur Vir acutissimus animadvertisse, non alias illas esse Curvas, a primis diversas, sed earundem partes ab aliis aliisque punctis inchoatas. Quod sic brevi demonstro. Esto, in ipsius Fig. 11. paraboloides sua Semicubicalis, cujus vertex A, latus rectum AD, quod sit, verbi gratia, 9, (quae nempe recta, in quadratum interceptae diametri ducta, solidum efficiat Cubo ordinatimapplicatae aequale,) sitque Semibasis EF. Formenturque ad mentem suam ES, ER, EL, Secunda, Tertia, & Quarta, ab ilia Prima derivatae. Exponatur autem Parabola G\, cujus Latusrectum GH sit 4, (nempe | rectae AD.} Sumptisque (in Diametro) GK aequali lateri-recto, & GY ejusdem quadrupla, continuentur KQ, & Yd, quarum utraque sit semibasi EF aequalis, & ordinatim-applicentur KI, QP, YT, OX. I ostendit E2 19 aequalem,) E2 78

ostenderat: Neile's demonstration was described by Wallis in Tractatus duo, Oxford 1659, 92; Opera mathematica I, 551-2. This priority dispute came to a head again in 1673 when Huygens in his Horologium oscillatorium denied that Neile had fully succeeded in rectifying the curve. Wallis thereupon published a defence of Neile's claim along with further papers to this end from William Brouncker and Christopher Wren. See WALLIS, 'Primarn inventionem et dernonstrationern aequalitatis lineae curvae paraboloidis cum recta, anno 1657 . . . ' , in: Philosophical Transactions No. 98 (17 November 1673), 6146-9. 79 ab ... peractum est: HEURAET, Epistola de transmutations, curvarum linearum in rectas, in: Gcometria a Renato DCS Cartes, ed. Frans van Schooten, Amsterdam 1659, 517-20. Although the Epistola is dated 13 January 1659 (new style), we know that Heuraet in fact made his discovery in the summer of 1658. 80 Epistola: i.e. Wallis's Tractatus cpistolaris de cissoide et corporibus inde genitis, in: Tractatus duo, Oxford 1659, 75-121; Opera mathematica I, 542-69.

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7. WALLIS to DIGBY, 24 August/[3 September] 1660 His ita ad mentem suam constructis; Assume, tanquam ab ipso demonstrata, Curvae AE particulas, quantumvis minutas, (vel potius harum tangentes,) rectis Diametro Parallelis abscissas, respectivis in trunco Parabolico KIPQ ordinatim-applicatis proportionales esse; (nempe, curvae 5 particulas, sive harum Tangentes, ad correspondentes particulas basis, ita esse ut sunt respectivae ordinatim-applicatae in Parabola ad suum Latus-rectum:) Item, Curvae LE particulas, respectivis in Trunco Yr\9 ordinatim-applicatis similiter proportionales. His positis; AE Curva eousque deorsum continuetur donee basin ^v 10 aequalem habeat toti K6. Eaque ita in po~ divisa, ut in QY dividitur KO, erigantur inde perpendiculares; quarum altera occurret Curvae in E; occurrat altera in T.

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Manifestum est, ex suis demonstratis, ut AE Curva Trunco KIPQ, sic Curvam Av Trunco KIXO correspondere, & partes partibus: Adeoque ET Curvam Trunco QPTY, & Curvam TV Trunco YTX9-, & partes partibus respective. Sed, eidem YTX9 trunco similiter correspondet LE Curva, (quod ex 1 demonstratum E2 24

7. WALLIS to DIGBY, 24 August/[3 September] 1660 illo supra ostensum est,) & partes partibus. Ergo (per ipsius concessa & demonstrata) Curva LE eadem est atque TV. Et similiter ostendetur; si sumeretur [io dupla rectae GH (& av ut prius aequalis rectae EF,) esset TV Curva eadem atque RE. Sin /j,a rectae GH aequalis, esset TV eadem atque SE. Et in reliquis similiter. Non sunt igitur £"5, ER, EL, aliae ab AE Curvae, specie distinctae; sed, ejusdem continuatae, aliae atque aliae partes. Atque haec sunt, Vir Illustrissime, quae impraesentiarum monenda duxi. Caeterum Vale, Vir Illustrissime, Tuoque faveas, Observantissimo & Devinctissimo, Joh. Wallis. Ad meam Circuli Quadraturam quod spectat, quam (ex mea Arithmetica Infinitorum petitam) sub finem Epist. XXIII, sic designaveram81: Ut fac[296] tum ex quadratis numerorum imparium 3, 5, 7, 9, &c, in infinitum, ad factum ex iisdem quadratis unitate minutis: Sic Quadratum Diametri, ad aream Circuli. Puta, ut 9 x 25 x 49 x 81 x 121, &c, in infinitum; ad 8 x 24 x 48 x 80 x 120, &c, in infinitum. (Quae quadraturae meae nonnisi pars est; quatenus nempe ad numeros absolutes reduci possit.) Quod reponit82 D. Freniclius, Hanc aliam non esse quam Methodum approximandi, qualis est ilia Archimedis per inscriptas & circumscriptas; & ut nunquam perventuri sumus ad illud infinitum, ita nee ad perfectam Circuli Quadraturam hac via pertingemus: Omnino verum est, prout hie per numeros absolutos designatur. Sicut nee potest numerus surdus, puta Y/2, aliter designari in numeris absolutis, quam simili approximatione in infinitum; puta, per Unitatem cum annexis partibus decimalibus, ut 1.41421356 &c, (continuando radicis quadraticae extractionem in infinitum.) Nee tamen culpandus ille erit qui valorem numeri Surdi ^/2, numeris absolutis sic designandum dixerit: Quoniam ut numeris absolutis perfecte designetur (aliter quam per approximationem) numerorum natura non patitur; quique illud fieri postulet, postulat aSvvaTOV. Idemque & hie obtinet. Demonstraveram83 enim (Arithm. Infin. prop. 189, 27 radices E1 corr. ed. 31 fieri postulat, E2 81

designaveram: i.e. WALLIS-DIGBY 4/[14].III.1657/8 (Epist. 23). reponit: not ascertained. 83 Demonstraveram: i.e. WALLIS, Arithmetica infinitorum, 169-78; Opera mathemati82

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190, hujusque Scholio,) primus credo omnium, fortasse & solus; Rationem Quadrati ad Circulum Inscriptum, talem esse, ut nee numeris absolutis exprimi possit, nee etiam Radicibus Surdis (puta Quadraticis, Cubicis, Biquadraticis, &c.) sed neque ulla adhuc recepta Aequationum formula: Quippe ad hoc requiritur, ut numerus Impar in duos integros aequales dividatur; atque ut Aequationis formula reperiatur Laterali & Quadraticae intermedia; adeoque quae radices habeat plures quam unam, sed pauciores quam duas. Quorum utrumque est impossibile. Quod autem in radice Surda designanda fit; nempe, ut, quod exacte fieri non possit, nota aliqua insinuetur quasi factum; puta ^/2, vel -y/1 x 2; quo significetur terminus intermedius inter 1 & 2 in serie continue proportionalium 1, 2, 4, 8, &c, quae fit continua multiplicatione per communem Multiplicatorem 2; puta 1 x 2 x 2 x 2 &c: Idem hie faciendum ostendimus; nempe cum demonstratum sit, rationem Circuli ad Quadratum Diametri esse ut 1 ad d terminum intermedium inter 1 & | in serie 1, |, ^, ^, &c, quaefitex continua Multiplicatione (non quidem per eundem communem Multiplicatorem, ut in continue-Proportionalibus, sed) numerorum 1 x | x | x | &c, poterit ille (ad formam medii Proportionalis, inter 1 & 2, puta \/l x 2,) sic utcunque designari; mr11|| (vel alia forma simili.) Et propterea (prout latus ad diagonium quadrati est ut 1 ad -y/1 x 2, sic) Circulus ad Quadratum Diametri, ut 1 ad jnT 1 |. Quae vera est Circuli quadratura in numeris, quatenus ipsa numerorum natura patitur. Quam ad numeros absolutes (per continuam approximationem) sic reduci posse ut supra dictum est, ibidem demonstravimus84 Prop. 191. Quomodo autem in lineis exhibeatur, ostensum est85 ibidem Prop. 192, 193, 194. Quas autem memorat D. Fermatii rectas Curvis aequales, jam consideravimus.

10 v/2 aut /L x 2 E2 12 Multiplicatorem 2: Idem hie faciendum E2 16 eundem continuum Multiplicatorem E2 23 ut dictum est E2

ca I, 462-7. 84 demonstravimus: i.e. WALLIS, Arithmetica infinitorum, 178-93; Opera mathematica I, 467-76. 85 ostensum est: i.e. WALLIS, Arithmetica infinitorum, 193-8; Opera mathematica I, 476-8.

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8. WALLIS to HUYGENS, 31 August/[10 September] 1660 8.

WALLIS to CHRISTIAAN HUYGENS London, 31 August/fit) September] 1660 Transmission: W Letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 779, 4pp. (p. 3 blank) (our source).—printed: HUYGENS, (Euvres completes III, 126-8.

Reply to: HUYGENS-WALLIS [21J/31.III.1659/60 and HUYGENS-WALLIS [5J/15.VII.1660.

Accepi, Vir Nobilissime, binas a Te literas86 (alteras prid. Cal. Apr. alteras 15 Julii datas) quibus hactenus nihil responsi dedi: partim ob varias apud nos indies mutationes, quae animum alio avocarunt; partim quia nihil habui, nee dum habeo, quod Te dignum judicem, impertiendum. Ne tamen ingratus videar, aut neglectus arguar, rescribendum utcunque duxi. Priores quod spectat, quasque inibi memoras literas tardius ad Te quam vellem datas87; id hinc evenisse intelligo, quod quas ego Oxoniae eo qui indicatur die scripserim, Londini apud Bibliopolam aliquandiu haerebant; sed nee inde etiam ad Te ea quam sperabam diligentia perferebantur. Judicium vero quod de me meisque feceris, quanto candidius sit quam quod tulerint Galli, posteriores quas misisti literae notum faciunt. Quam autem memoras Narrationem Examinis88, non illam prius videram, aut de ilia inaudiveram quicquam, (necque ex nostris quern scio aliquis) quam Tu illam miseris. Literas vero quas ea memorat, non aliae sunt quam quarum ego in libro edito89 mentionem feceram; (si saltern unam excipias, quae hoc saltern petebat, ut indicare vellent num priores accepissent.) Ad quas omnes quum mihi ne tribus quidem verbis quicquam responsi dederint, (sed neque Examen illud, quod me spectabat maxime, transmitti curabant;) mirabar equidem; nee imputabam tamen (aliter, quam nuda facti narratione) me magis fortasse quam par erat fuisse neglectum. Quum vero 4 Te add. 15 edito add. 86

binas a Te literas: i.e. HUYGENS-WALLIS [21J/31.III.1659/60 and HUYGENS-WALLIS [5J/15.VII.1660. 87 literas ... datas: i.e. WALLIS-HUYGENS 24.XI/[4.XII].1659. This letter did not reach Huygens until [10]/20 March 1659/60. 88

Narrationem Examinis: i.e. [PASCAL], Recit de I'examen et du jugement des ecrits envoyes pour les prix proposes publique.me.nt sur le sujet de la, Roulette . . ., Paris 1658. 89 libro edito: i.e. WALLIS, Tractatus duo, Oxford 1659.

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in literis tandem D. Carcavii turn ad Te90, turn ad me91, (in posterioribus tuis hue transmissis92) tantas tragoedias excitari video, quamque sit irae impotens Vir Nobilissimus; mirari subit, quas sibi solent Nobiles Galli loquendi formulas indulgere, quamque a nostris diversas! (nisi forsan triobularem me nebulonem putaverint, quern eapropter corripiant quod Gallos ausim compellare.) Quicquid sit, negligenda certe mihi videntur isthaec omnia. Non moror enim quod ex Dettonvillio me desumsisse insinuet, quae de Cycloide scripseram;93 quum jamdudum Illustrissimum Brounkerum testem advocaverim94, qui per aliquot menses ante legerat isthaec omnia prout nunc extant scripta, (quodque amplius est, integrum ille calculum a capite ad calcem severe examinaverat,) et, magnam etiam (si memini) partem, prius impressa, quam prodiit illud Dettonvillii opus. Neque suum illud ex adverse mentiri me, me quicquam movet. Non novit me, Vir Nobilissimus, nee mores meos, qui hoc impingit. Quinam autem sint Errores illi, quos in meo sive Hobbii Elencho95, sive Infinitorum Arithmetical^, vel me vel amicorum meorum aliquem (utrum enim vult, non satis assequor) fassum esse asserat97 Robervallius, ego non intelligo; aut, quis sit ille amicorum. (Sed nee intelligo, quid hoc ad Rhombum; Quid ad Cycloidem ista?) Nondum enim adhuc mihi conscius sum, errorem ullum Geometricum me admisisse in utrovis libro, nedum fassum esse. (Sed nee in aliis a me editis; unum illud si excipias quod in Epistola98 ad Illustrissimum Brounkerum quae scriptis in Meibomium praefigitur revocavi.) Sed neque memini me ad Robervallium scripsisse unquam, (aut ad me ilium,) utut de illo nonnunquam| ad alios scripserim: [1] (unicam saltern si excipias Epistolam quam olim Gassendo, eove absente 11 calculum (1) severe (2) a 19 hoc | ad add.\ Rhombum 90

literis . . . ad Te: i.e. CARCAVI-HUYGENS [15]/25.VI.1660. literis ... ad me: i.e. CARCAVI-WALLIS VI?.1660. 92 in . . . transmissis: i.e. HuYGENS-WALLis [5]/15.VII.1660. 93 Non moror . . . scripseram: see CARCAVI-HUYGENS [15J/25.VI.1660. 94 advocaverim: Wallis claims that he sent Brouncker the first tract of his Tractatus duo, i.e. De cydoide, for examination in March 1659. See WALLIS, Tractatus duo, Praefatio, Sig. a4r; Opera mathematica I, 495, and WALLIS, Mechanicorum, sive tractatus de motu, pars secunda, Oxford 1670, 458-9; Opera mathematica I, 858-9. 95 Hobbii Elencho: i.e. WALLIS, Elenchus geometriae Hobbianae, Oxford 1655. 96 Infinitorum Arithmetica: i.e. WALLIS, Arithmetica infinitorum, Oxford 1656. 97 asserat: see CARCAVI-HUYGENS [15J/25.VI.1660: 'Monsieur de Roberval m'a bien dit avoir . . . II ne 1'a point fait imprimer.' 98 Epistola: i.e. WALLIS-BROUNCKER 5/[15].XII. 1656. This is the epistle dedicatory 91

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8. WALLIS to HUYGENS, 31 August/[10 September] 1660 Robervallio, indifferenter inscripseram"; quam periisse autumo100; vel, si exstet, nihil sane horum continet, aut etiam continere potuit;) ut mirum sit, quid ille ex literis meis ad Robervallium scriptis101 potis sit depromere quod hue spectet. Sed mittamus ista: et, quae de Torricellio dicta sunt: aliaque quae sequuntur. Quorum nihil, credo, est quod Te moveat. Ad hasce vero Carcavii literas nihil hactenus respondendum putavi, aut etiamnum puto. Non, quod Nobilissimum Virum neglexerim: sed, quoniam, quum eo stilo scriptae sint qui non admodum deceat ingenuos viros, cuique vix responderi posse videatur quin in aperta jurgia delabamur; condonare malim Nobiliss. Viro, ultra quam par est effervescenti, quam irritare. (quod et literis ad Digbaeum Equitem Parisios scriptis insinuavi). Quid interim sit, cur Pascalium a Dettonvillio videatur distinguere, non intelligo: Hos enim pro eodem (Te primum indicante) hactenus habui. Sin erratum sit, erroris fontem habes. Fermatii quern memoras libellum novum102 nuper vidi; quo eandem, quam prius turn nostri turn vestri etiam curvam aequaverant rectae, contemplatur. Quas autem in Dissertatione sua curvas alias inde derivatas et rectis comparatas, specie diversas existimat; non aliae sunt (aut ego admodum fallor) quam ejusdem curvae aliae atque aliae partes. In primaria siquidem, deorsum continuata, reperientur secunda tertia aliaeque in infinitum. Recta utique axi primariae parallela, quae inde distat | lateris recti, designat punctum quo incipit secundaria (deorsum in infinitum continuanda:) quaeque ab hac tantundem distat, tertiam ostendit; quaeque tantundem ab hac, quartam: et sic deinceps. Quod ubi examinaveris, facile deprehendes. Quod et literis ad Digbaeum Equitem scriptis demonstravi. Schotenii nostri mortem doleo. Quae de Cissoide ipse scripseris, lubenter videro. Quid Slusius praestitit103, ignoramus; neque enim quos miseris libros 17 alias specie diversas del.\ inde 27 lubenter (1) d breaks off (2) videro. to WALLIS, Adversus Marci Meibomii ... tractatus elenchticus, Oxford 1657. "Epistolam ... inscripseram: i.e. WALLIS-GASSENDI 31.VIII/[10.IX].1655. Cf. WALLIS-BROUNCKER 16/[26].X.1656 and WALLIS-BROUNCKER 20/[30].X.1656. 100 quam periisse autumo: Wallis was wrong in believing that this letter to Gassendi had gone missing. 101 ex literis . . . scriptis: Carcavi was possibly referring to the letters WALLISBROUNCKER 16/[26].X.1656 and WALLIS-BROUNCKER 20/[30].X.1656. 102 libellum novum: i.e. [FERMAT], De linearum curvarum ... dissertatio geometrica, Toulouse 1660. 103 praestitit: presumably in Sluse's Mesolabum, Liege 1659. 29

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9. FELL to YATE, 5/[15] February [1660/1 ?] nacti sumus. Quae de D. Pauli Nelii tubis sciscitaris104; partim, credo, Illustr. Brounkerus tibi corani indicavit; partim, spero, ubi in Angliam trajeceris, videbis. Interim Vale, Vir Nobilissime, Tuoque faveas 5

Londini, Aug. 31. 1660.

Observantissimo, deditissimoque Joh. Wallis.

Quae posthac ad me mittere dignaberis, inscribas licet D. Samueli Thomson105, Bibliopolae Londinensi in Caemiterio Paulino: Nam D. Underbill106 mortuus est. Dominum Dicas107, (cui dicis libros missos fuisse) quaerebam, sed 10 Parisiis jam agere intelligo; Collega suus Martinius108, negat (saltern non agnoscit) se accepisse libros ullos mini inscriptos.

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Eruditissimo Nobilissimoque Viro, D. Christiano Hugenio de Zulichem, Hagae Comitis.

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9. JOHN FELL to THOMAS YATE 5/[15] February [1660/1?] Transmission:

C Letter sent: OXFORD University Archives WPj/16/1, f. 92r-92v. On f. 92V in unknown hand: 'Letter of Bsp: Fell concerning Encroachments by the Town.'

7 D. (1) Th breaks off (2) Samueli 104

sciscitaris: cf. HUYGENS-WALLIS [21J/31.III.1659/60; WALLIS-OLDENBURG 4/[14]. X.1673. 105 Thomson: i.e. Samuel Thompson. 106 Underhill: i.e. Thomas Underhill (d. 1660), London bookseller at the 'Bible' in Wood Street and at the 'Anchor and Bible' in St Paul's Church Street. 107 Dicas: i.e. Thomas Dicas (d. 1669), London bookseller established at the 'Bell' and at the 'Hen & chickens' in St Paul's Churchyard. In 1660, he joined James Allestree (d. 1670) and John Martin. 108 Martinius: i.e. John Martin (fl. 1649-80), publisher and bookseller established at the 'Bell' in St Paul's Churchyard, publisher to the Royal Society. 30

9. FELL to YATE, 5/[15] February [1660/1 ?] The background to this and subsequent letters are numerous grievances of the University in respect of the city of Oxford, which apparently emerged following the failure of the mayor to take the oath 'to observe and keep all manner of lawfull liberties and customes of the University of Oxford'. During Convocation on 15 March 1660/1 ten grievances of the University of Oxford to the king were read, including the refusal of city officials to make the yearly oblation on St Scholastica's day, which was 10 February. See WOOD, Life and Times I, 370-2. In the course of pursuing the University's petition to the king, Cooper, Wallis, and Yate were required to spend considerable periods of time in London. See WOOD, Life and Times IV, 65.

Sir

Since my last I have considerd of several other greivances, occasiond by the towns very insolent and apparent encroachments on our priviledges, besides those you have breviats of. For instance the denying the homage on St. Scholasticas day which will be on the 10 of this moneth, in reference to which I have desird the vice-chancellor to give the maior109 an admonition before hand, that he may not pretend ignorance as he did in the case of the Oath. Secondly the Commission for making the river of thames navigable wherein we are grossely abusd, the profit in a manner coming wholy to the town and the Charges to us. Sly the licensing of Vintners which not only the wine office but the town have now usurpt. 41y the licensing of ale houses which they have engrost' contrary to our Charter & at this day have licensed about 700 as I am credibly enformed. Sly the proclaiming of the Market which they first assumed in 49 and think now they have prescription for. 61y the placing of the Market. This you know grounds a present contest between us & them. 71y the corporation of the brewers should depend on the university & take an oath to them, which is now left off. Sly licensing of Apothecarys &; Printers is usurpt by them. As lastly the royalty of having felons goods. I have spoken to Dr Wallis, to draw up with all speed, our pleadings to each of the fore mentioned breviats; which I suppose are all material & worth our insisting on. He tells me a memorable case of one Painter110 a Bayliffe of this town who in Queen Eliz: time took upon him to walk in the night: and being brought into the vicechancellors court & there sentenc't to pay twenty pounds (having it seems walkt ten times) he refusing to pay was put in prison, and having 24 pounds (1) it being (2) (having 109

maior: i.e. Sampson White. case of one Painter: for the case of the bailiff Thomas Painter cf. Wallis's Notes relating to the allowance of the claim in the Exchequer enclosed in WALLIS-JENKINS 21/[31].I.1667/8. 110

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10. WALLIS to DILLINGHAM, 16/[26] March 1660/1

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taken out his habeas corpus cum causa and removed himself, upon the universitys defence was cast, and fain to pay his fine. The particular of this I shall send you speedily, & suppose it may be a leading case, in all our other contests; and a better way then to insist upon a councel table order. As to the businesse of Justices of peace; the vicechancellor has ever since the late Act bin our, and the present Vice-chancellor111 owns himself as our. But the businesse of all our priviledgd persons, and other incidental occurrents cannot be dispatcht by any single man. By the Charter of Hen: 8112 which was confirmd by act of Parliament in Queen Eliz: time we are to have two besides him; named by our Chancellor, and this I think we should stand upon: I shall by my next give you a more particular account, be pleasd to excuse this very rude & hasty one, from Sir

Feb: 5.

Your assured friend & faithful servant J. Fell.

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For the reverend his honourd Friend Dr Yate, at Mr George Hills house in Hemlock Court in Sheer lane These D.D. London.

[92V]

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WALLIS to THEOPHILUS DILLINGHAM Oxford, 16/[26] March 1660/1 Transmission:

W Letter sent: OXFORD University Archives WP//3/15/9/30, 4 pp. (pp. 2 and 3 blank). On p. 4 in Wallis's hand: 'Letter to Cambridge. Mar. 16. 1660./I. To Dr Theophilus 1 himself (1) to the Kings bench (2) , upon 111

present Vice-chancellor: i.e. Paul Hood. Charter of Hen: 8: i.e. the charter of Henry VIII of 1 April 1523 ('Wolsey's charter'); cf. the Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II]. 1667/8. 112

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10. WALLIS to DILLINGHAM, 16/[26] March 1660/1 Dilingliam.' At top of p. 1 in Wallis's hand: 'To Dr Tlieopliilus Dillingham Master of Clare-hall Cambridge.' Answered by: DlLLINGHAM-WALLIS 1/[11].IV.1661.

Sir

I am desired by Mr Vicechancellor of Oxford113, in the behalf of this University, to request the favour from some of your University, of a particular information concerning the practise about the businesse of Alehouses & Victuallers. If I am not mistaken; the Government of Bread Wine Bear & other Victualls, having been anciently in the Mayor & Bailifs, & afterwards (perhaps) in a mixt jurisdiction of the Chancellor & Mayor jointly; about the beginning of Rich. 2. by a Grant in Parliament (upon the seisure of the Town liberties) that you should have Custodiam Assisae Panis Vini et Cervisiae, ac etiam plenam potestatem inquirendi et cognoscendi de Fore-Stallatoribus et Regratariis omnibus et super his punitionem debitam faciendi, ac etiam gubernationem correctionem et punitionem praemissorum et aliorum victualium quorumcunque, simul cum finibus forisfacturis et amerciamentis ex eis provenientibus, sicut Cancellarius et Scholares Universitatis Oxon ipsa habent in sua Villa Oxon et suburbiis ejusdem. (and by a Charter of 5° Ric. 2. in pursuance of that Parliamentary grant,) the Government of Victualls were settled on the University; & that ever since then have without interruption injoyed the licensing of all Taverns, Innes, Ale-houses, & Victuallers; & (I think) take

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12 [In left margin in Wallis's hand:] Rot. Parl. de anno 5° Ric. 2. memb. 49. in Tur. Lond.

2 Oxford, (1) to write to some of your (2) in 3 from some of your University, add. 4 practise |in your University del] about 5 the (1) power of the (2) Government of Bread Wine Bear fe other Victualls, (a) being (b) having been anciently in the (oa) Town, & then (66) Mayor 7 the (1) M breaks off (2) Chancellor & Mayor jointly; |it was del] about 16 by add. 113

Mr Vicechancellor of Oxford: i.e. Paul Hood (c.1585-1668), B.D. Lincoln College, Oxford 1617, D.D. 1623. Rector of Lincoln College 1620-68. Vice-chancellor of the University of Oxford 1660. 33

10. WALLIS to DILLINGHAM, 16/[26] March 1660/1

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the Recognizances of Butchers & Victuallers not to kill or dresse flesh in Lent, (but do not return those or the Recognizances of Alehouse-keepers, to the Quarter Sessions;) & grant licenses to sick persons yet for eating flesh, & to buchers to kill for such. If it be thus (as I presume it is) we should desire the favour of a Certificate thereof (or so much of it as you find so to bee) particularly & clearly set forth, (under the hands of Mr Vicechancellor or your Register or both;) in reference to a hearing at the Councell Table (betwixt this University & the City of Oxford,) which we expect very suddenly (within a fortnight or thereabouts;) Which being a businesse of common concernment to both Universities (to strengthen the hands each of other) hath, the rather induced mee to give you this trouble at present (having no acquaintance with any to whom I might more fitly apply myself in this particular; & having had experience formerly of your readynesse in this kind,) assuring you of a like readynesse from hence upon any the like occasion, so far as shall bee in the power of Sir

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From my house in Cat-street Oxon March. 16. 1660./1.

Your affectionate friend & humble servant John Wallis.

The shortnesse of the time makes mee request your answer as soon as can bee, directed either to Dr Hood our Vicechancellor or myself here; or at

1 Victuallers (1) not to dressin breaks off (2) for (3) not 2-3 (but do ... Sessions;) add. 5 should |{—} del.\ desire 5 a (1) Testimoniall to this purpose (or at lest (2) Certificate thereof (a) so (6) or so 6 find (1) thus (2) so 8 Table | which we add. and del.\ (betwixt 9 Oxford,) (1) within a (2) which we expect 12 rather (1) persuaded (2) induced 12 present (1) (not knowing to whom better to apply myself, as more acquainted & having had (2) bee breaks off (3) (not being acquainted with (4) (having no ... having had 14 your (1) forwardnesse in this (2) readynesse 15 upon any the like occasion, add.

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11. WALLIS: Humble Petition to Charles II, [March? 1661] lest, if it admit any delay, to be left for mee at Mr Clendons114 house in the Strand over against the New Exchange London.

11. WALLIS: Humble Petition to Charles II for a Grant of the Next Vacant Prebend in Christ Church, Oxford [March? 1661] Transmission:

W Paper sent: KEW The National Archives PRO SP 29/33, No. 66, 1 p. Wallis was appointed chaplain-in-ordinary to the king at the end of 1660 and was involved in negotiations between Presbyterians and Anglicans in London in early 1661. It is probable that he applied for the post of prebendary at the height of this activity around March.

To The Kings Most Excellent Majestic; The humble Petition of John Wallis Dr in Divinity, and Professor of Geometry in the University of Oxford,

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Sheweth, That your Petitioner, incouraged thereunto by your Majesties gracious favour, doth humbly crave, A Grant under the Great Seal, of the next Prebends place in Christchurch Oxon which shall be voyd; as an Addition to his place of Geometry Professor. And your Petitioner shall ever pray &c. 114

Mr Clendons: i.e. John Clendon, q.v.

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THEOPHILUS DILLINGHAM to WALLIS Cambridge, I/[II] April 1661 Transmission:

C Letter sent: OXFORD University Archives WP//9/R/7b/6, 2 pp. On p. 2 in unknown hand: 'Apr. 1. 1661. Dr Theoph. Dillinghams Letters to Dr Wallis. No. 6.' Reply to: WALLis-DiLLiNGHAM 16/[26].III.1660/1. The present letter enclosed a paper by Dillingham concerning the University of Cambridge's practice in respect of alehouses and victuallers. This was evidently in response to a request made by Wallis in his letter of 16 March 1660/1.

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I have sent you here inclosed a paper which I scribbled over this morning in answer to yours115 concerning our practise about the business of Alehouses, & Victualers. I am very sorry that I have been incumbred with so much business as I could not answer yours before this time, & should bee much troubled if your hearing bee past before these come to you. I have according to your desire added the Vicechancellers116 hand to it & have allso sent you a copy of a letter concerning Lent which it may bee your concernment at this time if you shall need anything els which I can procure for you as to this concernment or any other. I shall with much chearfulness bee very ready to serve your university or your self, remaining Your very loving friend Theoph. Dillingham

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8 may bee |may bee del. ed.\ your 115

yours: i.e. WALLIS-DILLINGHAM 16/[26].III.1660/1. Vicechancellers: i.e. Henry Feme (1602-62), who was vice-chancellor of the University of Cambridge in both 1660 and 1661; DNB. 116

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13. COOPER to WALLIS, 23 April/[3 May] 1661

13.

BENJAMIN COOPER to WALLIS Oxford, 23 April/[3 May] 1661 Transmission:

C Letter sent: OXFORD University Archives WPj/16/1, f. 70r-71av, lower half of f. 71ar-71av missing (f. 70V and 71ar blank). On f. 71av beneath address in Wallis's hand: '23 Apr 1661 Mr Cooper the Register letter'.

Sir

I conceive their will be noe necessity of my attending upon the University buisinesse at Whitehall the next Saturday; and therefore I sent up to you by Dr Lamplugh117 the Originall Order under Secretary Nicholas118 hand, least otherwise it might not have come safe to you. I left the Act of Convocation with Dr Yates119, which, I thinke agrees allmost in every particular with this which is enclosed. I made what enquiry I could to find out what estates the Bayliffs seized on of Stevenson120 and Clarkes121 who murdered themselves, concerning the first I could get noe information, though his friends were asked about it; of the last you have a particular; and allsoe the time when the present Bayliffs undertooke to place the Markett, which as I remember is what you left me in trust withall. I am

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Your very humble servant Ben: Cooper Oxon April the 23: 1661

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This morning your Children were very well.

10 about (1) , (2) it; 117

Dr Lamplugh: i.e. Thomas Lamplugh (1615-91), fellow of the Queen's College, Oxford. Created D.D. in 1660, he was later made bishop of Exeter and finally archbishop of York; DNB. 118 Nicholas: i.e. Sir Edward Nicholas (1593-1669), secretary of state to Charles I and Charles II; DNB. 119 Dr Yates: i.e. Thomas Yate, q.v. 120 Stevenson: not identified. 121 Clarkes: not identified. 37

14. [HOBBESJ: La duplication du cube, [June? 1661]

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These For the Reverend Doctor John Wallis at Mr Clendon's house a Tallow Chandler next door to the signe of the Horse-Shoe over against the New-Exchange London

[71av]

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[THOMAS HOBBES]: La duplication du cube [June? 1661] Transmission:

E1 Printed paper. Exemplar (with, handwritten corrections): CHATSWORTH The Devonshire Collection Hobbes MSS, letter 85, 2 pp. (our source). E2 WALLIS, Hobbius Heauton-timorumenos, Oxford 1662, 129-32 (English translation), figure on p. 130 (our source for the figure). Hobbes published La duplication du cube anonymously axound June 1661. He claime that it was printed in Paris, although Wallis describes this as pretence (Hobbius Heautontimorumenos, 127-8). It was reprinted with replies to objections by Wallis and Rooke as an appendix to Dialogue physicus (1661). Amended versions appeared in an appendix to Problemata, physica, London 1662 (Opera, philosophica IV, 378-82), and in Seven Philosophical Problems, printed in London 1682 (English Works VII, 59-62). The acronym is solved by Hobbes in the Dialogus physicus, London 1661, as 'Un Autre Que Roberval' (Opera philosophica IV, 295). As he explains in Seven Philosophical Problems, he had the paper printed in Paris 'on purpose to see what objections would be made to it by our professors of algebra here' (English Works VII, 59). As confirmed in WALLis-BaouNCKER? 23.VI/[3.VII].1661, the printed paper contained a diagram, supplemented here from the English translation in Wallis's Hobbius Heauton-timorumenos. The handwritten corrections to the present exemplar probably originate from Hobbes himself, who is known to have subjected the demonstration to numerous revisions. At the foot of the reverse is a refutation (in Hobbes's hand) entitled 'La Refufation monstrante que la droit YZ produite ne tombe sur P: c'est a dire que en la Triangle DYP la coste PD n'est pas egale a DC, mais plus court.' This is possibly the refutation referred to by Hobbes in Seven Philosophical Problems (English Works VII, 59).

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14. [HOBBESJ: La duplication du cube, [June? 1661]

(E1} La duplication du cube Par V. A. Q. R. Une ligne droite etant donnee, trouver entre elle & sa moitie deux moyennes proportionelles. Soit donnee la droite AB, dont le quarre ABCD, soit coupe en quatre quarres egaus par les deux droites EF, GH, qui s'entrecoupent au centre du quarre ABCD, au point I. Ainsi que les quatre cotes soient divises en deux parties egales aus quatre points E, F, G, H. II faut done trouver deux moyennes proportionelles entre DC & DF. Je tire les diagonales AC, BD, & decris les quatre quarts de cercle ABD, BCA, CDB, DAC, dont les arcs coupent lesdites diagonales en K, L, M, N. Ausquels points les arcs sont coupes, chacun en deux parties egales, ce qui est asses concu. Je produis BA, CD, aus points O & P jusques a tant qu'elles soient egales a AB, DC, chacune a chacune, Et ayant decrit le quart de cercle ADO, & tire la diagonale AP (qui coupera 1'arc DO en deux parties egales au point Q,} Et etant produite de 1'autre part en R, marquera BR egale au sinus droit de 45 degres, c'est a dire a la semidiagonale BI. Et par consequent SD est 1'exces de la plus grande extreme AD au dessus de la semidiagonale AS. Je coupe cette SD en deux parties egales en T. En AD produite, je pren DV egale a DF, & faisant T le centre & TV semidiametre, je decris le cercle VXYZ, coupant DC en X, DA en Y, & la droite RS produite en Z, Et dis que les deux droites DY, DX, sont les deux moyennes proportionnelles demandees entre DP, egale a DC, & DV, egale a sa moitie, DF. Car tirant les lignes droites VX, XY, Tangle VXY (dans le demicercle) sera droit; Et la droite XT, tiree & produite jusqu'a la concavite du cercle VXYZ tombera sur Z, parce que ST, TD, sont egales, & par [2] consequent) SZ, egale a DX, Et XZ sera le diametre du cercle VXYZ.

26 egale a AB corr. 26-27 , DF add.

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L'angle done XYZ dans le demicercle est droit; Et en tirant les droites YZ, VX, on fait VXYZ rectangle, & ses cotes VX, YZ paralleles. Or, si la droite YZ produite, tombe sur P, touts la ligne P, Z, Y, sera droite & parallele a VX, & les angles alternes YPX, VXP seront egaus. Aussi les angles YPX, & XYD, seront egaus, & les trois triangles rectangles PDY, YDX, XDV, seront semblables. Et par consequent, les quatre droites PD, DY, DX, DV, seront en la mesme raison continuelles. II faut done demonstrer que la droite YZ produite, tombe sur P. Soit tiree PV & coupee en deux parties egales en a. Soit tiree aussi la droite ab parallele a AV coupant PD en c. Soit encore tiree, Td, parallele a PD, coupant ab en d; & divisee, dc, en deux parties egales en g; sur le centre g, a la distance, ga, soit decrit le demicercle ahb, coupant PD en h, & ab en b. Cela fait, les deux droites tirees, ah, bh, feront un angle droit en h. Or ac est la moitie de DV, Et parce que dg, gc, sont egales, db, sera aussi egale a la moitie de DV, & ab sera la moitie de YV. Comme est done PD a DY, c'est a dire a la composee de DS & SY, ainsi est PC (la moitie de PD) a cb la composee des moities de DS & 5T, & par consequent Pb etant produite tombera sur Y. Et les droites hb, ha seront les moities des droites, XY, XV, Et XY etant divisee en deux parties egales en i, la figure Yihb sera rectangle, & Yb sera parallele a XV. Or yZ, est parallele a XV. Done, YZ, produite tombera sur P. Et (par ce qui a este demonstre) les quatre droites PD, DY, DX, DV, sont en une mesme raison continuelle. J'ay done entre une ligne droite donnee & sa moitie, trouve deux moyennes proportionelles. Ce qu'il falloit faire. Consect. Un Cube qui a pour cote la plus grande de ces deux moyennes, est le double du Cube qui a pour cote la moitie de la plus grande extreme. Car la raison de Cube a Cube est tripliquee de la raison de cote a cote; Et la raison de PD a DV est tripliquee de la raison de PD a DY.

1 tirant la (1) droite VZ (2) droites YZ, VX corr. ed. 40

15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661

(E2, figure)

15. WALLIS to [WILLIAM BROUNCKER ?] Oxford, 23 June/[3 July] 1661 Transmission:

W Letter sent: CHATSWORTH The Devonshire Collection Hobbes MSS, letter 51, 2 pp. (our source).—partly printed: ScHUHMANN, Hobbes. Une chronique, Paris 1998, p. 172. El First edition: Thomas Hobbes, Dialogue physicus, London 1661, 32-3; Opera philosophica IV, 288-91 (partly). 41

15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661 E2 Second edition: Thomas Hobbes, Problemata physica, London 1662, 119-20; Opera philosophica IV, 379-80 (partly). It would appear that after Wallis had received a copy of Hobbes's printed paper La duplication du cube he sent Brouncker the present refutation for his consideration. Further refutations were produced by Brouncker himself—see WALLIS, Hobbius Heautontimorumenos, Oxford 1662, 144—and by the professor of geometry at Gresham College, Lawrence Rooke, whose calculation is quoted by Hobbes in Problemata, physica, London 1662, 124; Opera philosophica IV, 382. Cf. Walter Pope, The life o f . . . Seth, Lord Bishop of Salisbury, London 1697, 118-19.

Oxoniae. Junii 23. 1661. Illustrissime Vir,

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Tradebatur mihi hesterna nocte, sub horam vespertinam, impressa pagina, ex Gallia (ut ferebatur) allata, et quidem idiomate Gallicano; Cui Titulus La Duplication du Cube, par V. A. Q. R.122 cum adjuncta Scheniate lineoso. Linearum vero magnam partem, et constructionis non exiguam, superfluam esse plane, nee ad problematis vel Constructionem vel Demonstrationem quicquam spectare, statim deprehendi: Rem ipsam autem infectam esse; et Demonstrationem paralogismo laborare.123 Quae visum est subjecta pagina breviter indicare; Meque interim profiteri Dominationis vestrae Observantissimum, Joh: Wallis. Summa dictorum in Pseudodiplasiasmo Cubi nupero, (rescissis quae turn in Schemate turn in Constructionibus sunt superflua,) haec est. Exposita AD recta, continuetur ad V, ut sit DV semissi rectae AD aequalis. Centro A, distantia AD, scribatur DO circuli quadrans, bisectus in Q: et QS, rectae AD perpendicularis. Bisecta vero SD, in T; centre T, ducatur per V circulus VXYZ: Cui occurrat DX (rectae AD perpendicularis) in XAD, in Y; et QS in Z.

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Affirmat, Rectas DY, DX, medias esse proportionales inter AD et DV. 122

La Duplication ... V. A. Q. R.: i.e. [HOBBES], La duplication du cube, [June? 1661]. Rem . . . laborare: cf. HuYGENS-MoRAY [25.XJ/4.XI.1661; HUYGENS, (Euvres completes III, 384. 123

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15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661

Quod sic conatur demonstrare. Ductis rectis VX, XY; erit angulus VXY in semicirculo rectus. Ductaque XT et continuata, (propter bisectam SD in T) occurret circulo in Z. Adeoque ducta Y Z] erit XYZ angulus in semicirculo rectus; ipsaque YZ rectae XY parallela. Producatur XD ad P, ut sit DP rectae AD aequalis. Si itaque YZ producta rectae PD occurrat in puncto P; erunt (propter similia triangula PDY', YDX, XDV,) rectae DP, DY, DX, DV, continue proportionales. Et consequenter (propter Cubos in ratione laterum triplicata) Cubus lateris DY subduplus [sic enim dicendum erat, non Duplus] cubi lateris DP, sive DA. Atque hactenus recte. Rectam vero YZ continuatam, puncto P occurrere; sic probare (frustra) contendit. Ducta PV, et bisecta in o; ducatur ab rectae DY parallela, rectam DP occurrens in c. Rectaeque ab perpendicularis Td. Bisecta vero dc in g; centro g ducatur per a, semicirculus ahb, rectae CD occurrens in h, rectaeque ab in b. Ideoque, propter turn ca aequalem semissi rectae DV turn eg semissi TD, erit ab semissi VY aequalis. Adeoque juncta Pb et continuata, occurret puncto Y. Ductisque bh, ba, rectis; erit an43

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15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661 gulus bha rectus; ipsaeque bh, ha, semissibus rectarum YX, XV, aequales et parallelae. Quae quidem vera sunt. Sed non item et sequentia. Nempe 5

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Bisecta itaque YX in i, junctaque ih: erit Yihb rectangulum; et Yb rectae XV parallela. Sed et YZ eidem XV parallela est. Ergo et YZ (sicut ipsa Yb) producta occurret puncto P. Haec ille. Sed male. Sequitur utique ex praedictis Yihb Parallelogrammum esse; sed non item, Rectangulum; (adeoque nee Yb rectae XV parallelam.) Non enim (quod ipsi fraudi fuit) quia bha angulum rectum esse ostenderat, ideo bhi rectum esse sequitur; nisi et simul demonstrarat, Rectam ah continuatam, ad punctum i pertingere. Quos neque demonstrat ille, neque verum est. Et quidem tantum abest ut ex argumento quicquam de mediis proportionalibus concludatur, ut, quantacunque sumatur DP (sive ipsi AD aequalis, sive major sive minor in quacunque ratione,) reliquis manentibus, eadem demonstratione non minus concludatur, easdem DX, DY', medias esse proportionates, inter eandem DV atque hanc DP quamlibet. Quod autem non modo indemonstratum sit, sed neque verum, quod affirmat, nempe DX, DY, medias esse proportionales inter DV, et DP rectae AD aequalem,) sic Demonstro.| Ponamus DV — 1; adeoque DA vel DP — 2. Cum itaque sint ejusdem circuli, turn AD radius, turn AS sinus graduum 45; erit AS = ^2; et SD = 2-^/2; et TD = 1-^2; adeoque TV = 2- ^2; et DY = 3-^2; et DX = •/: 3 — -/2. Ideoque tribus DV, DX, DY, quarta proportionalis (quam quidem abscindet YZ recta ad rectam DP continuata) erit 3 — ^/2 in \f: 3 — ^2 :, hoc est, 1,997 fere; minor quam DP = 2. Adeoque YZ producta occurret rectae DP, non quidem in puncto P, sed in puncto aliquot inter P et D. Et consequenter, cum sit XYZ angulus rectus, erit XYP recto major. Verum itaque non est, vel rectam YZ continuatam, ad punctum P pertingere, vel PYX aut bhi angulum rectum esse, vel hi eandem esse rectam atque ah continuatam, aut rectae VX parallelam, vel denique rectas DX, DY, medias esse proportionales inter DV, et DP rectae DA aequalem. Quod demonstrandum suscepi. 23 AS (1) radius (2) sinus 24 TV jhoc est TX del] = 2 - |^2; 44

[2]

16. FRENICLE to WALLIS, [October/November ? 1661]

16.

BERNARD FRENICLE DE BESSY to WALLIS [October/November? 1661] Transmission: C Part copy of missing letter sent (quoted in FRENlCLE-DlGBY [10J/20.XII.1661): LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 971, 1 p. (our source).—printed: HUYGENS, (Euvres completes IV, 44; HENRY, 'Recherches sur les manuscrits de Fermat', 695.

Answered by: WALLIS-FRENICLE XI/XII7.1661. Frenicle probably proposed this problem to Wallis in October or November 1661. Wallis subsequently sent Frenicle a solution (WALLIS-FRENICLE XI/XII7.1661), to which the latter then replied (FRENICLE-WALLIS [10J/20.XII.1661). Cf. FERMAT-[CARCAVI] XII?.1661; HUYGENS, (Euvres completes IV, 2-3; HENRY, 'Pierre de Carcavy', 351. Fermat's short account of the exchanges between Frenicle and Wallis was among various pieces which Carcavi sent enclosed with CARCAVI-HUYGENS [22.XII.1661]/!.1.1662; HUYGENS, (Euvres completes IV, 1.

Problema. Invenire duo Triangula Rectangula in numeris ita constituta, ut laterum circa angulum rectum differentia sit in utroque eadem; & quod in altero est majus duorum laterum circa angulum rectum, sit in reliquo Hypothenusa.

17. WALLIS to BERNARD FRENICLE DE BESSY [November/December? 1661] Transmission: w Part copy of missing letter sent (quoted in FRENlCLE-DlGBY [10J/20.XII. 1661): LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 971, 1 p. (our source).—printed: HUYGENS, (Euvres completes IV, 44-5; HENRY, 'Recherches sur les manuscrits de Fermat', 695. Reply to: FRENICLE-WALLIS X/XI7.1661. Answered by: FRENICLE-WALLIS [10]/20.XII.1661. This letter contained Wallis's reply to the problem Frenicle had posed him probably in October or November 1661. It was no doubt sent through the hands of Digby as was Frenicle's subsequent reply. The title was clearly supplied by Frenicle.

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18. FRENICLE to WALLIS, [[10]/20 December 1661] Solutio Clarissimi D. Wallisii.

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Esto duorum TViangulorum alterum BAG. alterum BCE. Sitque BC = 5 + x. BA — 5 — x. (ut sit 5 semisumma a;, semidifferentia laterum BC. BA.) Adeoque BCq. = 52 + 2.5x + x2, BAq. = 52 - 2.5x + x2 & horum differentia ACq. = 4.5ar, qui cum numerus quadratus esse debeat, oportet 5.x esse inter se, ut numeri quadrati.

Esto igitur 5 = a2, x = e2. Saltern 5 = ba2, x = be2. Ergo BC = ba + be2. BA = ba2 - be2. BCq = &V + 2.b2a2e2 + &2e4. BAq = b2a4 2.b2a2e2 + 62e4. Adeoque ACq. = 4.62a2e2. & AC = 2bae = AD. BD = ba2 - be2 - 2bae = BS. 6C = 2be2 + 2bae = CE. & CEq = 4.62e4 + 862ae3 + 462a2e2. Adeoque BEq = &V + 562e4 + 6.62a2e2 + 8.62ae3. Qui quum numerus quadratus esse debeat (etiam per b2 divisus) Quaerendum restat Quomodo investigandi erunt duo numeri a.e. ita constituti ut a4 + 5.e4 + 6.a2e2 + 8.a.e3 sit numerus quadratus. Interim suspicor (propter 8.ae3) num non casus sit impossibilis. Sed Nihil pronuncio. 2

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BERNARD FRENICLE DE BESSY to WALLIS [[10]/20 December 1661] Transmission:

C Part copy of missing letter sent (quoted in FRENlCLE-DlGBY [10J/20.XII.1661): LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 972, 2 pp. (our source).—printed: HUYGENS, (Euvres completes IV, 45; HENRY, 'Recherches sur les manuscrits de Fermat', 696. 10-11 CEq = 4.62e4 + 862ae3 + & 2 aV corr. ed. 46

19. WALLIS to BOYLE, 30 December 1661/[9 January 1662] Reply to: WALLIS-FRENICLE XI/XII7.1661. Frenicle sent the present response to Wallis's solution to the problem he had set as an enclosure to FRENICLE-DIGBY [10J/20.XII.1661; HUYGENS, CEuvres completes IV, 379. Digby probably forwarded it directly to Wallis. Cf. FERMAT-[CARCAVI] XII?.1661; HUYGENS, CEuvres completes IV, 2-3. A copy of the exchanges was made for Robert Moray who then sent them to Huygens with MoRAY-HuYGENS 30.1/[9.II]. 1661/2; HUYGENS, CEuvres completes IV, 34-6. See also MORAY-HUYGENS 24!/[3!I]. 1661/2; HUYGENS, CEuvres completes IV, 25-8.

Ad Clarissimi D. Wallisii Solutionem Responsum proponentis. Si absque alia conditione proponantur investigandi duo numeri a. e. ita constituti ut a4 + 5.e4 + Q.a2e2 + 8.ae3 sit numerus quadratus; facillima erit hujus Problematis Solutio. Sit namque a. quilibet numerus puta 2. e — 2.a. nempe 4. Erit a4 + 5.e4 + 6.a2e2 + 8.ae3 — 2704. numero quadrato [2] cujus radix 52. | Attamen non sufficit quaestioni ad quam solvendam numerus a. excedere deberet numerum e. in quo casu non ita faciles sunt inventu hi duo numeri a. e. In his autem perquirendis stat omnis quaestionis nodus.

19. WALLIS to ROBERT BOYLE Oxford, 30 December 1661/[9 January 1662] Transmission:

E1 First edition of missing letter sent: BOYLE, Works (1744), V, 511-12.—reprinted: BOYLE, Correspondence I, 473-5. E'2 Second edition: BOYLE, Works (1772), VI, 453-5 (our source). The present letter enclosed part of Wallis's Hobbius Heauton-timorumenos, the printing of which had not been completed at the time. This work takes the form of an epistolary tract addressed to Boyle.

Oxon. Dec. 30, 1661.

Sir, This, I suppose, may be at you time enough to wish you a happy new 47

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19. WALLIS to BOYLE, 30 December 1661/[9 January 1662]

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year; and many more to ensue. It brings with it a part of that in print124, which you had before an account of in writing125 (for it was thought fit by friends that it should also appear in this dress.) You might have had the whole ere this, had the press been as much at leisure, and as diligent (or rather not so much at leisure, but as diligent) as I could wish. I am now upon another work; as hard almost as to make Mr. Hobbes understand a demonstration. It is to teach a person126 dumb and deaf to speak, and to understand a language127: of which if he could do either, the other would be more easy; but knowjing neither, makes both the harder. [454] And though the former may be thought the more difficult, the latter may perhaps require as much of time. For, if a considerable time be requisite for him, that can speak one, to learn a second language; much more for him, that knows none, nor hath so much as the advantage of speech. You may think, perhaps, that it is not a piece of mathematics, to teach either speech, or language; as Mr. Hobbes, that the attendants of Jupiter were not found out by algebra. But though I am in this fully of his opinion, yet I must in that except to yours, at least thus far, that I find it therein no small advantage to have been versed in mathematics. I was the more willing to attempt it, because the person was represented as very ingenious and apprehensive (and at least so much of a mathematician, as to limn very well, being taught it by some of the best masters in London.) And I was the more confident that the defect was not in the organs of speech, (though possibly not so pliable as in a child, to the forming of unacquainted sounds) not only upon the common presumption, that the defect of speech in deaf persons is but an accidental consequent of their want of hearing: but also because he could once speak (though so long since that he does not remember it) till that about five years of age, having by accident lost his hearing, he thereupon lost his speech also: not at once, but gradually; that is, he was about half a year in losing it. He had, before he came, learned to write, I mean, as an English scrivener writes Greek; of which he knows neither the sound nor sense; and thereby hath saved me so much labour as the teaching him an alphabet: but hears either so little, or not at all, that I cannot, as I hoped, make any 124

tliat in print: i.e. WALLIS, Hobbius Heauton-timorumenos, Oxford 1662. account of in writing: probably in a now missing letter to Boyle. 126 person: i.e. Daniel Whalley. 12r lt is to teach. . . . language: see Wallis's account of this in SCRIBA, 'Autobiography', 41, and in WALLIS, Defence of the Royal Society, London 1678, passim. 125

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19. WALLIS to BOYLE, 30 December 1661/[9 January 1662] advantage of it. He cannot, from the back-side of my house, (which is a little more than a stone's cast, and no obstacle between) hear St. Mary's great bell ring; nor, in Christ Church choir, hear the organs. Close at his ear he can hear a sound, but not a voice, (unless I should rather say he feels it:) I mean, he discerns a noise, but not the articulation; and of a smart sound, rather than a loud. When a coach at night rushing close by the window, I perceived he discerned it: asking whether he heard it, he signified no, but he felt it, by the shaking of the ground. He hath now been with me a fortnight, and somewhat more. In which time, as to the language, he hath already learned many words, and somewhat of the Syntax. And, as to speech, hath pronounced all the sounds of our language (even those of L and R, and those of th in thy and thigh, which the French and others complain of as most difficult, and can hardly attain unto) which secures me, that the organs of speech are not indisposed to the forming of articulate sounds: and at some of these he is very ready, though he cannot at pleasure command them all. If you ask what my conjecture is as to the issue, or what I do design in it; I must confess, that as to the matter of speech, though I doubt not but he may come to speak any thing, yet I do not expect that he shall make the like advantage of it as those that hear: because, that neither hearing himself nor others, he will be subject to forget or mistake in forming sounds; and not to correct those mistakes, because he hears them not. For as one, who knows very well how to write, and hath a good command of his hand, yet if he want either sight or light, will hardly write well; the like must be expected in a deaf man's speaking; for, as then the eye guides the hand, so here, the ear the tongue. But as to the language, I know not but that he may, by writing, both express his own, and understand the conceptions of others, as well as other men; and so converse with men, as we do with the ancients, or persons distant, which is no small advantage [455] in human affairs: and may very much supply the defect of speech. | You may please to acquaint Sir Robert Moray128 with this adventure, who is himself so much a virtuoso, that he is the more inquisitive what others are doing; and will not allow me to be unemployed: but whether he will infer that I am busy, or was much at leisure, I cannot tell. Nor that I further trespass on your present affairs, than to say, that I am,

128

Moray: i.e. Robert Moray, q.v. 49

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21. WALLIS to BOYLE, 20 February/[2 March] 1661/2 your honour's affectionate and very humble servant, John Wallis.

20. ROBERT BOYLE to WALLIS 4 or 5/[14 or 15] January 1661/[1662] Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 10 (where the date given is 5 January) and 27 (where the date given is 4 January). Answered by: WALLIS-BOYLE 14/[24]!II.1661/2 (i). In this letter, as in BOYLE-WALLIS 26!I/[8!II]. 1661/2, Boyle requested an account of Wallis's progress in teaching Daniel Whalley how to speak.

21. WALLIS to ROBERT BOYLE Oxford, 20 February/[2 March] 1661/2 Transmission:

E WALLIS, Hobbius Heauton-timorumenos. Or a consideration of Mr Hobbes his Dialogues in an epistolary discourse, addressed, to the Honourable Robert Boyle, Esq., Oxford 1662. Written in the form of an epistolary tract addressed to Boyle, Wallis's Hobbius Heautontimorumenos is ostensibly a reply not only to Hobbes's ExamAnatio et emendatio mathematicae hodiernae (London 1660), but also to his recent Dialogue physicus, sive de natura aeris (London 1661). In the latter, Hobbes had attacked the statutes and scientific practice of the Royal Society as well as Boyle's New Experiments PhysicoMechanicall (Oxford 1660). In addition, he had sought to reply to Wallis's and Rooke's refutations of his duplication of the cube, publishing his solution as an appendix. 50

22. BOYLE to WALLIS, 26 February/[8 March] 1661/[1662]

22. ROBERT BOYLE to WALLIS 26 February/[8 March] 1661/[1662] Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 10 and 27. Answered by: WALLIS-BOYLE 14/[24].III.1661/2 (i).

23. WALLIS to ROBERT BOYLE Oxford, 14/[24] March 1661/2 (i) Transmission:

El First edition of missing letter sent: Philosophical Transactions No. 61 (18 July 1670), 1087-97 ('A Letter of Dr. John Wallis to Robert Boyle Esq, concerning the said Doctor's Essay of Teaching a person Dumb and Deaf to speak, and to Understand a Language; together with the success thereof: Which Letter though written many years since, was but lately obtain'd to be inserted here, it being esteemed very well worth to be preserv'd and communicated for Publick Use') (our source).—reprinted: BOYLE, Correspondence II, 11-18. E2 Latin translation of E1: Miscellanea curiosa medico-physica Academiae naturae curiosorum sive ephemeridum medico-physicarum germanicarum curiosarum annus primus, Leipzig 1670; Appendix sen addenda curiosa omissorum ad annum primum miscellanorum medico-physicorum, Breslau 1670, 11-20. E3 Richard BOULTON, The Theological Works of the Honourable Robert Boyle, Esq; epitomiz'd, London 1715, I, 292-9. E* German translation of E2: AMMAN, Abhandlung von der Sprache, und wie Taubstumme darin zu unterrichten sind. Nebst zwei Briefen des Dr. Joh. Wallis ... vom Unterrichte der Taubstummen. Aus dem Lateinischen iibersetzt . . . von Dr. L. Graflhoff ..., Berlin 1828, 103-18. Reply to: BoYLE-WALLis 4 or 5/[14 or 15].1.1661/2 and BOYLE-WALLIS 26.II/[8.III]. 1661/2.

Sir,

1 did acquaint you a while since,129 That (beside the consideration of 2 the consideration I had in hand E3 129

I did ... since: i.e. in WALLIS-BOYLE 30.XII.1661/[9.I.1662]. 51

23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) 13

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°, which I had in hand;) I had undertaken another Task, (almost as 131 hard as to make Mr understand Reason,) to Teach a per son Dumb [1088] and Deaf, to speak and to Understand a Language. Of which if he could do either, the other would be more easy: But his knowing neither, makes both harder. And though the Former may be thought the more difficult; the Latter may perhaps require as much of Time. For if a considerable Time be requisite, for him that can speak One, to learn a Second Language; much more for him that knows None, to learn the First. I told you, in my last, that my Mute was now at least Semivocalis, whereof because you desire a more particular Information, I thought my self obliged to give you this brief Account of that whole Affair: that you may at once perceive, as well, upon what considerations I was induced to Attempt that Work, and what I did propose to my self as Fesible therein, as what Success hath hitherto attended that Essay. The Task it self consists of Two very different parts; each of which doth render the other more difficult. For, beside that which appears upon the first view, To teach a person who cannot Hear, to Pronounce the Sound of Words: There is that other, of teaching him to Understand a Language, and know the signification of those words, whether spoken or written, whereby he may both express his own sense, and understand the Thoughts of others: without which latter, that former were only to speak like a Parrot; or to write like a Scrivener, who understanding no Language but English, transcribes a piece of Latin, Welsh, or Irish; or like a Printer of Greek or Arabick, who knows neither the sound nor signification of what he printeth. Now, though I did not apprehend Either of these impossible; yet, that each of them doth render the other more hard, was so obvious as that I could not be Ignorant of it. For, how easily the understanding of a Language is attain'd by the benefit of Discourse, we see every day; not onely in those, who knowing one Language already, are now to learn a second; but (which doth more resemble the present case) in Children, who as yet knowing none, are now to learn their First Language. | [1089] For it is very certain, that no Two Languages can be so much different the one from the other, but that the knowledge of the one will be subservient to the gaining of the other: not only because there is now a 1 (almost . . . Reason,) omitted in E3 130

of : i.e. of the reply to Hobbes's Dialogus physicus, sive de natura aeris, which Wallis published under the title Hobbius heauton-timorumenos in 1662. 131 Mr : i.e. Thomas Hobbes.

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23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) common Language, wherein the Teacher may Interpret to the Learner the signification of those Words and Notions which he knows not, and express his own Thoughts to him; but likewise (which is very considerable,) because the common Notions of Language, wherein all or most Languages do agree, and also so many of the Particularities thereof as are common to the Language he knows already, and that which he is to learn, (which will be very many,) are already known; and therefore a very considerable part already dispatched, of that work which will be necessary for the teaching of a First Language, to him who as yet knows none. But to this disadvantage (of teaching a First Language,) when that of Deafness is super-added, it must needs augment the difficulty: since it is manifestly evident from Experience, That the most advantageous way of teaching a Child his First Language, is that of perpetual Discourse; not onely what is particularly addressed to himself, as well in pleasing divertisements, or delightful sportings, (and therefore insinuates it self without any irksom or tedious labour,) as what is directly intended for his more serious Information: But that discourse also which passeth between others; where, without pains or study, he takes notice of what Actions in the Speaker do accompany such words, and what Effects they do produce in those to whom they are directed; which doth, by degrees, insinuate the intendments of those words. And, as that Deafness makes it the more difficult to teach him a Language: so on the other hand, that want of Language, makes it more hard to teach him how to speak or pronounce the Sounds. For there being no other way to direct his Speech, than by teaching him how the Tongue, the Lips, the Palate, and other Organs of speech, are to be applyed and moved in the Forming of such sounds as are required; to the end that [1090] he may, by Art, pronounce those| Sounds, which others do by Custome, they know not how; it may be thought hard enough to express in writing, even to one who understands it very well, those very nice Curiosities and Delicacies of motion, which must be observed (though we heed it not,) by him, who without help of his Ear to guide his Tongue, shall form that variety of Sounds we use in speaking: Many of which Curiosities are so nice and delicate, and the difference in forming those Sounds so very subtile, that most of our selves, who pronounce them every day, are not able, without a very serious consideration, to give an account, by what Art or Motion our-selves form them; much less to teach another how it is to be done. And if, by writing to one who understands a Language, it be thus difficult to give Instruction, how, without the help of Hearing, he 53

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23. WALLIS to BOYLE, 14/[24] March 1661/2 (i)

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may utter those Sounds, it must needs increase the Difficulty, when there is no other Language to express it in, but that of Dumb signs. These Difficulties (of which I was well aware) did not yet so far discourage me from that undertaking, but that I did still conceive it possible that both parts of this Task might be effected. As to the First of them; Though I did not doubt but that the Ear doth as much guide the Tongue in speaking, as the Eye doth the Hand in writing, or playing on the Lute: and therefore those who by accident do wholly lose their Hearing, lose also their Speech, and consequently become Dumb as well as Deaf; (for it is in a manner the same difficulty, for one that Hears not, to speak well; as for him that is blind, to write a fair hand:) yet, since we see that 'tis possible for a Lady to attain so great a Dexterity, as, in the dark, to play on a Lute, though to that variety of nimble motions, the Eyes direction, as well as the Judgment of the Ear, might seem necessary to guide the Hand; I did not think it impossible, but that the Organs of Speech might be taught to observe their due Postures, though neither the Eye behold their Motion, nor the Ear discern the Sound they make. And as to the other; That of Language might seem yet| more possible. [1091] For, since that in Children, every day, the Knowledge of words, with their various Constructions and Significations, is by degrees attained by the Ear; so that, in a few years, they arrive to a competent ability of Expressing themselves in their first Language, at least as to the more usual Parts and Notions of it; Why should it be thought impossible, that the Eye (though with some disadvantage) might as well apply such Complication of Letters or other Characters, to represent the various Conceptions of the mind; as the Ear, a like Complication of Sounds? For though, as things now are, it be very true that Letters are, with us, the immediate Characters of Sounds, as those Sounds are of Conceptions: yet is there nothing, in the nature of the Thing it self, why Letters and Characters might not as properly be applyed to represent Immediately, as by the Intervention of Sounds, what our Conceptions are. Which is so great a Truth, (though not so generally taken notice of,) that 'tis Practiced every day; not onely by the Chineses, whose whole Language is said to be made up of such Characters as to represent Things and Notions, independent on the Sound of words; and is therefore differently spoken, by those who differ not in the Writing of it: (like as what, in Figures, we Write, 1, 2, 3, for One, Two, Three- a Frenchman, for example, reads Un, Deux, Trois:) But, in part, also amongst our selves; as 54

23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) in the Numeral Figures now mentioned, and many other Characters of Weights and Metals, used indifferently by divers Nations to signifie the same Conceptions, though expressed by a different Sound of words: And, more frequently, in the practice of Specious Arithmetick, and operations of Algebra, expressed in such Symbols, as so little need the Intervention of Words to make known their meaning, that, when different persons come to express, in Words, the sense of those Characters, they will as little agree upon the same Words, though all express the same sense, as two Translators of one and the same Book into another Language. [1092] And, though I will not dispute the Practical possibility) of introducing an Universal Character, in which all Nations, though of different speech, shall express their common Conceptions; yet, that some Two or Three (or more) persons may, by consent, agree upon such Characters, whereby to express each to other their sense in Writing, without attending the Sound of words; is so far from an impossibility, that it must needs be allowed to be very Fesible, if not Facile. And, if it may be done by new-invented Characters; why not as well by those already in use? Which though to those who know their common use, may signifie Sounds; yet to those that know it not, or do not attend it, may be as immediately applied to signifie Things or Notions, as if they signified nothing else: And consequently, so long as it is purely Arbitrary, by what Character to express such a Thing or Notion; we may as well make use of that Character or Collection of Letters, to express the Thing to the Eyes of him that is Deaf; by which others express the Sound or Name of it to those that Hear. So that, indeed, that shall be, to Him, a Real Character, which expresseth to Another a vocal Sound; but signifieth, to Both, the same Conception: Which is, To understand the Language. To these Fundamental Grounds of Possibility in Nature, I am next to add a Consideration which made me think it Morally-possible; that is, not impossible to succeed in Practice. And, because I am now giving an Account to one who is so good a Friend to Mathematicks, and Proficient therein, I shall not doubt but this Consideration will have the force of a great swasive. Considering therefore, from how few and despicable Principles the whole Body of Geometry, by continual consequence, is inforced; if so fair a Pile, and curious Structure may be raised, and stand fast upon so small a Bottom; I could not think it incredible, that we might attain some considerable success in this Design, how little soever we had at first to begin upon: and, from those little Actions and Gestures, which have a kind of Natural significancy, we might, if well managed, proceed gradually 55

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to the Explication of a compleat Language, and withal, direct to those curiosities of Motion and Po|sture in the Organs of speech, requisite to [1093] the Formation of a Sound desired; and, so to effect both parts of what we intend. My next Inducement to undertake it, was a consideration of the Person132, (which, in a work of this nature, is of no small concernment;) who was represented to me as very Ingenious and Apprehensive, (and therefore a fit subject to make an Essay upon;) and so far at least a Mathematician as to Draw Pictures; wherein, I was told, he had attained some good ability, which did induce me to believe that he was not uncapable of the Patience, which will be necessary to attend the Curiosity of those little varieties in the Articulation of Sounds; being already accustomed to observe and imitate those little Niceties in a Face, without which it is not possible to Draw a Picture well. I shall add this also, That, once, he could have spoken, though so long ago, that (I think) he doth scarce remember it. But having, by accident, when about five years of age lost his Hearing, he consequently lost his speech also; not all at once, but by degrees, in about half a years time: which though it do confirm what I was saying but now, How needful it is for the Ear to guide the Tongue in Speaking, (since that Habit of Speaking, which was attained by Hearing, was also lost with it,) and might therefore discourage the undertaking; yet I was thereby very much secured, that his want of Speech was but a consequent of his want of Hearing, and did not proceed originally from an Indisposition in the Organs of Speech to form those Sounds. And though the neglect of it in his younger years, when the Organs of Speech, being yet tender, were more pliable, might now render them less capable of that Accurateness which those of Children attain unto: (whereof we have daily experience; it being found very difficult, if not impossible, to teach a Forraigner well in years, the Accurate pronouncing of that Sound or Language, which in his tender years he had not learned:) yet, if he can attain to speak but so well as a Forraigner, at his years, may learn to speak English; what shall be farther wanting to that Accurateness which a| Native from his Childhood attains unto, may, to an indifferent [1094] estimate, be very well dispensed with. Having thus acquainted you with those Considerations which did induce me to attempt it; least you may think I build too confidently thereupon, and judge me guilty of too much vanity, in promising my self a 132

Person: i.e. Daniel Whalley. 56

23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) greater success than can, in reason, be hoped for; It will next be necessary to give you some account, what measure of Success I might propose to my self as probable, in such an undertaking. And as to the first part of it, (that of Speaking;) Though I did believe, that much more is to be effected than is commonly thought Fesible, and that it was possible for him so to speak as to be understood; yet I cannot promise my self, that he shall speak so Accurately, but that a Critical ear may easily discern some Failures, or little differences from the ordinary tone or pronunciation of other men; (since that we see the like every day, when not Forraigners only, but those of our own Nation in the remoter parts of it, can hardly speak so Accurately, as not to discover a considerable difference from what is the common Dialect or Tone at London.) And this not onely upon the consideration last mentioned; (concerning the Organs of Speech less pliable to those Sounds to which they were not from the first accustomed;) but especially upon that other consideration, concerning the Ears usefulness to guide and correct the Tongue. For as I doubt not, but that a Person, who knows well how to Write, may attain, by custome, such a Dexterity, as to write in the Dark tolerably well; yet it could not be expected, that he should perform it with the like Elegancy, as if he saw the Motions of his hand: so neither is it reasonably to be expected, that he who cannot Hear, though he may know how to speak Truly, should yet perform it so Accurately, as if he had the advantage of his Ear also. Nor can I promise, nor indeed hope, that how Accurately soever he may learn to speak, he shoud be able to make so great Use of it as others do. For since that he cannot hear what others say to Him, as well as [1095] express his own Thoughts | to Them; he cannot make such use of it in Discourse as others may. And though it may be thought possible, that he may, in time, discern, by the motion of the Lips, visible to the Eye, what is said to him; (of which I am loth to deliver a positive judgment, since much may be said conjecturally both ways;) yet this cannot be expected, till at least he be so perfectly Master of the Language, as that, by a few Letters known, he may be able to Supply the rest of the Word; and by a few Words, the rest of the Sentence, or at least the sense of it, by a probable conjecture, (as when we Decipher Letters written in Cipher:) For, that the Eye can actually discern all the varieties of Motion in the Organs of Speech, and see what Sounds are made by those Motions, (of which many are Inward, and are not exposed to the Eye at all,) is not Imaginable. 57

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But as to the other Branch of our Design, concerning the Understanding of a Language: I see no reason at all to doubt, but that he may attain This, as perfectly as those that Hear; and that, allowing the like Time and Exercise, as to other men is requisite to attain the Perfection of a Language, and the Elegance of it, he may Understand as well, and Write as Good Language as other men; and (abating onely what doth directly depend upon Sound, as Tones, Cadencies, and such Punctilio's,) no whit inferiour to what he might Attain to, if he had his Hearing as others have. And what I speak of him in particular, I mean as well of any other Ingenious person in his Condition; who, I believe, might be taught to use their Book and Pen as well as others, if a right Course were taken to that purpose. To tell you next, What Course I have hitherto used towards this Design, it will not be so necessary. For should I descend to Particulars, it would be too Tedious; especially since they are to be used very differently, and varied as the present Case and Circumstances do require. And, as to the General way, it is sufficiently Intimated already. As to that of Speech; I must first, by the most significant signs I can, make him to understand, in what Posture and Motion I would have him apply his Tongue, Lips, and| other Organs of Speech, to the Forming of [1096] such a Sound as I direct. Which if I hit right, I confirm him in it: If he miss, I signifie to him, in what he differed from my Direction; and, to what Circumstances he must attend to mend it. By which means, with some Trials, and a little Patience, he learns first one, then another Sound; and, by frequent Repetitions, is confirmed in it; or (if he chance to forget) Recovers it again. And for this Work, I was so far prepared before hand, that I had heretofore, upon another occasion, (in my Treatise De Loquela1^3, prefixed to my Grammar for the English Tongue134",) considered very exactly (what few Attend to) the Accurate Formation of all Sounds in Speaking, (at least as to our own Language, and those I knew:) without which, it were in vain to set upon this Task. For, if we do not know, or not consider, how we Apply our own Organs in Forming those Sounds we Speak, it is not likely, that we shall, this way, Teach another. As to that of Teaching him the Language: I must (as Mathematicians 133

Treatise De Loquela: i.e. WALLIS, De loquela sive sonorum formatione tractatus grammatico-physicus, [Oxford 1653]. 134 Grammar for the English Tongue: i.e. WALLIS, Grammatica linguae anglicanae, Oxford 1653.

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23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) do from a Few Principles first granted,) from that little stock (that we have to Begin upon) of such Actions and Gestures as have a kind of Natural significancy, or some Few Signs, which himself had before taken up to express his Thoughts as well as he could, Proceed to Teach him, what I mean by somewhat else; and so, by steps, to more and more: And this, so far as well I can, in such Method, as that what he Knows already, may be a step to what he is next to Learn; as, in Mathematicks, we make use, not of Principles only, but Propositions already demonstrated, in the Demonstration of that which follows. It remains now, for the Perfecting the Account which at present you desire of me, only to tell you, what Progress we have already made. Which, had not your Desires commanded from me, I should have respited a while longer, till I might have made it somewhat Fuller. He hath been already with me somewhat more than Two Moneths. In which time, though I cannot be thought to have Finished such a Work; [1097] yet the Success is not so little, | as to Discourage the Undertaking: but as much as I could hope for in so short a time; and more than I did Expect. So that I may say, the Greatest difficulty of Both Parts being almost over; what Remains, is little more than the work of Time and Exercise. There is hardly any Word, which (with deliberation) he cannot Speak; but, to do it Accurately, and with Expedition we must allow him the Practice of some considerable Time, to make it familiar to him. And, as to the Language; though it were very Indifferent to him who Knew none, which to begin withal; yet, since it is out of Question, that English, to him, is like to be the most Useful and Necessary; it was not adviseable to Begin with any other. For though he can Pronounce the Latine with much more Ease, (as being less perplexed with a multitude of concurring Consonants;) yet this is a Consideration of much less concernment than the other. To this therefore having applyed himself, he hath already Learned a great many Words, and, I may say, a considerable part of the English, as to Words of most Frequent use: But the whole Language being so Copious, though otherwise Easy, will require a longer Time to Perfect what he hath Begun. And this, Sir, is the full History of our Progress hitherto. If you shall hereafter esteem our Future Success, worthy your taking notice of, You may Command that, or what else is within the power of 4 teach him somewhat else; E3

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24. [WALLIS:] Note on letter to Boyle, 11/[21] July 1670

Oxford, March 14. 1661/2.

Sir, Your Honours very humble Servant, John Wallis.

24.

[WALLIS:] Note on letter to Boyle Oxford, 11/[21] July 1670 Transmission:

W Draft note: LONDON Royal Society Early Letters Wl, No. 110, 1 p. (our source). On reverse in Wallis's hand: 'To Mr Oldenburg. July 11. 1670. Oxford. Sir' (deleted). El First edition of draft note: Philosophical Transactions No. 61 (18 July 1670), 1098-9 ('A Letter of Dr. John Wallis to Robert Boyle Esq, concerning the said Doctor's Essay of Teaching a person Dumb and Deaf to speak, and to understand a Language; together with the success thereof: Which Letter though written many years since, was but lately obtain'd to be inserted here, it being esteemed very well worth to be preserv'd and communicated for Publick Use').—reprinted: BOYLE, Correspondence II, 18-19. E2 Latin translation of El: Miscellanea, curiosa medico-physica Academiae naturae curiosorum sive ephemeridum medico-physicarum germanicarum curiosarum annus primus, Leipzig 1670; Appendix sen addenda curiosa omissorum ad annum primum miscellanorum medico-physicorum, Breslau 1670, 20-2. E^ German translation of E2: AMMAN, Abhandlung von der Sprache, und wie Taubstumme darin zu unterrichten sind. Nebst zwei Briefen des Dr. Joh. Wallis ... vora Unterrichte der Taubstummen. Aus dem Lateinischen ilbersetzt ... von Dr. L. Grafihoff . . . , Berlin 1828, 118-19. The present note represents a postscript to WALLIS-BOYLE 14/[24].III.1661/2 (i). It was written by Wallis in form of an editor's note to be appended to the version of that letter which was printed in the July 1670 issue of Philosophical Transactions. He probably handed the manuscript to Oldenburg together with a copy of the letter itself.

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The Person to whom the foregoing Letter135 doth referre, is Mr Daniel Whalley, (son of Mr Whaley136, late of Northampton, & Mayor of 135

foregoing Letter: i.e. WALLIS-BOYLE 14/[24]!II.1661/2 (i). Mr Whaley: presumably Peter Whalley (d. 1656), mayor of Northampton

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24. [WALLIS:] Note on letter to Boyle, 11/[21] July 1670 that Town.) Hee was, (soon after the Date of this Letter) on the 21th day of May, 1662, present at a meeting of the Royal Society137, (of which the Register of that days proceedings takes particular notice,) and did in their presence (to the great satisfaction of the Company) pronounce distinctly enough such words, as by the Company were proposed to him; and, though not altogether with the usual Tone or Accent, yet so as easyly to be understood. About the same time allso (his Majestic having heard of it, & being willing to see him,) hee did the like several times at Whitehal, in the presence of his Majestic138, & Prince Rupert139, & divers others of the Nobility; though hee had then imployed but a small time in acquiring this ability. In the space of One Year, which was the whole time of his stay with Dr Wallis; hee had read over a great part of the English Bible, & had attained so much skill as to expresse himself intelligibly in ordinary affairs; to read and understand letters written to him, & to

4 presence, to their great satisfaction pronounce E1 4 pronounce (1) very distinctly (2) distinctly enough W 7 understood (1) , (—} did it appear to the company that (2) . \[here you may adde, if you please, what the Register says of it.] del] (a) In the space of One year (which was the whole time of his stay with Dr Wallis) hee had (6) About W understood: Whereupon also the said Doctor was, by the same Assembly, encouraged to pursue what he had so ingeniously and so successfully begun. About E1 8 did (1) it (2) the like several times (a) in his Ma breaks off (b) at Whitehal, W 9 Majestie, (1) of th breaks off (2) of his Highness the Duke of York, (3) & Prince W Majestie, His Highness Prince E1 12 Dr Wallis; (1) hee (2) hee had read over a great part of the English Bible, & W 13 himself (1) intelligly, in most (2) intelligibly in W 14 to |read and add.\ understand W to understand E1 (1646 and 1655) and member of parliament for burgesse of Northampton (1654-5). 137 Hee . . . Society: cf. BIRCH, History of the Royal Society, I, 84: 'Dr Wallis brought with him the young man born deaf and dumb, and made him pronounce several words; and was desired to continue his practice upon him'. 138 his Majestie: i.e. King Charles II. 139 Prince Rupert: i.e. Prince Rupert (1619-82), son of Elizabeth, Queen of Bohemia, and Frederick V, Elector Palatine. He fought on the Royalist side during the civil wars and was appointed Commander-in-Chief in 1644. After the Restoration, he rose to first lord of the Admiralty (1673-9). Elected fellow of the Royal Society in March 1664. DNB. 61

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write answers to them; though not elegantly, yet so at lest as to be understood: and in the presence of many Forraigners (who out of curiosity have come to see him) hath oft times not onely read English & Latine to them, but pronounced the most difficult words of their languages (even the Polish itself) which they could propose to him. Since that time, though he have not had the opportunity of making much further improvement, for want of an Instructer; hee doth yet retain what hee had attained to: or, wherein hee may have forgot the niceness requisite in the pronunciation of some sounds, doth easyly recover it with a little help. Nor is this the onely person on whom hee hath shewed this effect of his skill; But he hath since done the like for another140, (a Young Gentleman of a very good Family & a fair Estate,) who did, from his Birth, want his Hearing. On this occasion, I thought it very sutable to give notice of a small Latine Treatise141 of this same Author, first published in the year 1653, intituled De Loquela (of Speech;) prefixed to his Grammaire142 of the English Tongue (written allso in Latine.) In which treatise of Speech (to which he refers in this discourse, & on confidence hee durst undertake that difficult task) hee doth very distinctly lay down the manner of forming all Sounds of Letters usual in Speech, as well of the English as of other Languages. Which is, I think, the first book ever published in that kind; (For though some writers formerly have here and there occasionally sayd something of the formation of some particular Letters; yet none, that I know of, had before him undertaken to give an account of all:) Whether any, since him, have with more judgement and accurateness performed

1 though, not elegantly, yet add. W 1 yet so as to be E1 1-2 understood: (1) And since th breaks off (2) and in the presence of (a) divers (b) many W 6 making (1) any great (2) much further W 9 recover it (1) at (2) with W 15 first published . .. Speech;) add. W 20 all (manner of del] Sounds W 22 some (1) have (2) writers formerly have (a) now an breaks off (b) here W 24 all:) (1) And I question add] whether (3) Whether W 140

another: i.e. Alexander Popham, son of Anne Wharton by her first marriage. Latine Treatise: i.e. WALLIS, De loquela sive sonorum formatione tractatus grammatico-physicus, [Oxford 1653]. 142 his Grammaire: i.e. WALLIS, Grammatica linguae anglicanae, Oxford 1653. 141

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25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) the same, I will not take upon mee to determine. In his Grammaire of the English Tongue (to which this, of Speech, is prefixed;) hee hath so briefly & clearly given a true account of this language, as may be very advantageous not onely to strangers for the easy attainment thereof, but even to the English themselves for the clear discovering (which few take notice of) the true genius of their own language.

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25. WALLIS to ROBERT BOYLE [Oxford], 14/[24] March 1661/2 (ii) Transmission:

W Letter sent: LONDON Royal Society Boyle Letters 6, f. llr-12v (our source). At bottom of f. 12V at 90° in Boyle's hand: 'The Businesse about the Bible.'—printed: BOYLE, Correspondence II, 7-10.

Sir

I have, as you desired, considered of Mr Chilinski's143 businesse144, & perused the Papers & Letters by him produced, (many of the particulars being allso otherwise known to myself to be true as he relates them.) And the case, according to the best estimate I can make of it, stands thus. 1654

Mr Samuel Boguslaus Chylinski, being sent out of his own country (of Lithuania) for that purpose, studied Divinity in the University of Franeker in order to fit him the better for the service of the church at home, for the space of Two Years, being so long supplyed by means allowed him from those Churches.

1 upon me at all to determine. El 1 In (1) that (2) his W 2 so (1) cl breaks off (2) briefly W 3 given an account E1 4 attainment (1) of (2) thereof W 5 discovering (1) (of what (2) (which few W 143

Chilinski's: i.e. Samuel Boguslav Chylinski, q.v. businesse: i.e. Chylinski's efforts to publish the Bible in Lithuanian. See MADDISON, Life of the Honourable Robert Boyle, 111. 144

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The Wars145 then breaking forth & wasting those Churches, and hindering all supplies from thence, he was afterward forced to subsist on the Benevolence of such as pittyed his condition. Hee came to Oxford with Letters of Recommendation from the University of Franeker, and there for some years hee lived piously & studiously, on like Benevolence of divers there. In this condition, considering how hee might best imploy his study for the service of those Churches, he thought of Translating the Bible into the Lithuanian Language; into which (though the Gospel have been there received for about 300 years) it was never yet translated. In June 1657, he began the Translation of the New Testament; & finished it the year following; But, having nothing but Benevolence to live on, was not able to get it Printed146. In September 1658, hee began the Translation of the Old Testament, & finished it about the end of the next year. His Translatinge being finished, divers in Oxford, of the most eminent persons there, approving this design, gave him ample Testimonialls, as well of his life & conversation, as of approving so good a work, recommending the promoting of it: And many of them subscribed towards the Impression of it divers summes of many (which were afterwards payd, & imployed therein,) but far short of what were necessary for the compleating of it. In March following, hee began the Impression at London, (hoping that other Benefactors would afterwards be found, for the perfecting of it;) and carried it on, as far as the monies collected, and his own credit would reach. | In September following, the monies collected being all imployed in that work & there being farther due to the Printer above 100', (about one

1 then (1) beginning (2) breaking 2 on (1) such (2) the 16 His (1) being wellnigh (2) Translatinge being 17 Testimonialls, (1) of his (2) as well 20 were |most of them del] afterwards 23 following, |(the Translation being before that time compleated,) del] hee 28 100(, (1) p breaks off (2) (about one (a) third (b) halfe 145

Wars: i.e. the Russo-Polish war (1654-6). Printed: Chylinski's Lithuanian translation of the New Testament was not printed until 1958 as Biblia litewska Chyliriskiego. Nowy testament, Poznan 1958. 146

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[IF] 1660

25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) halfe part of the Bible being then printed:) The Printer was not willing to proceed further; but required Bond of him for the rest of the mony due for what was allready done; which was given him. In this condition, applying himself to divers of the London Ministers 1660 soliciting farther contributions for that work, they Recommended him to the Lord Mayor147 & Court of Aldermen; who so far resented his condition, & that pious work, that they were willing to Recommend his case to the King for a Publike Collection & made an Order of the Court to that purpose. 1660 Having thus far promoted this affair upon his private account, & the interest of those friends that were willing to favour him in it: When the Petition to his Majestie was drawn, & ready to be signed; as the Desire of the Lord Mayor, the Court of Aldermen, & the Ministers of the City, & the day appointed for that purpose; Mr John de Kraino Krainski148, coming over as an Agent from those Churches to desire Relief for them, occasioned a stop therein for the present. 1660 This Agent, suspecting that it might prove disadvantageous to his desired supplies if this collection for the Translation should proceed; & yet understanding how far it was promoted, & that so many considerable persons were allready ingaged in it, that he could not well hinder it; pretended a readynesse to promote it (hoping thereby the better to carry on his own businesse;) suggesting (though unknown to Mr Chylinski) that those Churches had procured this Translation, & caused it to be thus far printed; & desiring a Collection for the carrying it on, and for the further Relief of those distested Churches.

1 then add. 2 to (1) pro breaks off (2) proceed 14 John |de add.\ Craino Crainski corr. 17 might add. 19 was (1) proceed, & how (2) promoted, fe that so 20 he add. 24 desiring (1) supply (2) a Collection 24 on, (1) as we (2) and 147

Lord Mayor: i.e. Thomas Alleyn (d. 1690), lord mayor of London 1659-60. John de Kraino Krainski: i.e. Jan Krainski (1625-85), Calvinist priest. He lived in exile in London 1660-6, and raised funds there to aid the Lithuanian Calvinist church. 148

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25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii)

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Meanwhile, Mr Chylinski not finding a supply of means here to carry on his Impression, did, in March following, go over into his own Country, & acquainted the Synod there, what he had done, & how far hee had proceeded, who approved the work & incouraged him to goe on; as appears by the order of the Synode to that purpose, bearing date Sept. 1. 1661. In his Absence the Agent (whose design was, rather to hinder than promote the Work, as appears by his own Letters, & the whole sequele, though he pretended otherwise) had upon suggestion that this was the Churches Work (who as yet knew nothing of it) and the Translation procured by them obtained Letters Patents for a publike Collection, for carrying on this Impression, & for the Relief of those Churches; bearing Date July 12. 1661. In November following, Mr Chylinski returning to London (having heard of a Collection granted for that work,) and willing to go on with the Work; the Printer upon credit of those Letters Patents, was willing to go forward with| the Impression; & proceeded therein as far as the Psalmes. The Agent understanding it; in stead of promoting the Work, or taking care to pay what was due to the Printer for what was allready done; gives expresse order to the Printer not to proceed. Whereupon the work is at a stand. The Printer, thus Inhibited, writes a Letter to Mr Chylinski, demanding his mony due, & threatening to Arrest him, if he did not by the next day put in Bail to Answer an Action commenced against him: The Agent having disclaimed the Work. Mr Chilinski repairing to the Agent with this Letter, told him that hee must be forced to complain to his Majestie & the Counsell for relief, if he did not otherwise take care of it.

1 carry add. 4 on add. 6 rether corr. ed. 7 sequele, (1) ) had upon sugg breaks off (2) though 9 the (1) Translated (2) Translation 13 following, (1) when Mr Chylinski returned (2) Mr Chylinski returning 14 work, |but not knowing upon what suggestion, del.\ and 14 go add. 18 pay what add] was 22 not add. 26 & the Counsell add.

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1660./ 1.

1661.

1661.

[12r]

25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) The Agent, though first he told him that before he should be able to make friends there, to have it represented to the Counsel, it would cost him as much time as himself had allready spent in getting the Letters Patents; Yet, thinking it necessary to prevent such Complaint; & being to Petition the Counsell for other Monies, obtained an Order (amongst other Payments) for 50^ to be payd to the Printer in part of his debt, (whereas he might as well have obtained an Order for the whole, if he had desired it;) & for allowance of 4f per mensem to the Translator for his Subsistence. The Printer having received this 50', (& 20' more upon another Petition of his own,) being not yet satisfied for the whole of what is allready due, (nor the Agent willing to Undertake or Own the Work,) Mr Chilinski can neither have his Bond delivered; nor can the Work of the Impression go on (being still inhibited by the Agent;) nor can he get any of that 4' per mensem for his Subsistence; the Treasurers refusing to pay any, without an Acquittance signed by the Agent, which is not to be had; And the Agent now gone out of Town, to solicite Collections in other places, is not likely to return for divers Monthes. And the Agent, more effectually to retard the Work; would now have Mr Chylinski go back again into his own Country, to have the Translation examined & reviewed there: Sometimes pretending, that he was not imployed by those Churches in this work, and therefore not the Translator intended in the Letters Patents; Sometimes, that he will send or hath sent himself for other Translators; Sometimes, that there is more necessity of relieving their poor, than printing the Bible; Sometime, wishing, that he may perish, & his work with him: Notwithstanding that he had before suggested to his Majestie, (& thereupon obtained the Letters Patents,) that those Churches had procured this Translation, &, as a Testimony

1(1) The Agent, thinking it necessary though with opprobrious words |he add.\ told (2) The Agent, though 1 to (1) find (2) make 2 Counsel, (1) he (2) it 11 what (1) was (2) is (a) due (6) allready 13 nor |nor del. ed.\ can 20 Translation (1) re breaks off (2) examined 21 not |not del. ed.\ imployed 24 that (1) the (2) there 28 that (1) the (2) those 67

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26. BOYLE to WALLIS, 5/[15] April 1662

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thereof, a Copy of so much thereof as| was then Printed, was presented [12-] to his Majestie. Mean while, the Person who upon his own charge, with the supply of his private friends, & his own Study & Credit, hath hitherto carried on that work, wants subsistence to live on, (having been forced to pawn what he hath;) & the work of Printing the Bible in that Language, which seemes to be the first Intention of the Letters Patents, (the Surplusage onely of that Collection being ordered for the relief of the Poor in those Churches) is actually hindred. Having thus given you account of what I take to be the true State of the businesse; I know not well, (considering this Disagrement beween the Agent & him,) what is more advisable for perfecting the Intended Edition of the Bible in that Language, (which the Letters Patents do expressely direct,) than that by Order of his Majestie & Counsel, or by such ways as they may think fit so much out of the monies collected for that purpose, be payed to Mr Chilinski or the Printer (upon the Acquittance of both of them) by the Treasurers immediately (without dependence on the Agent,) as shall be necessary for that work. I have nothing farther to adde to it, than that I am Your Honours very humble servant J. W.

20

March. 14. 1661./2.

26.

ROBERT BOYLE to WALLIS 5/[15] April 1662 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 10 and 28.

5 (having (1) pawned (2) been 11 well, (1) what in this (2) (considering 13 do (1) di breaks off (2) expressely direct,) (a) that (b) than 16 the (1) T breaks off (2) Printer

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27. WALLIS to MORAY, 7/[17] April 1662 In this letter Boyle thanked Wallis for the account of his progress in teaching Daniel Whalley to speak which he had provided in WALLIS-BOYLE 14/[24].III.1661/2 (i). It would appear that he also expressed the wish to see this demonstrated. Cf. MORAY ? et al.-WALLis 8/[18].V.1662.

27.

WALLIS to ROBERT MORAY Oxford, 7/[17] April 1662 Transmission:

El First edition of missing letter sent: WALLIS, Cono-Cuneus: or, the Shipwright's circular wedge, London 1684, sig. A2 r -A2 v , Epistle dedicatory; printed as additional treatise I to WALLIS, Algebra, London 1685 (our source). E2 Latin translation of El: WALLIS, Opera mathematica II, 681. Answered by: MORAY-WALLIS IV/V.1662. The present letter enclosed Wallis's theoretical exposition of geometrical sections of the shipwright's circular wedge, suggested to him by models of these which Peter Pett had demonstrated to Brouncker and Moray in his presence in London.

To the honourable Sir Robert Moray, Kt.

Sir, Since I came home from London, I have taken some time to consider of those Solids and Lines made by the Sections thereof; proposed to Consideration (to my Lord Brouncker and your self, at your Lodgings, where I was also present) by Mr. Pett149, one of His Majesties Commissioners for the Navy, and an excellent Shipwright. The Bodies proposed to consideration were all of this form. On a plain Base, which was the Quadrant of a Circle, (like that of a Quadrantal Cone or Cylinder) stood an erect Solid, whose Altitude (being arbitrary) was there double to the Radius of that Quadrant; and from every Point of its Perimeter, streight Lines drawn to the Vertex, met there, not in a Point (as is the Apex of a Cone), nor in a parallel Quadrant (as in a Quadrantal Cylinder), but in a streight Line or sharp Edge, like that of a Wedge or Cuneus. On which consideration, I thought fit to give it the 149 Pett: i.e. Peter Pett (1610-70?), Navy Commissioner, fellow of the Royal Society from September 1662 onwards, DNB. Cf. BIRCH, History of the Royal Society, I, 85.

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27. WALLIS to MORAY, 7/[17] April 1662

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name of Cono-Cuneus, as having the Base of a Cone, and the Vertex of a Cuneus. By the various Sections of this Solid, in several Positions, he did (rightly) conceive, that divers new Lines must arise, in great variety, different from those arising from the Section of a Cone. Some of which he supposed might be of good use in the Building of Ships; in order to which it was, that he proposed them to Consideration. Now because he judged it troublesom (as indeed it would be) first to form such Solids, and then cut them by Plains in such Positions as he desired; he had (for avoiding that trouble) ingeniously contrived this Expedient. He caused divers Boards, of a good solid Wood, to be exactly planed, some of an equal thickness, some meeting in a sharp edge; those of the former, he caused to be glewed together in a parallel Position; those of the latter sort, he caused so to be glewed together, as that their sharp edges met in one common Angle. And having thus formed several Solids, of Boards thus glewed together, he then caused them to be wrought into such a form as that before described: Which being done, he then caused the Glew| to be dissolved in warm Water, whereby the several Boards, falling [A2V] asunder, did exhibit, in their several faces, the respective Sections of those Solids. And such were those he shewed us; which being put together, made up such Solids; and taken asunder, shewed the several Sections of them. I do not intend at all to disparage the ingenuity of that Contrivance, which was indeed very handsom, and neatly performed, but do withall suppose, that it would not be unpleasing to your self, or him, to see those Lines described in Piano, which would arise by such Section of the Solid. That therefore is the work of these Papers, to represent the true nature of such Lines, and the ways to draw them, without the actual Section of a Solid. Which I have the rather undertaken, because this is a Solid which I do not know that any other have before considered. And because this may be a Pattern; according to which, other Solids of like nature may be in like manner considered if there shall be occasion. If beside these Sections which he hath already considered, there be any other Sections of this or other the like Solids which he shall conceive useful to his purpose; the same may in like manner be represented (without the actual Section of such Solids) by Lines thus described in a Plain. But which of them may be most advantageous to his design, I do not pretend to understand so well, nor can with so much certainty affirm; as, 70

28. MORAY to WALLIS, April/May 1662 that I am, Sir,

Oxon, Apr. 7. 1662.

Your very humble Servant, John Wallis.

28.

ROBERT MORAY to WALLIS April/May 1662 Transmission:

Manuscript missing. Existence: Mentioned in and answered by WALLis-MoRAY 6/[16].V.1662. Reply to: WALLIS-MORAY 7/[17].IV.1662.

29. WALLIS to ROBERT MORAY Oxford, 6/[16] May 1662 Transmission:

W Letter sent: LONDON Royal Society Boyle Letters 5, f. 170r-171v (f. 170V and 171r blank) (our source). Postmark on f. 171V: 'MA/7'.—printed: BOYLE, Correspondence VI, 428-9. El First edition of letter sent: BOYLE, Works (1744), V, 512. E2 Second edition: BOYLE, Works (1772), VI, 455. Reply to: MoRAY-WALLis IV/V.1662.

Sir,

5

You will excuse mee for not returning you my thanks for yours150, (so full of civility & respect,) when you understand that I had nothing else to send with it that might bee worth giving you that trouble: And yet pardon this at present, which waits on you with this onely addition, that 6 not add. 9 which (1) onely (2) waits 150

yours: i.e. MoRAY-WALLis IV/V.1662. 71

30. MORAY? ET AL. to WALLIS, 8/[18] May 1662

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I have thoughts of waiting on you myself next week. At which time I may possibly present you with somewhat as acceptable as the last151; but in another kind. For I shall bring with niee (I suppose) a person152 on whom I have made an attempt lately of teaching him to Speak, though he cannot Hear; wherein your Curiosity may possibly make you desirous of understanding what progresse I have made. Which being of such a nature as not to be so satisfactory in a representation to the Eye as to the Ear, I shall forbear the Narrative, & refer it till you may receive a personal account thereof from himselfe, & from, Sir

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Oxon. May. 6.1662.

15

Your very humble servant John Wallis. [171V]

For my honoured Noble Friend, Sir Robert Moray, at his lodgings in the Privy Garden at White-hall. 30.

ROBERT MORAY ? ET AL. to WALLIS 8/[18] May 1662 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 28. According to Wallis, members of the Gresham Society, i.e. the future Royal Society, requested that he provide them with direct knowledge of his language instruction to a deaf-mute, of which they had previously received a report. This possibly took the form of a reply to WALLIS-MORAY 6/[16].V.1662. On 14 May 1662, Wallis gave members of the Society an account of his progress in teaching Daniel Whalley how to speak. At their request he presented the young man to them at the next meeting on 21 May 1662. 8 may add. 151

last: probably WALLIS-MORAY 7/[17].IV.1662, in which Wallis enclosed his theoretical exposition of geometrical sections of a shipwright's circular wedge. 152 person: i.e. Daniel Whalley.

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31. WALLIS to TITUS, 12/[22] June 1662

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WALLIS to SILAS TITUS 12/[22] June 1662 Transmission: Manuscript missing. Existence and date: Referred to in WALLIS, Algebra, 225-7; WALLIS, Opera mathematical, II, 245-6. In this letter, Wallis sent his proposed solution to a problem which evidently originated from Pell, and which Titus, 'a very Ingenious Person . . . and very well accomplished in Mathematical and other Learning' (WALLIS, Algebra, 225), had communicated to him. The task concerned was to find the numbers a, &, c, given that aa + be — 16, bb + ac = 17, and cc + ab = 18. Having been informed by Titus of the origin of the problem, Wallis drew up his solution in general terms according to Pell's method. The mathematical content of Wallis's solution is to be found in the pages of the Algebra referred to above.

32. ANNE WHARTON to WALLIS 30 September/[10 October] 1662 Transmission: C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 130ar-130bv (f. 130av and f. 130br blank). The address is in another hand.

Sir

Your undertakeinge of the trouble of my soone153 I must needs ackolidge154 as a great obligation, since you make such waighty obrecti(ons)155 as the lose of your time and study besides the incovenence of your attendance. But the high caracter I have receaved of you, from severall persons that amonght many of your comendations this is not the smalest that you make a consience of what you undertake hath maid me more importunate 153 Your undertakeing . . . soone: i.e. Wallis's attempt to teach Anne Wharton's deafmute son Alexander Popham how to speak. Alexander was an offspring of her previous marriage to Edward Popham (16107-51). Cf. SCRIBA, 'Autobiography', 42. 154 ackolidge: i.e. acknowledge. 155 obrecti(ons): i.e. objections.

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33. BROWNE to CHYLINSKI, 3/[13] October 1662

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with you then I should have beene with any one else the sume of mony that you name for his di(et) and his mans with the reward for your instructions and care of him I suppos to be a 100 pound a yeare; I confese it was 40 pound more then was given Dr Holder156 upon the same account; I shall desire you would eithere send an abaitment of the sume by this messenger or refere it to Dr Bathurst157 who will be as impartiall to you as to her as is You sarvant Anne wharton

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Sepe 30. If you thinke fitt I shall waite one you the beginning of the next weeke

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[130bv]

These for the Reverend Dr Wallis in Catte-Street in Oxford.

33.

RICHARD BROWNE to SAMUEL BOGUSLAV CHYLINSKI [London], 3/[13] October 1662 Transmission:

c Copy of missing letter sent in Chyliiiski's hand: OXFORD Bodleian Library MS. Add. D. 105, f. 12V (noted on CHYLINSKI-WALLIS 4/[14].X.1662 and superscribed with 'Copia Literarum Innotescentiarum'). 4 account; (1) but becos (2) I 156

Holder: i.e. William Holder (1616-98), divine and writer, fellow of the Royal Society (from January 1661 onwards), DNB. He published Elements of Speech: An Essay of Inquiry into the Natural Production of Letters; together with an Appendix to instruct Persons Deaf and Dumb, London 1669, and later accused Wallis and Oldenburg of failing to do justice to his own attempt at teaching speech to Alexander Popham. Wallis, according to whom Holder's efforts were largely unsuccessfull, wrote his Defence of the Royal Society, and the Philosophical Transactions, Particularly those of July, 1670. In Answer to the Cavils of Dr. William, Holder, London 1678, in reply. 157 Bathurst: i.e. Ralph Bathurst (1620-1704) physician and divine. Fellow and (from 1664 onwards) President of Trinity College, Oxford. Elected fellow of the Royal Society in August 1663. DNB. 74

34. CHYLINSKI to BROWNE, 4/[14] October 1662 Answered by: CHYLINSKI-BROWNE 4/[14].X.1662.

Dominus Chylinski. Translator of the Lithuanian Bible is to attend the Lords of the Provie Councell Satterday the 4.th of October at the Councell Chamber at Whitehall at 3. of the clocke after dinner. 3d October 1662.

Richard Browne.

34. SAMUEL BOGUSLAV CHYLINSKI to RICHARD BROWNE [London], 4/[14] October 1662 Transmission:

C Copy of missing letter sent in unknown hand: OXFORD Bodleian Library MS. Add. D. 105, f. 13r (written on CHYLINSKI-WALLIS 4/[14].X.1662 and superscribed by Chylinski with 'Copia Responsi mei'). The signature is in Chyliriski's hand. Reply to: BaowNE-CHYLiNSKi 3/[13].X.1662. Worthy Sir,

5

Receiving a note158 or surnons this very morning from your Worship; whereby I was required to give my attendance on the Lords of his Majesties Privy Councill this present Satterday afternoone. And not being fitted for the same: Humbly make bold to crave and intreate the suspending of my said Attendance, untill Satterday next, being the Eleventh day of this instant October: Not only for that my warning was not a full day; But also in that I had not the benefit of advising with such of my friends as I hold both grave and fit to direct in so weighty an Affair. This being accquainted and made known to his Majesties said Lords, and the suspending thereof endeavoured by your Worship I crave leave to subscribe my self, October the 4th 1662.

Your Worships most faithfull servant Samuel Boguslaus Chylinsky. Translator of the Lithuanian Bible.

158

note: i.e. BROWNE-CHYLINSKI 3/[13].X.1662.

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35. CHYLINSKI to WALLIS, 4/[14] October 1662

35. SAMUEL BOGUSLAV CHYLINSKI to WALLIS London, 4/[14] October 1662 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 12r-13v. On lower half off. 12V Chyliriski's copy of BROWNE-CHYLINSKI 3/[13].X.1662, and on f. 13r copy in unknown hand of CHYLINSKI-BROWNE 4/[14].X.1662.

Reverendissime in Christo Jesu vir, Fautor gratiosissime.

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Ex quo R. T. vidi, coram consilio S. R. Majestatis adhuc non comparui, neque enim consultum fuit e re mea et operis mei, quin illud fecissem hactenus, nisi immediate ab ipsomet Consilio ad praeteritam Petitionem meam prosequendam fuissem vocatus, a quo quia nonnisi hodie fui accitus comparere, ad probandum, quod nuper in Petitis meis, contra Adversarium meum159, proposui. Quia autem illud idonee absque interventu eorum, qui me noverant sex ab his annis et ultra in Anglia, praestare non possum, (non quod se res secius habeat,) sed quia ad negotium intelligendum multuni conducet operis quod ago, et Serenissimae R. Majestatis Consilii Honoratissimi, si aliquem habeam, qui interpretem meum agat. Distuli hodiernam comparitionem ad proximum diem Sabbathi160, eo fine potissimum, ut oculatum testem haberem, qui melius eis negotium meum exponat, quam ego ipsemet, aut adversarius meus Delegatus. Ideoque R. T. per omnia sacra, per amorem in me, et opus meum, rogatum volo, ut in gratiam mei illud oneris in se suscipere velit, venire nempe circa illud tempus et mihi in laboranti negotio assistere, quodsi confio, quod expeto R. T. ut proram et puppim negotii et fortunae meae agnoscam non solum ipse, sed etiam sequentia tempora, ni illud fiat, hoc sciat pro certo, ultimum incurret totum hoc Biblicum negotium, pericujlum, quod quia nee [12-] ullus pius, nee R. T. optaret et aptat, ideo rogo ut illam molestiam in se insumat et sumptus, quo me et opus meum eripiat ex charybdi. Hoc cum a Te Pientissimo Viro DEi rogo, ni hoc felici omine finio, DEum inditurum et R T. animum et Consilio S. R. Majestatis saluberrimum Consilium, ne 7 ad (1) respondendum (2) probandum 159 160

Adversarium meum: i.e. Jan Krainski. Cf. WALLIS-BOYLE 14/[24].III.1661/2 (ii). Distuli . . . Sabbathi: see CHYLINSKI-BROWNE 4/[14].X.1662 76

36. WALLIS to DILLINGHAM, 21/[31] October 1662 opus intercidat, quod concernit promotionem gloriae DEi ex Regni in his terris, Cujus providentiae nutu Te committo et maneo Dabam Londini 4. Sbris. 1662

Reverendae Tuae dignitatis humillimus cultor cum opere meo periclitante S. Boguslaus Chylinski.

5

[13V] These

To the Right Reverend Dr John Wallis. in Oxford. Whith care and speed I pray.

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WALLIS to THEOPHILUS DILLINGHAM 21/[31] October 1662 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by DlLLINGHAM-WALLIS 27.X/[6. XI].1662. In this letter, Wallis requested that Dillingham supply information on cases where privileges of the University of Cambridge were allowed in the Exchequer. Such cases would serve as precedents to which the University of Oxford could also refer.

37. THEOPHILUS DILLINGHAM to WALLIS Cambridge, 27 October/[6 November] 1662 Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 71br-71cv (f. 71bv and 71cr blank). On bottom off. 71br Wallis has noted: 'This is Dr Dillingham (Vice Chancellor 34. (1) 7bris (2) Sbris.

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37. DlLLlNGHAM to WALLIS, 27 October/[6 November] 1662 of Cambridge) his answere to one of mine desiring an account of what precedents they had, |had del] of Privileges allowed in Exchequer; it being denyed us in the case of Mr Wilkins161 the Bedle, for want of a precedent.', and underneath this: 'We had it afterwards allowed in Lichfields162 case, upon a solemn argumen: Judge Hales163 being then Lord Chief Baron.' Reply to: WALLis-DiLLiNGHAM 21/[31].X.1662.

Sir

I received yours164 of the 21 instant in answer to which I have sent you the inclosed paper, which is as much as I could in so short a time finde out for your purpose. I conceive the words of our charters will as well 5 beare us out against the proceeding of the Exchequer as of any other of the Westminster Courts. Nullus judex se intromittat justiciarii ad placita, et alii judices quicunque me thinks should serve for all. Sir though I am very full of business at this time yet I thought my self obliged to make some speedy returne to those than last bee: as I 10 remember I received letters from you about half a yeare since, which I could not then presently usherre an answer to, and afterwards thought it too late. I now crave your pardon for that neglect, & remaine Your loving friend to serve you Theoph: Dillingham 15

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from Clare hall in Camb. Oct: 27. 62. [71cv]

To his much respected friend Dr John Wallis Geometry professor in the University of Oxford these with speed

5 as Iwell as del. of 161

Wilkins: i.e. Timothy Wilkins (1617-71), esquire bedell of divinity at Oxford (from 1657 onwards) and brother of John Wilkins, q.v. 162 Lichfields: i.e. Solodell Lichfield (d. 1671), yeoman bedell of law at Oxford (from 1635 onwards) and esquire bedell of arts (from 1666 onwards). 163 Hales: i.e. Sir Matthew Hale (1609-76), justice of common pleas (1654), lord chief baron of the Exchequer (1660), lord chief justice of the King's Bench (1671), DNB. 164 yours: i.e. WALLis-DiLLiNGHAM 21/[31].X.1662.

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38. HOBBES to WALLIS, end of 1662 38.

THOMAS HOBBES to WALLIS end of 1662 Transmission:

El First edition: [HOBBES], Mr Hobbs considered in his loyalty, religion, reputation, and manners. By way of a letter to Dr Wallis, London 1662. E2 Second edition: HOBBES, English Works IV, 413-40. Written in the form of an epistolary tract addressed to Wallis, Mr Hobbs considered replies to Hobbius Heauton-timorumenos, in which Wallis had accused Hobbes of having written his Leviathan in support of Oliver Cromwell.

39. WALLIS to JOHANNES HEVELIUS Oxford, 9/[19] February 1662/3 Transmission:

W Letter sent: PARIS Bibliotheque Nationale Nouv. acq. latines 1641, f. 110r (verso blank). Answered by: HEVELIUS-WALLIS [25.XII.1663J/4.I.1664.

Oxonii Febr. 9. 1662./3. Praeclarissime Vir, Accepi hodie, dono tuo, gratissimum opus165, De Mercurio in Sole viso: et grates rependo nostras. Et quidem eo magis gratulor successus tuos166, quod Tibi faverit hac in re clementius caelum quam Nostratibus. Utut

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opus: i.e. HEVELIUS, Mercurius in Sole visus Gedani, Anno Christiana MDCLXI. d. III. Maj, St. n. Cum aliis quibusdam rerum Coelestium observationibus, rarisque phaenomenis. Cui annexa est Venus in Sole pariter visa, Anno 1639, d. 24 Nov. St. V. Liverpoliae, A Jeremia Horoxio, nunc primum edita, notisque illustrata. Quibus accedit succincta novae Historiola illius, ac mime Stellae in collo Ceti, certis anni temporibus dare admodum affulgentis, rursus omnino evanescentis. Nee non Genuina Delineatio, Paraselenarum et Paraliorum quorundam rarissimorum, Danzig 1662. 166 successus tuos: cf. HUYGENS, CEuvres completes XXI, 872. 79

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40. WALLIS to BOYLE, 27 March/[6 April] 1663

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enim ex Nostris aliqui conati fuerint id ipsum observare; vix tamen, ob caelum nubilum, id datum erat. Sub hora tamen prima post meridiem eodem die, (nempe Apr. 23. 1661, stylo nostro; qui vobis est Maii 3.) D. Hugenius (qui turn Londini fuit167) et cum illo aliqui, jam turn Mercurium Solis discum intrasse notaverunt168; sed, supervenientibus nubibus, statim occultari et Mercurium et Solem etiam. (Quod eo ipso tempore contingebat, quo, in celeberrima Magnatum corona, Corona Regia Serenissimi Caroli II. capiti turn primum imponebatur.169) Sed et horrenda Tonitrua vesperi sequebantur. Verum non dubito, quin ab ipso D. Hugenio Tibi distinctius, quid ipse observaverit, indicatum fuerit. Numqui autem ex Nostratibus alii quam qui cum illo erant, observaverint, non audio. Video interim et Horroxii170 nostri Veneris observationem, Tuae Mercurii subjunctam esse. Spero hinc utriusque Syderis motum accuratius imposterum restitutum iri. Deum Opt. Max. precor, ut longam tibi et vitam et valetudinem concedat; quo diu, bono publico, his invigilare possis observationibus. Vale. Tui observantissimus, Job. Wallis. 40.

WALLIS to ROBERT BOYLE Oxford, 27 March/[6 April] 1663 Transmission:

E1 First edition of missing letter sent: BOYLE, Works (1744), V, 512.—printed: BOYLE, Correspondence II, pp. 71-2. E2 Second edition of missing letter sent: BOYLE, Works (1772), VI, 455-6 (our source).

6 ipso add. 15-18 his invigilare ... Joh. Wallis. at 9(f in left margin 16r

Londini fuit: Huygens was in London from 23 March 1660/1 until about 17 May 1661. 168 notaverunt: Huygens was with the instrument-maker Richard Reeves (fl. 1641-79) when he made the observation of the transit of Mercury across the sun. See HUYGENS, (Euvres completes XV, 72-3, XVI, 307, and XXII, 575. 169 imponebatur: On the coronation of Charles II on 23 April 1661 see The Diary of John Evelyn, ed. DE BEER, III, 278-84; HUYGENS, (Euvres completes XXII, 575. lr °Horroxii: i.e. the astronomer Jeremiah Horrox (16177-41), DNB. 80

40. WALLIS to BOYLE, 27 March/[6 April] 1663 Oxon, March 27, 1663. Sir,

The inclosed171 is in obedience to your commands laid upon me, when I waited on you at London.172 If it be too large, you may extract out of it as short a one as you please; and if it may seem in ought too short, there is scope of enlarging it as far as you shall think fit. The design is so ordered as to obviate the inconveniencies, and to reserve as much of convenience as well may be. The other command173, which I received in order to Dr. Pocock174,1 have endeavoured to observe so far as was in me. I acquainted him with your desire; but he tells me, that to give account of all the longitudes and latitudes in Abulfeda, is, in a manner, to transcribe the whole book; for it contains little else but the longitude and latitude of places, with some very brief descriptions of them in two or three lines, and not digested into distinct tables, but to be collected out of the text. But he tells me, that Mr. Clark175 is designing somewhat out of him and other geographers compared; which perhaps may better satisfy the desire of the gentleman, than a bare account of Abulfeda alone. I hoped to have given a speedier account of both, had not somewhat else so often interposed. Your goodness, I trust, will excuse the delay, and accept the endeavours of,

Sir, your honour's very humble servant, John Wallis. Since I had written thus much, Mr. Hyde176 (the under library1

The inclosed: The enclosure is missing. when .. . London: Wallis was in London during the first half of March. 173 other command: Apparently, Wallis was to request of Edward Pococke that he provide an account of the longitudes and latitudes in Abulfeda's Geography. 174 Pocock: i.e. Edward Pococke (1604-91), regius professor of Hebrew and Laudian professor of Arabic at Oxford. DNB. 175 Clark: i.e. Samuel Clarke (1625-69), orientalist, M.A. in 1648 at Merton College, Oxford, since 1658 architypographus and upper bedell of the civil law at the University of Oxford. His surviving papers indicate that he was contemplating a new edition of Abulfeda's Geography. See WOOD, Athenae Oxonienses III, 882-5; TYACKE (ed.), History of the University of Oxford IV, 490; DNB. 176 Hyde: i.e. Thomas Hyde (1636-1703), since 1659 underkeeper and from 1665 to 1701 chief librarian of the Bodleian library, Laudian professor of Arabic (from 1691 onwards) and regius professor of Hebrew (from 1697 onwards). In 1665, he published an edition of Ulug Beg's catalogue of fixed stars. DNB. m

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keeper) tells me, that if you please, he will collect out of Abulfeda the longitudes and latitudes, and| transmit to you (I suppose he will expect [456] some gratuity for his pains.) He hath lately, for the bishop of Exeter177, transcribed out of Uleg Beig1^8 the longitude and latitude of all the fixed stars, according to his observations. If you think it tanti, and gave me such order, I will desire him to undertake it.

41. WALLIS to JOHANNES HEVELIUS Oxford, 30 March/[9 April] 1663 Transmission:

W1 Draft of letter sent: LONDON Royal Society Eaxly Letters Wl, No. 2, 4 pp. At top right of p. 1: 'read May 4: 64. entered LB. 1. 131.' At bottom of p. 4 Oldenburg has noted at 90°: 'A Copy of Dr Wallis's letter sent to M. Hevelius and mention'd in the letter next foregoing, about the New Staxr in Cete'. W'2 Letter sent (?): OXFORD Bodleian Library MS. Add. D. 105, f. 14r-15v (our source). w1 Copy of W1: LONDON Royal Society Letter Book Original 1, pp. 131-2. w2 LONDON Royal Society Letter Book Copy 1, pp. 153-4. Answered by: HEVELius-WALLis [25.XII.1663]/4.I.1664. A year after the present letter was sent, Wallis forwarded to Oldenburg the original draft as an enclosure to WALLIS-OLDENBURG 30!V/[10.V].1664, after the Royal Society had expressed an interest in John Palmer's observations of the Star of Cetus. This interest was apparently generated by Wallis's letter to Hevelius of 5/[15].IV.1664, which had been read at the meeting of the Society on 13 April 1664. The endorsement on p. 1 of the draft W1 is possibly incorrect: According to Birch, the letter read at the meeting on 4 May 1664 was WALLIS-OLDENBURG 30.IV/[10.V].1664 (BIRCH, History of the Royal Society I, 422). However, Birch is clearly mistaken when he dates the present letter 30 March 1664, and indicates that it was read at the meeting of the Society on 13 April 1664. As the summary he gives makes clear, the letter read at that meeting was actually WALLIS-HEVELIUS 5/[15].IV.1664 (BIRCH, History of the Royal Society I, 414). W2, which is a fair copy of the draft W1, has been folded and sealed; it is possibly the letter actually sent or may have been substituted by a second letter incorporating new or other information. lrr

bishop of Exeter: i.e. Seth Ward. Uleg Beig: i.e. Ulug Beg (1394-1449), Usbek astronomer, grandson of Tamerlane and himself Great Khan from 1447 to 1449. In 1420, he founded an observatory near Samarkand. 178

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41. WALLIS to HEVELIUS, 30 March/[9 April] 1663 Praeclarissime Vir, Post perlustratum, quern ad me misisti, Tuum, de Mercurio a Te in Sole conspecto, Tractatum179 perquam gratum; una cum eis quae subjunxisti; (inter quae Horroxii nostratis de Venere tractatum180, quern nimium diu publico negatum saepius conquestus sum, tandem comparere gaudeo:) Duo saltern sunt quae Te monendum duxi. Alterum, de mira Ceti stella; de Fixis reliquis, alterum. Nempe Stellam illam, ab Anno Dom. 1639, et deinceps; a Nostratibus (quod Te nesciisse video) multoties observatam fuisse; nunc comparere quidem, nunc disparere. Et primo quidem (quantum scio) a D. Joh. Palmero181 (Viro doctissimo, et Astronomicis Observationibus, ut per alia licet negotia, intento; munere tamen Ecclesiastico occupato:) deinde ab aliis; praesertim D. Samuele Fostero182 Astronomiae olim Professore in Collegio Greshamensi Londini; qui me, dum vixit, ea de re compellavit aliquoties jam ante multos annos. Quid autem Palmerus obiter, in libro183 ab ipso ante aliquot annos edito, de hac stella monuit184; exscriptum huic subjungam. Sed et propediem missurus sum ad ilium, tractatum quern a Te accepi tuum, ut, quum ilium perlegerit, possit, si videbitur, fusius, quae observavit, ipse aliquando impertire. Stellarum vero Fixarum cum videam Te Longitudines et Latitudines denuo instaurandas tantum non polliceri; Monendum duxi, extare apud nos, in Manuscriptis Persicis, Tabulas Astronomicas, ab Uleg-Beig, Rege

10 Joh. add. 13 Samuele add. 13 olim add. 15 jam ante multos annos add. 19 ipse Tibi aliquando Wl 179

Tractatum: i.e. HEVELIUS, Mercurius in Sole visus, Danzig 1662. See WALLISHEVELIUS 9/[19].II.1662/3. 180 tractatum: i.e. Horrox's Venus in Sole visa, which Hevelius had printed in his Mercurius in Sole visus. 181 Palmero: i.e. John Palmer (1612-79), rector of Ecton in Northamptonshire 1641-79, archdeacon of Northampton 1665-79, pupil of Samuel Foster; conducted astronomical observations. 182 Fostero: i.e. Samuel Foster (c.1600-52), mathematician, professor of astronomy at Gresham College in 1637 and from 1641 onwards, DNB. 183 libro: i.e. PALMER, The Catholique planisphaer. Which Mr. Blagrave calleth the mathematical Jewel ..., London 1658. 184 monuit: i.e. PALMER, Catholique planisphaer, 100-1. 83

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olim et Astronomo insigni, institutas; et, inter illas, Fixarum loca, prout ab ipso suisque qui sibi aderant Astronomis fuerant observata; in urbe Samarkanda, quae turn sibi turn avo proavove suo Tamerlano natalis erat: Neque enim Ptolemaei aliorumve Tabulis (quas a vero non raro deviare notaverat) satis fidendum esse duxit. Aptantur illae Fixarum Tabulae, ad initium Anni Hegirae 841; hoc est, ad mensem Julium Anni Aerae Christianae 1437. Quas quia fieri potest ut non videris, curabo, si id desideraveris, ex Persico in Latinum versas, ad Te transmitti185. Interim Vale, Praeclarissime Vir, studiaque, quod facis, Astronomica in utilitatem publicam promovere perge. Oxonii; Martii 30. 1663.

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Tui observantissimus, Joh. Wallis.

In libro186 quodam, Anglico sermone, a Dno. Johanni Palmer, Ecclesiasta [141 Ectonensi in agro Northamptoniae Angliae, edito Londini, anno 1658; cui Titulus; Planisphaerium Catholicum, (or Mr Blagrave's Mathematical Jewel, described, &c;) lib. 4. cap. 34. pag. 100, 101. inter alia haec occurrunt; quae latine versa sic sonant. Novembris 14, 1639 (stilo veteri;) Stellam observavi, tertiae Magnitudinis in Corde Ceti: Quam inde stellam non vulgarem esse cognovi, quod earn ante nunquam conspexeram, nee illam potui in Tabulis Ptolemaei, Tichonis, aut aliorum invenire. Erat ejus Ascensio Recta, grad. 30, 13'. Declinatio Australis gr. 4. 50'. prout observavi modo post tradendo, cap. 44. Quum illam primo observaveram, anno dicto, 1639; nee ullam potui illius mentionem invenire in Tabulis Ptolemaei,

1 olim in India et W1 3 Samarkanda, (1) qui (2) quae 4 Tabulis ( (1) quae (2) quas 20 nunquam notaveram, W1 185si ... transmitti: Hevelius did in fact request a copy of Ulug Beg's catalogue of fixed stars; this was sent by Oldenburg together with a diploma testifying his election to the Royal Society and a letter from Wallis (WALLIS-HEVELIUS 5/[15].IV.1664) on 21 May 1664. See OLDENBURG-HEVELIUS 27.V/[6.VI].1664; OLDENBURG, Correspondence II, 204-5. 186 libro: i.e. PALMER, Catholique planisphaer.

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41. WALLIS to HEVELIUS, 30 March/[9 April] 1663 Copernici, Tichonis, aut Magini188; hoc indicabam amicis meis non vulgaribus, Doctor! Johanni Twisden189; tune in Kantio; et Do. Samueli Foster, Astronomiae Professori in Collegio Greshamensi, turn Londini; qui eandem quam ego, quoad stellae locum, observationem fecerunt: Et convenimus omnes, lucem ipsius turn continuo crescere, ut Mense Decembri 1639, tertiam magnitudinem superaverit; nullam autem sensibilem habere parallaxin. Cumque in animo habuerim, brevem aliquam ejus insinuationem, Latino sermone, in lucem mittere; quo Astronomi transmarini, juxta atque nos, illius observationi attenderent: Literis indicavit D. Foster, se stellam hanc in imaginibus Bayeri, Anno 1616 editis190, depictam invenisse. Anno sequente, 1640, in manus meas pervenit, a Dno. Twysden transmissus, Tractatus de Stella hac turn nuper editus a Phocylide191 quodam, Logices Professore Franaquerae; cujus observationes nostris fuerunt consentaneae. Existimavit autem ille, Stellam hanc ex Ecclipsi niagna Lunari, quae Decemb. 10. 1638. in anterior! pede H contigerat, originem habuisse: Verum in hoc ab illo dissentientes fuimus. Sententiam ejus videre est, in libri sui pag. 197. Haec stella, apparet saepe, iterumque disparet. Est aliquando tertiae magnitudinis; aliquando, quartae. Saepe illam observavi in hemisphaerio Orientali; in Occidentali raro. Dis1 Copernici, Stadii187, Tichonis Wl 1 aut (1) aliorum invenire; (2) Magini; 4 quam ego, (1) observati breaks off (2) quoad 9 ejusdem insinuationem Wl 12 indicabat Wl 187

Stadii: i.e. Jan Stade (1527-79), Belgian astronomer and mathematician. Magini: i.e. Giovanni Antonio Magini (1555-1617), Italian astronomer and mathematician. 189 Twisden: i.e. John Twysden (1607-88), physician, fellow of the College of Physicians in London from 1664 onwards. In 1659 he published some of the papers left by Samuel Foster as Miscellaries or Mathematical Lucubrations. DNB. 190 imaginibus Bayeri, Anno 1616 editis: possibly a reference to the Uranometria of the German lawyer and astronomer Johann Bayer (1572-1625), which first appeared in Augsburg in 1603. 191 Tractatus . . . a Phocylide: i.e. Jan Fokkes HOLWARDA (Johannes Phocylides) (1618-51), ITaz/<7eA77i/o<j EKAetTrrtKr/ 6i,av"/dS,ovaa, Franeker 1640. 188

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paret nonnunquam per plures continuas septimanas. Hoc anno, usque ad Febr. 2. 1656./7. (stylo Angliae) non potui illam observare. Quibus otium suppetit studiis hisce vacandi, commodum erit si huic observanda intendant, et vicissitudinum causis investigandis. In quern finem, publicam hanc notitiam indicare visum est. Cap. 44. haec sequuntur.

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[151

Anno Dom. 1639. Observabam Mandibulam Ceti, Genam Ceti, et peregrinam hanc stellam Cor Ceti, (cujus mentionem fecimus cap. 34.) in eadem esse linea recta. Et Radio Astronomico, qui tune ad manum erat, observavi, Cor Ceti, a Mandibula, distare grad. 13. 4'. Atque hinc colligit, Ascensionem rectam, et Declinationem, ut supra. Longitudinem, grad. 26. Latitud. gr. 16. 10'. Praeclarissimo Doctissimoque Viro, D. Johanni Hevelio, tradantur, Dantisci.

[151

42.

ROBERT BOYLE to WALLIS August/September 1663 Transmission:

Manuscript missing. Existence: Mentioned in and answered by WALLIS-BOYLE 10/[20].IX.1663. This letter accompanied, or was sent about the same time as, a copy of Boyle's Some Considerations touching the Usefulnesse of Experimental Natural Philosophy, London 1663, on which the author requested Wallis's comments.

7 visum erat. W

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WALLIS to ROBERT BOYLE Oxford, 10/[20] September 1663 Transmission:

El First edition of missing letter sent: BOYLE, Works (1744), V, 512-13—reprinted: BOYLE, Correspondence II, 108-9. E'2 Second edition of missing letter sent: BOYLE, Works (1772), VI, 456 (our source). Reply to: BOYLE-WALLIS VIII/IX.1663. Enclosures: A physician's account of the illness of Rebecca Wallis (and Wallis's comment thereon); Wallis's comments on Boyle's Usefulnesse of Experimental Natural Philosophy.

Oxon, Sept. 10, 1663.

Sir, I have with very great content and satisfaction gone through that book192 of yours, wherein you have so very charitably obliged mankind. You will easily believe, though I should not tell it you, that in so short a time it were not possible for me to consider so much variety of good matter, suitably to the worth of it. Which though it be true, yet doth not arise from want of respect to it, but from mine impotence and utter inability to pass a particular judgment of chemical processes, being so nothing of a chemist myself. And therefore, the animadversions, which you invite me193 to make, you may be secure, will not concern any thing of that kind. The papers inclosed194 come so rudely, because I could not judge what is in them to be worth writing twice, which I fear you will have reason to think not worth reading once. And if after (or before) you have once read them, you will please to let them help make a fire under one of your learned furnaces, it will be a greater honour than they deserve. For by that means, they may give some light to some of your works. But your mention of some specifics you have for distempers in genere nervoso, minds me of what I thought to have discoursed with you, but neglected, when I had the opportunity concerning some very odd and 192

book: i.e. BOYLE, Some Considerations touching the Usefulnesse of Experimental Natural Philosophy, London 1663. 193 invite me: i.e. in BOYLE-WALLIS VIII/IX.1663. 194 papers inclosed: i.e. Wallis's comments on Boyle's Usefulnesse of Experimental Natural Philosophy.

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44. WALLIS to BOYLE, 10/[20] September 1663, enclosure (i)

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dangerous symptoms in a relation of mine (my brother's wife195) so good a person, and so useful, that her life and health are very well worth the preserving. In order to which I have transcribed the account196 of her case, as her physician (a good chemist, and successful practitioner, the same I mention in the inclosed papers) did upon my request draw it up to send to Dr. Willis197; together with what afterwards I did myself observe. If you have any arcanum, which you judge proper in the case, by communicating the medicine (though you keep your secret) you will not only do a great work of charity to her, and in her to many more, and very much oblige, Sir, your honour's very humble servant, John Wallis.

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WALLIS to BOYLE 10/[20] September 1663, enclosure (i): A physician's account of the illness of Rebecca Wallis (and Wallis's comment thereon) Transmission:

E1 First edition of missing enclosure sent: BOYLE, Works (1744), V, 513-4.—reprinted: BOYLE, Correspondence II, 109-12. E2 Second edition: BOYLE, Works (1772), VI, 456-8 (our source). Enclosure to: WALLis-BoYLE 10/[20].IX.1663.

25° Julii, 1663. 15

Uxor Dom. Wallis Ashfordiensis, Cant, habitus mediocris, temperamenti melanophlegmatici, quadregesimum agens annum, victu euchymo ut plurimum utens, ab incunabulis plerumque valetudinaria existens, rarissime cum morbo ullo acuto (exceptis variolis) tentata, semperque pro scorbutica 195

wife: i.e. Rebecca Wallis (d. 1677), wife of Henry Wallis (c.1620-66). account: i.e. the enclosed physician's account. 19r Willis: i.e. Thomas Willis (1621-75), physician, Sedleian professor of natural philosophy at Oxford (since 1660), elected fellow of the Royal Society in 1661, DNB. 196

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44. WALLIS to BOYLE, 10/[20] September 1663, enclosure (i) suspecta; post varia & innotata symptomata leviora, quae annis praeteritis quandoque passa fuit, sub autumni superioris initio, dolors, lateris dextri, pleuritico hand multum dissimili, prehensa est: cui, phlebotomia praesidiisque aliis levibus, pacato, paulatim successit dolor acutissimus, mordicans & pungens, sub mucronata cartilagine, quandoque ad umbilicum fere, quandoque ad costas nothas dextras, quandoque ad ilia dextra se porrigens; una vel altera post pastum hora delapsa, aegrotam invadens: quern gradatim supervenit oris ventriculi (post sumptum cibum, ut antea) superioris morsus acerbissimus, animi fere deliquium secum trahens, per circuitus revertens, cum respiratione frequenti, & fere intercepta, oscitatione, ructatione, vellicatione, & saepissime totius quasi ventriculi convulsions,; [457] nunquam\ illam jejunum invadens, neque post sumptum panem neque frumentacea edulia inhausta; sed tantummodo quum victitasset carnes fere quascunque: unde per totam hyemem, frumentaceo & triticeo victu, vinoque generoso, contenta remansit; saepissime etiam per mensem ampliusve temporis, tali regimine, immunis evasit, solitis fungens muneribus domesticis, placideque nodes trahens, satisque hilariter inter vicinos se gerens; paroxysmis praedictis (quando invasere) magis magisque sensim minuentibus. Vere ingrediente, omnia mitiora facta, praedictis symptomatibus in totum fere evanescentibus. Ingruente aestate, corporis illius status non procul a naturali & eucratico videbatur, eduliisque quibuscunque & placuit vescens, sineque molestia ferens. Attamen sub mensis praeteriti calce subito post pastum paroxysmo praeteritis dissimili correpta est; cum dolore hypochondrii dextri, turn gravativo turn pungitivo, cum frequentissima respiratione, & metu suffocationis, febriculam secum habente, cum urinis coloratis, anxietate insueta, subitaneoque virium lapsu, ut vix loqui vel se de loco in locum movere potuit, loquela interim gemitibus insolitis interrupta, & summotam, in ultimum vitae discrimen adstantibus redacta videbatur, ita ut per plures dies in cubiculo vel lecto se continebat, unoque eodemque statu per quinque vel sex dies permansit, usitato interim victu utens uti placebat; sine appetitus dejectione, ventriculi retentricis vel expultricis vitio, vel coctricis imbecillitate; videbatur autem coctio depravata, sicque se habebant facultates stomachi per totum morbi decursum. Hoc insuper notandum, quod medicamenta purgantia nunquam ferre possit. Nunc temporis, omnia leviora sunt facta, cibos (uti arrident) capit, sine ullorum praedictorum symptomatum insultu, vitamque trahit (quam antea) beatiorem, sine dolore vel tristitia. Haec sunt praecipua, quae in nostra aegrota se produnt. 22 ut placuit E1 89

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This is verbatim the account, which I had from her physician198 (her brother) who had diligently watched her, as you may discern by the so particular account he gives. Upon discourse with him, he tells me, he had sometime thought the cause of those strange symptoms to have been in the coats of the stomach (because digestion in the stomach, and other faculties there, seemed not to be much impedited;) since that it was rather in the ferment of the stomach, but from whence injected (from the pancreas, or whence else) was dubious. Her head not molested with fumes or aches, nor is she wont to be feverish, or otherwise (save as is related) discomposed. Her stomach he complains of as troubled with viscous matter, or such, as that he can hardly make any medicine fasten on it, which he would willingly move by vomit, but dares not, knowing how deadly sick she hath been some years since with that physic. Yet once he ventured on a gentle vomit, but durst not stir from her all the while it worked; and is not willing to venture again. And some times a pipe of tobacco doth not only repel an approaching fit, but bring up matter seemingly as clear as water, but (whether presently, or after some time, I remember not) if stirred with a stick is ropy and viscous like the whites of eggs. The most of evacuating physic, that he useth, is now and then some very gentle pills, scarce more than to keep the body soluble. But (beside what others I know not) the chief remedy he applies (and with good success) is an arcanum (which my other papers mention) much of kin to spirit of harts-horn; but, he saith, more prevalent than either that or soot (for he useth them all very frequently with his patients;) and I suppose it is some animal spirit, but from some other animal (yet not spirit of blood neither) of which she hath taken abundance, I think constantly, about an hour or two before dinner, and before supper. To which he attributes what of preservation or cure is wrought. Dr. Willis199 (upon the account above) hath delivered his judgment of it to be a dissaffection in genera nervoso; and adviseth spirits of harts-horn, or of soot, which agrees well enough with the judgment and course already taken. He tells me of a peculiar juice he apprehends to be conveyed by the nerves, quite different from the circulation of the blood. [458] While I was in the country with her (after the date of the paper, which gives account of it) she had ever and anon rather attempts of fits than perfect fits, but severe enough, if she had not known worse, if at any time she did eat either more freely, or more hastily than ordinary (for she is 198

physician: almost certainly John Nowell (d. 1670/1), brother of Rebecca Wallis. At his burial, Nowell's occupation was noted as 'physition'. 199 Willis: i.e. Thomas Willis.

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45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) forced to be very moderate, and very deliberate in eating.) Any sudden surprise of passion (whether joy, grief, or other) is very apt to discompose her. These fits keep her very faint and feeble, though not sick. If they return with the winter, as they were last winter, they must needs be very dangerous: if but as they are, they must weaken, and must needs threaten very ill consequents.

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WALLIS to ROBERT BOYLE 10/[20] September 1663, enclosure (ii): Comments on Boyle's Usefulnesse of Experimental Natural Philosophy Transmission:

W Notes sent: LONDON Royal Society Boyle Papers 44, f. 23r-27v (our source).— printed: BOYLE, Correspondence II, 112-19. Enclosure to: WALLIS-BOYLE 10/[20].IX.1663.

I know you will smile; when you shall see mee descant upon words of one so much master of them, & so well skilled in proprieties. Yet here & there one, you will give mee leave to animadvert upon, that you may at lest be perswaded I have read the book200. Your Conducive (which I have sometime stuck at, because such terminations are usually deduced from Supines, according to which analogy it should be Conductive] which yet, because we make use of Conduct & Conduce, both from Conduco, but in different senses, would not so well serve your turn; for which cause Conducing hath been wont to be more used:) I am now reconciled to: Because I find Nocivus (though possibly not by Tully201, or very ancient Latinists.) used in that form from Noceo, instead of Nocitivus, which were analagous; for which I suppose Tully 7 see add.

13 of (1) both (2) Conduct 200 book: i.e. BOYLE, Some Considerations touching the Usefulnesse of Experimental Natural Philosophy, London 1663. 201 Tully: i.e. Cicero.

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45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) would rather say Nocuus and Noxius; &, in the compound, Innocuus (not Innocivus) & Innoxius. And in the same form wee use Coercive; though yet I question whether Coercivus be to be found in any good Latine Author; but rather Coercitivus, or Coerctivus, as there is Coercitio, and Coerctio, (from coercitum or coerctum) & instead of Coerctio, is creapt 5 in coertio, & from thence Coercio, whence coercivus for coerctivus. Which two words, nocive & coercive, (beside which I do not| remember any other [23V] in that form) which use hath made passant rather than analogy, will the rather justify conducive, from Conduce, that conductive may accompany 10 his cognate Conduct. p. 140. part. 2. But Aperitive (though modern Physicians use it) I should not choose to use, because Apertive may as well bee sayd, which is Analogous; Aperio making the supine not Aperitum, but Apertura, whence Apertio, Apertura, &c. And so I find you use it p. 155. 1. 12. Apertive. part. 1. p. 2. 1. 7. Perfectionated, I think might as fully have been 15 expressed by the ordinary word Perfected; as well because it looks like a needlesse affectation of a new coyned word; as allso because Perfectionari will hardly be found in Latine; but they would content themselves with Perfici, or compleri, to which answer the English, Perfected & compleated. p. 62. 1. 25. / ignore not, that; though analogous inough; yet might as 20 significantly, & more usually, bee expressed by / am not ignorant, that. p. 28. 1. 5. And not altogether ignor'd, had been as fully sayd, not altogether unknown. p. 24. That excellent quotation202 out of Seneca, de Otio sapientis; because I found the sense impedite, made me consult the Author (but with 25 somewhat the more difficulty, because I could in my Seneca find no such Title; but at length found it to cohere with that other, De Vita Beata;) Where I find the obscurity to arise from omission (in the transcribing) of a line or more. For it should thus be read| [24r] Ut scias illam spectari voluisse, non tantum aspici, vide quern

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4 or (1) Coerct breaks off (2) Coerctivus 5 of (1) Coerctio, (2) Coerctio, 9 Conduce, (1) because (2) that 12 use, (1) be (2) because 13 making (1) the (2) the supine (a) Ape breaks off (b) not 25 impedite, (1) I consulted (2) made me consult 26 in add. 202

quotation: i.e. SENECA, De oiio V, 4. 92

45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) nobis locum [dederit. In media nos sui parte constituit, et circumspectum omnium nobis] dedit. Nee erexit tantummodo hominem; sed etiam, ad contemplationem factum, ut ab ortu sidera in occasum labentia prosequi posset, et vultum suum circumferre cum Toto, (meaning the Universe, not his whole Body, as the translator takes it,) sublime fecit ei caput, et collo flexibili imposuit. where, ad contemplationem factum, must agree with Caput which comes after, not be referred to hominem before; which if intended it should have been facto, because in this clause the structure is changed by the verb fecit ei~ so that it should be, sed etiam ad contemplationem facto, fecit ei. & so that I should rather have rendered it. That you may know shee intended not onely to be seen, but to be looked upon, consider the place shee hath assigned us. She hath placed us in her midst, & given us a prospect round about on every thing. Nor hath shee onely made man in stature upright; but, having designed his head for contemplation, to the end he might follow with his Eye the moving stars from east to west, & turn about his face, together with the universe (or, as the World turns about;) she hath placed it high (or set it in the highest place) and on a flexible neck (a neck fitted to turn about:) or, shee hath made it his highest part, & set it on a neck fitted to turn about. Whereas, had, Vultum circumferre cum toto, been meant of turning his face together with his whole body, that which follows, of a flexible neck, had not been to the purpose. Then in the close; aliquid ipso mundo Antiquius, might I think be as well rendered somewhat more Venerable than the World itself; according as that phrase Quo nihil [24^] Antiquius is wont to be used.| p. 30.1. 1. disprovided; why not, unprovided! Disprovided, as it sounds somewhat harsher being lesse used, so it seems rather to input more than is intended; viz. not barely a want of provision, but a being despoiled of provision, which it is supposed to have had. p. 10. 1. 3. are beholding; I should rather say beholden. From Hold, (teneo, oblige;) the passive participle is Holden, (tentus, obligatus;) & Beholden is but the same, with a usual prefix be, frequent with verbs & 8 where, (1) ad factum (2) ad 12-15 rendered it. (1) Nor shee hath not onely (2) |That you . . . every thing. add.\ Nor 29 viz. (1) a being deprived (2) not barely a want of provision, but a being (a) deprive breaks off (2) despoiled

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especially participles; as besmeared, befouled, bespattered, bespangled, and the old womens country proverb, Blessed is the Corps that is berained, & the Bride that is beshined, so bepissed, bespread, & an hundred more (in which be seemes to import upon.} Beholden being the same sense precisely with Obliged, not Obliging. But this note, I perceive, might be spared; because afterwards I find it usually so written. p. 2. penult, inconsiderablest instanse. p. 59. I. 1. incomparably transcendenter degree, p. 60. 2. glorify God exactlyest: & some other seemingly harsh comparatives & superlatives, though analogous inough: I should rather (for the most part) expresse by the pariphrasis of more, & most, (as in divers cases the Latins do by magis et maxime; for they do not vary all Adjectives according to the degrees of comparison; most inconsiderable instanse. incomparably more transcendent, glorify God most exactly. &c. Yet p. 2. I. 7. I am content you should say Lovelyest & most improving; to avoid repeating most twice; though, if alone, I should rather say most lovely. [25r] p. 59. 1. 18. All equal, because all infinite. Quaere, whether that consequence will allways hold. An infinite number of men, will have more eyes, yet but infinite. Anni futuri to eternity, are in number infinite, yet the Days are more, & the Hours yet more, yet but infinite. And yesterday, or at the worlds beginning, were more future than now are. A Line supposed infinitely long, being in any point divided, each part will be infinite, yet lesse then the whole. p. 69. 1. 28. depends upon the pressure of the Air, the table of Errata, addeth against the suckers chest. Which emendation I think were better spared; for it is rather the pressure of the Air upon the stagnant liquor. p. 73. end. I am not satisfyed, that the sagacity of Foxes, Bees, &c is not Homogeneous with the like in man, (as well as sense, & local motion, digestion, &c.) notwithstanding what is to be sayd for the immortality of rational souls. But whether to call it Reason, or no; is onely lis de nomine: Depending upon what you shal arbitrarily Desire, to be the meaning of the 14 [In left margin in Wallis's hand:] p. 2. 1. 7.

1 besmeared, (1) befould (2) befouled 3 more add. 29 fee.) (1) whether (2) notwithstanding 30 to (1) call (2) call

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45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) word. I should sooner assent to somewhat like Plato's Anima Mundi203, than to think Beasts to be meer Engines, not cogitant (Ingenia sine ingenio.) For what if it should be sayd, that there were, equally originall, two Substances, Matter & Soul, of which according to different proportions of the one & the other & various textures, several degrees of Animalls do arise; & upon the dissolution thereof, the component parts be variously V [25 ] disposed of into other concretes. | p. 75. 1. 13. Meteora, I find translated p. 121. Meteors: I should rather have renderd Sublime Bodies; for the Greek word is wont to be so used; (though the English bee not;) & it is here evidently meant of the Stars, & Sun, moon, &c. which we are not wont in English to call meteors. part. 2. p. 32. 1. 10. purification, read petrification. part. 2. p. 114. An Instrument for weighing liquors. Monsieur Monconits204 shewed us at Oxford a very ingenious one; I suppose you saw it at London. p. 151. Spirit of Soot. A good Chymist & practitioner of my acquaintance, commends it as an excellent specifick in a Pleurisy, upon his own experience; & he says he knows none better. p. 152. Stone-horse dung, I have heard commended as a specifick in a Pleurisy; & used with successe. p. 153. 1. 9. Ancient, I find corrected in the Errata, Antient: yet, of the two, I should prefer the former spelling. For I suppose it comes from Antiquus or Anticus (in the same forme as Posticus,) whence allso our English word Antick (for odde kind of shapes & gestures,) And as from Physicus, Physicien, &c (for so the French form their termination from

1 Mundi, (1) that (2) than 2 Engines, \& del] not 3 should (1) say, (2) be sayd, 4 Soul, (1) by (2) of 11 which . . . meteors, add. 19-20 p. 152. . . . successe. add. 22 from (1) Antiq breaks off (2) Antiquus 23 allso (1) A breaks off (2) our 25 their (1) lat breaks off (2) termination 203

Plato's Anima Mundi: see PLATO, Timaeus 34B-37C. Monsieur Monconits: i.e. the French aristocrat Balthasar de Monconys (1611-65), who was in England in May and June 1663. See Journal des voyages de M. de Monconys, vol. 2, Lyons 1666, 49-50, 54. 204

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45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) the Latine cus,) so from Anticus, Anticien, (so that the last t might be spared,) thence Ant'den, & (by a further Syncope of t before another consonant) Ancien, rather than Antien, or Antient. And upon the same account I choose to write Physician, Mathematician, &c. rather than 5 -tian. p. 155. Millipedes' I have been told (by one who knew her) were much used & with good successe, by an old Woman for her eye indangered by Catarracts. p. 336. et alibi. Scent. Scituate. Why not Sent, Situate"? For that 10 comes from Sentio; this, from Situs.] p. 382. Spirit & Salt of Soot, & of kin to those of Harts horn & Urine. The Chymist I mentioned above; give mee this difference (amongst others) observed in his practise; between Sp. of Soot, & another Arcanum of his which I have reason to think neerer of kin to that of Harts-horn, though he assures mee it is not the same: That of Soot he finds excellent in 15 Pleurisies, but will not at all fix an Epileptick fitt: in which his other (though perhaps not so good at pleurisies) is extremely successeful. And from my own experience of it (who have commonly some of it by mee) I have to divers given of it in such fitts (whom it hath never failed, at lest after twice or thrice taking, to cure) but at another time (not having of 20 it) I gave spirit of soot, which had no successe. p. 174. Your mention here, of Malleable Glasse, makes mee here mention what I meant to have put by itself. One, now with mee (but no Chymist) tells mee a story (how probable I cannot say) that an Apothecary about six weekes agoe carrying (I think, to a patient,) in his pocket 25 (where it was for 24 hours together) a Medicine, or what else, which he says to be prima solutio Turbeth Mineralis; when he went to take it out of his pocket, found the Glasse soft & flexible into any form. p. 177. I suppose I need not tell you that Distillers of Strong Water in London, do it with Sea-coal uncharred. 30 p. 178. Iron & Steel (I am told at least) if but put into a coach-mans

1 t (1) be (2) might 6 were (1) (—) (2) much 7 by an old Woman add. 19 at lest add. 24 how (1) probably (2) probable 25 agoe add.

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[26r]

[26V]

45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) box, rusty, will (with the moving of the Coach) in a few days become very bright. p. 285. Corall to sower humours in mans Body. I have been apt to think, that Magisteryes of Coral & Perl used in Convulsions of little Children (&, may be, the like of Ungula Aids &c) may do the good it doth, onely by being a prey for such sowr humours to fall upon, &, as by precipitation, disarm them. After this (though not to the present purpose) I shall tell you of an Observation I made. About a year since, turning an Armillary Sphere, I chanced to pinch, (between two of the heavy brasse circles,) the forefinger of my right hand very sorely about the midle space between the two uppermost joints; Which, having extricated leasurely, by turning back the movable circles, (for had I offered to snatch my finger away, as people are apt hastyly to do on a suddain pain, I had hurt it worse, than by induring the pressure for about a quarter of a minute;) pulling of my glove, I found, on the back side of the finger, (for the in-side was more defended with flesh, & met with a blunter edge,) the sharp edge of the brasse, had, without breaking the skin, made a dint as deep as the bone would suffer it, (&, I suppose, had bruised the bone itself,) as if a notch had been cut in a stick; & did not suddenly fill up. I applyed to it, as r [27 ] soon as I could get home, a plaister (as I remember) of Paracelsus: And continued it on for some weekes. For, though the pain were much sooner over, (saving a dull aking about the bone;) yet, a numnesse continuing, I thought it necessary to preserve it from the outward air. This numnesse (which is the thing for which I mention it,) though principally at the place pressed, did extend itself through the nail & so to the fingers end, especially on that side of the fore-finger against which my pen leaneth, though the hurt were rather on the other side, where the knuckle of the

1 rusty, (1) with (2) will 6 as add.

10 pinch, (1) (—) (2) (between 12 having (1) pulled (2) extricated 13 the (1) movabl breaks off (2) movable 14 apt (1) sud breaks off (2) hastyly 15 the (1) pain (2) pressure 19 itself,) (1) appearing (2) as 23 (saving (1) an (2) a 23 I that it corr. ed. 97

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middle finger beares against it. The bone though not quite ridde of the pain for a Month or more: the Numnesse continued much longer. But I observed it leasurable to shift forward from the place hurt toward the fingers end, over the joint & so towards the naile; (& even that in the pulp of the finger where the pen lyes, moved thither allso;) when it was come as high as the nail, (whereby the nail & parts immediately covered by it were in a manner wholly senselesse,) it still moved on-wards, till it came by little & little to be pared away with the nail. Which slow journey it was at least half a year in dispatching, I think I may say 3 quarters; or rather that there is some little symptome of numnesse about the edge of the nail even yet remaining. (You will excuse mee for being so teadious in relating circumstances, some of which might perhaps bee spared; because an errour on that hand you are wont to account lesse blamable than on the other hand an errour of omission.) If you ask, what I gather from this Observation; Perhaps not much; but yourself possibly may more. The| [27V] Numnesse, I presume, was from the pressure of some Nerves, or Tendons, or both. Whether the promoting of it to that part of the nail where it is pared off, were a thrusting forward of some hurt matter in those Nerves or Tendons; or of those nerves or Tendons themselves; I cannot determine. Because I am not yet clear what that substance is whereof the Nail is made; But it seemes to be a protrusion or succrescence of tendonous matter grown hard & callous, in the room of which succeeds new & fresh made: So that the very substance of what was before deeper in the finger, seems to be thrust forth & pared off at the fingers end. And the benummed matter being thus quitted, the next succescent, is clear of it. Whether hurts further off, as in the Arm or higher, (& the like of the leg or thigh) will be thus protruded ad extrema, as to the Nails of the Fingers, or Toes; may be yet of further inquiry. And, whether the Gout, & some such diseases, may not be occasioned by the impediting of some such protrusions. The rather because in the Nodosa podagra, 'tis sayd that at joints (where such protrusions are aptest to be hindred) hard chalky matter doth oft break forth; which had it kept on its course till it had been pared off with the Nail, had been inoffensive. But I forget myself, in tyring you beyond measure: Having at first intended, onely in 1 quite add. 4 that | part del] in 5 allso;) (1) & (2) when 7 moved (1) onw breaks off (2) on-wards, (a) (it) (b) till 18 were |from del] a

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46. WALLIS for the ROYAL SOCIETY, 24 September/[4 October] 1663 half a dozen lines, to set down what I had observed to happen; & leave it to your thoughts.

46. WALLIS for the ROYAL SOCIETY 24 September/[4 October] 1663 Transmission:

W Paper sent: LONDON Royal Society Early Letters Wl, No. 3, 2 pp. (our source). At top of p. 1 in Oldenburg's hand: 'An account of a new way of hatching pigeons read before the Society Oct. 14. 1663.' Beneath this: 'Entered LB: 2: 303.' wl Copy of paper sent: LONDON Royal Society Register Book Original 2, p. 303. w'2 LONDON Royal Society Register Book Original 2 (copy), p. 165. w3 LONDON Royal Society Register Book Copy 2, p. 49. E First edition of paper sent: BIRCH, History of the Royal Society I, 313-14 (partly). The paper was read at the meeting of the Royal Society on 14 October 1663; BIRCH, History of the Royal Society I, 313.

Sept. 24. 1663. Keeping some tame Pidgeons about our house, wee had a pair hatched on the 13th of July last, the day of our publike Act205 at Oxford. The next morning wee went from home; and, having been absent allmost 7 weekes, wee found, upon our return, this pair of Pigeons, well grown & lively; and making a nest as others do when they are ready to lay Egges & sit. But not finding them to lay, (as indeed at that age it could hardly be expected) wee did (about the 2d or 3d of this instant September) take two Egges, newly layd, from another pair, & put under these young ones; Who, though too young to lay Egges of their own, did sit on these very diligently, by turns, (as the Cock & Hen pigeon use to do:) Which they were so desirous to do, that they would often fight for the nest, or pull one another off, or at lest sit waiting with much patience for the nest. And did on Munday last, Sept. 21. hatch two young ones, (themselves

3 Sept. 24. 1663. del. (probably by Oldenburg) 205

Act: The annual public Act or commencement ceremony was held in St Mary's Church (until the opening of the Sheldonian Theatre in 1669) in the second week of July, after the end of the academic term. 99

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being that day 10 weekes old;) and are now brooding them as older pigeons do. That pair from whence these Egges were taken did within 4 or 5 days after lay again, & do now sit, allmost ready to hatch. That pair from whence these two young breeders were hatched, have allready layd & brought their eggesj to good, 7 times since March last. [2] And the Hen of their first brood, hatched in March, hath with another Cock (her own mate being stollen, or killed, or otherwise miscarried wee know not how) layd, & hatched, & is now brooding her young. J. Wallis.

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JOHANNES HEVELIUS to WALLIS Danzig, [25 December 1663]/4 January 1664 Transmission:

C Copy of missing letter sent: PARIS Bibliotheque Nationale Nouv. acq. latines 1641, f. lll r -lll v . Reply to: WALLis-HEVELius 9/[19].II.1662/3 and WALLIS-HEVELIUS 30.III/[9.IV].1663. Answered by: WALLIS-HEVELIUS 5/[15].IV.1664. This letter was brought to London, together with HEVELius-OLDENBURG [25.XII. 1663J/4.I.1664, by Johannes Koopman, a relative of Hevelius. See OLDENBURG, Correspondence II, 136-8.

Praeclarissimo Viro, Dn. Joh. Wallisio, amico perquam Colendo. Oxonii Vir Praeclarissime

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Binas Tuas literas206, die 9 Febr. et 30 Martii anno elapso Oxonii datas, optime mihi redditas esse scias; ad quas autem citius, profecto respondere haud potui, ob varias occupationes, tarn publicas, quam privatas, inprimis 1 brooding |& feading del. them 7 first add. 9 is add. 206

Binas Tuas literas: i.e. WALLIS-HEVELIUS 9/[19].II.1662/3 and WALLIS-HEVELIUS 30.III/[9.IV].1663. 100

47. HEVELIUS to WALLIS, [25 December 1663J/4 January 1664 speculationes diversas Coelestes, quibus diu noctuque distringor. Initio gratius habeo magnus, quod commercium istud literarum, aliquandiu ob temporis injuriam ab utroque nostrum intermissum iterum instauraveris; agnosco exinde summam Tuam erga me propensionem, praesertim quod de nonnullis ad res Uranicas spectantibus, bono publico, simul certiorem me reddideris207, utpote inter alia de longitudinibus et latitudinibus Fixarum, a quibusdam Astronomis Regis Uleg-Beig annotatis. Quarum observationum si particeps fieri possem in latine idiomate, nihil mini unquam gratius accideret208; spero enim haud parum emolument! inde ad Astronomiam redundaturum; cujus incrementa et Tu nobiscum sine omni dubio exoptas. Ego quid praestiterim, aliquot annorum spatio in isto negotio, tarn circa Fixas Universas hactenus cognitas, quam plurimas neglectas, nudis tamen oculis optime deprehensibiles, suo tempore, ex Machina nostra Coelesti209 percipies; quam, prima occasione, si Altissimo ita visum fuerit, publici faciam juris; dummodo prius Cometographiam [11F] nostram210, quae jam sub praelo sudat, penitus| expedivero. Quid vero apud vos agatur in re literaria ex quid Illustriss. Societas Regia nuper fundata a Serenissimo Rege Britanniae moliatur, animus perquam scire gestit; cumprimis aliquanto prolixius mini exponas rogo, fundationem illius; et quot, quibusque Literatis collegium istud constet? num certus eorum detur numerus? quisnam Praesidis fungatur officio an saepius conferant? et quidnam praecipue intendant? Denique harum literarum latorem, Nobil. Job. Koopman211 affinem meum charissimum majorem in modum Tibi commendo; fac, quaeso, ut intelligat etiam apud Exteros, quos invisendi gratia profectus est, nostram commendationem aliquod habuisse pondus. Vale ex annum hunc, quern ut spero, optoque feliciter es ingressus, cum DEO secundissime absolve. Dabam Gedani Anno 1664 die 4 Januarii St. Greg.

12 Universas (1) tarn (2) hactenus 24 fac, quaeso, add. 20r

reddideris: see WALLIS-HEVELIUS 30.III/[9.IV].1663. nihil... accideret: cf. HEVELIUS-OLDENBURG [25.XII.1663]/4.I.1664; OLDENBURG, Correspondence II, 136-8. 209 Machina nostra Coelesti: Part I of Hevelius's Machina Coelestis was in fact not published until 1673; part II appeared in 1679. 210 Cometographiam nostram: Hevelius's Cometographia did not appear until 1668. 211 Koopman: i.e. Johannes Koopman, probably the brother of Catherina Elisabetha Koopman, whom Hevelius married in 1663 after the death of his first wife. 208

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49. WALLIS to BROUNCKER?, ? March 1664 (ii) Tuus Officiosa voluntate Job. Hevelius Cons. Gedanensis ac p. t. Jud. Vet. Civ.212

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WALLIS to WILLIAM BROUNCKER ? ? March 1664 (i) Transmission:

Manuscript missing. Existence and date: Mentioned in BIRCH, History of the Royal Society I, 404. This now missing letter was the first of two letters from Wallis to Brouncker (concerning the translation and printing of Ulug Beg's astronomical tables by the Society's printers), which Brouncker produced at the meeting of Royal Society on 30 March 1664. Since the Society had requested to Wallis on 2 March that he procure the translation of Ulug Beg's catalogue, the two letters were clearly written some time in March 1664.

49.

WALLIS to WILLIAM BROUNCKER ? ? March 1664 (ii) Transmission:

Manuscript missing. Existence: Mentioned in BIRCH, History of the Royal Society I, 404. This was the second of two now missing letters of Wallis to Brouncker, which Brouncker produced at the Royal Society's meeting on 30 March 1664; see WALLIS-BROUNCKER? ?.III.1664(i). 212 ac . . . Civ.: i.e. ac pro tempore Judex Veteris Civitatis. The Old-City or Altstadt of Danzig had a separate administration from the Rechtstadt.

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50. WALLIS to HEVELIUS, 5/[15] April 1664

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WALLIS to JOHANN HEVELIUS Oxford, 5/[15] April 1664 Transmission:

w Copy of missing letter sent: LONDON Royal Society Early Letters Wl, No. 5, 4 pp. (our source). At top of p. 1 in Oldenburg's hand: 'A Copy of Dr Wallis's letter to Monsr Hevelius'.—printed: OLDENBURG, Correspondence II, 166-8 (Latin original), 168-70 (English translation). Reply to: HEVELIUS-WALLIS [25.XII.1663]/4.I.1664. Answered by: HEVELIUS-WALLIS c.31.VIII/[10.IX].1664. Enclosure to: WALLis-OLDENBURG 6/[16].IX.1664. This letter was sent together with Wallis's transcript of Ulug Beg's catalogue of fixed stars via Oldenburg as an enclosure to WALLIS-OLDENBURG 6/[16].IV.1664. It was read at the meeting of the Royal Society on 13 April 1664. Cf. BIRCH, History of the Royal Society I, 414, where a correct summary is given, but the letter itself is mistakenly dated March 30, 1664. Birch evidently meant WALLIS-HEVELIUS 30.III/[9.lV].1663, the draft of which was sent to Oldenburg a year later, but clearly this was not the letter actually read. Oldenburg forwarded the letter to Hevelius as an enclosure to OLDENBURGHEVELIUS 11/[21].V.1664 (OLDENBURG, Correspondence II, 182-8, 183, 186; cf. OLDENBURG-HEVELIUS 27.V/[6.VI].1664, ibid., 204-5).

Praeclarissime Vir, Literis tuis213 gratissimis, 4° Jan: Gedani datis, (a Societate Regia Londini mihi transmissis nuper) quo maturius respondeam, id facit, Quod, jam ante quam eas acceperam, curaveram id, quod petis214, fieri: nempe, Catalogum Fixarum stellarum, cum earum Longitudinibus et Latitudinibus ab Uleg-Beig Rege, Astronomisque suis, Samarcandae ante ducentos annos observatis, ex Persico in Latinum verti: Non dubius quin id Tibi gratum esset futurum, et publico cessurum bono. Num tota ejus Institutio Astronomica, (Cujus hie Fixarum Catalogus unum absolvit Caput,) cum illius motuum Tabulis, Latina itidem net; nondum ausim polliceri: Non despe213

Literis tuis: i.e. HEVELIUS-WALLIS [25.XII.1663]/4.I.1664. petis: see HEVELIUS-WALLIS [25.XII.1663]/4.I.1664. Hevelius had also expressed his desire for a copy of Ulug Beg's catalogue of fixed stars in a letter written on the same day to the Secretary of the Royal Society: HEVELIUS-OLDENBURG [25.XII.1663]/4.I.1664; OLDENBURG, Correspondence II, 136-7. At the meeting of the Royal Society on 2 March 1663 it was ordered that John Wilkins write to Wallis to procure the requested Latin translation of Ulug Beg's catalogue. See BIRCH, History of the Royal Society I, 390. 214

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ro tamen quin et public! aliquando juris net. Interim, ut quern cum his accipies215 Fixarum Catalogum, summa cura (saltern quod numeros spectat, quorum maxima hie habenda ratio) ex tribus collatis Codicibus M.SS Persicis fideliter expressum habeas: id operis mihi demandabam, ut, quod propria manu ex Interpretis Autographo216 transcripseram Apographum, idem ad Persicorum Codicum numeros ipse Exaniinaverim, et (sicubi opus erat) emendaverim. Gratulor interim Tibi studiisque tuis, qui, (post niagna hactenus a Te praestita,) Fixarum loca, novis porro Observationibus, vigiliisque tuis instituendis, certius adhuc statuminanda polliceris: Adeo Tu indefessus Astronomiae in bonum publicum instituendae, invigilare et nullos non Labores devorare, dolce putas. De admiranda Ceti stella (qua de Commentarium ipse scripseris217) subinde nunc apparente nunc disparente: quam et jam inde ab Anno 1639 observavit Johannes| Palmerus218 nostras; Eoque monente, post ilium [2] alii; Nihil habeo quod illis addam, quae Anno praeterito Literis ad Te missis impertiebam; nisi quod (quae itidem res miranda est) monet ille, se per annos aliquot, utut illud curiose inquisiverit, non potuisse stellam illam in Hemisphaerio occidentali unquam conspicere; etiarn quum in Hemisphaerio orientali conspicienda fuerat: sed postquam illam in Ortu conspexisset, et prope Meridianum evectam notaverit; quum eo loci pervenerit, inopinate disparuit stella, non ultra conspicienda. Verum quidem est, quod, ut ut pluribus annis illud in cassum quaesiverit; tandem etiam ultra Meridianum, occidentem versus, viderit aliquoties: raro tamen. Illustrissimam denique Societatem Regiam Londini institutam ad Naturalem Cognitionem promovendam; quod spectat: Cujus Originem et Studia, petis, ut ego Tibi saltern paucis Exponerem219: Quamquam illud brevi (Credo) scripto in eum finem edito220, indicabitur; non gravabor interim votis tuis hac in re, aliquatenus obsequutum ire. Solebant jam a 24 Londini dictani corr. Oldenburg 215

quern . . . accipies: Wallis's transcript of Ulug Beg's catalogue lias not survived. Interpretis Autographo: i.e. Thomas Hyde's manuscript translation, as emerges from WALLIS-OLDENBURG 6/[16].IV.1664. 217 scripseris: i.e. the annex to Hevelius's Mercurius in sole visus Gedani: 'succincta novae historiola illius, ac mirae stellae in collo Ceti, certis anni temporibus clare admodum affulgentis, rursus omnino evanescentis'. 218 Palmerus: i.e. John Palmer. 219 Illustrissimam denique . . . paucis Exponerem: cf. Wallis's account on the foundation of the Royal Society in SCRIBA, 'Autobiography', 39-40. 220 scripto in eum finem edito: i.e. Sprat's History of the Royal Society, which in fact was not published until 1667. 216

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50. WALLIS to HEVELIUS, 5/[15] April 1664 multis annis, Viri aliquot docti ingenuique, qui sibi invicem familiariter noti erant, nunc pro data occasione, nunc etiam statis temporibus, incerto numero, prout contigerit, sive Londini, sive Oxonii, sive etiam alibi, non raro convenire; (ubi et inihi aliisque ex nostratibus, Tibi saltern de nomine notis, non raro adesse contigit;) Qui, prout erant in variis studiorum generibus, pro cujusque Vita ratione exercitati; de studiis suis, Experimentis, observatis, Inventisque, sive suis, sive aliorum, etiam ubivis terrarum, familiariter colloqui, aut etiam Experimenta coram instituere solebant. Cumque ante Triennium, post auspicatissimum serenissimi Regis Caroli secundi, in Angliam suam, reditum, varii etiam ex Nobilibus, qui| et [3] literis favebant, eisque incumbebant, erantque turn eruditi turn eruditionis promovendae propensi, inque rebus alii aliis gnari, se adjungere non dedignati fuerint: in numerum jam majorem provecti, solebant statis temporibus, stato Loco, Londini convenire; atque de rebus istiusmodi, non certa methodo, aut strictis cancellis coerciti, libere (ut res tulerit) pro data occasione, calculum suum quisque, ut sibi visum fuerit, in commune conferre. Quod quum aliquandiu factum fuerat; placuit tandem Serenissimo Regi, jam ante Biennium, et quod excurrit, Diplomate suo Regio Societatem sancire, qui eisdem rebus ultra promovendis incumbant: Non certo numero determinatam; sed, praeter Consilium (inter quos et Praeses aliisque ex Officiariis) quod constat ex membris 21, (prima vice, ab ipso Rege in Diplomate nominatis; deinde singulis annis, stato die221, communi sodalium suffragio eligendis, ex suo numero, in annum sequentem,) qui rebus societatis in commune prospiciant; alii, quot videbitur, Sodales in Societatem, adsciscantur; (communibus Sodalium suffragiis electi saltern | praesentium calculis comprobati,) turn Nostrates, turn etiam exteri, qui terras has inviserint: Ut, nunc temporis, sit ex numero 120 circiter222, conflata Societas; Magnatibus, Nobilibus, Theologis, Medicis, Jurisconsultis, Mathematicis, Mercatoribus, aliisque. Conveniunt autem (quibus vacare contigerit) saltern singulis septimanis semel, stato die et loco; atque de rebus omne genus naturalibus libere philosophantur; puta, Physicis, Chymicis, Anatomicis, Mathematicis, Astronomicis, Opticis, Mechanicis, Staticis, Nauticis, aliisque, quae aut naturam rerum indagantibus usui esse [4] posse videbantur, aut utilitati publicae subservire. Quaeque hue conferre| possint, Experimenta sua, et Observata; vel Conjecturas etiam et con221

stato die: i.e. 30 November, St. Andrew's day. Cf. The Record of the Royal Society, London 1912, 23-4. 222 ex numero 120 circiter: The president, council, and original fellows in fact numbered 119. By the time the present letter was written, a further 27 fellows had been elected. 105

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silia, rationesque suas quisque contribuit: Atque Experimenta varia, quae veritati vel comprobandae vel indagandae commoda videantur, vel coram instituuntur, vel (si saltern majore aut molimine aut tempore indigeant quam ut illud fiat,) selectis aliquot ex suo numero curanda demandantur; qui explorata reliquis referant. Sed et de ejusmodi rebus, cum aliis alibi terrarum viris eruditis, datis acceptisque literis, turn sua sensa communicant, turn exquirunt aliorum. Atque haec fere sunt quae de Regia Societate nostra tibi exponenda desiderabas: saltern si illud adjecero; Praesidis munus jam subire Honoratissiniuni Dominum Vicecomitem Brouncker; virum turn alias eruditum atque ingenuum maxime, turn in rebus speciatim Mathematicis, (Astronomicis, Geometricis, Staticis, Nauticis, &c,) sagacissimum; deque Te, Tuisque studiis optime sentientem. Quern mini literis tuis commendatum vis Affinem tuum223, credo, Oxonium nondum appulisse; neque ilium Londini me vidisse contigit; Quern si comperero, curabo ut intelligat quanti faciam tuani amicitiam, et quern lubenter Tibi sim, Tuisque, quaelibet humanitatis officia praestiturus. Vale. Dabam Oxonii. 5°. Aprilis, 1664

Tui et amantissimus et observantissimus Johannes Wallis.

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WALLIS to HENRY OLDENBURG Oxford, 6/[16] April 1664 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 4, 4 pp. (our source). At top right of p. 1: 'read April 13. 1664. entered LB. 1. 126.' On p. 4 Oldenburg lias noted above address at 180°: 'Dr Wallis's letter to M. Oldenburg concerning (1) M Horrox's Manuscripts. (2) the Translation of Uleg-beg, and the digesting of M. Horrox's Papers. Oxford April 6. 1664.'—printed: OLDENBURG, Correspondence II, 162-4; BIRCH, History of the Royal Society I, 412-13 (detailed summary). w 1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 126-9. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 146-50. Answered by: OLDENBURG-WALLIS 21.IV/[1.V].1664. Enclosure: WALLis-HEVELius 5/[15].IV.1664. 223

Affinem tuum: i.e. Johannes Koopman. 106

51. WALLIS to OLDENBURG, 6/[16] April 1664 Together with the present letter, Wallis sent his letter to Hevelius of 5/[15].IV.1664 (enclosing his transcript of Hyde's translation of Ulug Beg's catalogue of fixed stars) to be transmitted to Hevelius. He also returned Horrox's astronomical papers which he had taken with him from London, having been asked to peruse them at the Society's meeting of 16 March 1664 (BIRCH, History of the Royal Society I, 395). The letter was read to the Royal Society in the meeting of 13 April 1664; cf. BIRCH, History of the Royal Society I, 412-13.

Sir,

That I may at once give you an account, of the tasks imposed on mee by the desires of the Royall Society (whose commands I shall very readyly obey;) I did presently upon my return to Oxford speak with Mr Hyde (as I was desired by the Society) concerning the translating, & printing of Uleg-Beg in Persian & Latin, to know whether he were willing, & upon what terms to undertake it. Who (as I have formerly signified224 to our Noble President, My Lord Vicount Brounker) tells mee that as to the part concerning the fixed Stars, which he hath allready translated, hee is willing to take the care of the Impression gratis, if your Printer will undertake the charge thereof; provided it be printed at Oxford; because otherwise hee cannot attend the Presse, by reason of his attendance on the Library here. For the rest of it, if the Society desire it, hee will bee willing to serve them in translating & taking care of the Impression in like manner; & would willingly refer himself to them to gratify him for his pains as they shall think fit: But being pressed farther to make a particular proposall (according as my instructions were;) hee thinks hee may deserve 20 or 30^, & some copies to present his friends: But if the Society think it too much, hee will however bee ready to serve them, & refer himself absolutely to their pleasure. Hee addes moreover, that hee is willing, if hee undertake it, to go in hand with it as soon as may bee, because hee knows not how soon hee may be otherwise diverted. And, if it bee so agreed on, hee would willingly put that which is done allready, presently into the Presse, that this may be printing while the rest is in translating. And truly, if I may have liberty to interpose my opinion, I 11 thereof; (1) For (2) provided 15 him (1) as (2) for 22 hee (1) is (2) knows 224 signified: presumably in one of the two (now missing) letters WALLIS-BROUNCKER? ?.III.1664(i) and WALLIS-BROUNCKER? ?.III.1664(ii).

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51. WALLIS to OLDENBURG, 6/[16] April 1664

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think his labour may well deserve the reward, if the Society bee so well in stock as to spare the mony: Considering, that| hee must not onely [2] translate, but allso transcribe the whole Persian copy for the Presse; & then (which is not the lest part of the task) attend & correct the Presse all the while; which in a book of that Bulk (for it will make a good big quarto Book) & in such a language, hee will find to be a troublesome work. But, I suppose, hee doth rather look upon the Credit of the work, than the summe proposed, as his reward. In the mean time (which was a second part of my task,) I have transcribed that part of it which conteins the Catalogue of the Fixed Stars, out of his Translation; which I had before compared, as to numbers, with the three Persian books, very accurately. And I did the rather do it myself, because I might so bee the better satisfyed of the accurate agreement of the numbers with the Originall; which is of great concernment in a work of this nature, & very subject to mistake by an ordinary transcriber. And this Transcript (with a Letter225 to Hevelius in answer to his226 to mee) you will receive together with this Letter, that so it may be forthwith transmitted to Monsieur Hevelius; who I presume would be willing to see this sooner than the Impression can bee expected. As to the other particular, concerning those peeces227 of Mr Horrox, which I was desired228 to peruse, for the satisfaction of the Society, whether they be proper to be printed. I have therein allso observed their commands; & have desired my Collegue Dr Wren229 to do it also. And upon perusall of them; our opinion is, that, as to the English piece, it

1 labour (1) will (2) may 5 all the while add. 12 myself, (1) because I might (2) because 16 (with . . . mee) add. 18 Hevelius; (1) would (2) who 23 desired (1) Dr (2) my 225

Letter: i.e. WALLIS-HEVELIUS 5/[15].IV.1664, which enclosed the (now missing) transcript of Hyde's translation of Ulug Beg's catalogue of fixed stars. 226 his: i.e. HEVELIUS-WALLIS [25.XII.1663]/4.I.1664. 22r peeces: i.e. astronomical papers of Jeremiah Horrox, an edition of which was published by Wallis as Horrox's Opera posthuma in 1673. 228 desired: at the meeting of 16 March 1664; cf. BIRCH, History of the Royal Society I, 395. 229 Wren: i.e. Christopher Wren, q.v. 108

51. WALLIS to OLDENBURG, 6/[16] April 1664 will not be proper to print; because it contains onely broken, incoherent things, set down onely for his own memory as they came to mind; and are, for the most part, brought into the Latin pieces in their proper places; & sometimes with retractations and alteration upon second thoughts, of what hee had at first set down in the English. The Latin pieces, wee look upon as the beginning & attempt of an excellent work, for the restitution of Astronomy; & which serves to shew [3] how great a losse it was that hee died so soon,| who gives evidence by this, (& that piece230 allready printed concerning his observation of Venus seen in the Sun,) performed by him when hee was so very young; with how much diligence & sagacity hee was like to have applyed himself to those studies had hee lived. But these pieces are very imperfect, & not near finished; Yet contain in them the hints of many very good things; & which deserve to be preserved. 'Tis true, that it may seem now needlesse to confute Lansberg231, & perhaps unseasonable, when the World is allready so well satisfyed that his tables do not answer the commendations that himself gives them. But, though this seem to be the direct intention of the title, yet is it the best part of what is intended in the work. For it undertakes so to shew Lansbergs errors, as withall to substitute the true Hypothesis, & true measures, (from Observation,) instead of them. And upon that account; especially, if his Observations be allso preserved, which these peeces do testify hee made in great numbers, & with great care: and his Tables, (which Dr Wren tells mee hee hath seen in Mr Jonas Moors232 hands;) they may together with these Latin pieces, be printed to very good purpose. But it will then be necessary, that these two Latin

1 incoherent (1) p breaks off (2) things 2 to my del. mind 11 have (1) performed (2) applyed 12 pieces (1) , (or piece rather (2) axe 14 to (1) persuade the world that (2) confute 18 in (1) it; which (2) the work. 21 if | his add] Observations be | allso add] preserved, 230

piece: i.e. HORROX, Venus in Sole visa, Danzig 1662. Hevelius had this work printed as annex to his Mercurius in sole visus, Danzig 1662. 231 Lansberg: i.e. Philip van Lansberge (1561-1632), church minister in Antwerp and Goes. He had published Tabulae Coelestium Motuum perpetuum (Middelburg 1632), founded on the principles of the Copernican hypothesis. 232 Moors: i.e. Sir Jonas Moore (1617-79), mathematician and surveyor, elected fellow of the Royal Society in 1674, DNB. 109

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52. WALLIS to OLDENBURG, 7/[17] April 1664

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pieces be both digested so as to be brought into one (for they are but two attempts of the same thing,) which may well bee, by beginning with the Prolegomena of the lesser piece; & then proceeding with the other in order, to the beginning of the 5th disputation: Which 5th disputation (which is the same in substance with what follows in the lesse peece, after the Prolegomena,) may admit of some insertions out of the lesser piece so far as it goes. And a Preface must be prefixed by the Publisher, to give an account why it comes out so late, & why so imperfect. And the whole must be compared with the papers out of which it is transcribed; for it is manifest by the grosse mistakes (of which I have mended some hundreds as I read it) that it was transcribed by one who did neither well understand Astronomy nor Latin. I have, with this account, returned you the books themselves, to be restored to the Society from whom I had them. And rest Your freind to serve you John Wallis.

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Oxon April. 6. 1664.

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[4]

For my very worthy friend Mr Henry Oldenburgh, in the Palmal in St James's fields.

52.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] April 1664 Transmission: Manuscript missing. Existence and date: Mentioned in BIRCH, History of the Royal Society I, 412. This letter is mentioned as one of two letters sent by Wallis to Oldenburg from Oxford, which were read at the meeting of the Royal Society on 13 April 1664. While the other letter, WALLIS-OLDENBURG 6/[16].IV.1664, is summarized in detail in BIRCH, History of the Royal Society I, 412-13, only the date of the present letter is given. It was possibly only a short addition to the letter sent the day before.

2 by (1) beginin breaks off (2) beginning

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HENRY OLDENBURG to WALLIS 21 April/[l May] 1664 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 30.IV/[10.V]. 1664. Reply to: WALLIS-OLDENBURG 6/[16].IV.1664. At the meeting on 13 April 1664, Oldenburg had been instructed to return to Wallis the thanks of the Royal Society for the care and pains taken over the treatise by Ulug Beg and to desire him 'to gratify the learned world with digesting the Latin pieces of Mr. Horrox in such a manner, as he had advised himself in his letter of April 6' (BIRCH, History of the Royal Society I, 414).

54.

WALLIS to HENRY OLDENBURG Oxford, 30 April/[10 May] 1664 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 6, 4 pp. (p. 3 blank) (our source). At top right of p. 1: 'Read May 4. 64. Entered LB. 1. 130.' On p. 4 Oldenburg has written to the right of address: 'Dr Wallis's letter |to M. Oldenburg add] concerning the (1) Starr in Cete (2) Translation of Uleg-beg, and the New Starr in Cete etc.' Postmark on p. 4: 'MA/2'.—printed: OLDENBURG, Correspondence II, 177-8. wl Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 130-1. w'2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 151-2. Reply to: OLDENBURG-WALLIS 21.IV/[1.V].1664. Answered by: OLDENBURG-WALLIS 4/[14].V.1664. The present letter, which enclosed Wallis's copy of WALLIS-HEVELIUS 30.III/[9.IV].1663 and a printed sheet for Brouncker, possibly a specimen of Ulug Beg's catalogue, was read at the meeting of the Royal Society on 4 May 1664; BIRCH, History of the Royal Society I, 422.

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54. WALLIS to OLDENBURG, 30 April/[10 May] 1664 Oxon. Apr. 30. 1664. Sir,

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I understand by yours233, how much I am obliged to the Royall Society for the kind interpretation & reception of the service which I the rather owed them here, since that by my so much absence I am so little in a capacity of serving them at London. I have, according to your intimation spoken to Mr Hyde, who promiseth mee to go on with the translation of Uleg-Beig with as much speed as his necessary attendance, on the publike Library here, will permit him. But hee is of opinion, (& I am very much of his mind, being well acquainted with the inconveniences of the Authors absence from the presse, even when the work is in our language,) that it will not be possible, in Persian to have the book printed at London, in his absence which transcribes & translates it, without very frequent mistakes, & to the great prejudice of the work. And it makes mee suspect there may possibly be a mistake in Mr Thompson's234 apprehension, as thinking onely of a Latine Translation, & not of the Persian to be printed with it: which will alter the case very much. Which is to be explicitely mentioned (if it have not been allready) lest when you think yourselves agreed, you be still as far off as at first. Or, if it bee the Counsells intention, onely to print the Latine; then is the mistake on my side: & you may please to rectify it. To print the Latine onely, will bee by much the cheaper; but to publish the Original with it (if it may bee conveniently) will be the more honourable for the Society: & I did apprehend that to bee their intention, though (upon a review of your letter, upon this consideration,) I find it not explicitely so sayd. [2] As to the Papers of Mr Horrox, which Dr Wren & myself have perused; if the Society (as you intimate) desire our farther care therein: one, or both of us, will be ready to serve them in it: Onely it will then be necessary that wee have the Originall papers to peruse allso, because the 4 service (1) I (2) which I the rather 10 mind, (1) ) that (2) being 11 presse, (1) though even the (2) even when the 16 printed (1) allso (2) with 20 you (1) p breaks off (2) may 22 with it add. 27 intimate) (1) will impose that (2) desire 233 234

yours: i.e. OLDENBURG-WALLIS 21.IV/[1.V].1664. Thompson's: i.e. Samuel Thompson. 112

54. WALLIS to OLDENBURG, 30 April/[10 May] 1664 transcripts are very faulty. Concerning that Starr in Cetus; since the Society hath thought fit to take so much notice, of the intimation I mentioned in my letter235 to Hevelius; I thought it not amisse (to save myself the labour of transcribing, & you the trouble of seeking Mr Palmers book236, if it be not at hand,) to send you the copy (which I kept by mee) of a former letter237 of mine to Hevelius; that you may communicate that or transcribe out of it as you shall see occasion: Onely desiring you to return the Originall, to

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Your very humble servant John Wallis. My very humble service to those Noble & worthy persons, whose remembrance of mee your last mentions. I sent inclosed to my Lord Brounker, (directed to Dr Goddards238 Lodgings in Gresham College,) a Specimen of one sheet printed here239; which should have been received before the date of yours Apr. 21. [4] To my very worthy friend Mr Henry Oldenburgh, at the Lady Ranalagh's house in the Palmall, in St James's fields, London.

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6 copy (1) of (2) (which 8 return (1) mee the Originall. (2) the Originall, to 235

letter: i.e. WALLIS-HEVELIUS 5/[15].IV.1664, which had been read to the Society on 13 April 1664; cf. BIRCH, History of the Royal Society I, 414 (date mistakenly given as that of WALLIS-HEVELIUS 30.III/[9.IV].1663, a copy of which was enclosed to the present letter). 236 book: i.e. PALMER, The Catholique planisphaer, London 1658; cf. WALLIS-HEVELIUS 30.III/[9.IV].1663. 23r former letter: i.e. WALLIS-HEVELIUS 30!II/[9!V].1663. 238 Goddards: i.e. Jonathan Goddard (1616-75), since 1655 professor of physic at Gresham College, founder member of the Royal Society, DNB. 239 Specimen of one sheet printed here: possibly of Ulug Beg (not found).

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56. WALLIS to OLDENBURG, 7/[17] May 1664

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HENRY OLDENBURG to WALLIS 4/[14] May 1664 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 7/[17].V.1664 and WALLIS-OLDENBURG 14/[24].V.1664. Reply to: WALLIS-OLDENBURG 30.IV/[10.V].1664. As emerges from Wallis's replies and the Society records, Oldenburg reported on matters arising at the meetings on 27 April and 4 May 1664. He apparently described at length a paper on desiderata in music by John Birchensha, read at the former meeting, in which the author undertook 'to bring the art of music to that perfection, that even those, who could neither sing nor play, should be able, by his rules, to make good airs, and compose two, three, four, or more parts artificially' (BIRCH, History of the Royal Society I, 418). As instructed by the Council at the meeting on 4 May, at which Wallis's letter of 30 April was read, Oldenburg thanked Wallis and Wren for their offer to digest Horrox's papers and agreed to send them as requested the originals as soon as they should be obtained from Jonas Moore. In accordance with the instructions of the Council, he also asked Wallis to press Hyde to continue the translation of Ulug Beg, the proposal already having been issued at the earlier meeting of the Council that only the Latin translation of the text be printed.

56.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] May 1664 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 7, 4 pp. (our source). At top right of p. 1 in Oldenburg's hand: 'Dr Wallis's Letter to Mr Oldenburg concerning Musick, and the Translation of Uleg-Beg.' Beneath date: 'read May 18. 64. Enterd LB. 1. 149'. On right margin of p. 4 at 90° to address in unknown hand: 'Dr Wallis's letter to Mr Oldenburg concerning Musick and the translation of Uleg-beg May 7th. 1664'. Postmark on p. 4: 'MA/9'.—printed: OLDENBURG, Correspondence II, 179-81. w 1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 149-51. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 173-6. Reply to: OLDENBURG-WALLIS 4/[14].V.1664. The letter was read to the Royal Society at the meeting on 18 May 1664; BIRCH, History of the Royal Society I, 423-4.

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56. WALLIS to OLDENBURG, 7/[17] May 1664 Oxon. May. 7. 1664. Sir

I thank you for the large account you gave mee in your last240, of what the Society hath been of late intertained with241, as to the business of Musick. I do not profess myself to be a Master in it; & therefore what I have been formerly unsatisfyed in, I have been willing to impute to my imperfect knowledge; not having applyed myself so particularly to that, as to some other parts of Mathematicks. But that there is a great defect in the Theory thereof (at lest as to those either Ancient or Modern Musicians that I have read) I think is without question. Arising mostly from hence; that many of those who have applyed themselves to the Theory of it, have either been but Transcribers, or else but imperfect Geometers. (Out of which I must yet except Euclides Sectio Canonis242; which is perfectly Geometricall. And what Ptolemy's Harmonicks243 are, I know not; as not having seen them.) The Distances of Diapason, Diapente, Diatessaron, & Tonus; or, as they are now called Eights, Fifths, Fourths, & a Note; are well stated by Pythagoras his notion of them; as answering the proportions, 2 , |, |, |. And consonant hereunto, the compounds of them; as Diapason & Diapente, Diapason & diatessaron, dis-diapason &c; that is |, f , f , &c. But their Hemitonici, & Dieses, &c. are very obscurely at lest, if not imperfectly delivered: wanting much of Illustration, & (I doubt) somewhat of Correction. That there bee many Distances, or Diastemata, of which no notice is taken; is very evident: (&, that some of them are not lesse Musical, than divers of those allready observed:) Which it will not be hard for one that is both a good Musician (in the Theory) & a good Geometer (as to the nature of Proportions) to supply. And the person244 you mention (though I know him not) hath, I suppose, well considered it. I am apt to think allso, that the sounds wee sing, are not the same 16 Fifts corr. ed. 240

last: i.e. OLDENBURG-WALLIS 4/[14].V.1664. intertained with: i.e. with a paper on desiderata in music by John Birchensha, read at the meeting on 27 April 1664; BIRCH, History of the Royal Society I, 418. 242 Euclides Sectio Canonis: i.e. [EUCLID], Sectio canonis. 243 Ptolemy's Harmonicks: i.e. PTOLEMY, Harmonicorum, sen de musica lib. Ill, contained in Aristoxeni . . . Harmonicorum elementorum libri III, ed. Antonius Hermannus Gogava, Venice 1562. In 1682 Wallis edited Ptolemy's Harmonics in Greek and Latin: Claudii Ptolemaei harmonicorum libri tree: Ex codicibus manuscriptis undecim; nunc primum Graece editus, Oxford 1682. 244 person: i.e. John Birchensha (fl. 1664-72), musician, who was largely engaged in the study of the mathematical basis of music, DNB. 241

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wee prick (according to the scale as it is now supposed to be divided) in divers cases; but somewhat different from them: the Ear, in these niceties, guiding the Voice, better then the Scale (if a little erroneous) can do. But I leave these considerations to those who have more particularly considered the matter, than I have yet done. And I suppose it very reasonable that the Gentleman be incouraged in his design; especially if such a person as your letter represents him. His censure of the Condition wherein Musick stands at present; I concur with: His Propositions are good.| [2] Mr Hobbes his notions you mention, about the Compression of Air; are but the same which hee hath formerly published245: And I shall hardly trouble my thoughts further concerning them. Concerning Uleg-beg's translation & publishing: I beleeve that Hevelius will be willing to print it; but it must then bee onely in Latine: For, beside that it can not bee correctly printed without the Translators presence, or one who doth well understand the language; I may very well presume that at Dantsick they have neither Letters nor Printers for such a work. But, if the Persian cannot bee done; the Latine will be better then nothing. Though this fate commonly do attend such editions, that if the Originals be not at first published, wee can scarce expect it afterward. Of which wee have example in Apollonius, Serenus, & Pappus; which, because Commandinus246 did not set forth in Greek as well as Latine; the Greek hath not been published to this day. I might allmost say the same of Euclide, Ptolemy, & Archimedes;of which wee have no Editions Greek & Latine; & the Greek that is, none of the best. What you intimate concerning Mr Hydes saving himself much of his pains; is very kind: But will not at all be acceptable to him: because it 7 wherein (1) it (2) Musick 16 not Printers corr. ed. 18 if (1) they (2) the Originals 20 Serenus, add. 23 no (1) Greek Editions comparable to (2) Editions 245

published: probably HOBBES, Dialogue Physicus, sive de Natura Aeris ..., London 1661. 246 Commandinus: Federigo Commandino published Latin editions of Apollonius's Conies (Apollonii Pergaei conicorum Libri quattuor ..., Bologna 1566), of Serenus's Section of the Cylinder and Section of the Cone (Se.re.ni Antinensis Philosophi Libri Duo: Unus de Sectione cylindri, alter de Sectione coni . . ., Bologna 1566), and of Pappus's Mathematical Collections (Mathematicae Collectiones, Pisa 1588). He also edited several works of Euclid, Ptolemy, and Archimedes in Latin.

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56. WALLIS to OLDENBURG, 7/[17] May 1664 puts him beside his chief aim in it: Which is, by publishing the Persian, to represent himself so to his friends from whom hee may expect incouragement, as that they may thereby be induced the rather to do something for his advantage. So that although, I presume, hee will not refuse to serve the society, even in the other way; yet I know hee had rather take the rest of the trouble gratis, than that the Persian should not be printed. To which I shall adde but this one thing more; that, if the Latin be once printed without it, the publishing of the Originall is for ever destroyed: For, when the world is once furnished with the Latin; no Printer will undertake the Persian, because he can then have no expectation of vending [3] the Impression: And, as it would therefore be more honourable for the| Society to publish it with the most advantage: so, if otherwise, they may possibly incur the same blame which Commandino doth from some; who, though they much commend his care, in translating &; publishing divers of the most considerable Mathematicians, yet they say, he ought to have published the Greek with them; for want of which, wee have it not to this day. But I shall not prescribe to the Society, who can better judge of their own conveniences, & see the difficulties which the Printers interpose, than I can, especially at this distance. What they shall think fit to determine on, I shall bee ready in my capacity to be subservient to them, as being their and

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Sir

Your very humble servant John Wallis. [4] For my very worthy friend Mr Henry Oldenburgh, at the Lady Ranalagh's house in the Palmal in St James's fields London.

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10 no (1) opp breaks off (2) expectation 14 though, add. 14 publishing (1) the (2) divers 18 & (1) d breaks off (2) see

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57. WALLIS to OLDENBURG, 14/[24] May 1664

57.

WALLIS to HENRY OLDENBURG Oxford, 14/[24] May 1664 Transmission:

Wl Letter sent: LONDON Royal Society Early Letters Wl, No. 8, 8 pp. (our source). At top right of p. 1 beneath date: 'read May 18: 64. entered LB. 1. 152'. At top of p. 8 Oldenburg has written above address at 180°: 'Another letter of Dr Wallis to M. Oldenburg, concerning Musick.' — printed: OLDENBURG, Correspondence II, 190-201. W2 Copy of letter sent with additions and postcript: LONDON Royal Society Boyle Papers 40, f. 23r-30r. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 152-64. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 177-92. w3 Copy of revised version (based on W2) with further additions, modifications, and enlarged postscript: DUBLIN Marsh's Library Z3.4.24, 19 pp. (our source). Reply to: OLDENBURG-WALLIS 4/[14].V.1664. In this letter Wallis proposes improvements to the tonal scales by employing the theory of proportions in the determination of intervals. The copy in Dublin (w 3 ) has been made from a later revised version with substantial additions and modifications. As follows from the postscript and from WALLISOLDENBURG 25.V/[4.VI].1664, Wallis apparently prepared this version some time after the original letter was sent, having consulted in the meantime works of modern authors on music theory such as Kepler. The revised version possibly was destined for Wallis's correspondents in Dublin. For the first part up to § 13, the major additions and modifications of w 3 are given in the apparatus for W1 , while the substantially reworked ending of w3 (§ 14 following) is printed completely.

(Wl) Oxon. May. 14. 1664.

Sir, 5

Though (as I sayd in my last247) I do not look upon myself as so much a Master in Musick, but that much of the Obscurity or Dissatisfaction which I have mett with, may well be imputed to my own imperfect skill, & the few thoughts I have imployed in that Theory; nor my opinion so authentick, as that upon my own authority I shall presume to innovate any thing which hath been, for so long a time, so generally received: Yet 4 (as I sayd in my last) missing in W2 5 Obscurity (1) & (2) or 247

last: i.e. WALLIS-OLDENBURG 7/[17].V.1664.

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57. WALLIS to OLDENBURG, 14/[24] May 1664 since you seem to invite mee to it, I shall venture to give you a brief account of my present unpolisht thoughts: which whether they will be worth pursuing or not, I shal leave to better Judges. To this purpose; I shall select an Octave out of the Scale, or Gam-ut: suppose, from E la mi, to e la mi: which is wont to bee sung by these notes La Fa Sol La Mi Fa Sol La.

Whereof all, but Fa, are supposed to rise a full Tone, or (as they call it) a whole Note: and, Fa, an Hemitone, or Half a Note. So that, between the Eight Sounds, are Seven Intervalls, containing Six Tones; or rather, Five Tones & Two Hemitones. But forasmuch as it is confessed (& is by Euclide demonstrated248) That Six whole Tones, are more than an Octave or Dia-pason; Because that I (the Proportion of a Tone) being six times compounded, is more than |; (the proportion of a Dia-pason:)

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Therefore, allowing to all the Tones, their full measure; they do professe, by the Hemitone, to mean, not precisely Haifa Tone, but, somewhat lesse than half a Tone; (viz. so much as the Dia-tessaron is more than two Tones; or, Dia-pente, than Three Tones, which is fff 0 Yet S^ve ^ the name of Hemitonium, because it is very near it.

11 demonstrated in his Harmonicks w3 16 measure; (1) do (2) they 248

by Euclide demonstrated: cf. [EUCLID], Sectio canonis, prop. 14.

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57. WALLIS to OLDENBURG, 14/[24] May 1664 Three Tones. Dia-pente. Hemitone.

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Two Tones. Dia-tessaron. Hemitone. That is, proxime. which is less than Half a Tone, or That is, proxime. And this I thought necessary expressely to mention; that I might not seem to wrong the Ancients. For though, possibly, some of our modern composers may not heed it; the Ancients did clearly discern, that a Diatessaron, was something lesse than Two Tones & an half; and, a Dia-pente, than Three and an half. This being the constitution of the Ancients, in their Genus Diatonicum; (according to which Euclide proceeds in Sectione Canonis; & which the Moderns do also follow:) Without saying any thing to their Genus Chromaticum, and Enharmonium; I shall shew wherein I think it might be amended with good advantage; or rather, some Mis-takes rectified. (For indeed, my opinion is, that the Voice doth make truer Musick, than what the Notes, according to this Hypothesis doe direct.)| [2] And here (without laying a new Foundation) I shall proceed upon the Hypothesis of Pythagoras; that the Diastemata or Intervalls of Sounds, are conveniently expressed by Rations or Proportions: to wit, in what Proportion one Sound is more Acute or more Grave, than another compared with it. As if the Acuteness of the one be sayd to be Double, Treble, Quadruple, Sesquialter, Sesquitertian, &c, to that of the other. By what methods this may be determined most conveniently; I will not take upon mee to define: As, whether a Hammer, half so heavy, will make a sound, Twice so Acute; Or, a String, Stretched with a Double Weight, will make a Sound, Doubly Acute: Or, A string, half so long, will sound, Twice as High; & the like in other proportions. I shall not, I say, here define, whether all or any of these ways will determine it; Because, I conceive, it may be proper inough for the Society, to examine the truth of these Hypotheses, by good severe Experiments, rather than to take them

7 though; (1) s breaks off (2) possibly 13 the Modern Composers do generally follow.) W2 22 one add. 26 twice so Grave w3 26 String, add. 31 take (1) it (2) them

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the moderns have followed:) w3

57. WALLIS to OLDENBURG, 14/[24] May 1664 merely upon the authority of those that relate them. Nor shall I strictly inquire into the Physicall cause of Unisons. Harmony, or sweetnesse, between some sounds, & of Discord in others: As, whether two strings that are Unisons, be therefore Harmonious, because (supposing them to have one common beginning) the Vibrations of the one doe exactly answere Octaves. those of the other: And, next unto those, Octaves; because that the vibrations of the more Acute, being twice as many in the same time, every Vibration of the more Grave or slower string, is coincident with every second Vibration of the quicker or more Acute: And, in Fifths, because, the Quickness of the Fifths. Vibrations being in the one Sesquialter to that in the other, every Second Vibration of the one, is coincident with every Third of the other; Which may be therefore lesse harmonious than Octaves (because of a lesse frequent coincidence:) Yet Fourths. more than Fourths, where, the celerity of Vibrations being in Sesquitertian proportion, every Fourth of the one, will be coincident with every Third of the other: Yet will this be much more Harmonious than a Tone, whose proportion beTone. ing Sesquioctave, the coincidence will not be more frequent than for every Ninth of the one, to answer to every Eighth of the other. Whether this, I say, be the true Physicall cause of Harmony & Discord in Sounds, (though it be a very specious & promising account of it,) I will not positively determine; because I would not, even here, exclude a strict examination by Experiment. But that this, or somewhat equivalent hereunto, is the true Cause & Measure of Harmony & Discord in Sounds, we must needs grant; or else, that we are yet to seek for a Foundation to build upon. I shall therefore lay this for a foundation to proceed upon, (at lest till a better be discovered,) That, upon the proportion of Celerity in the Vibrations; or at lest (which, since Pythagoras, hath been generally received) upon the proportion of Acuteness & Gravity (whence soever this Proportion doth arise) dependeth the Harmony or Discord of Sounds. Next, That these proportions, according to which such coincidence 12 that (1) of (2) in 33 Acuteness (1) or (2) & Gravity ( (a) when s breaks off (b) whence 35 which (1) that (2) such 121

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will be more frequent, (& the fewest Vibrations not coincident,) are the more Harmonious; & the contrary, either more Discordant, or at lest less Harmonious. Upon these principles, I thus proceed. 1. First it is manifest, that, of all sounds compared, UniUnisons. sons are most Harmonious. For the Proportion of the Celerity of Vibrations, (or of what else shall be found to be substituted in the room thereof) being as 1 to 1, that is, Equall: the Vibrations of the one & the other will bee all coincident respectively. My meaning is, (which I interpose by way of Caution,) that it will so bee, upon supposition that in both strings they have one common beginning; or, which is aequivalent, in case any two Vibrations are coincident, all the rest respectively will bee so too. And with this Caution I would be understood all along. | [3] 2. It is allso manifest, that, next unto Unisons, those are Octave, or Dia-pason. most Harmonious, where the proportion is Double, or as 2 to 1. For then every Vibration of the slower will be coincident with every second vibration of the swifter. That is, of every Three vibrations (two in the one String, & one in the other,) there will bee Two coincident. And to this it is that the Ancients do accommodate their Octave or Diapason: So tuning the two Extremes of the Octachord, that the Acutenesse of the one be Double to that of the other; & those sounds which do thus differ, are called Octaves: Not as if the number 8 had any thing peculiar in the proportion; but, because this proportion tooke-in the whole compass of the Octachord, which they esteemed to contain a perfect Systeme: & the Sounds or Places considered therein, as it was then divided, were in number Eight. And conformable hereunto are the names of Dia-pente, and Dia-tessaron, Fifths & Fourths, as extending to

9 bee (1) (—) (2) all 15 along; which I once mention for all. w3, W2 23 the (1) (Oct) breaks off (2) Acutenesse 28 considered add. 29 number 8. To which eight sounds, were fitted, in the Harp, eight strings, fe conformable w3, W2 30 Dia-tessaron, (1) Forths (2) Fifths

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57. WALLIS to OLDENBURG, 14/[24] May 1664 so many of those Strings or Sounds: which would have been otherwise named, had the Sounds or Places, considered within this compasse, been more or fewer. And this of Octaves, is, of all Intervalls, the most Harmonious. Such is the Interval! La La; from E la mi to e la mi. A Fifth, or 3. The next Harmonious Intervall, (within the compasse Dia-pente. of an Octave, for of such I am now speaking,) is in the Sesquialter proportion, or as 3 to 2. Wherein every Second Vibration of one String, is coincident with every Third of the other. So that of every Five vibrations, (three in the one, & two in the other,) Two are coincident. This they called a Diapente, or Fifth; because it extended to Five of those Eight sounds in a perfect Systeme. Such is the Intervall La mi, (from E to b:) or La La, (from a to e.) And is supposed to contain Three Tones & an Hemitone. A Fourth, 4. The next to this, (I mean, within the compass of an or DiaOctave; & so in the rest,) is the sesquitertian proportion; or tessaron. that of 4 to 3. Wherein every Fourth vibration of the one, is coincident with every Third of the other. So that of every Seven Vibrations, (that is, four in the one, & three in the other,) there bee Two coincident. This they called a Dia-tessaron, or Fourth: as extending to Four of those Eight Musical Sounds of which they took notice in the Octachord. Such is the Intervall La La, (from E to a) or Mi La, (from b to e:) And is supposed to contain Two Tones & an Hemitone. 5. It is allso manifest, that these two Intervalls of Diapente & Dia-Tessaron; as they are both Harmonious; so, being put together, they make a Dia-pason. For, if the two Proportions of 3 to 2, & of 4 to 3, be compounded; the result is that of 2 to 1. Or, if the Proportion of 2 to 1, be abated by that of 3 to 2; the result is that of 4 to 3: Or, if abated by that of 4 to 3; the result is that of 3 to 2. As is manifest by the Definition & Doctrine, of compounding proportions. So that, the Diapente & the Dia-tessaron, are each the others complement to a Dia-pason: and that of a Dia-pason, equal to both those put 1 which (1) (—) (2) would 123

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57. WALLIS to OLDENBURG, 14/[24] May 1664 together. If, from E, we rise a Diapente to b, La Mi; the remainder from b to e, Mi La, is a Dia-tessaron. If, from E, we rise a Dia-tessaron to a, La La; the remainder from thence to e, La La, is a Dia-pente. If from E to b, we rise a Dia-pente, La Mi; and again, from b to e, a Dia-tessaron, Mi La; the whole from E to e, La La, will be a Dia-pason. Thus, in Numbers, 5 If we express the more Grave Sound of an Octave or Dia-pason, by 6; &, consequently, the Higher by 12, (in a double proportion to it;) the number 9, which is to 6, as 3 to 2, or a Dia-pente; will be to 12, as 3 to 4; which are therefore a Dia-tessaron: And the number 8, which is to 6, as 4 to 10 3, (a Dia-tessaron;) will be to 12, as 2 to 3, a Dia-pente. Which is the entendment of that frequent proposition in Musick; That a Dia-pente & a Dia-tessaron, make a Dia-pason. | [4] A Tone. 6. The Difference or Distance, between a Dia-pente & a Dia-tessaron; that is, between the Proportions of 3 to 2, & of 4 to 3; is that of 9 to 8, or the SesquiOctave. For if that of 3 15 to 2, be abated by that of 4 to 3; the result is that of 9 to 8. And this Distance they call a Tone. As, La Mi; (from a to b.) For, that from E to b, being a Dia-pente, & that from E to a, a Dia-tessaron: that from a to b, is the Intervall or Difference between them. 20 Thus far (I think) & no farther, Pythagoras did proceed: viz. to determine the Intervalls of Dia-pason, Dia-pente, Dia-tessaron, and Tonus;

4La(l) La (2) Mi 5 W2, w3 in margin:

15 Sesqui-Octave. So that of every 17 Vibrations (to wit 9 on the one string, fe 8 on the other) Two are coincident. For if w 3 17 Distance (1) {—} (2) they 19 the (1) Diff breaks off (2) Intervall 21 to (1) the (2) determine (a) the Proportions of the Intervalls (&) the Intervalls 124

57. WALLIS to OLDENBURG, 14/[24] May 1664 by the Proportions of |, | |, & |. And, consequently to determine Four of the Eight Sounds whereof their Octachord consists: in this manner.

The four Sounds, La, La, Mi, La, being thus determined by their proportions to one another: The other four, Fa, Sol, Fa, Sol, are neither by Pythagoras in like manner determined, (for ought appears to us;) nor have any of his followers to this day, (whom I have yet met with,) followed his steps in designing them: But have very inconveniently divided the two Tetrachords, La Fa Sol La, and Mi Fa Sol La: Making La Fa, and Mi Fa, Hemitones; and Fa Sol, Sol La, (in both places,) perfect Tones, equal to that of La Mi, from a to b. I shall therefore first shew the Inconvenience of such a Division; from hence, That, whereas it is manifest, that between the proportions of | and |, there be many which are more Harmonious than | (which is that of a Tone) though less Harmonious than | (that of a Dia-tessaron,) such as |, |, |, |, &c; For these, they have made no provision, nor taken any notice of them: Which yet are undeniably Musicall, and (as wee shall after shew) of frequent use, at lest some of them; Whereas those which they have put in stead of them, are much more Dissonant, than that of a Tone. And then I shall after shew, How they might with much more advantage be divided. Half a Tone. 7. I say therefore, that if La Fa, & Mi Fa, be Hemitones: & that if by an Hemitone, they mean, precisely Half a Tone: then is this proportion absolutely Un-Musicall. For the Tone being |; the Hemitone must be ^/| or -4^; (for this, twice compounded, makes \/| x ^/| = |.) Now the number 3, being 6 of his Followers, (as low as Boetius his time and a good while after,) persued his steps W2 of his fellows extant, as low as Boetius's time, fe a good while after, persued his steps w 3 22 an add.

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absolutely incommensurable to ^/8 (the surd root of 8;) it is not possible, that, Vibrations in such a proportion, if they have one common beginning, can ever after be coincident. And the like is to be understood of all proportions between terms incommensurable. Or if, by Hemitone, they mean that proportion of | | | beThe common Hemitone. fore described: the terms are then indeed commensurable: but there must passe (after their common beginning) 499 vibrations (that is, 256 upon the one string, & 243 on the other,) before any two will be again coincident. Which is very far from Harmonious. 8. The Intervall Fa La, (from F to a, or c to e,) if it be a A perfect Ditone, or perfect Ditone (as they suppose it to bee,) that is, Two intire Two Tones.

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Tones; the proportion will be as 81 to 64: and there must (after the common beginning) passe 145 vibrations, before any Two be again coincident. Which therefore must be a much harsher Intervall than that of a Tone. Whereas yet, in experience, Thirds are found to be a good Concord. | [5] Tone fe 9. The same is to be sayd of the Intervalls La Sol, & Mi Hemitone. Sol, (from E to G, & b to d:) which are imperfect Thirds; and are supposed to consist of a Tone & Hemitone: And, consequently, (supposing the Hemitone as before) that proportion will be, as 32 to 27, (being compounded of | & iff 0 & therefore 59 vibrations before there be a new coincidence. Yet experience tells us, that Imperfect Thirds, make no ill musick. This division therefore of the Tetrachord, being so incongruous, & not answering to experience; (for certainly Thirds, as they are wont to be sung, make better Musick than this Hypothesis will afford:) I shall next shew how the Tetrachord may be divided with more advantage, & more complying with experience: prosecuting, therein, the same steps, as in the division of the Octachord into Dia-pente & Dia-tessaron. 7 then add. 10 is add. 24 new add. 26 musick. Adde allso, that by this reconing, Imperfect Thirds should be better musick then perfect Thirds, (because 59 are fewer then 145) which the ear doth contradict, w 3

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57. WALLIS to OLDENBURG, 14/[24] May 1664 The greater 10. I say therefore, that (in pursuance of the former prinDia-trion: ciples) next to | (the proportion of a Dia-tessaron) succeeds or perfect that of |, the Sesquiquartan: when every Fifth vibration of Thirds.

the one, will be coincident with every Fourth of the other. So that, in Nine vibrations, Two will be coincident. Which though it be less Harmonious than that of a Dia-tessaron, or |; yet is much more Musical than a Tone, or |. And this I would make the Intervall Fa La, (from F to a, or c to e.) which is not perfectly a Ditone, that is || — | x |; but comes very near it, & is much more Harmonious. Imperfect 11. And, consequently, the Hemitone Mi Fa, or La Fa; Hemitone. from E to F, or b to c; (which is confessed not to be precisely Half a Tone,) will be, not | | | (as it is thought to bee,) but (which is nearer to an Harmony) j|. For if | the proportion of a Dia-tessaron, be abated by | the proportion of a Dia-trion; the result is ^f for the residue of that Intervall; which wee are content at present to call an Hemitone, as being near it. Tone fe Hemitone.

12. The Interval La Fa, (from a to c;) consisting of La Mi (a perfect Tone) & Mi Fa (that imperfect Hemitone;) is |: (which results from the composition of | and j|:) which is, though less Harmonious than |, yet more than |. And the same will bee the proportion of La Sol, or Mi Sol, (from E to G, or b to d,) in case we make Fa Sol a perfect Tone. As allso, Sol Mi, from G to b.

5 Vibrations (5 in the one string, fe 4 in the other) two will w3 9 near it, (f being equall to f|,) & is much more Harmonious: For here the coincidence returns at 9 Vibrations on the two strings; & there, not ill 145. w3 14 j|: So that in 31 Vibrations, (16 on one string, fe 15 on the other) two will be coincident. For if w3 14 if add. 20 & l l f - | x l f - l M - f S o tllat in U Vibrations (6 on the one string & 5 on the other) two will be coincident. Which is, though w3 23-128,1 perfect Tone of f . And Sol Mi (from G to b) will then be f ; equall to that of Fa La, (from F to a, or c to e) above mentioned. || 13. Then w3, W2

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57. WALLIS to OLDENBURG, 14/[24] May 1664 13. Then, for the Dividing of Fa La, (from F to a, or c to e;) if wee make Fa Sol a perfect Tone, (which is not inconvenient,) then is ^ the proportion of Sol La. For if we abate |, by |, the result is -^p. So that the Simple Intervalls (or Diastemata) with the principall of their Compounds (or Systemata) within the compass of an Octave, will stand in this order: Imperfect Tone.

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14. There be yet Two other Superparticular proportions, | & |, which being more Harmonious than a Tone |, it may be expected to have a place allotted them: Which may be done by interposing a Sound between Fa and Sol, (which may be called, Fa sharp') which may so divide the upper Dia-tessaron Mi La (from b to e,) or the Lower La La, (from E to a,) in such manner, as that, with one of its terms, it may make the proportion |; with the other, |; which, being compounded, make |, or |.| [6]

3 proportion: (So that of Nineteen Vibrations, 10 in one string & 9 in the other, Two will be coincident) for if w 3 4 (1) And Intervall Sol Mi, from G to b. (2) So that 16 or

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57. WALLIS to OLDENBURG, 14/[24] May 1664 15. There be allso some Superpartient proportions, (as |, | &c,) which I take to be very Harmonious, & do not want their use in Musick; (though I know that Euclide, in the beginning of his Sectio Canonis, excludes249 all as Dissonous, which are not either Multiple or Superparticular; but upon a reason which to mee doth not seem conclusive.) But they are such as either do not occur within the compass of a single Octave, (as |, |, |, |, |, ip, &c, which are greater than |:) Or, could not be expressed, without inserting the sound we mentioned, between Fa & Sol; (as |, from E, to such a sound between c &; d, as shall be to that in b, in the proportion |; for | x | = |: or |, from b, to such a sound interposed betwixt F, G, whose Interval from a shal be |;for| x | = |: or ^, from the same interposed sound, to e; for | x | — ^:) Or, not without too much confusion in the scheme; (as |, from G to e; |, from E to c; |, from E to d.) On which therefore I shall in this place insist no further. 16. Having thus determined the single Intervalls, within the compasse of a Dia-pason: It will be easy to determine (by compounding two or more proportions) all compounded Intervalls, whether within a Single, or Double, or Treble, Dia-pason, or further yet, if there be occasion. As for Example; In a Dis-diapason, from E to ee; the severall proportions will be these.

1 proportions, (and Multiple-Superparticular,) as W2 5 to mee add. 7 f , f, add] f , f , f, (1) fee, (2) f , 10 f: Or f , (from EE to the same Sound.) Or f , W2 12 for | x | = ^-. Or |, from a Sound between F, G, (whose Intervall from a shall be |) to c: For | x | = |. Or could not be expressed without too much confusion W2 15 the simple Intervalls, W2 18 further (1) , if (2) yet, 249

excludes: cf. the opening remarks to [EUCLID], Sectio canonis.

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To these, wee are to adde further; such others as shall arise upon altering the Key by Flats and Sharps. With which I shall not at present trouble you; having thought it sufficient to give this Specimen. Nor will it be necessary (because it is obvious) to insinuate, that 5 what is here fitted to the Octave from E to e, upon supposition that b is not Flat; doth (upon altering the Key) agree to that from a to aa, in case a Flat be in b: Or, to that from D to d, in case a Flat be allso in e la mi. I might adde also, that, in dividing the Dia-trion, Fa Sol La; whereas I make Fa Sol, |; and Sol La, ^: it had been free 10 to do the contrary; but, of the two,| that seemed to mee the [7] more eligible. Indeed it had been proper inough, to have made three Intervalls; to wit, twice ^; and a middle one between: which would bee |^: Which middle Intervall might indifferently be added sometime to the one, sometime to the other, 3 trouble the Scheme; W2 4 to intimate, W2 7 in e. W2 10 the most elligibles. W2

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57. WALLIS to OLDENBURG, 14/[24] May 1664 as occasion should serve: In like manner as an Octave, or Diapason, is divided into two Dia-tessaron's and a Tone between; which Tone being added to this or that of the Dia-tessaron's, makes it a Dia-pente. But I was willing to innovate as little as was possible. I have therefore onely, abating the Tone Sol La by |jj, as much in lieu thereof increased the imperfect Hemitone Mi Fa, or La Fa. But, how much this one Amendment, hath made the Intervalls more Harmonious; will be very evident to any one who shall but take the pains to make a like Scheme of all the Intervalls in a Dis-dia-pason, according to the received Hypothesis, (as I have here done according to this restitution,) and compare it with mine. I do not descend to the particularities of smaller Intervalls; nor to dispute the Names of these: Because I doubt I may have sufficiently tired you allready: And, for the same reason, many other things must be omitted. If what hath been sayd, be thought reasonable; it will be easy to adde more: If not; there is too much allready. How far this may agree with Mr Birchinsha's Thoughts or Notions concerning the same subject; I cannot tell. For your Letter intimating onely the Generall, both as to the Defects in Musick; & as to the Proposalls hee makes of supplying those Defects: it doth not at all appear from thence what his particular thoughts are, & by what methods he intends a restitution. To conclude; I shall the rather hope you will pardon mee the present trouble I give you in so prolix a Letter; because you in part occasioned it, by that of yours to,

Sir, Your very humble servant. John Wallis. 5 Tone (1) Fa (2) Sol 11 anciently received W2 16 sayd (1) allready, be [thought add] considerable (2) , be 18-23 How far . . . intends a restitution, missing in W2 25 in add. 29 John Wallis. || Post-script. Since I had written this, I find that our Modern Writers of the Theory of Musick (in this, fe part of the last Century,) have, upon like Principles, divided the Monochord much after the same Proportions as I have done: Though our Practicall Musicians, generally, take no notice of it. W2

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These For my very Worthy Friend, Mr Henry Oldenburgh, in London.

[8]

(w;3, ending) 14. There be yet two other Superparticular Proportions, g & f , (the [13] Sesqui-Sexton & the Sesqui-Septiman, or as we use to express them in English, Once with a Sixth part, & once with a seventh part") for which, 10 being more Harmonious then a Tone or |, it may perhaps be| expected [14] we should allow a place. Which may be done, if need be, by interposing a sound between Fa and Sol, (which may be called Fa sharp) which may so divide the Upper Dia-tessaron, Mi La, (from b to e,) or the Lower, La La, (from E to a,) in such manner, as that, with one of its terms, it may make the proportion of |; with the other, |: which being compounded 15 make |, or |.

And it may be done in either of these forms, according as you please to make the greater of the two, in the Lower or higher place.

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But because there will be a necessity to rest somewhere, (since proportions may be thus broken in infinitum;) & because experience teacheth us that the voice cannot well express less Intervalls, in a continued sequence, then those of the eight sounds in a Dia-pason, (However, by leaps, it may fall upon intermediate Sounds, denoted usually by Flats & sharps' amongst which possibly, these may be some, though not attended:) Therefore, I think, we may well rest in the number of Places for|merly receiv'd, without introducing these new proportions. [15] And that the rather, because these cannot be design'd without taking 132

57. WALLIS to OLDENBURG, 14/[24] May 1664 in another Prime number, (viz: 7.) besides those of 2. 3. & 5, which serve to Design all those Antecedent proportions, for though | & |, be expressed in smaller numbers, than |, ^, or j|, yet these being not Prime, but composite numbers, may be resolved into such as are designable, by 2. 3. 5, without taking in any other prime number. For 8 is but 2 x 2 x 2, & 9 = 3 x 3 , & 10 = 2 x 5 , & 15 = 3 x 5 , & 16 = 2 x 2 x 2 x 2 . Whereas 7 being a Prime number, cannot be so resolved. I am content therefore at present, to leave out the proportions of |, & |, as less Harmonious. 15. There be allso some superpartient proportions, as | |, &c: which I take to be very Harmonious, &; doe not want their use in Musick; at least some of them, (though I know that Euclide, in the beginning of his Sectio Canonis, excludes250 all as Dissonous, which are not either Multiple, or super-particular, but upon a reason which, to me, doth not seem conclusive.) But they are such as either doe not occurre within the compass of a single Octave; As |, |, |, |, ^, &c. which are greater than |. Or could not be expressed without inserting such a sound as we mentioned between Fa &; Sol, As |, from E, to such a sound between c & d, as shall be to b, in the proportion of |, (for § X g = | 0 & I & ; I (which are allso more than an Octave) from EE, & from B, to the same sound: And, |, from b to such a sound interposed between F & G, whose intervall to a shall be |, (for |x| = | : ) & ^ , from the same interposed sound to e, (for | x | = ^ : ) & | , from a Sound between F & G (whose intervall from a should be |) to c (for | x | — |:)fesuch others. [16]

Or, at lest could not be well expressed, without | making too much confusion in the scheme; as |, from G to e; &; |, from E to c; & |, from E to d, &c. On which therefore I shall here insist no further. 16. Having thus determined the single Intervalls, within the compass of a Dia-pason- it will be easy to determine (by compounding two or more proportions) all compounded Intervalls, whether within a single, or a double, or Treble Diapason' or further yet if there be occasion. As for example; in a Dis-diapason, from E to ee; The severall proportions will be these.

250

excludes: cf. the opening remarks to [EUCLID], Sectio canonis.

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57. WALLIS to OLDENBURG, 14/[24] May 1664

17. To these we are to add further, such others as shall arise, upon altering the Key of Flats, & sharps; Whereby divers of the Intervalls which here seem Harsh are mollified & sweetned. as for Instance A Flat in 6, d, bb, dd, which instead of a full Tone of |, (as now) shall be supposed to 5 make them rise but a Hemitone of ^f, will make their intervalls from F, which here are ||, f|, ||, ^, to become, |, f , f , ^, & 6, /, which now is || will be |. A sharp in 6, d, bb, dd, which shall| be supposed to raise [17] them an Hemitone of ^f, will make their Intervalls from F, which now are ||, ||, ||, ^, to become |, |, |, ^, And 6, / which now is || will 10 become |. And the like of many Others. With which I shall not trouble you at present, having thought it sufficient to give you this Specimen.

15

18. Nor will it be necessary (because tis obvious) to insinuate, that what is here fitted to the Octave from E to e, upon supposition that b is not Flat; doth (upon altering the Key) agree to that from a to aa, in case a Flat be in b: or to that from D, to d, in case a flat be also in e la 134

57. WALLIS to OLDENBURG, 14/[24] May 1664 mi. And the like Otherwhere according as the place of mi varies.

19. I might add also, that in dividing the Dia-trion, Fa Sol La; whereas I make Fa Sol, |; & Sol La, 1°-: it had been free to doe the contrary; But, of the two that seemed to me the more Eligible. Indeed it had been proper enough to have made three Intervalls, (for so many must be understood) to wit, twice ^, & a middle one between them, which would be |^, Which midle intervall might indifferently be added, sometime to the one, sometime to the other, as occasion should serve: In like manner as an Octave, or Dia-pason is divided, into two Diatessarons & a Tone between them; which Tone being added to this or that of the Dia-tessarons, makes it a Dia-pente. But, as there, when we rise on eight at Twice it is usuall first to rise a Fifth, & thence a Fourth (putting the bigger Intervall in the lower place) so I choose here to make [18] Fa Sol f; & Sol La %.\

So, from b to e, the Intervalls may be thus disposed, (with or without that sound between Fa, & Sol, for the proportions of | & |;)

Not that the voice can in a continued sequence well distinguish all these sounds; (as neither can it, those commonly designd by flats & sharps:) but that it may (as in flats & sharps is usuall) by leaps fall 135

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on each of these, from some further Interval!. 20. But being willing to innovate as little as possible, I have therefore onely abated the Tone Sol La by |^, as much increased in leu thereof, the Imperfect Hemitone, Mi Fa, or La Fa. But, how much this one Amendment hath made the Intervalls more Harmonious, will be very evident to any one, who shall but take the pains to make a like scheme of all the Intervalls in a Dis-dia-pason, according to the anciently receivd Hypothesis, (as I have here done according to this) & compare it with mine. I doe not descend to the particularities of smaller Intervalls; nor to dispute the Names of these: Because I doubt I may have sufficiently tired you already. And for the same reason, many other things must be omitted. If what hath been sayd, be thought reasonable; it will be easy to ad more: If not; there is too much already. How far this may agree with Mr Birchinsha's thoughts or Notions (of which your Letter speaks) concerning the same subject; I cannot tell. For your Letter intimating only the Generalls, both as to Defects in Musick, & as to the proposals hee makes of supplying those defects: it doth not at all appear from thence, what his particular thoughts are, & by what Methods he intends a restitution. To conclude; I shall the rather hope you will pardon me the present trouble I give you in so prolix a Letter; because you have in part occasioned it, by that of yours, to

Sir, 25

Oxon. May 14. 1664.

Your very humble servant, John Wallis.

Post-script.

30

Since I had written this; I find that our modern Writers of the Theory of Musick (in this & part of the last Century) have, upon like Principles, divided the Monochord much after the same proportion as I have done: (Though our practicall Musicians generally take little notice of it) And Ptolemy, before them, in his Harmonicks251, which when this letter was written I had not seen.) Tis therefore a wonder, how it come to be lost in

251

Harmonicks: i.e. PTOLEMY, Harmonicorum, sen de musica lib. Ill, contained in Aristoxeni ... Harmonicorum elementorum libri III, ed. Gogava, Venice 1562. Wallis published his own edition of Ptolemy's Harmonics in 1682.

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58. WALLIS to OLDENBURG?, 16/[26] May 1664 succeeding ages; or that he should be quite deserted till now of late: For Keppler252 is the first whome I find to have recourse to that notion.

58. WALLIS to HENRY OLDENBURG ? Oxford, 16/[26] May 1664 Transmission:

Manuscript missing. Existence and date: Mentioned in BIRCH, History of the Royal Society I, 426. In this letter, which was read at the meeting of the Royal Society on 18 May 1664, Wallis described an unusual fall of hail-stones at Oxford. According to Wallis, the hail-stones were like lumps of ice 'very transparent towards the sides, but more opake towards the middle, of very irregular shapes, and of the bigness of | of an inch in diameter'.

59. WALLIS to HENRY OLDENBURG Oxford, 25 May/[4 June] 1664 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 9, 2 pp. (our source). At top right of page 1 in Oldenburg's hand: 'Another Letter of Dr Wallis to M. Oldenburg, concerning (1) the same subject (2) Musick.', and beneath date: 'enter'd LB. 1. 173.' Postmark on p. 2: 'MA/25'.—printed: OLDENBURG, Correspondence II, 202-3. w 1 Copy of letter sent: LONDON Royal Society, Letter Book Original 1, pp. 173-4. w2 Copy of wl: LONDON Royal Society Letter Book Copy 1, pp. 200-2. This letter represents an addition to WALLIS-OLDENBURG 14/[24].V.1664, which Wallis wrote after having consulted modern authors on music theory, including Mersenne.

Oxon. May. 25. 1664.

Sir, Since my last253 to you (about 10 days since) I have taken occasion to consult some Modern Writers of the Theory of Musick. Which had I done 252

Keppler: cf. KEPLER, Harmonices Mundi Libri V, Linz 1619, lib. III. last: i.e. WALLIS-OLDENBURG 14/[24].V.1664.

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a little sooner, I should have omitted all or much of what I then wrote. Not that I see cause to retract what was then sayd, concerning the Proportion of sounds, & the Distribution thereof. But, because I find some of them have been before mee. For though from Pythagoras downwards, till a good while after Boethius254, (how long after, I am not certain;) no other Intervalls (for ought I find) were taken in to the Diatonick Systeme, but Tones of |; & Hemitones of |||; (& such as were compounded of these:) And, that even in our days, the Scale, &; Instruments fitted thereunto, (at lest, as used by ordinary Practicioners,) seeme to take notice of no other but Equal Tones, & Equall Hemitones: Yet I, since, find, (by what I meet with in Mersennus255 & some others,) that our Modern Theorists (at lest since Keppler256) admit a double Tone, (a Greater & a Lesser,) viz. one of |, another of ^, (whose difference, |jj, they call a Comma; & their Aggregate, |, a Ditone;) &, consequently, an Hemitone, or Limma, of j|; much different from the Pythagorean, of |||. Which are the very same Intervalls with those of mine; & are in the same Methode disposed: and very possibly may have been first discovered, upon a like inquiry with that of mine. Which though it do confirm me in the notion; yet I should not have thought it necessary to propose, had I been aware, it had been so far allready prosecuted, by those whom I had either not before consulted, or not lately; &, but very superficially if at all: as having imployed but very little time, & very few thoughts, (as I did at first premise,) in this Theory. However, it is (at most) but so much labour lost, in doing a second time, what had been done before: & perhaps you will not be displeased to see that drawn together in a clear & brief Synopsis, which might have cost as much time to recollect out of others: & the Methode so clear, as that it may, with the greatest ease, be pursued, (according to Pythagoras his 23 is (1) but (2) (at 254

Boethius: i.e. BOETHIUS (c.480-524), De instiiuiione musica libri quinque. Cf. especially Caput XVI (De consonantiis proportionum, et tono, et semitonio) and Caput XVII (In quibus primis numeris semitonium constet) of Liber primus (FRIEDLEIN (ed.), Boethii De institutione musica libri quinque, Leipzig 1867, 201-4). 255 in Mersennus: probably MERSENNE, Traitez des consonances, des dissonances, des genres, des modes, & de la composition, 119-20, contained in the collection Harmonie universelle, contenant la theorie et la pratique de la musique, 2 vols., Paris 1636-7. See also MERSENNE, La verite des sciences, contre les septiques ou Pyrrhoniens, Paris 1625, 349-84. 256 Keppler: i.e. KEPLER, Harmonices Mundi Libri V, Linz 1619, of which especially chapter VII of book III (De proportionibus harmonicis) is concerned with the ratios of tones and semitones.

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60. WALLIS to BLANDFORD, 13/[23] June 1664 first notion,) to as many subdivisions of proportion as you please. But, how far it will be fit to prosecute; I shall leave to those who have spent more thoughts upon it than I have done. And rest, Your very humble servant John Wallis.

5

[2] These

For my very worthy friend, Mr Henry Oldenburgh, at the Lady Ranalagh's house in the old Palmal, in St James's fields. London.

10

60. WALLIS to WALTER BLANDFORD London, 13/[23] June 1664 Transmission:

W Letter sent: OXFORD University Archives WP/3/10/17, f. 13r-13v and f. 30V (address). The background to the present and subsequent letters is an accusation of libel made by Edmund Bradford, a manciple of Balliol College, against William Thackwell, an attorney at the Court of Common Pleas, around the beginning of 1664. On the warrant of the vice-chancellor, Walter Blandford, Thackwell was arrested and, refusing to put in a security, committed to prison in Oxford. Since Thackwell was able to provide first a writ of privilege and then a Habeas corpus from the Court of Common Pleas, his case raised the question of the limits to the rights and privileges of the University itself, and in particular of the chancellor's court. Blandford, Wallis, Jenkins, and Yate variously attended to the matter in London.

Lond. Jun 13. 1664. Mr Vicechancellor, I did not give you an account by the last post of our Affairs in the businesse of Thackwell's Habeas Corpus cum causa at the Common Pleas: Because I understood that Dr Jenkins intended then to do it by letter & to be 15 cum causa add. 16 by letter add.

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with you on the Munday to do it more fully in person. It is thus, in brief; As soon as we came to town, wee waited on my Lord Chancellor257 who seemed at first much surprised when wee told him that the Body was not brought up, but onely a Return258 in Writing, which wee showed him: But when he had considered of the presidents which had passed in former times, (which he did not think could have been showed;) hee was better satisfyed. Hee promised us to speak to the Judges the next morning about it, & hee did so. On which day, viz. Friday, the first day of the Term, the Return from the University, in the Chancellor's name, was put into the Court, with desire it might be read; which was done, & the Court ordered it to be filed, & that the other party should have a copy, & that day sevenight appointed to hear Counsell on both sides. But soon after when our Counsell Serjeant Mainard259 was gone from the Barre; Mr. Justice Archer260 called for the Return again, & observing that the Body was not there, nor any expresse returne made concerning it, but onely of the Cause: Serjeant Hollaway261 presently offered that by Friday next the Body should be hear & the Return amended: though wee did ernestly desire him not to make any such motion, for that wee could not assure them that the University would send up the Body, nor had wee any commission to make any such motion, but onely that the present Return might bee received. And when, upon those words of his the Court had altered their former rule, & ordered the Body to be here by Friday next, & the Return to be amended: wee desired Serjeant Mainard 8 day, (1) the Return and the seal of (2) viz. ... the Term 10 read; |& filed, del.\ which 11 should |have del. ed\ have 15 that the (1) Return had spoke (2) Body 17 amended: (1) wherein we app breaks off (2) though 18 wee (1) desi breaks off (2) did 21 of his add. 25r

Lord Chancellor: i.e. Edward Hyde (1609-74), first earl of Clarendon, lord chancellor (1658-67), and chancellor of Oxford (1660-7). DNB. 258 Return: i.e. the University's return to the Court of Common Pleas, dated 8 June 1664. See Oxford University Archives WP/3/10/17, f. 3. 259 Mainard: i.e. Sir John Maynard (1602-90), judge and politician. Elected to the King's Bench 1648. DNB. 260 Archer: i.e. Sir John Archer (1598-1682), justice of the common bench since 1663. DNB. 261 Hollaway: i.e. Charles Holloway, senior (d. 1679), serjeant-at-law (from 1660 onwards) and Oxford lawyer.

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60. WALLIS to BLANDFORD, 13/[23] June 1664 to come to the Barr again, & to move that the first Rule might stand, which hee did; but could not prevail with the Court then to alter that second Rule. My Lord Chief Justice Bridgeman262 was not this day in 263 [13V] Court: But My Lord Chancellor| sent Mr Wren with us to waite upon him; desiring his opinion of those presidents wee had shewed him. My Lord chief Justice, did with a great deal of patience & favour hear the Return read; & those presidents with other things that wee had to say in this case: And made exceptions to divers particulars in the Return, (which wee have endeavoured since to amend:) & in particular, that the writ requiring Corpus, cum causa, wee must, beside the cause, either bring the Body, or at lest make an expresse return concerning it, corpus non potestas habere, quia, &c. for that part of the writ must have something sayd to it. In summe; observing that the Presidents had not (for ought approved) a direct Judiciall determination; (the letters from the Counsell Table, &c, being extra-judiciall;) if upon search he could find that the cause had been Judicially allowed us, wee need not doubt but it would be so again: If it had not been yet so determined; it must admit of a Debate, which wee should have with all fairnesse & indifferency to both parties. Wee have since, by the direction of Serjeant Mainard, had the return amended, so as to be put in next Friday, in case the University shall not yet think fit to return the Body: But in case you shall think fit so far to comply with the writ, as to return the Body; it will not bee difficult to put it into such a form. And what directions you shall further give us; shall to my ability be afforded with fidelity & diligence by Sir

I think Dr Jenkins may be with you as soon as this, however, I thought it 11 make (1) a, particular (2) an expresse 13 ought add. 14 determination; |but del] (the 17 so add. 20 in case the (1) Body be not (2) University 24 us; shall (1) be afforded (2) to my Bridgeman: i.e. Sir Orlando Bridgeman (16067-74), lord chief justice of common pleas (1660-8), lord keeper of great seal (1667-72). DNB. 263 Wren: i.e. Matthew Wren, q.v.

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Your very humble servant John Wallis.

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61. JENKINS to WALLIS, 15/[25] June 1664 necessary to give this account, in case any thing may yet stay him a little longer.

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You will please to excuse the rudenesse of the writing, because I thought I should had time (which I have not) to transcribe it fairer, & have kept this Copy for myself. [3(T]

For the Right Worshipfull Walter Blandford, Dr. in Divinity, & Vice-chancellor of the University of Oxford, 10 at Wadham College.

61. LLEWELYN JENKINS to WALLIS Oxford, 15/[25] June 1664 Transmission:

W Letter sent: OXFORD University Archives WP/3/10/17, f. 14r-14v (verso blank) and f. 29V (address). On f. 29V beneath, address a note from John Hopkins to Wallis: 'Sir. if you please I will meete you at S(erjean)t Holloways Chamber at 5. a clock, without faile Jo: Hopkins264.' As BLANDFORD-WALLIS 16/[26].VI.1664 makes clear, Jenkins effectively wrote the present letter for Blandford as a means of replying to WALLIS-BLANDFORD 13/[23].VI. 1664.

15. Jun. 64.

Sir.

15

I am commanded by Mr v. chancellor to returne you his hearty thankes for your Laboure, & your Care of the University affaire now in your Hands. He sends up onely a Parchment returne at this Time; and the rather 4 4 4 5

(which I have not) add. transcribed corr. W. fe |have add] kept myself. |But del]

264 Jo: Hopkins: probably John Hopkins (17th century), sometime solicitor to the university of Oxford.

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61. JENKINS to WALLIS, 15/[25] June 1664 for that my L. Chancellor265 (when I took my leave of his Lordship on Sunday Evening,) was pleased to tell me, he had received an Accompt of all that had pass'd when we wayted on my L. ch. Justice266; and that he did approve of my goeing down for the seal to be afHx'd to the Returne; as now amended with a Quia: his Lordship dismissing me with these words - we will contend as far as we may &c. but not giveing order in the least to bring up the Body. If we could gaine but 2 or 3 Court-dayes respite, I am persuaded I should be able justly to dismisse the Cause for want of prosecution on Bradfords part, who is now in Hampshire. Nay, Thackwell pretends that he would willingly plead, but his wife will not allow him to venture himself upon the Justice of this Place. You will stumble at the word Levata in the Returne: but 'tis, as 'twas dictated by Sir John Maynard267. Mr v. chancellor writes to Sir John, by Dr Yates268, who is his Intimate Acquaintance. I wayted last night on Mistris Wallis269 whom I found very well, nor did I forgett to deliver unto her Mrs Lichfields270 service, & message. You will receive in the same Box with the Returne, Thackwells Affidavit, and the last Rule of Court. I wish you all successe on Friday. And if it be still insisted on that the Body be brought up - give me leave (of my own head onely) to offer it unto your Consideration whither you & Mr Principall271 of Bras, should not wayt on my L. Chanc. to know his Lordships sence about communicating this affaire to the Convocation. Which if it be done 'tis more then probable they will Appeal to the K. in his Councell: & there we must not think to appear without my L. Chancellors Patronage. Sir I am your most humble servant L. Jenkins [29V]

For the Reverend Doctor Wallis Dr in Divinity, at Mr Clendon's 5 words - (1) will (2) we 10 not add. 17-25 you all ... L. Jenkins add. at 9(f in left margin 18 still (1) persisted th breaks off (2) insisted 265

L. Chancellor: i.e. Edward Hyde. L. ch. Justice: i.e. Orlando Bridgeman. 267 Maynard: i.e. John Maynard. 268 Yates: i.e. Thomas Yate, q.v. 269 Mistris Wallis: i.e. Wallis's wife, Susanna. 270 Mrs Lichfields: i.e. Ann Lichfield, Oxford printer. 271 Principall: i.e. Thomas Yate. 266

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63. BLANDFORD to WALLIS, 16/[26] June 1664 house, between the Horseshooe, & the halfe moon-Taverne in the Strand, London.

62.

WALLIS to WALTER BLANDFORD 15/[25] ? June 1664 Transmission: Manuscript missing. Existence and date: Referred to in and answered by BLANDFORD-WALLIS 16/[26].VI. 1664.

63. WALTER BLANDFORD to WALLIS Oxford, 16/[26] June 1664 Transmission: C Letter sent: OXFORD University Archives WP/3/10/17, f. 15r-15v (verso blank) and f. 28V (address). Postmark on f. 28V: 'JUN/17'. Reply to: WALLis-BLANDFORD 15/[25]?.VL1664.

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Sir

I must begg your pardon for answering your first272 by another hand273. Your second274 I received but too late to insert the clause into the returne. If you cannot interline it, I hope it will be admitted by word of mouth, or at least that you will gaine time to alter the forme. I cannot finde which of the two be more ancient, Bradfords cause or his privilege, he & Dr.

1 house, (1) near (2) between 272

first: i.e. WALLIS-BLANDFORD 13/[23].VI.1664. another hand: i.e. JENKINS-WALLIS 15/[25].VI.1664.

273

274

second: i.e. WALLIS-BLANDFORD 15/[25]?.VI.1664. 144

64. WALLIS to BLANDFORD, 18/[28] June 1664 Savage275 being both out of Towne: but this appeares that he was enterd into the Matricula above halfe a yeare since, & long before that a servant of the College. I thanke you for your paines, & doe hope they will prove successefull; but how severe soever our fate may be, 'twill be your honour as well as our satisfaction, that Si Pergama dextra etc.276 may be applyed to you. I pray by the next let me understand the issue & your advice upon it. My service to Mr Principal277 of Brasenose. I am Sir Wadh. Coll. June 16: 1664

Your affectionate freind & servant Walt: Blandford

[28V] For the Reverend Dr. Wallis at Mr Clendens house over against the New Exchange in London

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64.

WALLIS to WALTER BLANDFORD 18/[28] June 1664 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-BLANDFORD 20/[30].VI.1664. Answered by: BLANDFORD-WALLIS 21/[31].VI.1664. In this letter, Wallis evidently gave Blandford an account of recent developments in the Thackwell case.

275

Savage: i.e. Henry Savage (16047-72), master of Balliol College, Oxford (from 1650 onwards), DNB. 276 Si Pergama dextra etc.: i.e. VERGIL, Aeneid II, 291. 2rr Principal: i.e. Thomas Yate.

145

65. WALLIS to BLANDFORD, 20/[30] June 1664

65.

WALLIS to WALTER BLANDFORD London, 20/[30] June 1664 Transmission:

W Letter sent: OXFORD University Archives WB/3/10/17, f. 16r-16v (verso blank) and f. 27V (address). Answered by: BLANDFORD-WALLis 21/[31].VL1664.

Sir

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I gave you an account by Saturday's post278, of our affairs. To which I can adde no more at present, but that I thought it necessary to dispatch away my man with the inclosed Rule of Court, so soon as I could get a copy of it, which was not till this afternoon. What I can do more in the businesse as it now stands, I cannot tell. For when, wee have gone so far in justifying our allegations, as that the result is, that a Parliament would never have passed an act in those terms explicitely, which is acknowledged to bee done in effect; I know not what to answer. If the University think fit to proceed further: I humbly desire that some persons of more interest than myself may be imployed therein: Myself not pretending to any more, than what reason may infer from our Charters & their confirmation. Whether it be fit that wee should injoy, what hath been granted: is not for mee to argue. I am Sir

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London. Jun. 20 1664.

Your very humble servant. John Wallis.

These For the Right Worshipfull

[271

6 have (1) so (2) gone 6 so far add. 8 explicitely, (1) if they (2) which 9 If (1) any (2) the University 12 what (1) the (2) reason 13 it add. 278

Saturday's post: i.e. WALLIS-BLANDFORD 18/[28].VL1664.

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66. JENKINS to WALLIS, 21 June/[l July] 1664 Walter Blandford Dr. in Divinity, Vicechancellor of the University of Oxon.

66. LLEWELYN JENKINS to WALLIS Oxford, 21 June/[l July] 1664 Transmission:

C Letter sent: OXFORD University Archives WP/3/10/17, f. 17r-17v (verso blank) and f. 26V (address).

21. Jun. 1664

5

Sir, Though Thackwell had informed me that Bradford was gone into Hampshire, yet upon Information from Dr. Savage279 I find that he hath been forced to keep close all this while in his Coll. for fear of being arrested here; he having procured an Arrest from Westminster, & having Catchpoles posted in every Corner to strike him so that the Cause cannott be dismisst upon that accompt. We have made a Returne to this 4th habeas Corpus and added in it that Thackwell did take an oath in Court to give in his Personall Answer without favour &c. which amounts in summary proceedings to a Litis Contestatio; which barres the Plea of Priviledge. He pretends he knew not the meaning of that oath, so having hinted it in this Returne. I leave the Councell to manage it. There is none here of opinion to returne the Body. Sir, I am your most humble servant.

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L. Jenkins Much of the Cause of Action hath been given since Bradford's Relation to Dr Savage. 9 all this while add. 11 so that . . . that accompt. add. 17-21 Councell . . . Dr Savage, add. at 9(f in left margin 17 is add. 279

Savage: i.e. Henry Savage (16047-72).

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67. BLANDFORD to WALLIS, 21 June/[l July] 1664 [26V]

For the Reverend Dr Wallis, D.D. 67.

WALTER BLANDFORD to WALLIS Oxford, 21 June/[l July] 1664 Transmission: C Letter sent: OXFORD University Archives WP/3/10/17, f. 18r-18v (verso blank) and f. 25V (address). Reply to: WALLIS-BLANDFORD 18/[28].VI.1664 and WALLIS-BLANDFORD 20/[30].VI. 1664.

Sir

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I am sorry such expresse words of our Charters, & so strongly enacted should signify no more, but it seemes the Court is resolved, & I finde the University as resolute as they, nor is it in my power to cast in any allay; so that what will be the issue or how high it may goe I know not; but if we cannot be protected (as formerly we have been) I am ready to suffer, & shall much rather chuse to doe it then suffer our privileges to be betrayed in my time. Their Attachment I see is against the Judge280 of the Court, but then they must make such, my Lord Chancellour281 or me; for as for Dr. Jenkins, he being but the Assessor cannot certainly be the man, & that I suppose will be his plea on thursday next. I am glad you tell me that you had not spoken with my Lord Chancellour at the writeing of your last; for the helpe we hope thence would not then have left you so destitute of advice. We have sent a new returne, desireing you to manage it as well as you can, & to give us the successe. For matter of direction, Dr. Jenkins hath undertaken to send it. I pray returne not till you heare further from Your affectionate servant Walter Blandford.

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June 21.

280 281

Judge: i.e. John Archer. Lord Chancellour: i.e. Edward Hyde. 148

68. WALLIS to BLANDFORD, 23 June/[3 July] 1664 [25V] For the Reverend Dr. Wallis at Mr. Clendons house over against the New Exchange in London

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68. WALLIS to WALTER BLANDFORD London, 23 June/[3 July] 1664 Transmission:

W Letter sent: OXFORD University Archives WP/3/10/17, f. 21r-22v. Answered by: BLANDFORD-WALLIS 27.VI/[7.VII].1664.

Mr Vice-chancellor; Sir,

London, June. 23. 1664.

I can give you no very good account of your affairs. But this it is. I had, signifyed to my Lo. Chancellor282 by Mr Wren283 (not having opportunity of timely access myself) the proceedings of last Saturday284; suitable to the account285 sent you. And did after forbear awhile to wait further on him, hoping for some account from Oxford by the Carrier on Tuesday night or the post on Wednesday morning. Mean while the Lo. Chief Justice Bridgeman had shewed him Bradfords Libell, at which My Lo. Chanc. as well as the Judges were much offended; as arraigning their proceedings at Common-Law, in your Courts of Civil. My Lo. Chanc: thereupon sending for mee, I had newly, before the receit of your Return, received a chiding (or rather the University by mee) from his Lordship concerning the unreasonableness of admitting such Libells in our Courts: It was easy to reply, that no Court can prevent the commissioning of frivolous actions; nor was

12 the Carrier on add. 13 the post on add. 282

Lo. Chancellor: i.e. Edward Hyde. Wren: i.e. Matthew Wren, q.v. who was Hyde's secretary. 284 Saturday: i.e. 18/[28] June. 285 account: i.e. WALLIS-BLANDFORD 18/[28].VL1664. 283

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68. WALLIS to BLANDFORD, 23 June/[3 July] 1664

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the jurisdiction ever a whit the lesse because the action, when the merits of the cause comes to be considered, will deserve to be cast out; which wee thought, would be the issue of this action, if the court proceeded. But nothing of this would satisfy. But that the body must be brought up; & that wee are madmen if wee do it not. What discourses to this purpose had passed between my Lo. Chancellor and the Judges, (who had more then once been considering of it,) I cannot give account. But the Result, it seems, was to that pupose: & that wee ruine our privileges & our selves, (Dr Jenkins especially,) if wee comply not. Which is the unanimous discourse, both of all Judges, & all the Lawyers, (for ought I hear,) to a man. I have nothing of Reason more to urge than what hath been overruled allready: And, if I had; yet there is one answere, in every bodyes mouth will serve for all, that the courts at Westminster are the Interpreters of Privileges, Charters, & Acts of Parliament: & in their Judgement wee must acquiesce. I confesse, when I found it carryed so strongly & so highly against us; I could have wished that the Return from Oxford had been: That the Court there upon the merits] of the cause, had [211 seen reason to dismisse the cause, & that the prisoner was at large: which was the onely expedient that I would think of, & was suggested to mee in Dr. Jenkins Letter286. When I saw the Return last night to be otherwise: I found a hard game to play. However, to manage it the best wee could, Dr Yates & I went presently to wait on my Lo. Chancellor; & notwithstanding the chiding I had at (noon) wee resolved to venture another at night; & had it. And would not decline a third for you, if that would do your work: (for wee ow the University a greater service than to undergo a chiding for them.) After much had been sayd, to convince us, How unreasonable it was for us at Oxford, because we pretend to understand some things better then other men, to think that we understand the law better: How much he was ashamed of the businesse, and, How great a danger wee run of ruining the University, & ourselves, & incurring a Praemunire, & so the greatest penalties next to that of life; Dr Jenkins especially, who is looked upon as the Judge of the Court, & manager of all proceedings: The Result was, That hee could not protect us, nor would advise us; & that 1 action, (1) which (2) when 13 will serve for all, add. 20 When I (1) found (2) saw 28 How much ... businesse, add. 286

Letter: presumably JENKINS-WALLIS 21.VI/[1.VII].1664. 150

68. WALLIS to BLANDFORD, 23 June/[3 July] 1664 if we expected to be protected in such things as no honest man could do it; he would rather throw up his Chancellorship. Which I mention, not by way of reflection on his Lordship; (for he hath been really ready to do us all kindnesses otherwise, save onely as to our refusing to return the body:) but that you may see how unwilling hee is to be further pressed as to this point. One of your Letters wee had shewed him; & meant to have shewed the other allso, but that wee found hee had scarce the patience to read the first. At length, when we told him that another Return was sent up, of a like Tenour with the former, Hee would by no means allow it; as that which the Court could by no means indure, & must look up as multiplied reproaches in the face of the Court; after they had given their Judgement [22r] of the former. When wee| asked, if then wee should make no appearance or return at all: Hee answered, It was better to make none, than such a one: But bid us goe to my Lo. Chief Justice & make the best apology wee could, why wee had not so soon obeyed the order, & see if hee would be persuaded to respite it till further time. Having this expresse order from our Chancellor, not to put in that return; wee judged it necessary for us to observe it. And accordingly endeavoured to wait on the Lo. Chief Justice, but it was by this time so late (being about 6 of the clock before that Return came to our hands & now about 9;) that he was not to be spoken with till 6 of the clock this morning. At which time we found him (as heretofore) ready to receive us with all civility & kindnesse; but was peremptory that the body must be returned: All that could be done, he sayd (considering the nature of the Libell, on which hee layd great weight) was, to let it dy by degrees: for which end they would keep him for some time upon Bayl & afterwards release him: And the lesse was spoken of it in open Court, the better. Hee asked, if wee could undertake that the Body should yet bee brought up: Wee told him wee could undertake nothing, because wee had no Instructions so to do: But wee thought it probable,

3 Lordship; (1) ({so} as if I would insinuate a failure of that protection we might expect from him (2) (for he 5 how (1) little (2) unwilling 10 as add. 15 could, (1) for (2) why 15 not (1) as yet (2) so 20 & now about 9; add. 28 yet add. 29 no (1) Commission (2) Instructions

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68. WALLIS to BLANDFORD, 23 June/[3 July] 1664

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that within a few days wee should hear, either that it would bee sent up: or that the cause was dismissed by the court, & the Body at large; asking whether this Court would not be otherwise satisfyed. Hee told us; It was not for him to give that advice; but seemed to intimate that it would be so: And promised that upon such a suggestion in Court, (when the other side should move for the Attachment,) the Rule for this day should be continued till Tuesday next, which is the day before the Term ends. And accordingly this morning; when such a motion was made by the other side; Serjeant Holloway287, replyed, that the Body was sent for, & it was believed it would be here shortly. And the Lo. Chief Justice, added, that hee knew the businesse & did suppose that by Saturday next the Body would be at large: & would therefore make no new order but continue the last till Tuesday next. Which is the result of this days work. The reason why I did; both to my Lo. Chanc. & Lo. Chief Justice, interpose that of dismissing the cause in your court, (which perhaps had been well to have been done sooner,) was because| if one of them must be done (as I do not [22V] see how it can be avoyded) I thought it the better expedient hee be set at liberty by a sentance of your Court, (whereby the matter of bringing up the Body is waved,) than that it should bee done here by a sentance of this court over-ruling yours. These being the best terms that we can yet obtain, & not hoping to get better (the Tyde running so strongly against us, save one person, that I know of, to stand by us:) I can offer no opinion in that case, other than what I intimated the last time288; which is, to call a court (which you may do at any time) & dismiss the cause, either upon the insuffiency of the Libel, or want of prosecution or what other reason will suggest itself. Which if you shall not think fit to do, I must

1 either (1) that the body was dismissed (2) that it 2 dismissed |by add. ed.\ the 5 promised jthat add.\ upon 7 which is ... ends. add. 12 new add. 14 of |the del.\ dismissing 17 expedient (1) that it be done by (2) hee 20 yet add. 22 us, (1) that (2) save 26 fit add. 28r

Holloway: i.e. Charles Holloway. last time: presumably WALLis-BLANDFORD 20/[30].VI.1664.

288

152

68. WALLIS to BLANDFORD, 23 June/[3 July] 1664 ernestly beseech, that it may be henceforth managed by other hands, of greater interest: for feare, that the debate is not now what our Charters are, & how well confirmed; (which is the work wherein most properly my assistence might be subservient;) but rather; that, whatever our charters be, & however confirmed, the Body must be had here, since that the Opinion of the Court is such. I should have added; that in the behalf of Dr Jenkins wee have endeavoured to satisfy both my Lo. Chanc. & Lo. Chief Justice, that he cannot be responsible as Judge of the Court; since that hee is no more than an Assessor, whom the Vice-chancellor may call to him, or let alone at pleasure; may to day make up of one, to morrow of another; & who hath nothing but his labour for his pains: And therefore, it would be very hard, if, for what advise he gives gratis according to his judgement, & which the Vice-chancellor is at liberty to follow or not to follow at his pleasure; he should be looked upon as responsible, for all the proceedings of that Court. If, in so doing, I have done disservice to yourself in translating the danger, what ever it be, from him to you: I trust, you will excuse me therein, as having done but what (if I understand you aright) your own directions have injoyned mee; & which, I trust, will not prove prejudicial! to either. The Attorny on the other side, threatens to return the Penalties into the Exchequer, (of 50', 100', 200^;) & preferring Indictments of Praemunire; But, I think, that if yet the body be either brought up, or set at large; wee shall not be much in danger of those threatenings; but that the Court will, on Tuesday next, take off all contempts. I shall adde onely, that in case you dismiss the cause in your court; it will be necessary that it be Authentically certifyed hither, or some body here to make Affidavit of it, before Tuesday: that we may then move thereupon, to have all the contempts & penalties taken off.

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I am, Sir

Your very humble servant. John Wallis. I suppose, you may receive a letter to the same purpose from Mr

18 but add. 20 The (1) other (2) Attorny 22 either (1) set at laxge (2) brought 29-154,1 I am . . . College, add. at 9(f in left margin

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69. BLANDFORD to WALLIS, 27 June/[7 July] 1664 Principal289 of Brasen-nose College.

69. WALTER BLANDFORD to WALLIS Oxford, 27 June/[7 July] 1664 Transmission:

C Letter sent: Oxford f. 20r (address).

University Archives WP/3/10/17, f. 23r-23v (verso blank) and

Reply to: WALLis-BLANDFORD 23.VI/[3.VII].1664. Answered by: WALLIS-BLANDFORD 28.VI/[8.VII].1664.

Sir.

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I am sufficiently sensible, & so is the University too (whatever the successe be) of the great paines you have taken in the affaire you are employed about, & am heartily glad I have sent you such an account as will putt an end to it &; give you liberty to returne. But before you come off I hope you will advise Dr. Yates & take the best care you can that mine & my Assessors290 trouble doth not begin, by getting penalties taken off, & all future Actions prevented if possible. I shall not fill this paper with complaints, nor speake more of the busynes which you have enough of allready, but shall onely desire you if you stay in London to continue the kindenesse of sending an account to Sir

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June 27. 1664

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For the Reverend Dr. Wallis at Mr Clendans house over against the New Exchange London

Your very affectionate servant Walt. Blandford. [20V]

289

Principal: i.e. Thomas Yate. Assessors: i.e. Llewelyn Jenkins, q.v.

290

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70. WALLIS to BLANDFORD, 28 June/[8 July] 1664

70.

WALLIS to WALTER BLANDFORD London 28 June/[8 July] 1664 Transmission:

W Letter sent: OXFORD University Archives WP/3/10/17, f. 24r-24v and f. 19V (address). Reply to: BLANDFORD-WALLIS 27.VI/[7.VII].1664.

Sir After the receipt of yours291 yesterday Dr Yates & my self did endeavour to wait on my Lo. Chancellor292 & Lo. Chief Justice293. But could not do either, by reason that their occasions would not permit it. This morning wee waited betimes on my Lo. Chief Justice, & acquainted him that the cause was dismissed & the prisoner at large. Who is satisfyed therewith. And as to any consequences of contempts or penalties, his advise is, not to move the court in it, but himself will take order that we suffer no prejudice by it & that nothing of record shall remain whereby it may appear that there hath been any clashing with the University. I asked, whether wee might not then have our return out again; that, if it cannot do us any service, it may do us no disservice: Hee tells us, it shall bee done, & he will give order to the officers to that purpose: And Dr Yates hath taken that upon himself, to call for it. And truly my Lo. Chief Justice all along hath used us very civilly; (which wee did, in your name, thank him for) & hee doth promise a great readyness to do any kindnesse for the University; though he do still insist, that this return was not a good return, upon some exceptions, which, though I think they might in good measure be answered, I did not think it a time now to dispute.

9 remain (1) against us, but all (1) whereby 11 if it (1) may (2) cannot 15 hee add. 16 University (1) . I did (2) ; though 17 he add. 291

yours: i.e. BLANDFORD-WALLIS 27.VI/[7.VII].1664. Lo. Chancellor: i.e. Edward Hyde. 293 Lo. Chief Justice: i.e. Orlando Bridgeman. 292

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70. WALLIS to BLANDFORD, 28 June/[8 July] 1664

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Next, I waited on my Lo. Chancellor, to give him an account, both of the state of the businesse in your Letters, & of the discourse with my Lo. Chief Justice. With whom (being just then going to Westminster) I could have very little time more then to tell him what I have here related. To which I had scarce any return, but that it was a bad business: & hee asked whether you had received his letter294 of Saturday; of which I could give him no account. I left it allso with Mr Wren, to mindj him, that wee [24-] must crave his Lordships assistance in case Thackwell should attempt an action of false Imprisonment or the like: And Dr Yates, I suppose, (who was not then with mee,) doth intend when hee waits on my Lord, to press the same. This being the best end wee could make of the businesse (wherein though wee could not get it overruled for us, wee have avoyded the making a Precedent against us,) I thought fit to give this speedy account of it by your servant. And shall follow it as soon as I can: which I find cannot be till Friday; because I cannot before Thursday have place in a coach here, (nor could I then, just upon the Terms ending, if I had not secured it a week before.) I am Sir

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London; Tuesday morning, June 28. 1664.

Your very humble servant John Wallis.

For the Right Worshipfull Walter Blandford, Dr in Divinity, Vice chancellor of the University of Oxon.

[191

3 Justice. (1) This being the best end (2) With 3 then add. 9 (who ... mee,) add. 13 have add. 14 speedy add. 15 can: (1) possible (2) which 294

letter: i.e. HYDE-BLANDFORD 25.VI/[5.VII].1664. 156

71. JENKINS to WALLIS, early-mid July 1664

71. LLEWELYN JENKINS to WALLIS early-mid July 1664 Transmission:

Manuscript missing. Existence: Referred to in WALLIS-BLANDFORD 18/[28].VIL1664.

72. WALLIS to WALTER BLANDFORD London, 18/[28] July 1664 Transmission:

E First edition of (missing) letter sent: WYNNE, Life of Sir Leoline Jenkins, vol. II, 654-5.

Lond. July 18. 1664. Mr. Vice Chancellor, The Return you sent was yesterday put into Court, and upon the reading of it, as I remember, these Objections were taken to it. 1. That it was in the Chancellor's Name, whereas the Writ was directed to the Judge of the Court. 2. That the Cause was not particularly enough expressed. 3. That the Privileges there claimed were unreasonable, as that it should extend to all Places of England, and that it should be of all Pleas, &c. except Freehold, &c. 4. That it pretended to be a Supersedeas to that Court. To the 1st it was answered, That the Chancellor was the Judge of the Court, and that others acted but as his Substitutes in his Absence, and all Process and Sentences were in his Name. To the 2d, That actio injuriarum & damni dati, was the Style of the Action in this Court, and answering to an Action of the Case at Common Law. To the 4i/&, That it was only a Prayer that the Court above would supersede their Writ. In all which the Court, I think, was satisfied; but as to the Words of the Charters, though they were shewn to be confirmed by Acts of Parliament, and made as effectual, as if verbatim recited; and though it was not denied, 157

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73. HEVELIUS to WALLIS, c.31 August/[10 September] 1664

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but the Words of them did fully reach the Case, yet, I know not why, the Court insisted to have the Body of the Defendant, and made a Rule that a new Habeas Corpus be issued out, sub poena 200/. and Cause to be shewn next Thursday, why an Attachment should not go against the Judge of the Court, meaning Dr. Jenkins. Now I shall not presume to interpose my Advice, but what was suggested in Dr. Jenkins's Letter295 to me, viz. That he thought there would appear Cause enough upon the Merits of the Matter, and the Want of prosecuting by the Plaintiff to dismiss the Cause in your Court, and set the Party at Liberty, I think may be a good Expedient; and I wish the Course he advised had been taken sooner, for the Contents of the Libel to me do not seem a good Ground of an Action; so that I believe, if the Defendant do but appear, and alledge the [655] Insufficiency of it, the Cause may be dismissed without further Trouble; and therefore whether you may not think fit to do so, by calling a Court before further Process come down, I submit to your Consideration, and am yours, John Wallis.

73. JOHANNES HEVELIUS to WALLIS c.31 August/[10 September] 1664 Transmission:

Manuscript missing. Existence: Mentioned in WALLIS-OLDENBURG 29.X/[8.XI].1664. Reply to: WALLis-HEVELius 5/[15].IV.1664. According to WALLIS-OLDENBURG 29.X/[8.XI].1664, this letter was a rather short reply to Wallis's letter of 5/[15] April 1664. It was presumably sent with HEVELIUSOLDENBURG 31.VIII/[10!X].1664; OLDENBURG, Correspondence II, 214-18.

295

Letter: i.e. JENKINS-WALLIS eaxly-mid VII.1664.

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74. CONSTANTIJN HuYGENS for WALLIS, 7/[17] September 1664

74. CONSTANTIJN HUYGENS for WALLIS Oxford, 7/[17] September 1664 Transmission: C1 Draft epigram: THE HAGUE Koninklijke Bibliotheek KA XLIIIc, f. IT (our source).— printed: HUYGENS, (Euvres completes VI, 252. C2 Fair copy of C1 (without place and date): THE HAGUE Koninklijke Bibliotheek KA XLIIIc, f. 29 r . Constantijn Huygens wrote this epigram on the occasion of his meeting Wallis during his visit to Oxford in September 1664. Cf. CONSTANTIJN HUYGENS for WALLIS 6/[16].X.1664 and WALLIS-HUYGENS 31.VIII/[10.IX].1668.

Wallisium salvere jubet de Gente Batava Addicta studiis gente Zulichemius. Ipse Zulichemius, quondam, quern magna Poetam Vestra Poetarum credidit esse manus, Cum puer oxonii musas tentavit, et uni Oxonium gratas reddidit omne vices. Ipse iterum Vada vestra senex et sanus adire Gaudet, et, ut spes est, non gravis hospes erit. Sin gravis, et proprio est ingratus nomine, saltern Ut patrem Hugenii ferte mathematici. Oxonii ^ Sept. 1664.

2 (1) ( } plebe (2) Addicta studiis gente 3 (1) Hugenium dixere Patres (2) Ipse Zulichemius quondam 7 (1) Hie (2) Ipse iterum Vada vestra senex et (a) amicus (6) sanus 8 (1) Gestiit; at brevis et (2) Gestit, (3) Gaudet, et, ut (a) (esto) (6) spes est, 9 (1) Si (a) { } certe minus hac sit nomine, paucis Hugenii Patrem ferte mathematici. (b) gravis est, (c) fuerit, saltern (2) Sin 10 Hugenii Patrem ferte C2

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75. WALLIS to OLDENBURG, 21 September/[l October] 1664

75. WALLIS to HENRY OLDENBURG Oxford, 21 September/[1 October] 1664 Transmission:

W Letter sent: LONDON Royal Society Eaxly letters Wl, No. 10, 4 pp. (our source). At top of p. 1 in Oldenburg's hand: 'Dr Wallis's letter to M. Oldenburg concerning Horrox's papers.', and at top left: 'read Sept: 28: 64. entered LB. 1. 195.'—printed: OLDENBURG, Correspondence II, 231-2. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 195-6. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 225-7. Answered by: OLDENBURG-WALLis 29.IX/[9.X].1664. Together with the present letter, Wallis apparently sent the text which he had prepared from the astronomical papers of Jeremiah Horrox. The letter was read at the meeting of the Royal Society on 28 September 1664; cf. BIRCH, History of the Royal Society I, 470-1.

Sir,

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I have, at length finished that task, concerning Mr Horrox'es papers296, which were first committed to mee: Having not onely compared the copies with the Originalls, to correct the many errors in transcribing; But have digested all the severall pieces into one body, as well as the things would bear: supplying out of one what in others was wanting. Which as it was a businesse of more trouble then at first view it might seeme to be, so hath it taken up much more time than was expected. I have, to that purpose, made bold with the Transcripts by taking them in pieces, to make up one body of them, to save mee (as much as might bee) the pains of transcribing

8 time (1) that (2) than 296

task . . . papers: i.e. astronomical papers of Horrox which Wallis, at the request of the Royal Society, examined and prepared for printing; cf. WALLIS-OLDENBURG 6/[16].IV.1664. Wallis's edition of the papers did not appear until nine years later: HORROX, Opera Posthuma, viz. Astronomia Kepleriana defensa. ... Excerpta ex epistolis ad Crabtraeum suum. Observationum coelestium catalogue. Lunae theoria nova. Accedunt G. Crabtraei . . . Observationes Coelestes . . . Adjiciuntur J. Flamstedii de Temporis Aequatione Diatriba. Numeri ad Lunae Theoriam Horrocianam, ed. J. Wallis, London 1673. It had already been presented to the Royal Society on 30 October 1672 (BIRCH, History of the Royal Society III, 58).

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75. WALLIS to OLDENBURG, 21 September/[l October] 1664 the whole. (Which I made the lesse scruple to do, because, the Originalls being preserved, wee need be the lesse solicitous of the transcripts.) And where this would not sufficiently do the work; I have written into the Margins, or papers inserted, those things which out of the other pieces were necessary; sometimes transcribing whole sheets, & inserting them in their most convenient places. So that Mr Horrox having at lest in six severall ways begun to cast that one work which is here imperfect; that which I have put together is a Cento of them all. Wherein yet I have not (I think) omitted anything material in any of them. And though the [2] same things may sometimes come to be twice repeated, yet I| thought that a lesse error, than either to omit any thing necessary, or to break his texture too much and so give a piece of my own rather than his. I have prefixed an epistle297 to it giving an account both of the person (so far as I could gather from the papers) & of the thing. I think it were desirable that Dr Worthington298 be consulted in it, who is the best able (so far as I know) to give account of the matters of fact; & to adde to, or amend, any thing in that epistle. I think he may allso give an account who is that Mr Crabtree299 of Manchester, with whom Mr Horrox did so much correspond; & whether hee be living or when he dyed. As likewise what became of the Observations which were made by both of them, & of Mr Crabtree's letters to Mr Horrox. Those300 of Mr Horrox to him, I received from you, &; shall carefully peruse: I find (which I am glad of) that many of Mr Horroxes Observations are therein preserved: which I purpose to

I whole. (1) And where this would not (2) (Which 4 of (1) other copies (2) the other pieces (a) were (fit) to be preserved (6) were II his (1) textur breaks off (2) texture 16 adde (1) or (2) to, 17 who |is add] that Mr Crabtree |of Manchester add] , with 29r

epistle: i.e. an epistle dedicatory addressed to the president of the Royal Society (cf. BIRCH, History of the Royal Society I, 471). 298 Dr Worthington: i.e. John Worthington (1618-71), sometime master of Jesus College, Cambridge, DNB. Worthington was at the time of the present letter in possession of the astronomical papers of Horrox, which he had acquired from the heirs of William Crabtree. 299 Mr Crabtree: i.e. William Crabtree (1610-44?), astronomer, correspondent, and collaborator of Jeremiah Horrox, DNB. 300 Those: i.e. letters of Horrox to Crabtree, which were edited as part of the Opera posthuma of 1673.

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76. OLDENBURG to WALLIS, 29 September/[9 October] 1664

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gather out, to be annexed to the end of what I now send, if the Society shall think fit to print it; as I suppose they may: These Observations being good part of the materialls whereof hee intended to have made up that work which he hath now left so imperfect. I perceive, by these letters, that there were many other papers well worth preserving; & tables calculated in good measure towards the compleating| of the Work. But [3] what is become of them I know not; but think it worthy to be carefully inquired after. When I have perused the letters, & made extracts of them, I shall return you those allso; in the mean time, I thought fit to send you what is done, & rest Your friend & servant John Wallis.

Oxon. Sept. 21. 1664.

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These for Mr Henry Oldenburgh, in the Palmall. in St James's fields. London.

[4]

76. HENRY OLDENBURG to WALLIS 29 September/[9 October] 1664 Transmission: Manuscript missing. Existence and date: Mentioned in BOYLE-OLDENBURG c.2/[12].X.1664; OLDENBURG, Correspondence II, 244-6, 246.

Reply to: WALLS-OLDENBURG 21!X/[1.X].1664. At the meeting of the Royal Society on 28 September 1664 it was ordered that 'the secretary return the thanks of the Society to Dr. Wallis, for his great pains and judicious digesting' of Horrox's papers. Wallis was furthermore to be informed that the printing of these papers was to be left to the consideration of the Council (BIRCH, History of 1 out, (1) & annex (2) to be annexed 3 good part of add. 3 have (1) made (2) made up that (a) piece (6) work 6 measure (1) for (2) towards 7 it (1) worth the carefull (2) worthy to be carefully 12 Sept. (1) 3. (2) 21. 162

77. CONSTANTIJN HUYGENS for WALLIS, 6/[16] October 1664 the Royal Society I, 471). The letter was enclosed with OLDENBURG-BOYLE 29.IX/[9.X].1664 and forwarded by Boyle to Wallis.

77. CONSTANTIJN HUYGENS for WALLIS

between Boulogne-sur-Mer and Montreuil-sur-Mer, 6/[16] October 1664 Transmission:

C1 Draft epigram: THE HAGUE Koninklijke Bibliotheek KA XLIIIc, f. 27V (our source).— printed: HUYGENS, (Euvres completes VI, 252. C2 Fair copy of Cl (without address, place, and date): THE HAGUE Koninklijke Bibliotheek KA XLIIIc, f. 29r. Constantijn Huygens wrote this epigram after having visited Wallis in Oxford in September 1664. Cf. CONSTANTIJN HUYGENS for WALLIS 7/17.IX.1664 and WALLIS-HUYGENS 31.VIII/[10.IX].1668.

Ad Doct. Wallisium Profess. Oxoniensem. Wallisium salvere iterum301 jubeo atque valere, Gratia pro multis multa sit officiis. Nil nisi verba fero, male verba cadentia; sed quae Credier ex imo pectore nata velim. Si poscit meliora, rogo meliora ministret, qui docet et mulos congrua verba loqui. In via a Bolonia302 Monstrolium303 versus. 6. Octob. 1664.

6 Ex imo credi C2 301

iterum: see CONSTANTIJN HUYGENS-WALLIS 7/[17].IX.1664. Bolonia: i.e. Boulogne-sur-Mer. 303 Monstrolium: i.e. Montreuil-sur-Mer. 302

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80. WALLIS to OLDENBURG, 29 October/[8 November] 1664

78. HENRY OLDENBURG to WALLIS 13/[23] October 1664 Transmission:

Manuscript missing. Existence and date: Mentioned in OLDENBURG-BOYLE 22.X/[1.XI].1664; OLDENBURG, Correspondence II, 269-70, 270. According to Oldenburg, he already in this letter informed Wallis of his intention to send him further papers of Horrox.

79. HENRY OLDENBURG to WALLIS 21/[31] October 1664 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 29.X/[8.XI]. 1664. This letter evidently accompanied a further packet of Horrox's papers, referred to in OLDENBURG-WALLIS 13/[23].X.1664. Oldenburg possibly also informed Wallis of the latest decision of the Royal Society regarding the printing. At the meeting of the Council on 5 October 1664 it was simply ordered that the printing be deferred until the president William Brouncker had perused the manuscript as produced by Wallis 'and given his judgment of the fitness of printing it' (BIRCH, History of the Royal Society I, 473).

80. WALLIS to HENRY OLDENBURG Oxford, 29 October/[8 November] 1664 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 11, 4 pp. (pp. 2-3 blank) (our source). At top of p. 1 in Oldenburg's hand: 'Extract of a Letter from Dr Wallis to Mr Oldenburg concerning Horrox's papers, and the Conjunction of O and ^.', and beneath this: 'read Jan: 18. 64-5 entered LB. 1. 217'. Postmark on p. 4: 'OC/31'.— printed: OLDENBURG, Correspondence II, 275-6.

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80. WALLIS to OLDENBURG, 29 October/[8 November] 1664 wl Copy of letter sent: LONDON Royal Society Letter Book Original 1, p. 217. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, p. 252. Reply to: OLDENBURG-WALLIS 21/[31].X.1664. The letter was read at the meeting of the Royal Society on 18 January 1664/5, together with Wallis's letter of 24.XII.1664/[3.I.1665] (cf. BIRCH, History of the Royal Society I, 7).

Sir

1 received the parcel304 of Mr Horrox's papers sent last week. And shal peruse them as soon as I have done the Extract out of his letters305. I like well of Dr Worthingtons sending into Lancashire for further information concerning Mr Horrox. And the mistakes of my conjectures I desire may be corrected upon better information. The Printing of his Venus in sole, together with the rest; I well approve of; & had thoughts to move it myself. Monsr Hevelius letter306 to mee, contained little more then words of respect & thanks for my readynesse to gratify him; & a desire that I would very particularly return his thanks to the Society for the respects they have shewed him. Hee intimates a great desire of having all Ulegbeg published but sayth nothing concerning his doing of it. The conjunction of O and $, Esquire Boyle & myself (with some assistants) indeavoured to observe as diligently as wee could. On Munday from about 9 a clock till past 2. for so long wee did, by intervalls, injoy the Sunne. On Tuesday from before 8 till after 4; enjoying the Sun that day pretty constantly. On Wednesday morning, from before 8 till towards 10. Wee did let in the sun (into a room very well darkened) through a good Telescope of 6 foot; & so as to make a discus on the Paper, of 9 inches diameter. But in all our Observation, we could not in the lest discover any thing of $; nor was there any Spot in the sun to be seen. So that wee can give you no other but of our Diligence: from

Oxon. Octob. 29. 1664.

Sir Your assured friend to serve you. John Wallis.

2 Mr Horrox's add. 11 all (1) Heve breaks off (2) Ulegbeg 304

parcel: i.e. the further packet of Horrox's papers, evidently sent with OLDENBURGWALLIS 21/[31].X.1664. 305 letters: i.e. letters of Horrox to Crabtree; cf. WALLIS-OLDENBURG 21.IX/[1.X].1664. 306 letter: i.e. HEVELIUS-WALLIS c.31.VIII/[10.IX].1664.

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82. CHYLINSKI to WALLIS, 19/[29] November 1664

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These For my worthy friend Mr Henry Oldenburg; at the Lady Ranalagh's house, in the Palmal near St James's London.

[4]

81. WALLIS to SAMUEL BOGUSLAV CHYLINSKI c.!7/[27] November 1664 Transmission: Manuscript missing. Existence: Mentioned in and answered by CHYLINSKI-WALLIS 19/[29].XI.1664.

82.

SAMUEL BOGUSLAV CHYLINSKI to WALLIS London, 19/[29] November 1664 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 16r-17v (f. 16V and 17V blank). Postmark on f. 16r: 'NO/19'. Reply to: WALLIS-CHYLINSKI c.!7/[27].XI.1664. This letter is again related to Chylinski's efforts to publish, the Bible in Lithuanian; cf. WALLIS-BOYLE 14/[24].III.1661/2 (ii) and CHYLINSKI-WALLIS 4/[14].X.1662.

Reverendissime in Christo Jesu Vir Fautor semper culte et colende. 10

Suavissimas Reverendae Tuae heri accepi responsorias307, plane eo puncto

30r

responsorias: i.e. WALLIS-CHYLINSKI c.!7/[27].XI.1664. 166

[17r]

82. CHYLINSKI to WALLIS, 19/[29] November 1664 temporis, quo me contulerum ad Albam-Aulam308, habiturus juxta iteratum promissum nuper Domini Browne309, audientiam, a Dominis Commissariis: sed oleum et operam etiam hac vice perdidi vel quia illis fessis majoris momenti aliquotiis, vacare non placuit tarn exiguae causae, vel quia neminem habebant qui eis in hoc negotio vellicaret aurem; quicquid sit, interim non me terque quaterque miserum. Quo nam convenientiori medio sequenti vice hoc in passu sit usurus, Illustris et Generosus Dominus Boyle prudentiae ejus reliqui, quern ut in gratiam mei magis ac magis movere non desistat, maximum in modum Te Reverendissime Vir, oro. Munificentia ilia qua me mactare dignatur Illustris et Generosus Dominus Boyle nondum devenit ad manus meas, interim earn gratus accipio, et DEum oro, ut det ei invenire misericordiam apud Dominum die illo, quod toties respicere dignatur ad Amictionem Josephi. Vale Reverendissime Vir, et votis porro responde macte animo Londini. d. 19. 9bris. 1664.

Tui

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clientis devotissimi S. B. Chylinski. Hasce inclusas postquam Reverenda Tua per legit, obsignare et reddere dignetur, cum dataria etiam a me salute Illustri et Generoso Domino Boyle.

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[16r] These For the Worshipfull Doctor John Wallis. Oxford. with care and speed I pray.

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1 iteratum add. 6 convenientiori add. 10 dignatur |Vir del] Illustris 308 309

Albam-Aulam: i.e. Whitehall. Browne: i.e. Richard Browne, q.v.

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84. BURSCOUGH to WALLIS, 24 November/[4 December] 1664

83. WALLIS to JOHN BURSCOUGH ? November 1664 Transmission: Manuscript missing. Existence: Mentioned in and answered by BURSCOUGH-WALLIS 24.XI/[4.XII]. 1664.

84. JOHN BURSCOUGH to WALLIS Stoke by Guildford, 24 November/[4 December] 1664 Transmission: C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 18r-18v. Reply to: WALLis-BuRSCOUGH ?.XI. 1664.

Sir

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I shall wayt for your order where I shall receive the 50', & upon receipt what you expect for security. As for the 200' more, you propose verie good wayes sutable enough to my occasions, & I should willingly pay of 60' per annum or upward save only this first year (which how chargeable it may prove I cannot yet divine) therefore would the person be content (of whom the money may bee taken up) with 6 per Cent this year, & after of annuity of 60' or upwards per annum untill the Principall & Interest (as you shall judg reasonable) be run up it would bee verie acceptable newes to mee. As for Dr Greenwood310, I am something afrayd my deep silence (of which I am asham'd) may have wrought some prejudice in his brest towards mee, especially hearing from Mr Pawling311 that Mrs Greenwood took it

6 divine (1) & (2) ) 8 per annum add. 310

Greenwood: i.e. Daniel Greenwood (16057-73), D.D., principal of Brasenose College, Oxford (1648-60), vice-chancellor of the University of Oxford (1650-2); WOOD, Fasti Oxonienses II, 157. 311 Pawling: not identified.

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84. BURSCOUGH to WALLIS, 24 November/[4 December] 1664 ill shee was not acquainted with my marriage aforehand, & I beleive hee lets not out moneyes without her knowledg, & you are acquainted with the Ladyes disposition; were not these considerations in the way (which yet might easily be remov'd by a candid interpretation that I have been immersd in matrimoniall affayres) I would say none next yourself like Dr Greenwood. It was my unhappiness that I was born up with expectations from others, or 'tis likely I might have made my application to you, before moneyes were out of your hands. But now I humbly crave your assistance in procuring that summ which I heartily desire to pay of to my Father for fear of discontents as soon as might bee; I have no reason to decline that person but only for fear hee should think I make my application to him only in a time of need & neglect him at other times a disposition I abhorr, especially towards a person unto whom I am so much oblieged; But I leave the matter with you to doe as seems good. And now Sir I am as for other favours so especially for your most friendly & prudent advice to return your heartiest thanks in my own & wife's name, who tho shee will not detract from the neatness of the sex, yet lookes upon her self as an exception from the generall rule, or at least as affayres stands with us is verie willing to stand as an Heteroclite, & at present will only consult for necessityes & afterwards as God blesses us for delight; but necessitated wee are to keep house, tho boarding would have been for sometime more eligible, & would on her part have been easily assented unto if it could have been with any tolerable convenience; Sed jacta est alea312 houshold stuff wee must have, & as the outside has been excessive chargeable, the inside wil be in its measure, husband wee things never so carefully. Some of the newes you imparted313 to mee I had some notice of before. But not the least of Mr Duckworths314 attempt on the rich widow, certainly hee has been reading of Socrates vertues, & intends to attempt imitation in one. But perhaps heel find patience a great deal more difficult in subjecto, than in idaea. I should bee glad to hear of Mrs Crosse's315 little great Belly 19 an add. 23 any tolerable add. 312

sed . . . alea: SUETONIUS, Caesar 32. imparted: i.e. in WALLIS-BURSCOUGH ?.XL 1664. 314 Duckworths: probably Richard Duckworth (fl. 1649-92), B.A. 1651, M.A. 1653, fellow of Brasenose College, Oxford. Became rector of Steeple Aston, Oxfordshire, in 1679, on the death of Daniel Greenwood, nephew of the principal of Brasenose of the same name. DNB. 315 Crosse: probably the wife of John Crosse, an apothecary, in whose house Boyle had 313

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85. OLDENBURG to WALLIS, mid-December 1664

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bee: I was a kind of Paranymph or what you please to Mr Bridegroom, unto both if you please my service; with our heartiest respects & service to your self & good Lady. And doe you ask what I have been doing? even no other then what you know who were adoing many years agoe. No more but that I am verie desirous to hear from you, as soon as possible, & wish the newes may bee such as that all attempts may bee made straight twixt my Father & Stoke Nov: 24. 1664.

Sir your most oblieged friend & servant J Burscough.

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I leave the affayr wholy to your management.

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These For the right worshipfull Dr Wallis D. D. & Geometry Professor in the University of Oxford

[181

85.

HENRY OLDENBURG to WALLIS mid-December 1664 Transmission:

Manuscript missing. Existence: Mentioned in and answered by WALLis-OLDENBURG ? 24.XII.1664/[3.I.1665]. Oldenburg reported on the comet recently observed in several parts of England. Accounts of the observations were read at the meetings of the Royal Society on 14 and 21 December 1664; BIRCH, History of the Royal Society I, 508.

1 Paranymph (1) to that (#) or what rooms. 170

86. WALLIS to OLDENBURG?, 24 December 1664/[3 January 1665] 86.

WALLIS to HENRY OLDENBURG ? Oxford, 24 December 1664/[3 January 1665] Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 13, 4 pp. (our source). At top of p. 1 in Oldenburg's hand: 'Extract of Dr Wallis's Letter to M. Oldenburg, about two Comets seen.', and at top right beneath date: 'read Jan: 18: 64: entered LB. 1. 218'. Postmark on p. 4: 'DE/26'.—printed: OLDENBURG, Correspondence II, 339-40. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 218-9. w 2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 253-5. Reply to: OLDENBURG-WALLIS mid-XII.1664. The letter was read at the meeting of the Royal Society on 18 January 1664/5, together with WALLIS-OLDENBURG 29.X/[8.XI].1664 and with Wallis's extracts from Horrox's letters to Crabtree (cf. BIRCH, History of the Royal Society II, 7), the preparation of which Wallis had already mentioned in his letter to Oldenburg of 29.X/[8.XI].1664. Although addressed to Boyle, it seems to be more plausible that the letter was intended for Oldenburg, as indicated by Oldenburg's endorsement. Oxford Dec. 24. 1664.

Sir, The Comet your last316 mentioned, I have had no hap to see, nor have now any hope. Till Monday last wee heard not of it. Monday night, Tuesday, Wednesday, & Thursday night, were all so cloudy & misty with us that though I did attempt it, I could not see it. Last night was a very bright Moon-shiny night, when wee had hopes of seeing it. And, to be sure not to miss it, I watched it (with divers more) in or on the Tower of the publike schooles, from eleven of the clock till all most six in the morning, but could see nothing of it: though wee had no disadvantage but the too much light of the Moon, & a misty obscurity round the horizon; which in this place we are seldom free from. We watched it the sooner, because by some

6 a add. 8 it, (1) we (2) I 316

your last: i.e. OLDENBURG-WALLis mid-XII.1664.

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86. WALLIS to OLDENBURG?, 24 December 1664/[3 January 1665]

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observations taken (as it is sayd) at Portsmouth317, wee are informed that it was Dec. 4. hor. 2. matut. South East; at 6 in the morning at Meridian, 18 degrees altitude. Decemb. 11. at the Meridian at halfhour past 5; and with much less altitude. It is sayd to have been near Cor hydrae, (that is, as I understand it, the former day;) the latter day by its distance given from divers fixed stars it was much more to the South west. Your observations at London give it yet lesse meridian altitude, about Dec. 16. which makes me think it to be now so low as that we shall not see it here. But instead of that, we have another Comet318, which| was first taken [2] notice of here, the last night, as a cloudy star, the moon being then very bright, for I did not take notice of it till the moon was a good bright. But viewing it by a little Telescope, I found the body much resemble that of a Comet; that is, cloudy; not bright & sparkling as stars use to appear in the Telescope. Which did a little surprise mee, as being a shrewd symptome of a Comet. I looked for it in my Planispheres (which are according to Ptolomyes, not Tychoes Catalogue) but could find no star in that place. But in the Globes (according to Ticho's catalogue) I find there a smal star, but wondered that it should appear in so clear a night when stars of the third magnitude were but dimly seen. Yet, I am mistaken, if I have not heretofore observed a like cloudy star about the same place. However I took it now to bee that little star (by whatever accident magnified) which I there found, out of form, about the midway between Taurus and Eridanus. This night, looking for it again between 5 & 6 of the clock, before the moon appeared to darken the smal stars; 6 it appeared a good fair Comet, of a cloudy body, much bigger to appear-

2 was (1) the {—} (2) Dec. 4 been (1) (abo) breaks off (2) near 5 day (1) it (2) by 16 Tychoes (1) cl breaks off (2) Catalogue) 18 wondered (1) if it (2) that 3lr observations ... at Portsmouth: cf. WALLIS-OLDENBURG 21/[31].1.1664/5 for a more detailed report on these observations. 318 another Comet: In fact it was the same comet which Wallis had observed before; cf. the account presented by Hooke to the Royal Society on 28 December 1664 (BIRCH, History of the Royal Society I, 511).

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86. WALLIS to OLDENBURG?, 24 December 1664/[3 January 1665] ance than last night, & with a long taile, which last night it had not; after the moon grew bright the body did appear lesse, & the tayl too, but did not vanish with the moons light, but continues (while I write this, about 9 of the clock) pointing almost directly to the two shoulders of Orion, the Comet being but a little below, a right line passing through the two [3] shoulders of Orion, & so to the star in the| mouth of Cete (in Tichoes globes) leaving the comet a little on the south side of the line. I judge it, by my Eye comparing it with its position to other stars, to be allmost in the Equator, (but rather a little south) having about 48, or 49, degrees of right ascension. Which is all I can yet say of it: or have time to say at present, more than to crave your pardon for a rude hasty scrible, & subscribe myself,

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Your affectionate friend, & very humble servant. John Wallis. [4]

These For the Honorable, Robert Boyle Esquire; at the Lady Ranalagh's house in the Palmal, near St James's London.

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1 than (1) the (2) last I last | night add] it 4 of (1) the two (2) Orion, (a) in a right with a (b) the 6 so (1) ({—) (2) (to the (3) to 7 a little add. 71 (1) juge (2) judge II present, (1) but (2) more

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87. WALLIS to OLDENBURG, 21/[31] January 1664/5

87.

WALLIS to HENRY OLDENBURG Oxford, 21/[31] January 1664/5 Transmission:

W Letter sent (original figure inserted in w 1 ): LONDON Royal Society Early Letters Wl, No. 14, 4 pp. (our source). At top right of p. 1 beneath date: 'read Jan: 25: 64-5 entered LB. 1. 220.' On p. 4 Oldenburg has noted beneath address: 'Dr Wallis's letter concerning (1) the Cornet (2) Mr Horrox's Hypothesis (3) Monsr Hevelius and Mr Horrox's Hypotheses of Comets; and his owne Observations of the Comet of 1664/5.' Postmark on p. 4: 'JA/24'.—printed: OLDENBURG, Correspondence II, 353-6; BIRCH, History of the Royal Society II, 11 (excerpt). wl Copy of letter sent with original figure of W inserted: LONDON Royal Society Letter Book Original 1, pp. 220-23 (figure inserted between p. 222 and p. 223) (our source for the figure). w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 255-61. The letter was read at the meeting of the Royal Society on 25 January 1664/5; BIRCH, History of the Royal Society II, 11. The figure originally included in the letter sent has been taken out and inserted in the Letter Book Original.

(W) Oxford Jan. 21. 1664./5. Sir, 5

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I thank you for the paper319 you sent mee concerning the Comet; & if you have occasion to write to the Author you may please to return my thanks to him. The Hypothesis concerning the Motion of Comets, which he seemes to look upon as first discovered by himself; I beleeve (though he declare it not) is but that which others before him have conjectured: Which is, that the true way of the Comet is a streight line, which being a Tangent to a Circle, is at the point of Contact nearest to us, & as it is farther from that point appears therefore both to move slower & appear lesse. The plain of the Circle, will bee easyly determined by a

319

paper: i.e. presumably Adrien Auzout's L'Ephemeride du Comete ... fait le 2 Janvier 1665, [Paris 1665], even if Oldenburg's endorsement (and BIRCH, History of the Royal Society II, 11) suggests Hevelius to be the author. Auzout had sent copies of his paper to Oldenburg in early January 1664/5 (cf. OLDENBURG, Correspondence II, 341-2).

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87. WALLIS to OLDENBURG, 21/[31] January 1664/5 few Observations. The apparent motion, will be estimated according to a line of Tangents. What variation it hath from such a position, may arise from the Earths change of place. How far the Calculation he gives, or the Observations of this Comets motion appear to justify that Hypothesis, I have not yet taken time to Examine. But since conjectures about such Hypotheses be now stirring: I thought it not amisse to revive that of Mr Horrox which I find among his papers. In some of his letters to Mr Crabtree, hee intimates his conjecture of their issuing forth of the Body the Sun, by a streight line. Afterward, upon examining the motion of some Comets; he addeth, to that of their coming from the Suns Body, some curvity of motion arising from it. At length seems to have pitched on such an hypothesis as the figure inclosed320 represents; making it issue out of the suns body, & by an Ellipticall figure (or near it) to return thither again: according to which hee hath traced the motion of that in Anno C. 1577 (according to Tycho's Observations as I suppose.) But this paper being his own hand; & (for ought I see) all that is remaining concerning this Hypothesis; I desire that it be carefully preserved: & either returned to mee, or at least lodged with the rest of his papers which you have. It may, if you please, be coppyed out; (which yet I had not time to do, since I thought of sending it to you:) & then inserted in the Extract of his letters, where he mentions the businesse of Comets. I can pass no judgement of the Hypothesis concerning the truth of it: But, (if it answer Observation,) it is very ingenious. Concerning the present Comet; I have seen but very little of Observations as yet, (save what I have made myself, which are not very considerable:) but such as they are I shall give you; because I perceive my Lord Brounker & Dr Wren are comparing what they have from severall places. From Porthsmouth, I saw an Observation321 of Decemb. 4, & Dec. 11. by whom made I know not; but it was sent hither from one Mr Bernard an Apothecary in London. To this purpose. 8 of (1) his (2) his 11 some (1) ell breaks off (2) curvity 26 you (1) . || From Ports breaks off (2) ; because 320

figure inclosed: i.e. the figure inserted in wl. Observation: An account of an observation of the comet in Portsmouth, made by the Earl of Sandwich, was read at the meeting of the Royal Society on 21 December 1664; BIRCH, History of the Royal Society II, 508. 321

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87. WALLIS to OLDENBURG, 21/[31] January 1664/5

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Dec. 4. h. 2. S.E. The meridian altitude gr. 18, at h. 6. mat. Dec. 11. It was in the meridian hor. 5, 30'. The Altitude not observed; but the distances from fixed stars was thus. From Cor Leonis gr. 44, 50'. From Cauda Leonis gr. 43, 32'. From Spica Virginis gr. 35, 0'. From the East star in D Corvi gr. 15, 16'. From the West star in the same D gr. 18. 50'. | It was near Cor Hydrae, & hath South Latitude gr. 23 or 24. [2] Its motion is West, with a motion of Trepidation toward the South. The form like a star of the first magnitude but something bigger. The Beard 3 Fathoms; pointing N.W. The Colour White. These Observations, I perceive, are not very accurate; for there is no point of the heavens answering to all those distances. But by them, as they are, it should seem to be Dec. 4. about the South ear of Crater, or between Crater & Corvus: And Dec. 11. in the body of Hydra (not very near Cor Hydrae, but) below the foot of Crater. Dec. 16. I hear it was seen at London over the constellation of Navis, about the top of the mast. Sometime after (but I know not what day) Dr Willis322 tells mee that he saw it (not knowing it to be a Comet) near Orions Girdle. Dec. 23, in the evening, was the first time that I saw it; when it was in the voyd space between Taurus & Eridanus. I have since observed it for many nights; but without any other Instrument than my Eye comparing it, by its distance from the stars about it, with the Celestiall globe; marking the place of it on the Globe every night as I observed it; According to which it was in these places. Dec. 24. It had Right Ascension, gr. 49, South Declination gr. |. the long tail pointing to the shoulders of Orion. Dec. 25. was cloudy; but in a short intervall I found it very fair with a much greater light & tayl than the night before. It was now manifestly more North; but I saw it not long inough to take particular notice of its place. Dec. 26, 27, were cloudy; I saw it not. Dec. 28. It was among the stars in the Whales head. Right Ascension gr. 35. North Declination gr. 5^. Its tail lay parallel! with that in medio

21 it (1) next (2) for 27 but (1) by a (2) in 33 with (1) it (2) that 322

Dr Willis: i.e. Thomas Willis (1622-75).

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87. WALLIS to OLDENBURG, 21/[31] January 1664/5 Oris, & Lucida mandibulae; & fell a little below the star in Rostro or extremis naribus; it extended as far as the two stars in Tauro which ly near together, & are next to it, (toward the nose of Taurus:) & sometime it seemed to reach beyond them. Dec. 29. It had Right Ascension gr. 33\. North Declination gr. 6|. Its tail covered that in Rostro vel naribus Caeti, (yet so as the star might be seen through it:) & went a good way beyond it. Dec. 30. It had Right Ascension gr. 31^. North Declination gr. 7|. a little above a little star there scituate, which was so near it as that my eyes (which are none of the best) could scarce discern that starr. The tail passed above that in Rostro Ceti, & covered another little star between that and the Comet (but so as that it might be seen through the tail of the Comet.) It reached toward a little star or two in Taurus, which are above those two mentioned day 28. Dec. 31. Jan. 1, 2, 3, were cloudy nights; and though I discerned still that it was going onward in the same way, yet I had hardly the opportunity of designing his place accurately. It's tail higher; but in a parallell position (or near it) to that of the former days. Jan. 4. It had Right Ascension gr. 27^. North Declination gr. 101. And was for Longitude about T 29^; with about 1 degree of South Lati[3] tude. | Jan. 6. It's Right Ascension gr. 25|. North Declination gr. 11. Longitude T 28 fere. South Latitude gr. 1. fere. Jan. 8. Its Right Ascension gr. 24|. North Declination gr. ll|. Longitude T 27. North Latitude gr. 2 fere. Jan. 9. Right Ascension gr. 23|. North Declination gr. 12. fere. Longitude T 26^. North Latitude gr. 2, paulo plus. Jan. 10. Right Ascension gr. 23|. North Declination gr. 12 paulo plus. Longitude V 26. North Latitude gr. 2^. Since that time, the nights were either cloudy, or inconvenient by reason of the Moon-shine; which considering the slownesse of the motion,

3 near add. 6 Rostro vel add. 9 a jlittle add. star 13 It (1) passed (2) reached 17 accurately. (1) \\ Jan. 4. It had (2) It's 30 nights (1) being (2) were

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87. WALLIS to OLDENBURG, 21/[31] January 1664/5 made mee lesse solicitous to note it's place. But as the dark evenings come on, I will endeavour (if I can see it) to note its place when the Evenings be clear. I shall adde to these observations but this note; that as for the time of them, they were taken about 8 of the clock in the evening. And that 5 I made use of Hondius323 his Globe printed Anno 1613; so that some allowance is to be made for what the fixed Starrs are moved in longitude since that time. And then, the estimations being made onely by the Eye; it is not to be thought they can be very exact; but may possibly miss a 10 quarter of a degree sometimes. You will excuse the bad writing by reason of my hast therein; & make such use of the Paper as you shall see fit. I am Sir

Your very humble servant John Wallis.

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You may please to present my service to My Lord Brounker, Mr Boyle, Sir Robert Murray324, the Bishop of Excester325, Dr Wren326, & others as you have occasion.

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These For very worthy friend Henry Oldenburgh Esquire; at the Lady Ranalagh's house in the Pall mall near St James's London.

[4]

9 be (1) very (2) very 17 Murray, (1) jtlie Bishop of Exeter, add] & others as you have occasion. (2) the 323 Hondius: i.e. Jodocus Hondius (1563-1611), Flemish engraver, who made several terrestrial and celestial globes. 324 Murray: i.e. Sir Robert Moray, q.v. 325 Bishop of Excester: i.e. Seth Ward. 326 Dr Wren: i.e. Christopher Wren, q.v.

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88. H. L. to WALLIS, 10/[20] April 1665

(w 1 , figure)

88. H. L. to WALLIS 10/[20] April 1665 Transmission: C1 Letter sent: OXFORD Bodleian Library MS. Add. D 105, f. 19r-20v (f. 20r blank).

Aprill the 10 1665

Noble Docter

in this misfortune of beeing sepperated from a person whose preservation, and well beeing is of so pretious a concernment to us, there is nothing so great a consolation to my husband, and my selfe as that she shall not only bee in a perfect sanctuary to secure her from all designes; when you have once received her: but in so eminent a schooll of vertue that I hope it will much contribute to the embellishment, and adorning of her mind: therfore 179

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I shall entreat you, as soone as you have received this leter, that your lady would bee pleased in person to goe and remove her to your house: with all the sweet expresions of thankfullnes to Mrs litchfield327 and her familly, one my behalfe imaginable: and with all the coniurations of a perpetuall silence of her name and though her weeke will not bee expired by two or three dayes. Yet I have writt to her mayde to satisfie for the whole fortnight: and because they shall not wonder at so suden a remove: bee pleased to say you have received a leter from her father, who hath made it so absolutly his request to you: that having preserved a long frendshipp and correspondancy with him: you knew not how to resist the giving him this satisfaction: when your lady and selfe have made them this declaration: I beseech you take her home with you: and let her not have any privat commerce nor whisper with any of that familly: and when she is with you, suffer not any one of that familly, nor any other, to have recourse to her, except, those of docter willises328, and Mr cross329, the apothicary: the which you may doe with this pretence, that she is indisposed: and that her parents have made it theyre suite to you to keep her as private and concealed as posibly you can and now that you may know the mysterie of all these precautions: and whether I have reson to have apprehensions or noe: she hath already corapted one or more of Mrs litchfields familly, to bee all together at her devotion, and to doe what ever she| desires, and [19V] for an instance of this truth: she had not bin above foure dayes there: but she convayed a private leter to a frend of hers in towne: the which my woman got a sight of: that she was now as hapie as she wished: for that she had made a conquest of a person in that house: the which I conceive is the mayde: that she would convay what leters she would to any place and had that fedelity for her that she could trust her with any secret: and that now she could bee accommodated with any thing she could desire and that her heart bledd she could not communicate to her some secrets, which could not bee commited to paper: and desired this frend when she writt to her: to derect her leters to this person: of what nature these secrets are I am not able to determine: but this I am sure of, it is beter preventing

327

litchfield: almost certainly Ann Lichfield (d. 1672), widow of Leonard Lichfield the elder (d. 1657), printer to the University of Oxford from 1657, and Wallis's neighbour. She carried on the business until it was handed on to her son, Leonard Lichfield the younger. 328 willis: i.e. Thomas Willis (1622-75). 329 cross: i.e. John Cross. Cf. BuascouGH-WALLis 24.XI/[4.XII].1664. 180

88. H. L. to WALLIS, 10/[20] April 1665 dangers then geting remedys which perhapps may arive to late: and if she have noe further traficke with this person, then only to get things which are hurtfull to her disease: it is reson enough to part them when you have her with you: you may easilly get a perfect assendant over her if you are not too famillier with her. But cary it with a cerimony full of distance: for then she will not get within you: and this was the artifice that docter willis and his wife observed when she was last at oxford: that it gained and preserved so absolut an empire over her: that they might impose upon her whatever they pleased: and because she loves flattery, bee pleased now and then to take occasion: to commend her for those vertues you would willingly inspire her with (for that will tie her up to the hight of exactness) somtimes tell her how hansomly you will represent her to her parents: and that within a short time you hope to see her so recovered, that you may bee the good instrument of preferring her to some considerable and advantagious mariage: these things performd with your incomperable ingenuity: and singuler dexterity of witt will so disarme her pasions, and turne them into gentlnes and softness of humor: that she will bee insensible of her imprisonment, grow weaned from that first society: and in fine grow beter prepared: to receive your most excellent impresions: but wee must bee sure to conceall all this from her: and her mayde: docter willis and every body else: for if it once breake forth: she will grow fuller of designes and act them a great deall the closer: her servant whom she hath beguiled will reproach her with it: and the people of the house if they know they are suspected will bee apt to rayle: and say ill faverd things: endevoring to lesson the errour among them: by finding of faults with us: therfore I coniure you to make the best use of these advertisments: and to conceall them: from her knowledge and her maides: for it will beget etternall quarells if it bee knowne: the performance of these requests will lay an unspeakable obligation on her that is Your affectionate frend and servant HL My service to your good Lady. Sir Geo:330 presents his hearty service to you both.

330

Sir Geo: not identified. 181

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91. WALLIS to BOYLE, 29 April/[9 May] 1665 To the most honoured Docter Walice at his house at oxford in catt street these present

[20-]

89.

ROBERT BOYLE to WALLIS early 1665 Transmission:

Manuscript missing. Existence: Mentioned in WALLIS-BOYLE 29.IV/[9.V].1665. It would appear that Boyle, who was in London during winter and spring 1665, suggested to Wallis that he in future send copies of his work so that he might comment on them.

90. WALLIS to ROBERT BOYLE 24 April/[4 May] 1665 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS-BOYLE 29.IV/[9.V].1665. According to Wallis, papers for Boyle were included in this letter. It is probable that these were connected with the Horrox edition on which he was working.

91.

WALLIS to ROBERT BOYLE Oxford, 29 April/[9 May] 1665 Transmission:

W Letter sent: LONDON Royal Society Boyle Letters 5, f. 172r-173v (f. 173V blank) (our source). Postmark on f. 172r: 'MA/1'.—printed: BOYLE, Correspondence II, 464-5. 182

91. WALLIS to BOYLE, 29 April/[9 May] 1665 El First edition of letter sent: BOYLE, Works (1744), V, 514. E2 Second edition: BOYLE, Works (1772), VI, 458.

[173r]

Oxford. Apr. 29. 1665.

Sir, Your Honour doth not expect, I should tell you that I have allready read over that excellent Book331 of yours which I received this day from your favour. But, I have allready perused so much thereof; & seen therein so much of worth, that I see great cause not onely for my self to thank you, (which I heartyly do;) but for mankind to do the like; for so much of cost and pains which you have sustained to enrich them. Which should they fail to do; you would thence have occasion to enrich your second Title, with one generall more, of subjects disposed to be Frozen: while yet I shall desire to have your honour of being ranked under your third Title, of subjects Indisposed; nor shall I think that Indisposition to be a Disease, but a Duty. Sir, I hope your Honour hath received those Papers which on Munday last I sent, with a Letter332 directed to your Honour, by Bartlet the Carrier. Which, because of frequent neglects in carriers, I take this occasion to mention: not that I think them of so much worth as that I should be solicitious about the miscarriage of them; but, that I would not be thought guilty of disobeying your commands, intimated in [172V] your Honours last333 to mee; &, because I am ambitious of| submitting all I do to your Honours judgement; Having had so oft experience both of your ability to judge, & of your inclination to be favourable to, 6 self to add. 10 one (1) particular (2) generall 12 of subjects Indisposed add. 18 in add. ed. 19 am add. 331

Book: i.e. BOYLE, New Experiments and Observations touching Cold, or an Experimental History of Cold begun. To which are added, an Examen of Antiperistasis, and an Examen of Mr. Hobbes 's Doctrine about Cold. Whereunto is annexed an Account of Freezing brought into the Royal Society by the learned Dr. C. Merret, a Fellow of it. Together with an Appendix containing some promiscuous Experiments and Observations relating to the precedent History of Cold, London 1665. This book had been presented to the Royal Society at its meeting on 12 April 1665; BIRCH, History of the Royal Society II, 28. 332 Letter: i.e. WALLis-BoYLE 24.IV/[4.V].1665. 333 your Honours last: i.e. BoYLE-WALLis early 1665.

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92. WALLIS to OLDENBURG?, 8/[18] May 1665

Sir Your Honours very humble & obliged servant, John Wallis. 5

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Sir If you have the opportunity of seeing Sir Robert Moray, you may please to present him my humble service; & assure him that as I am ambitious to observe all his commands, so I shall be particularly carefull of those concerning the Gentleman334 he recommended to mee in his last335. [172r]

For the Honourable, Robert Boyle Esquire; at the Honourable the Lady Ranalagh's house in the Palmal, near to St. James's London.

92. WALLIS to HENRY OLDENBURG ? Oxford, 8/[18] May 1665 Transmission:

Manuscript missing. Existence and date: Mentioned in BIRCH, History of the Royal Society II, 48. At the meeting of the Royal Society on 17 May 1665 a letter from Wallis, dated 8 May 1665 and probably addressed to Oldenburg, was read. In the letter, which was referred to Council for consideration, Wallis described his progress in editing the Horrox papers and made recommendations concerning the printing of the edition.

7 carefull add. 334

Gentleman: not identified. his last: not found.

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93. MATTHEW WREN to WALLIS, 30 May/[9 June] 1665

93. MATTHEW WREN to WALLIS London, 30 May/[9 June] 1665 Transmission:

W Letter sent: LONDON British Library MS. Add. 32499, f. 15r. Beneath the letter Wallis has drafted his reply. Answered by: WALLIS-MATTHEW WREN 4/[14].VI.1668. The background to the present letter would appear to be fears arising from the repressive legislation against dissenters, including Presbyterians, which parliament had introduced since 1664. Among this legislation, unjustly attributed to the lord chancellor as the 'Clarendon Code', was the Five Mile Act, passed in October 1665.

Sir,

These two inclosed Letters336 were intercepted at the Posthouse in Banbury & sent up to my Lord Chanceller337, who finding they indeavour to conceale themselves under a Cifre, believes it is for some ill design, there being now so many of them carried on by the Discontended Party in the Nation. He knows that to your Industrie & Sagacity the most accurate Cifres ly as open as a common Alphabet, & this Cifre being as He conjectures a very meane one will he supposes give you very little trouble to unmask it. He has commanded Me to send them to You with his desire that You will take the paines to decifre them, with as much speed as Your Occasions will permit You, & to use that Secrecy in it which belongs to an Affaire on this nature. I am

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Sir, Worcester house May 30th. 1665

Your most humble Servant M. Wren.

7 Cifres (1) are (2) ly as 10 take add. 336

inclosed Letters: These ciphers appear to be missing. Lord Chanceller: i.e. Edward Hyde (1609-74).

337

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95. OLDENBURG to WALLIS, 28 September/[8 October] 1665

94. WALLIS to MATTHEW WREN Oxford, 4/[14] June 1665 Transmission:

W Draft of (missing) letter sent: LONDON British Library MS. Add. 32499, f. 15r. The draft is written beneath, the letter to which it is a reply. Reply to: MATTHEW WREN-WALLIS 30.V/[9.VI].1665.

Oxford June 4. 1665. Sir,

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I have returned you herewith, the two letters338 which I received from you in Cypher to be deciphered, together with the exposition of them. Which though they prove to be of no great moment, at lest as they appear to mee: yet did they cost some time to discover. You may please, with them, to present my very humble service to your Lord339; from him who shal be glad to imbrace any opportunity to declare himself his Lordships, & Sir

Your very humble servant John Wallis.

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For my honoured friend Mathew Wren Esquire; at Worcester-house London.

95. HENRY OLDENBURG to WALLIS 28 September/[8 October] 1665 Transmission:

Manuscript missing. 338

two letters: i.e. the two letters originally enclosed with MATTHEW WREN-WALLIS 30.V/[9.VI].1665. 339 your Lord: i.e. Edwaxd Hyde. 186

96. OLDENBURG to WALLIS, 26 December 1665/[5 January 1666] Existence and date: Mentioned in OLDENBURG-BOYLE 28.IX/[8.X].1665; OLDENBURG, Correspondence II, 531-5, 532. This letter, with one for Seth Ward, accompanied presentation copies of Hevelius's Prodromus cometicus for each of the two Savilian professors which had arrived in London from Danzig at least ten days earlier. See OLDENBURG-BOYLE 18/[28].IX.1665; OLDENBURG, Correspondence II, 511-13. Cf. HEVELius-OLDENBURG [22.VJ/1.VI.1665 (ibid., 392-5) and HEVELIUS-OLDENBURG [2]/12.IX.1665 (ibid., 493-4). On this occasion Oldenburg employed the services of the bargeman Richard Moss to bring the package (which also enclosed items belonging to Prince Rupert and the Duke of York) to Oxford. It would appear, however, that the package failed to arrive. See OLDENBURG-MORAY 7/[17].X.1665 (ibid., 549-50), OLDENBURG-BOYLE 10/[20].X.1665 (ibid., 555-8), OLDENBURG-BOYLE 17/[27].X.1665 (ibid., 571-2), MORAY-OLDENBURG 19/[29].X.1665 (ibid., 574-6), and MORAY-OLDENBURG 29/30.X/ [8/9.XI].1665 (ibid., 582-3).

96. HENRY OLDENBURG to WALLIS 26 December 1665/[5 January 1666] Transmission:

Manuscript missing. Existence and date: Mentioned in OLDENBURG-BOYLE 30.XII.1665/[9.I.1666]; OLDENBURG, Correspondence II, 652-7, 652. In this letter Oldenburg discussed the printing of the Philosophical Transactions for January 1665/6. Since November 1665 the journal had been printed in Oxford by Leonard Lichfield. Oldenburg delegated a large part of the editorial work to Wallis. See OLDENBURG-BOYLE 19/[29].XII. 1665; OLDENBURG, Correspondence II, 645-8.

97. HENRY OLDENBURG to WALLIS 30 December 1665/[9 January 1666] Transmission:

Manuscript missing. Existence and date: Mentioned in OLDENBURG-BOYLE 30.XII.1665/[9.I.1666]; OLDENBURG, Correspondence II, 652-7, 653. Oldenburg, in his letter to Boyle of 30 December 1665, says that he intends to write to Wallis by the same post in order to inform him that Hooke would soon communicate

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98. COLLINS to WALLIS, 2/[12] January 1665/[1666] the results of his observations to those persons that had read and considered the recent works on comets by Hevelius and Auzout.

98. JOHN COLLINS to WALLIS [London], 2/[12] January 1665/[1666] Transmission:

C Draft of (missing) letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 184r-184v (our source).—printed: RIGAUD, Correspondence of Scientific Men II, 458-60.

Reverend Sir

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The unexpected favours340 I received from you when at Oxford cannot but make me mindfull of those obligations of thankes I ought to returne, as a Testimony whereof vouchsafe to accept of a few bookes I herewith send viz. Mengoli341 Geometria Speciosa, Antimi Farbii342 Opusculum Geometricum et Caravagii343 Mediolanensis Applicationum Doctrina which handles the Limits of such Aequations as have but two Nomes. Not that I thinke any mans workes can adde to your vast treasure of knowledge, but possibly may excite you to supply what they have omitted or amend what they have but perfunctorily performed. It hath pleased God still to reserve me as a Monument of his Mercy in the Land of the Living.

6 Geometricum (1) and (2) et 7 such add. 7 Nomes (1) as is obs breaks off (2) . Not 10 God jstill add. to reserve me |in these times of Contagion del] as 340

unexpected favours: Collins clearly had at least one meeting with Wallis at Oxford in late summer of the previous year. Among Wallis's papers is to be found a problem concerning the capacity of a wine or beer vessel which Collins proposed to him on 12 August 1665 and to which Wallis prepared on the same day an extensive solution (OXFORD Bodleian Library MS. Don. d. 45, f. 279r-277v (pagination reversed)). 341 Mengoli: i.e. Pietro MENGOLI (1625-86), Geometriae speeiosae elementa, Bologna 1659. 342 Farbii: i.e. Honore FABRI (1607-88), Opusculum geometricum, Rome 1659. 343 Caravagii: i.e. Petro Paulo CARAVAGGIO (1617-88), Geometria applicationum deficientium figura data specie, Milan 1659. 188

98. COLLINS to WALLIS, 2/[12] January 1665/[1666] But upon my returne hither I found wanting Mr Anthony Thompson344 and Mr Henrie Button345, two of the best Mathematicall Instrument Makers. And the quondam Servant of Mr Sutton, John Marke346, being now returned is about taking his Masters house; he desires the presentment of his most humble service. We hope he may prove as good a Workeman as his deceased Master, and that mite of knowledge I have attained to, I shall most willingly serve him with. Some New bookes are published in france as Pascal du Triangle Arithmetique347 &c and some others whereof I hope ere long to have an Account, from the hands of an Ingenious German Gent348 now residing at Paris and familiar I presume with Monsr Robervall349, who being on the Contemplation of Curves perchance will have proposed to his Consideration, the Curve in Wrights350 or Mercators351 Sea Chart, that represents the great Arch of a Circle, and with what Plaine such a Curve may be cutt out of the Cilinder in which Mr Wright supposes the Sphere inscribed; whether the said Meridian Line be

5 hope (1) ere (2) lie 5 as (1) a (2) good 6 and (1) this (2) that 6 have (1) is (#) attained to, (a) is at his intreaty (b) shall 6 to, |I add. ed.\ shall 7 Some (1) Newes (#) New 8 &c add 10 now odd. 12 Curve | arising from the degrees of the Meridian Line del. in 15 said add. 344

Thompson: i.e. Anthony Thompson (d. 1665), an instrument maker in Hosier Lane, Smithfield. See TAYLOR, Mathematical Practitioners I, 220-1. 345 Sutton: i.e. Henry Sutton (d. 1665), noted instrument maker with premises behind the Royal Exchange in Threadneedle Street. Like Thompson, he was presumably a victim of the Great Plague in London which had begun in April 1665. See TAYLOR, Mathematical Practitioners I, 220. 346 Marke: John Marke (fl. 1665-79), a former workman of Henry Sutton, who succeeded him on his death. He was admitted member of the Clockmakers Company in 1667. See TAYLOR, Mathematical Practitioners I, 252-3. Cf. COLLINS-PELL 28.VIII/[7.IX].1666; RIGAUD, Correspondence I, 115-18, 118. 347 Pascal du Triangle Arithmetique: i.e. Blaise PASCAL (1623-62), Traite du Mangle arithmetique, avec quelques autres petits traitez sur la mesme matiere, Paris 1665. 348 German Gent: not identified. 349 Robervall: i.e. Gilles Personne de Roberval (1602-75). 350 Wrights: i.e. Edward Wright (1558-1615). 351 Mercators: i.e. Gerhard Mercator (1512-94).

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not the same, or in the same ratio as Mr Gunters352 Logarithme Scale of Tangents beginning therein at 45d and counting each ^ degree a whole one; and the nature of that Curve that may be made by raising the degrees of that Meridian Line as Perpendiculars upon the| degrees of the [184V] Equinoctiall in that Chart, which are equall parts and the contrary; and this in Order to the more easy description of the great Arches Curve in that Chart, or for the supply of the Meridian line, it not yet being knowne (geometrically) to find the Rumbe betweene two places of knowne Latitudes and given diff. of Longitude. And if any thing come to my knowledge about these matters I should be willing to communicate it, and humbly crave the favour when your occasions draw you to London, to affoard me the cognisance thereof that I may have some further opportunity administred of expressing my Gratitude. In the interim if you shall vouchsafe a Line or two in returne it needs no other direction but to me as an Accomptant353 at the excise office in St Bartholomew Lane London who desires to be accounted an Admirer of your Worth and Your most affectionate Servitor Janu. 2d 1665

3 and (1) what Curve that may be (2) the 5 and the contrary add. 6 great add. 7 line (1) |or its table add.\ without which the Latitudes and difference of Longitude of two places being proposed to find the Rumbe (2) , it 7 knowne ( (1) without it (2) geometrically 10 matters |I add. ed.\ (1) shall presume (2) should be willing to communicate it |to you del. , and 12 the (1) knowledge (2) cognisance 12 have (1) f breaks off (2) some further opportunity (a) of demonstrating (b) administred of expressing 15 St add. 15 who (1) shall (2) desires 352

Gunters: i.e. Edmund Gunter (1581-1626), professor of astronomy at Gresham College (1619-26) and colleague of Henry Briggs (1561-1630). Together with Briggs and Napier, he was a central figure in the development of logarithms. 353 Accomptant: Collins was accountant at the Excise Office in London c.1660-70. 190

99. WALLIS to COLLINS, January/February 1665/6

99. WALLIS to JOHN COLLINS January/February 1665/6 Transmission:

Manuscript missing. Existence: Mentioned in and answered by CoLLiNS-WALLis 28!I/[10!II].1665/6. Wallis apparently sent Collins a mathematical exercise together with considerations on Wright's and Mercator's sea charts.

100. JOHN COLLINS to WALLIS [London], 28 February/[10 March] [1665/6] Transmission:

C Draft of (missing) letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 185r-185v (our source). On right margin of f. 185V at 90° in Collins's hand: 'To Dr Wallis feb. 28'.—printed: RIGAUD Correspondence of Scientific Men II, 460-2. Reply to: WALLis-CoLLiNS I/II. 1665/6.

Reverend Sir I have received yours dated the Instant354 with your Enclosed Exercise and Considerations about Mercators or rather Mr Wrights Sea Chart355, for the use whereof I returne most hearty thankes but withall am sorry I should put you to so much trouble about such a Subiect which I presume I as farre understood as those Authors left it, if not farther. This must redound to my Crime and your goodnesse who were pleased to 2 yours (1) with (a) the (6) the Papers your (2) dated 4 Chart (1) . I (2) , for the ... but withall 4 I add. ed.

6 , if not farther add. 354

yours dated the Instant: i.e. WALLIS-COLLINS I/II.1665/6. Collins has neglected to enter the date of this letter in the draft. 355 Considerations . . . Sea Chart: cf. COLLINS-WALLIS 2/[12]. 1.1665/6.

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divert your Studies from other sublime Speculations. You have discovered where the difficulty lyes &c. may it redound to your immortall fame to remoove it; not that I presume to desire you to reveale your discoveries in such Worthy Speculations to my selfe, but to be reserved for your owne Opportunity to communicate to the Learned; as many of which, as I ever had the happinesse to Converse with, account you willing Lampada tradere. The Latitudes of two places and their difference of Longitude given, to find the Rumbe geometrically (as farre as the nature of such a Probleme admitts) is very usefull and therefore desirable, because Seamen (expertus Loquor) are much for Delineations; hence the Common tables of naturall Sines may perchance be sufficient for marine Calculations (if others were wanting) and hence the Common Sea Chart with parralell IVtcrccitors I Chart. Meridians equally divided, may supply the want of Wright

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I confesse as a Sciolist I have adventured to my shame too farre in this Subiect and this made me not to send you my endeavours sooner as blushing at so Judicious a Censure. Such a Chart seemes to be an Evolution of the Sphere. For admitt a Sphere made by the Conversion of that Semicircle that is the Semibase of a Cilinder, unrowle it selfe in the Concave of the said Semicilinder (cut through the Axis) with the Axis of the Sphere erect to the Side of the Paralellogram, it then evolves the Sphere and on this Supposition it is not hard to delineate the Curve that any Rumbe Line, the Ecliptique or any other great Circle shall leave behind. But the geometricall Speculations about such Curves seeme sublime. Moreover that the Meridian Line of Mercators Chart should seeme (as it doth) to be the same with the Logarithme Tangents (viz. that the adding of naturall Secants should Constitute a Logarithme tangent,

I sublime add. 3-7 not that . . . Lampada tradere. add. in margin 3 that I (1) desire such of either (2) presume (a) to desire your undertakement or Discoveries of (&) or reveale (aa) the (&&) your discoveries in corr. ed. 6 with, (1) you (2) account 8 geometrically (1) is very desirable (2) (as 9 therefore add. II for (1) all (2) marine 12 Chart (1) (now (2) with 15 not (1) sooner (2) to send 26 tangent, (1) (in s breaks off (2) though 192

100. COLLINS to WALLIS, 28 February/[10 March] [1665/6] though to an unwonted ratio) is Mysterium aliquod grande proposed long since to Mr Briggs356 and Mr Gunter357, but not approoved or disprooved. See my Plaine Scale358 herewith sent, the Navigation part Page 117, 44, 45, 60, 61, 62. And admitting that the Meridian Line were the same with a Logarithme Tangent, it would not want a geometricall way to describe it by such Curves as pure Geometry will reiect. But in regard I hope to have the honour to attend you at your arrivall359 here, I cease enlargement, or to Cumber you with some latter thoughts of my owne about the said Chart, Curves or Meridian Line. At the request of Mr V [185 ] Sutton360| I wrote a despicable treatise361 of quadrants. His designe was to demonstrate himselfe to be a good workeman in cutting the Prints of those quadrants and thereby to obtaine Customers, mine to Improove the Prints by Vernish, which I was certaine I could accomplish 12 yeares since, to a better lustre then this I herewith send (togeather with a sheete of

3 See (1) the (2) my 6 as Iperchaunce del] pure 7 at your arrivall here add. 8 latter add. 10 wrote a (1) Treatise (2) despicable 10 was to (1) obt breaks off (2) demonstrate 11 cutting (1) by (2) the 12 to (1) accumulate C breaks off (2) obtaine Customers, (a) might (6) mine 13 since, (1) and as I h breaks off (2) to a 14 togeather add. 356

Briggs: i.e. Henry Briggs (1561-1630), first Gresham professor of geometry (15971619) and afterwards first Savilian professor of geometry at Oxford. 357 Gunter: i.e. Edmund Gunter. 358 Plaine Scale: i.e. COLLINS, The Mariners Plain Scale New Plain'd, London 1659. 359 arrivall: Wallis had apparently announced his intention to visit London. He was evidently present at the first meeting of the Royal Society following eight months' interruption due to the Great Plague on 14 March 1666. See BIRCH, History of the Royal Society III, 68 and OLDENBURG-BOYLE 17/[27].III. 1665/6; OLDENBURG, Correspondence III, 60-3. 360 request . . . Sutton: At Sutton's request—he apparently had no leisure to do it himself—Collins wrote a decription of three (in a later edition four) quadrants. Henry Sutton engraved the plates of these instruments, the principal of which had been designed by Thomas Harvey. 361

treatise: i.e. COLLINS, The Sector on a Quadrant, or, A Treatise containing the Description and Use of Three Several Quadrants ... of great use to seamen, London 1658.

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my Booke) which I now send and the which, Commaculated with Dirt or Inke, will be washed away without dammage. The first thing I published362 was about a quire of Paper concerning Merchants accounts, which upon latter thoughts I found my selfe unable to amend and was reprinted in May last, but by reason of the Contagious Death of the Bookseller not procurable as yet. And among these Luxuriancies I mett with a Dyalling Scheme of Mr Fosters363 and Commented upon that364; which it is too late to wish undone (as the Author dyed and left it) but what Iniproovements I have mett with on that subiect I shall be willing to manifest. I hope to be more wise and silent hereafter. Vouchsafe herewith also to receive and accept Broschius his larger treatise365 de numeris perfectis with your owne Papers, and to account me one of your most thankfull and Obliged Servitors and Admirers

1-2 Booke) \ ( 1 ) which (a) (will with) (6) I (oa) thinke (66) presume (aaa) will not be (bbb) scernes the iniury of Dirt or Inke (2) which Commaculated . . . (3) which I now send and the which, Commaculated . . . dammage add. \ . The 4 which (1)1 presume I shall never be able (2) upon (a) the (b) latter thoughts I found my selfe unable 5 last add. 5 Contagious add. 8 Author (1) 1 breaks off (2) dyed Sit) (1) and (2) but 9 subiect |I add. ed.\ shall 10 herewith |also add] to receive |and accept add] Broschius 12 with | the returne of del] your 362 first... published: i.e. COLLINS, An Introduction to Merchants Accounts, containing Five distinct Questions of Accounts, London 1653. 363 Dyalling . . . Fosters: i.e. Samuel Foster (d. 1652). A highly renowned diallist, Foster published The Art of Dialling in 1638. 364 Commented upon that: i.e. COLLINS, Geometricall Dyalling, ... Being a full explication . . . of divers difficulties in the works of learned Mr Samuel Foster . . ., London 1659. 365 treatise: i.e. Johannes BROSCIUS (Jan Brozek), De numeris perfectis disceptantis, Amsterdam 1638.

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101. OLDENBURG to WALLIS, early 1666

101.

HENRY OLDENBURG to WALLIS early 1666 Transmission:

Manuscript missing. Existence: Mentioned in and answered by WALLIS-OLDENBURG 3/[13].IV.1666. This is clearly only one of a number of now missing letters exchanged between Oldenburg and Wallis since late 1665; in it the secretary of the Royal Society would appear to have passed on the request for information on the manuscripts of Peter of Blois (Petrus Blesensis) held in the Bodleian and college libraries in Oxford. The unnamed correspondent in France from whom he had received this request was probably Henri Justel, who was evidently involved in preparations for a collected edition of the works of Petrus Blesensis. The latter's Opera omnia, edited by Pierre de Goussainville, was published in Paris in 1667.

102.

WALLIS to HENRY OLDENBURG Oxford, 3/[13] April 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 16, 4 pp. (p. 3 blank) (our source). On p. 4 beneath address in Oldenburg's hand: 'April 6. 1666.'—printed: OLDENBURG, Correspondence III, 79-80. Reply to: OLDENBURG-WALLis early 1666. This is the first of a number of letters in which Wallis describes his efforts in collecting information on manuscripts of Petrus Blesensis in English libraries.

Oxon. Apr. 3. 1666.

Sir, I have not been quite idle in your busyness of Petrus Blesensis366, though since your last367 to mee I have given you no particular account of it. I have caused our libraries here at Merton & Balliol Colleges to be searched. 366

Petrus Blesensis: i.e. Petrus Blesensis or Peter of Blois (c.H35-c.l204), French theologian and humanist. 36r your last: i.e. OLDENBURG-WALLIS early 1666.

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In Merton wee find two copies of Petr. Bles. de Utilitate Tribulationis; but find them both imperfect, & containing onely the 4th of those 11, or 12, utilities which are in that copy in the Bodlyan Library: which made mee put a stop to transcribing that of Merton College: And will take care it shal be transcribed out of the Bodlyan copy. The Epistola Aurea de Silentio &c. is in the Merton library; & if it do not appear to be allready printed amongst his Epistles, I will get it transcribed thence. In Baliol College, wee find a pretty large tract, bearing no other title (that I hear of) but Quales sunt. that being the constant Beginning of divers sections. That piece sayd to bee in Bibl. Vulni. (of which I could not tell what to make, & conjectured it might be Oriel Coll;) wee find to be Bibliotheca Lumleyana; being a library sometime belonging to the Lord Lumley368; but hath since been transmitted to Sir Robert Cottons369 Library, or the Earl of Arundels370; in one of which (I beleeve, the first) that book is now to be sought about London. Wherein, at this place I can do you no further service. The like I must say to that of Kays College371 in Cambrige,| for which [2] it will be requisite to imploy some body there. I find a difficulty in getting a transcriber, who can read those old hands: & I am almost afraid it will fall out to be my own task when all is done. The Tracts I find are not very large; & so the matter will bee the less. I was all the last week confined within doores with a great cold, & the troublesome effects of it in my Head, Face, & Throat; else, I think, wee should have been a little forwarder then at present I am. No more at present but that I am your humble servant Jo. Wallis. 15 to add. ed.

17 sought (1) at Lo breaks off (2) about 368

Lord Lumley: i.e. John, first Baron Lumley (15347-1609), high steward of the University of Oxford (from 1559 onwards) and member of the Society of Antiquaries, DNB. 369 Sir Robert Cottons: i.e. Sir Robert Bruce Cotton (1571-1631), antiquary, collector of books and manuscripts, DNB. 370 Earl of Arundels: i.e. Thomas Earl of Arundel (1586-1646), collector of antiquities. 371 Kays College: i.e. Gonville and Caius College, Cambridge.

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103. WALLIS to OLDENBURG, 23 April/[3 May] 1666 [4] For Mr Henry Oldenburgh in the Old Palmall near St James's London.

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WALLIS to HENRY OLDENBURG Oxford, 23 April/[3 May] 1666 Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 17, 4 pp. (p. 3 blank) (our source). Postmark on p. 4: 'AP/25'.—printed: OLDENBURG, Correspondence III, 97-9.

Oxon. April. 23. 1666. Sir,

I had sent you a fortnight since, a transcript of Petrus Blesensis his Tract, de Utilitate afflictionis372, out of a good Copy in the Bodlyan library, (compared with two imperfect copies in the Library of Merton College;) But that I was to wayt till I should have leave so to do. For though the Vice-chancellor373 had before told mee, hee was not against it, & that he beleeved that the rest of the Curators would not be against it neither, (in confidence whereof I took the pains to transcribe it;) Yet I did not think fit to send it away, till the Curators should have a meeting (which was not till last week) to whom I might propose it in common (as then I did) & have their leave for it; which they readyly yeelded. Which having obtained, I should presently have sent it to you, but that I have lately 9 of | the del. ed.\ a 16 in add. 17 & (1) had (2) have 18 sent (1) y breaks off (2) it 18 have (lately add] lighted upon another (perfect add] copy 372

de Utilitate afflictionis: i.e. the tract De utilitate tribulationis. Vice-chancellor: i.e. Robert Saye (d. 1691), provost of Oriel College, Oxford (from 1653 onwards), vice-chancellor of the University of Oxford 1664-6. 373

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lighted upon another perfect copy of the same tract, (of which the Printed Catalogue takes no notice) in the Bodleyan library; which though not so good as the former, I have yet compared with my transcript, & noted in the margin, the differences from what I had; which though for the most part they seem worse than what I had before, yet it hath served to confirm some of my conjectures for the restoring some faults in the former; & to supply two or three omissions. Which comparing cost mee wellnigh as much time as transcribing the first copy. It hath rendred my transcript the more blotted; but, I hope, not the less acceptable. I have found out allso (though the Catalog failed mee in it) a Treatise intituld Tabula Dictaminis (the same I suppose which is their Summa Dictaminis) which is about half as much again as this de Utilitate tribulationis; and I have allready transcribed good part of it; It proves to be a Treatise De modo dictandi (sen conscribendi) epistolas. Which though it do not (in our copy) expressely bear his name (but is Anonymus) yet is manifest inough, by divers circumstances, to be his. And, that I may at once give you an account of what with tumbling over a multitude of manuscripts I have found in the Bodleyan library; There is In one Volume (amongst other Treatises of other Authors) His Epistolae, in number about 163. In a second, De vita Job. In a third, Epistolae 173. De transfiguratione domini. De vita Job. De conversione Pauli. In a fourth, Epistolae 108.

1 (of . . . notice) add. 3 with (1) it with (2) my 7 comparing add. 8 as (1) the fir breaks off (2) transcribing the first copy. It hath rendred (a) the (b) my 13 I add. 13 it |allready del.\ ; It 19 Treases corr. ed. 24 Epistolae 173. add.

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103. WALLIS to OLDENBURG, 23 April/[3 May] 1666 Tabula Dictaminis. In a fifth, Epistolae 143. De vita Job. De transfiguratione domini. De Utilitate Tribulationis. In a sixth, De vita Job. In a seventh, De Utilitate Tribulationum Contra Judaeos. These Tracts are all printed, except that De Ulilitate Tribulationis, and Tabula Dictaminis. Unlesse, amongst his Epistles there may be some diverse from those printed: Which would be a long work to compare; those in our books, not being placed in the same order, either one with another, [2] or with the Printed Books. | Beside these there is in the Library of Baliol College, another (not printed) which bears no other title (that I know of) but Quales sunt] about as big as both those in the Bodleyan library. In Merton College there are Two imperfect copies of De Util. Tribulationis. His Epistola aurea de silentio servando (if at lest it be his) And contra usum legum both very short things. But there is beside these, a great Volume, De Remediis Conversorum; divided into two parts, each containing about six or seven bookes; The former shewing the Evills & dangers they be subject to, The latter the remedyes of them. I do not find this amongst the printed Works; but being so large as it is; it will be worth while that they (before they be at the pains or charge of transcribing) to be well advised whither this be not in print under the name of some other Author; & be not indeed (or at lest have not been

13 be add. 18 sunt] (1) welln breaks off (2) about 20 College (1) two (2) there 26 parts, (1) in (2) jeacli add] containing

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104. WALLIS to BOYLE, 25 April/[5 May] 1666

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thought to bee) a work of his. This being so large, I dare not undertake to transcribe it myself; & truely I know not where to find one to undertake it, that can read these old hands. This Letter I send you by the Post this morning, to give you notice, that I purpose to send you, by Moor374 in the afternoon, my transcript of that De Util. Trib.; at whose warehouse at the Saracens head without New-gate, you may find it directed to you. Which I do, lest the packet should ly there for want of calling for. I am Sir

Your affectionate friend & servant John Wallis.

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These For my very worthy Friend Mr Henry Oldenburg at his house in the Old Pell-mell, near St James's London

[4]

104.

WALLIS to ROBERT BOYLE Oxford, 25 April/[5 May] 1666 Transmission:

wl Copy of missing letter sent with corrections and additions in Wallis's and Oldenburg's hand: LONDON Royal Society Early Letters Wl, No. 18, 25 pp. (our source). At top of p. 1 in Oldenburg's hand: 'Dr Wallis's Discourse of The Ebbing and flowing of the Sea; as it was first written to Mr Boyle from Oxon April 25. 1666. and afterwards (1) read (2) communicated to the R. Society; where it was read May 16. 1666.', and beneath date: 'Entered R.B. 3: 123 Phil. Trans: 16.'—printed: BOYLE, Correspondence III, 141-56. w2 Copy of w1: LONDON Royal Society Register Book Original 3, pp. 123-43. w 3 Copy of w 2 : LONDON Royal Society Register Book Copy 2, pp. 251-73. E1 First edition of w1: Philosophical Transactions No. 16 (6 August 1666), 263-81 ('An Essay Of Dr John Wallis, exhibiting his Hypothesis about the Flux and Reflux of the 2 truely (1) not (2) I 374

Moor: i.e. the Oxford carrier. 200

104. WALLIS to BOYLE, 25 April/[5 May] 1666 Sea'). w 4 Latin translation of E1: LONDON Royal Society MS. 368, No. 1, pp. 1-29. At top left: 'read (1) Apr: 2. 68. (2) May 9. |or 16. add.\ 1666. English Entered R.B. 3. 123. Trans No. 16.' E2 First edition in Latin (based on a translation of El different from w4): POTTER, Tlv@a~[6pa<; nfTefiipvxos, sive Theses Quadragesimales in Scholis Oxonii publicis pro forma habitae ... Quibus ... adjecta est Cl. D. D. Wallisii ... Dissertatio Epist. ad D. Boyle De Fluxu & refluxu Maris, Leiden 1684, 126-71. E3 First edition of w 4 : WALLIS, Opera mathematica II, 739-49 (De aestu maris, hypothesis nova. Epistola prima). Written in the form of a letter to Boyle, Wallis's hypothesis of the tides clearly came soon into the hands of Oldenburg who produced it at the meeting of the Royal Society on 9 May, having already sought permission to publish it in his letter to Wallis of 5 May. In the meeting on 9 May it was ordered that the Society's clerk 'write a fair copy of this discourse' and 'make the schemes contained in it large, against the next meeting'. This copy is clearly w 1 ; BIRCH, History of the Royal Society II, 88. The hypothesis was discussed for the first time at the meeting of the Society on 16 May and objections brought forth by members were communicated by Oldenburg to Wallis on the following day. His reply, dated 19 May, was read at the meeting on 23 May where it produced further objections. These were likewise brought to Wallis's attention by Oldenburg in his letters of 29 and 31 May. Wallis in turn replied to these in his letter to Oldenburg of 2 June and was later able to participate personally in discussions, by being present at the meeting of 20 June. Wallis sent a slightly extended Latin translation (w 4 ) of this letter to Oldenburg in February 1667/8. At the same time, he also sent him translations of two others: that of 18/[28].VII. 1666, which was published as an appendix to the hypothesis in the same issue of Philosophical Transactions, and that of 24.VII/[3.VIII]. 1666, which was likewise printed in Philosophical Transactions No. 16. See OLDENBURGWALLIS 24.XII.1667/[3.I.1668], WALLIS-OLDENBURG l/[ll].II.1667/8, and WALLISOLDENBURG 8/[18].II. 1667/8.

Oxon April 25. 1666. For the Honourable Robert Boyle Esquire. Sir

You were earnest with me, when you last went from hence, that I would put in writing somewhat of that, which at divers times, these three or four years last past, I have been discoursing with your-self and others concerning the Common Center of Gravity of the Earth and Moon, in order to salving the Phaenomena as well of the Sea's Ebbing and Flowingas of some perplexities in Astronomical Observations of the Places of the Celestial Bodies. 1-2 Oxon . . . Esquire, del. for publication

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How much the World and the great Bodies therein, are managed according to the Laws of Motion, and Statick Principles, and with how much more of clearnesse and satisfaction, many of the more abstruse Phaenomena have been salved on such Principles, within this last century of Years, than formerly they had been; I need not discourse to you, who are so well versed in it. For, since that Galilaeo, and (after him) Torricellio, and others, have applyed Mechanick Principles to the salving of Philosophical Difficulties; Natural Philosophy is well known to have been rendered more intelligible, and to have made a much greater progresse in lesse than an hundred years, than before for many ages. The Sea's Ebbing and Flowing, hath so great a connexion with the Moons motion, that in a manner all Philosophers (what ever other Causes they have joyned with it) have attributed much of its cause to the Moon; which either by some occult quality, or particular influence, which it hath on moyst Bodies, or by some Magnetick virtue drawing the water towards it, (which should therefore make the Water there highest, where| the Moon [2] is vertical) or by its gravity and pressure downwards upon the Terraqueous Globe (which should make it lowest where the Moon is vertical) or by what ever other means (according to the several Conjectures of inquisitive persons,) hath so great an influence on, or at least a connexion with, the Sea's Flux and Reflux, that it would seem very unreasonable, to seclude the consideration of the Moons motion from that of the Sea: The Periods of Tides (to say nothing of the greatnesse of them near the New-moon and Full-moon) so constantly waiting on the Moon's motion; that it may be well presumed, that either the one is governed by the other, or at least both from some common cause. But the first, that I know of, who took in, the consideration of the Earth's motion, (Diurnal and Annual) was Galilaeo; who in his Systeme of the World, hath375 a particular discourse on this Subject: Which, from the first time that I ever read it, seemed to me so very rational, that I could never be of other opinion, but that the true Account of this great Phaenomenon was to be referred to the Earths motion, as the Principal cause of it: Yet that of the Moon (for the reasons above mentioned)

33 it: (1) (but (2) Yet corr. Oldenburg 3r5

hath: i.e. the 'dialogo quarto' of GALILEO, Dialogo sopra i due massimi sistemi del mondo, Florence 1632, 409-58; Opere VII, 442-89. 202

104. WALLIS to BOYLE, 25 April/[5 May] 1666 not to be excluded, as to the determining the Periods of Tides, and other circumstances concerning them. And though it be manifest enough, that Galilaeo, as to some particulars, was mistaken, in the account which there he gives of it; yet that may be very well allowed, without any blemish to so deserving a person, or prejudice to the main Hypothesis: For that Discourse is to be looked upon onely as an Essay of the general Hypothesis; which as to particulars was to be afterwards adjusted, from a good General History of Tides; which it's manifest enough that he had not; and which is in a great measure yet wanting. For were the matter of Fact well [3] agreed on, it is not likely | that several Hypotheses should so far differ, as that one should make the Water then and there at the Highest, where and when the other makes it at the Lowest; as when the Moon is Vertical to the place. And what I say of Galilaeo, I must in like manner desire to be understood of what I am now ready to say to you. For I do not professe to be so well skilled in the History of Tides, as that I will undertake presently to accommodate my general Hypothesis to the particular cases; or that I will indeed undertake for the certainty of it, but onely as an Essay propose it to further consideration; to stand or fall, as it shall be found to answer matter of Fact. And truly had not your importunity (which is to me a great Command) required me to do it, I should not so easily have drawn up any thing about it, till I had first satified my self, how well the Hypothesis would answer Observation: Having for divers years neglected to do it, waiting a time when I might be at leisure throughly to prosecute this designe. But there be two reasons, by which you have prevailed with me, at lest to do something. First, because it is the common Fate of the English, that out of a modesty, they forbear to publish their Discoveries till prosecuted to some good degree of certainty and perfection; yet are not so wary, but that they discourse of them freely enough to one another, and even to Strangers upon occasion; whereby others, who are more hasty and venturous, coming to hear of the notion, presently publish something of it, and would be reputed thereupon, to be the first Inventers thereof: though

23 neglected (1) it (2) to 31 and (1) dextrous, (2) venturous, corr. Wallis 203

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even that little, which they can then say of it, be perhaps much lesse, and more imperfect, than what the true Authors could have published long before, and what they had really made knowne (publikely enough, though] not in print) to many others. As is well known amongst us, as to [4] the businesse of the Lymphatic Vessells in Anatomy 376 , the Injection of Liquors into the veins of Living animals377; the exhibiting of a streight line equal to a Crooked378; the Spot in Jupiter379, whence his motion about his own Axis may be demonstrated; and many other the like considerable Inventions. The other Reason (which, with me, is more really of weight, though even the former be not contemptible) is, because, as I have been already for at least three or four years last past diverted from prosecuting the inquiry or perfecting the Hypothesis, as I had thoughts to doe; so I do not know, but like Emergences may divert me longer; and whether I shall ever so do it, as to bring it to perfection, I cannot determine. And therefore, if as to my self any thing should humanitus accidere380; yet possibly the notion may prove worth the preserving to be prosecuted by others; if I do it not. And therefore I shall, at least to your self, give some general account of my present imperfect and undigested thoughts. I consider therefore, that in the Tides, or the Flux and Reflux of the Sea, besides Extraordinary Extravagancies, or Irregularities, whence great Inundations or strangly high Tides do follow, (which yet perhaps may prove not to be so meerly accidental as they have been thought to be, but might from the regular Laws of Motion, if well considered, be both well accounted for, and even foretold;) There are these three notorious Observations made of the Reciprocation of Tides. First, the Diurnal Reciprocation; whereby Twice in somewhat more than 24. hours,

376

Lymphatic Vessells in Anatomy: i.e. the discovery of the lymphatic vessels by the physician George Joyliffe (1621-58), DNB. 377 Injection ... animals: i.e. the transfusion experiments of Thomas Willis (1621-75) and of the Oxford anatomist and physician Richard Lower (1631-91), elected fellow of the Royal Society in 1667, DNB. 378 exhibiting ... Crooked: i.e. the rectification of the semi-cubic parabola, independently discovered by William Neile and Hendrik van Heuraet. See WALLIS-DIGBY 24.VIII/[3.IX].1660. 379 Spot in Jupiter: i.e. the discovery of Jupiter's Great Red Spot by Robert Hooke in 1664. 380 humanitus accidere: ClCERO, Orationes Philippicae 1, 4, 10.

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104. WALLIS to BOYLE, 25 April/[5 May] 1666 we have a Floud and an Ebbe; or a High-water and Low-water. Secondly, the Menstrual; whereby in one Synodical period of the Moon, suppose [5] from| Full-moon to Full-moon, the Time of those Diurnal Vicissitudes doth move round through the whole compasse of the Nvx&r\\j,epov or Natural day of twenty four hours: As for instance, if at the Full-moon the full sea be at such or such a place just at Noon, it shall be the next day (at the same place) somewhat before One of the clock; the day following, between One and Two; and so onward, till at the Newmoon it shall be at midnight; (the other Tide, which in the Full-moon was at midnight, now at the New-moon coming to be at noon;) And so forward, till at the next full moon, the Full Sea shall (at the same place) come to be at Noon again: Again, that of the Spring-tides and Neap-tides (as they are called;) about the Full-moon and New-moon the Tides are at the Highest, at the Quadratures the Tides are at the Lowest: And at the times intermediate, proportionably. Thirdly, the Annual; whereby it is observed, that at sometimes of the year, the Spring-Tides are yet much higher than the Spring-tides at other times of the year: which Times are usualy taken to be, at the Spring and Autumne; or the two Aequinoxes; but I have reason to beleeve (as well from my own Observations; for many years, as of others who have been much concerned to heed it; whereoff more will be said by and by;) that we should rather assigne the beginnings of February and November, than the two Aequinoxes. For, at least all along the Coast of Rumney Marsh, in Kent, where the Sea is kept out by high Earth walls, that it do not overflow the Levell; the Inhabitants do always apprehend their Sheep most in danger of drowning (by the Sea's breaking the Wall, or coming over it) at Candlemasse stream, and Allhallow stream (as they use to call them,) that is, at those Springtides which happen about those times; and accordingly when they expect [6] these Spring-tides (especially,! if from the Tides foregoing they conjecture them likely to be very high; or that the Wind be high, and set so as to favour the Tide;) they be carefull to drive their sheep to Banks, or higher grounds: But all the rest of the year, reckon themselves pretty secure. Now in order to the giving account of these three Periods, accord-

12 again: (1) Secondly, (2) Again, corr. Oldenburg 13 called;) jtliat del] about 20 whereoff more will be said (1) below; (2) by and by; add. Oldenburg 23-32 For . . . secure, del. for publication

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ing to the Laws of Motion and Mechanick Principles; We shall first take for granted, what is now a days pretty commonly intertained by those, who treat of such matters; That a Body in motion is apt to continue its motion, and that in the same degree of celerity, unlesse kindred by some contrary Impediment; (like as a Body at rest, to continue so, unlesse by some sufficient mover, put into motion:) And accordingly (which dayly experience testifies) if on a Bord or Table, some loose incumbent weight, be for some time moved, and have thereby contracted an Impetus to motion at such a rate; if that Bord or Table chance, by some external obstacle, or otherwise, to be stopped or considerably retarded in its motion, the incumbent loose Body will shoot forward upon it: And contrary wise, in case that Bord or Table chance to be accelerated or put forward with a considerably greater speed than before, the loose incumbent Body, (not having yet obtained and equal Impetus with it) will be left behind, or seem to fly backward upon it. Or, (which is Galilaeo's instance381,) if a broad Vessel of Water, for some time evenly carried forward with the water in it, chance to meet with a Stop, or to slack its motion; the Water will dash forward and rise higher at the forepart of the Vessel: And, contrarywise, if the Vessel be suddenly put forward faster than before; the water will dash backwards, and rise at the hinder partj of the Vessel. So [7] that an Acceleration or Retardation of the Vessel, which carries it, will cause a rising of the Water in one part, and a falling in another: (which yet, by its own weight, will again be reduced to a Level as it was before.) And consequently, supposing the Sea to be but as a loose Body, carried about with the Earth, but not so united to it, as necessarily to receive the same degree of Impetus with it, as it's fixed parts do; The acceleration or retardation in the motion of this or that part of the Earth will cause (more or lesse, according to the proportion of it) such a dashing of the Water, or Rising at one part, with a Falling at another, as is that, which wee call the Flux and Reflux of the Sea.

35

Now this premised, We are next, with him, to suppose the Earth carried about with a double motion: The one Annual, as in BEC the great Orbe, in which the Center of the Earth B, is supposed to move about the Sun A. The other Diurnal, whereby the whole moves upon it's own Axis, and each point in its surface describes a Circle, as DEFG.

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381

Galilaeo 's instance: i.e. GALILEO, Dmlogo, 417-19; Opere VII, 450-1. 206

104. WALLIS to BOYLE, 25 April/[5 May] 1666

It is then manifest, that if we suppose, that the Earth, moved but by any one of these motions, and that regularly, (with an equal swiftnesse;) the water having once attained an equal Impetus thereunto, would still hold equal pace with it; there being no occasion, from the Quickening or [8] Slackening of the Earths motion, (in that part where the water| lyeth) for the water thereon either to be cast Forward or fall Backward, and thereby to accumulate on the other parts of the water: But the true motion of each part of the Earths surface being compounded of those two motions; the Annual and Diurnal; (the Annual in BEC being, as Galilaeo there supposeth382, about three times as fast as a diurnal motion in a great Circle, as DEF;) while a Point in the Earth's surface moves about it's Center B from G to D and E; and at the same time, it's Center B be carried forwards to (7; the true motion of that Point forwards is made up of both those motions; to wit, of B to C, and of G to E: but while G moves by D to E, E moves backward by F to G, contrary to the motion of B to C\ So that the true motion of E, is but the difference of BC, and EG: (for, beside the motion of B about the Center; G is also put forward as much as from G to E; and E put backward as much as from E to G:)

17 about add. Oldenburg 382

supposeth: i.e. GALILEO, Dialogo, 441; Opere VII, 473. 207

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so that the diurnal motion in that part of the Earth which is next the Sun, as EFG, doth abate the progresse of the Annual, (and most of all at F;) and in the other part which is from the Sun, as GDE, it doth increase it, (and most of all at D:} that is; in the day time there is abated, in the night time is added to the Annual motion, about as much as is GE the Earths Diameter. Which would afford us a Cause of two Tides in twenty four hours; the One upon the greatest Acceleration of motion, the Other upon it's greatest Retardation: And thus far Galilaeo's Discourse holds well enough. But then in this it comes short; that as it gives an Account of two Tides; so those two Tides are always to be at F and D- that is, at Noon and Midnight: whereas Experience tells us, that the Time of Tides, moves in a moneths Space through all the 24. hours. Of which he gives us noe| account. For [9] though he do take notice383 of a Menstrual Period; yet he doth it onely as to the Quantity of the Tides, greater or lesse; not as to the Time of the Tides: sooner or later. To help this, there is one (vid. Jo. Baptista Balianus^} who makes the Earth to be but a Secondary Planet; and to move, not directly about the Sun, but about the Moon, the Moon mean while moving about the Sun; in like manner as we Suppose the Earth to move about the Sun, and the Moon about it. But this, though it might furnish us with the foundation of a Menstrual Period of Accelerations and Retardations in the compound motion 17 [In left margin in Wallis's hand:] Vide Ricciol. Almagest. Nov. pag. (—} [deleted by Oldenburg and replaced by his own marginal note:] Vid. Riccioli Almagest, novum Tom. 1. lib. 4. cap. 10. n°. 111. pag. 216. 2.

9 Discourse (1) looks (2) holds corr. Oldenburg 17 this, \ ( 1 ) I am told of some body (2) there is one alt. Wallis] |(o) (whom I have not seen, nor do at present remember his name) (b) (vid.Jo. Baptista Balianus) alt. Oldenburg] who makes 383

take notice: i.e. GALILEO, Dialogo, 439f.; Opere VII, 471f. Balianus: i.e. Giovanni Batista Baliani (1586-1666), sometime commander of the fortress at Savona. His De motu naturali solidorum et liquidorum, Genua 1646, was directed against the system of Galileo. Wallis appears not to have had this text at hand, as he refers the reader instead to Giovanni Baptista RICCIOLI, Almagestum novum Astronomiam . . . promotam, Bologna 1651. 384

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104. WALLIS to BOYLE, 25 April/[5 May] 1666 of several parts of the Earths Surface; Yet I am not at all inclined to admit this as a true Hypothesis, for divers Reasons; which if not demonstrative, are yet so consonant to the general Systeme of the World, as that we have no good ground to disbeleive them. For 1. The Earth being undeniably the greater Body of the two (whereof there is no doubt to be made) it cannot be thought probable, that this should be carried about by the Moon, lesser than it self: The contrary being seen, not onely in the Sun, which is bigger than any of the Planets, which it carries about; but in Jupiter, bigger than any of his Satellites; and Saturne, bigger than his. 2ly. As the Sun by it's motion about it's own Axis, is with good reason judged to be the Physical cause of the Primary Planets moving about it; So there is the like reason to beleeve, that Jupiter and Saturne moving about their Axes, are the Physical cause of their Satellites moving about them, which motion of Jupiter hath been of late discovered, by the help [10] of a fixed Spot discerned in him; and we have reason to| beleeve the like of Saturne. Whether Venus and Mercury, (about whom no Satellites have been yet observed) be likewise so moved; we have not yet the like ground to determine: But wee have of Mars; from the Observations of Mr Hook, made in February & March last, and by him communicated385 to the Royal Society, & since printed386 in the Transactions, published

14 the add. Oldenburg 16 Whether \Mars, del] Venus 17 observed |to move del] ) 18-210, 3 determine: \ ( 1 ) But of Mars wee have (2) But wee have of Mars; from the Observations of Mr Hook, jmade add. in February fe March last, and (a) first communi breaks off (b) by him communicated to the Royal Society, & since printed in |(ao) the Philosophicall (bb) these (cc) the alt. Oldenburg] Transactions, published Apr. 2. 1666; consonant to the like observations of Jupiter, made by him in May. 1664, and since communicated to the |(aoa) Royal (bbb) same alt. Oldenburg] Society; & then published in print in |(aaaa) the Philosophical (bbbb) these (cccc) the alt. Oldenburg] Transactions, |(aaaaa) dated (bbbbb) of alt. Oldenburg] March 6. then next following. add. Wallis] Now 385

communicated: At the meeting of the Royal Society on 28 March 1666, Hooke presented a paper on his recent observation of several spots on Mars which had apparently changed their position. See BIRCH, History of the Royal Society II, 74-5. 386 printed: see [HOOKE], 'Some New observations about the Planet Mars, communicated since the Printing of the former sheets . . . ' , Philosophical Transactions No. 11 (2 April 1666), 198; see also [HOOKE], 'The Particulars of those Observations of the Planet Mars, formerly intimated to have been made at London in the Months of February and March A. 1665/6', Philosophical Transactions No. 14 (2 July 1666), 239-42. 209

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Apr. 2. 1666; consonant to the like observations of Jupiter, made387 by him in May. 1664, and since communicated to the same Society' & then published388 in print in the Transactions, of March 6. then next following. Now that the Earth hath such a motion about its own Axis (whereby it might be fitted to carry about the Moon) is evident by it's Diurnal motion. And it seems as evident that the Moon hath not; because of the same side of the Moon always turned towards us; which could not be, if the Moon carried the Earth about: Unlesse we should say, that it carries about the Earth in just the same Period, in which it turns upon its own Axis: Which is contrary to that of the Sun carrying about the Planets; the shortest of whose Periods, is yet longer than that of the Suns moving about its own Axis. And the like of Jupiter, shorter than the Period of any of his Satellites; if at least the Period of his conversion about his Axis, lately said to be observed, prove true. (Of Saturn we have not yet any Period assigned; but it's likely to be shorter, than that of his Satelles.) And therefore we have reason to beleeve, not that by the Moons motion about its Axis the Earth should be carried by a contemporary Period (whereby the same face of the Moon should be ever towards us;) but that by the Earths revolution about it's Axis in 24 hours, the Moon should be carried about it in about 29. dayes, without any motion on its own Axis: And accordingly, that the Secondary Planets about Jupiter and Saturne, are not (like their Principals) turned about their own Axes. And therefore I am not at all inclined to beleeve, that the Menstrual Period of the Tides with us, is to be salved by such an Hypothesis. In stead of this, that Surmise of mine, (for I dare| not yet, with [11] confidence give it any better name,) of what I have often spoken to you heretofore (and which hath occasioned this present account which I am now giving you,) is to this purpose.

15 his (1) Satellites.) (2) Satelles.) alt. Oldenburg 20 in jabout add. Oldenburg] 29. dayes, 24 salved (1) with (2) by 38r observations .. . made: At the meeting of the Royal Society on 4 January 1665, Hooke reported that he had observed a moving small spot in the largest of the three black belts of Jupiter on 19 May 1664. See BIRCH, History of the Royal Society II, 3. 388 published: see [HOOKE], 'A Spot in one of the Belts of Jupiter', Philosophical Transactions No. 1 (6 March 1664/5), 3.

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104. WALLIS to BOYLE, 25 April/[5 May] 1666 The Earth and Moon being knowne to be Bodies of so great connexion (whether by any Magnetick, of what other Tye, I will not determine; nor need I, as to this purpose;) as that the motion of the One follows that of the Other; (the Moon observing the Earth as the Center of it's periodick motion:) may well enough be looked upon as one Body, or rather one Aggregate of Bodies, which have one common center of Gravity; which Center (according to the known Laws of Staticks) is in a streight Line connecting their respective Centers, so divided as that its parts be in reciprocal proportion to the Gravities of the two Bodies. As for Example; Suppose the Magnitude (and therefore, probably, the Gravity,) of the Moon to be about an One and fourtieth part of that of the Earth; (and thereabouts Hevelius in his Selenography page 203. doth out of Tycho, estimate389 the proportion; and an exact certainty is not necessary to our present businesse:) And the distance of the Moon's Center from the Center of the Earth, to be about fifty six semidiameters of the Earth, (as thereabouts he doth there estimate it, in its middle distance; and we need not be now very accurate in determining the numbers; wherein Astronomers are not yet very well agreed.) The distance of the common Center of Gravity of the two Bodies, will be from that of the Earth, about a two and fourtieth part offiftysix Semidiameters; that is, about || or | of a Semidiameter; that is about ^ of a Semidiameter of the Earth, above [12] its Surface, in the Air, directly between the Earth and Moon.| Now supposing the Earth and Moon, joyntly as one Body, carried about by the Sun in the great Orb of the Annual motion; this motion is to be estimated, (according to the Laws of Staticks in other cases,) by the motion of the common Center of Gravity of both Bodies. For we use in Staticks, to estimate a Body, or Aggregate of Bodies, to be moved upwards, downwards, or otherwise, so much as its common Center of Gravity is so moved, howsoever the parts may change places amongst themselves. And accordingly, the line of the Annual motion, (whether Circular or Elliptical; of which I am not here to dispute,) will be described, not by the Center of the Earth (as we commonly estimate it, making the Earth

389

estimate: see HEVELIUS, Selenographia, 203.

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a Primary and the Moon a Secondary Planet,) nor by the Center of the Moon, (as they would do, who make the Moon the Primary and the Earth a Secondary Planet, against which we were before disputing:) But by the common Center of Gravity of both the Bodyes Earth and Moon, as one Aggregate. Now supposing ABCDE to be a part of the great Orb of the Annual motion, described by the common Center of Gravity in so long time as from a Full-Moon at A to the next New-moon at E; (which, though an Arch of a Circle or Ellipse, whose Center we suppose at a due distance below it; yet being but about ^ of the whole, may well enough be here represented by a streight Line:) the Center of the Earth at T, and that of the Moon at L, must each of them (supposing their common Center of Gravity to keep the Line AE) be supposed to describe a Periphery about that common Center, as the Moonj describes her Line of Menstrual [13] motion. (Of which I have (in the Scheme) onely drawn that of the Earth; as being sufficient to our present purpose; parallel to which if need be,

we may suppose one described by the Moon; whose distance is also to be supposed much greater from T. than in the figure is Expressed, or was 6 [In left margin in Oldenburg's hand:] (1) Fig. 2. (2) See Fig. 2: and 3.

4 both add. 12 at (1) S> (2) L, must each of them (a) be (6) (supposing 17 by (1) S) ; (2) the Moon; aft. Oldenburg

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104. WALLIS to BOYLE, 25 April/[5 May] 1666 necessary to expresse.) And in like manner EFGHI, from that New-moon at E, to the next Full-moon at /. From A to E (from Full-moon to New-moon,) T moves (in its own Epicycle) upwards from the Sun: And from E to /, (from New-moon to Fullmoon) it moves downwards, toward the Sun. Again, from C to G, (from last quarter to the following first quarter) it moves forwards, [14] according to the Annual motion; But from G forward to C, (from the first| Quarter to the ensuing last Quarter,) it moves contrary to the Annual motion. It is manifest therefore, according to this Hypothesis, that from Last quarter to First quarter (from C to G, while T is above the line of the Annual motion) its Menstrual motion in its Epicycle adds somewhat of Acceleration to the Annual motion; and most of all at E, the new-moon: And from the first to the last quarter (from G forward to (7, while T is below the line of the Annual motion,) it abates of the Annual motion; and most of all at I, or A the Full-moon. 3 [In left margin in Oldenburg's hand, deleted:] Fig. 2.

1 Figure, deleted:

6 moves \ ( 1 ) Backwards, (2) forwards, corr. Wallis\ |(o) contrary |(6) according corr. Wallis\ 8 moves (1) according (2) contrary corr. Wallis 11 quarter add. Oldenburg 12 Epicycle (1) Adds (2) adds alt. Oldenburg 16 at I, | Oldenburg above line, deleted: Fig. 3. or A \ Oldenburg above line, deleted: fig. 2.1 the

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So that in pursuance of Galilaeo 's Notion, the Menstrual adding to or detracting from the Annual motion, should either leave behind, or cast forward, the loose waters incumbent on the Earth, (and thereby cause a Tide, or Accumulation of Waters;) and most of all at the Full-moon and New-moon, where those Accelerations or Retardations are greatest. Now this Menstrual motion, if nothing else were superadded to the Annual, would give us two Tides in a month, and no more; (the one upon the Acceleration, the other on the Retardation;) at New moon and Fulmoon; and two Ebbs, at the two Quarters; and in the Intervals, Rising and Falling water. But the Diurnal motion superadded, doth the same to| this Men- [15] strual, which Galilaeo supposeth390 it to do to that Annual; that is, doth Adde to, or Subtract from, the Menstrual Acceleration or Retardation; and so gives us Tyde upon Tyde. For in whatsoever part of its Epicycle, we suppose T to be; Yet, because, while by its Menstrual motion the Center moves in the Circle LTN; each point in its surface, by its diurnal motion moves in the Circle LMN: whatever effect (accelerative or tardative) the Menstrual would give, that effect by the Diurnal is increased in the parts LMN (or rather IMn. the Semicircle) and most of all at M: but diminished in the parts NOL (or rather nOl) and most of all at O. So that at M and 0, (that 15 [In left margin in Oldenburg's hand:] See Fig. 4. 14 us (1) Tydes upon Tydes (2) Tyde upon Tyde 18 LMN: | And del] whatever 19 effect add. Oldenburg 21 rather (1) nOl (2) nOl 390

supposeth: see GALILEO, D^alogo, 440-1; Opere VII, 471-2. 214

104. WALLIS to BOYLE, 25 April/[5 May] 1666

is when the Moon is in the Meridian below or above the Horizon,) we are to have the Diurnal Tide or High-water, occasioned by the greatest Acceleration or Retardation, which the Diurnal Arch gives to that of the Menstrual: which seems to be the true cause of the Daily Tides. And withall gives an account, not onely why it should be every day; but likewise, why at such a time of the day; and why this time should in a month run through the whole 24 hours; Viz. because the Moons coming to the Meridian above and below the Horizon, (or as the Seamen call it, the Moons Southing, and Northing,) doth so: As likewise of the Springtides and Neap-tides. For, when it so happens, that the Menstrual and diurnal Accelerations or Retardations, be coincident, (as at New-moons [16] and Full-1moons they are,) the Effect must needs be the greater. And although (which is not to be dissembled) this happen but to one of the two Tides; that is, the Night-tide at the New-moon (when both motions do most of all Accelerate,) and the Day-tide at Full-moon (when both do most Retard the Annual motion;) Yet, this tide being thus raised by two concurrent causes; though the next Tide have not the same cause also, the Impetus contracted will have influence upon the next Tide; Upon a like reason as a Pendulum let fall from a higher Arch, will (though there be no new cause to occasion it) make the Vibration on the other side (beyond the Perpendicular) to be also greater: Or, of water in a broad Vessel, if it be so jogged, as to be cast forward to a good hight above its Levell; will upon its recoyling, by its own gravity, (without any additional cause) mount so much the higher on the hinder part. 215

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But here also we are to take notice; that though all parts of the Earth by its diurnal motion do turn about its Axis, and describe parallel Circles; yet not equal Circles; but greater near the Aequinoctial, and lesser near the Poles, which may be a cause why the Tides in some parts may be much greater than in others. But this belongs to the particular considerations; (of which we are not now giving an Account:) not to the general Hypothesis. Having thus endevoured to give an account of the Diurnal and Menstrual Periods of Tides; It remains that I endeavour the like as to the Annual. Of which there is, at least, thus much agreed, That, at some times of the year, the Tides are noted to be much higher, than at other times. But here I have a double task; First, to rectify the Observation; and then, to give an account of it. As to the First' It having been observed (grossely) that| those high [17] Tydes have used to happen about the Spring and Autumne; it hath been generally taken for granted (without any more nice observation) that the two Aequinoxes are the proper Times, to which these Annual high Tides are to be referred; And such causes sought for, as might best sute with such a Supposition. But it is now, the best part of twenty years, since I have had frequent occasions to converse with some Inhabitants of Rumney-marsh in Kent', where the Sea being kept out with great Earthen walls, that it do not at high water overflow the Levell; and the Inhabitants livelyhood depending most on grazing, or feeding Sheep; they are (as you may beleeve they have reason to be) very vigilant and observant, at what times they are most in danger of having their Lands drowned. And I find them generally agreed, by their constant Observations, (and Experience dearly bought,) that their times of Danger are about the beginning of February and of November: that is, at those Spring Tides which happen neer those times; to which the give the names of Candlemasse-stream and Allhallond-stream: And if they scape those Spring-tides, they apprehend themselves out of Danger for the rest of the year. And as for March and September (the two Aequinoxes) they are as little solicitous of them, as of any other part of the year. This I confesse, I much wondred at, when I first heard it; and suspected it to be but a mistake of him, that first told me, though he were 13 the (1) Observations (2) Observation 18 Times, (1) at (2) to

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104. WALLIS to BOYLE, 25 April/[5 May] 1666 indeed a person not likely so to be mistaken, in a thing wherein he was so much concerned: But I soon found, that it was not onely his, but a general observation of others too; both there, and elsewhere along the Sea-coast. And though they did not pretend to know any reason of it, (nor so much [18] as to enquire after it;)| yet none made doubt of it; but would rather laugh at any that should talk of March and September, as being the dangerous times. And since that time; I have my self very frequently observed (both at London and elsewhere, as I have had occasion) that in those months of February and November, (especially November} the Tides have run much higher, than at other times: Though I confesse, I have not been so diligent, to set down those Observations, as I should have done; Yet this I do particularly very well remember, that in November 1660. (the same year that His Majesty returned) having occasion to go by Coach from the Strand to Westminster, I found the water so high in the middle of Kingstreet, that it came up, not onely to the Boots, but into the Body of the Coach; and the Pallace-yard (all save a little place neer the West-End) overflowed; as likewise the Market-place; and many other places; and their Cellars generally filled up with water. And in November last, 1665, it may yet be very well remembred, what very high Tides there were, not onely on the Coasts of England, (where much hurt was done by it;) but much more in Holland, where by reason of those Inundations, many Villages and Towns were overflowed. And though I cannot so particularly name other years, yet I can very safely say, that I very often observed Tides strangly high about those times of the year. This Observation did for divers years cause me much to wonder, not onely because it is so contrary to the received opinion of the two Aequinoxes; but because I could not think of any thing Signal at those times of the year: as being neither the two Aequinoxes, nor the two Solstices nor [19] the Sun's Apogaeum and Perigaeum; (or| Earths Aphelium and Perihelium;) nor indeed, at contrary times of the year, which at least, would seem to be expected. From Alhallontide to Candlemasse being but three months; and from thence to Allhallontide again, nine months. At length it came into my mind, about four years since, that though there do not about these times happen any single Signal Accident, which might cast it on these times, yet there is a compound of two that may doe it: which is the Inequality of the Natural day (I mean that of 24. hours, 4 reason (1) for (2) of 10 dilligent, corr. 12 that (1) the King (2) His Majesty alt. Oldenburg 217

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104. WALLIS to BOYLE, 25 April/[5 May] 1666

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from noon to noon) arising at least from a double cause; either of which singly would cast it upon other times, but both joyntly on those. It's commonly thought, how unequal soever the length be of the Artificial days as contradistinguished to nights, yet that the Natural Day, reckoning from noon to noon, are all equal: But Astronomers know well, that even these dayes are unequal. For, this Natural Day is measured not onely by one intire conversion of the Aequinoctial, or 24 Aequinoctial hours, (which is indeed taken to be performed in equal times,) but increases by so much, as answers to that part of the Sun's (or Earths,) Annual motion as is performed in that time. For, when that part of the Aequinoctial, which (with the Sun) was at the Meridian yesterday at noon, is come thither again to day, it is not yet Noon (because the Sun is not now at the place where yesterday he was, but is gone forward about one degree, more or lesse.) But we must stay, till that place, where the Sun now is, comes to the Meridian, before it be now Noon. Now this Additament (above the 24 Aequinoctial hours, or intire conversion of the Aequinoctial) is upon a double] account unequal. First, [20] because the Sun, by reason of its Apogaeum and Perigaeum, doth not at all times of the year dispatch in one day an equal Arch of the Ecliptick; but greater Arches neer the Perigaeum, which is about the middle of December; and lesser near the Apogaeum, which is about the middle of June: As will appear sufficiently by the Tables of the Sun's Annual motion. Secondly, though the Sun should in the Ecliptick move alwayes at the same rate; Yet equal Arches of the Ecliptick do not in all parts of the Zodiack answer to equal Arches of the Aequinoctial, by which we are to estimate time: Because some parts of it, as about the two Solsticial Points, ly nearer to a parallel position to the Aequinoctial, than others, as those about the two Aequinoctial points, where the Ecliptick and Aequinoctial do intersect: whereupon an Arch of the Ecliptick, near the Solsticial points, answers to a greater Arch of the Aequinoctial, than an Arch equal thereunto, near the Aequinoctial points: As doth sufficiently appear by the Tables of the Suns right Ascension. According to the first of these causes, we should have the longest natural days in December, and the shortest in June, which if it did operate alone, would give us at those times two Annual High-waters. 11 For, (1) th breaks off (2) when 23 motion. (1) \\ Secondly, though the Sun should in (2) Secondly, though the Sun should in alt. Oldenburg 218

104. WALLIS to BOYLE, 25 April/[5 May] 1666 According to the second cause, if operating singly, we should have the longest dayes at the two Solstices in June and December, and the shortest at the two Aequinoxes in March and September; which would at those times give occasion to four Annual High-waters. But the true Inequality of the Natural Days, arising from a complication of those two causes, sometimes crossing and sometimes promoting [21] each other: though we should | find some increases or decreases of the Natural dayes at all those Seasons answerable to the respective causes (and perhaps of Tides proportionably thereunto:) yet the longest and shortest natural dayes absolutely of the whole year (arising from this complication of Causes) are about those times of Allhallontide and Candlemas (or not far from them) about which those Annual High-Tides are found to be: As will appear by the Tables of Aequation of Natural days. And therefore I think, we may with very good reason cast this Annual Period upon that cause, or rather complication of causes. For, (as we before shewed in the Menstrual and Diurnal) there will, by this inequality of Natural dayes, arise a Physical Acceleration and Retardation of the Earth's Mean motion, and accordingly a casting of the Waters backward or forward; either of which, will cause an Accumulation or High-water. 'Tis true, that these longest and shortest dayes, do (according to the Tables, some at lest) fall rather before, than after Alhallondtide and Candlemas (to wit the ends of October and January;) but so do also (sometimes) those high Tides: And it is not yet so well agreed amongst Astronomers, what are all the Causes, (and in what degrees,) of the Inequality of Natural dayes; but that there be diversities among them, about the true time: And whether the introducing of this New Motion of the Earth in its Epicycle about this common Center of Gravity, ought not therein also to be accounted for, I will not now determine. Having already said enough, if not too much, for the explaining of this general Hypothesis, leaving the particularities of it to be adjusted according to the true measures of the motions; if the General Hypothesis be found fit to be admitted. [22] Yet this I must adde, (that I be not mistaken,) that| whereas I cast the time of the dayly Tides to be at all places, when the Moon is there in the Meridian; it must be understood of open Seas, where the water hath such free Scope for it's motions, as if the whole Globe of Earth were equally covered with water: Well knowing, that in Bayes and In-land6 promoting |of del] each. 11 of |of del] Allhallontide

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104. WALLIS to BOYLE, 25 April/[5 May] 1666

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Channells, the position of the Banks and other like causes must needs make the times to be much different from what we suppose in the open Sea's: And likewise that even in the Open Sea's, Islands, and Currents, Gulfs and Shallows, may have some influence, though not comparable to that of Bays and Channells. And moreover, though I think that Seamen do commonly reckon the time of High-water in the Open Sea's, to be then, when the Moon is there in the Meridian (as this Hypothesis would cast it:) Yet I do not take my self to be so well furnished with a History of Tides, as to assure my self of it; much lesse to accommodate it to particular places and cases. Having thus dispatched the main of what I had to say concerning the Sea's Ebbing and Flowing: Had I not been already too tedious (this having run out into much greater length, than I thought it would have done:) I should now proceed to give a further reason, why I do introduce this consideration of the Common-Center of Gravity in reference to Astronomical Accounts. For indeed, that which may possibly seem at first to be an Objection against it, is with me one reason for it. It may be thought perhaps, that if the Earth should thus describe an Epicycle about the common Center of Gravity, it would (by this it's change of place) disturbe the Celestial motions; and make the apparent places of the Planets, especially some of them; different from what they would otherwise be. For though so small a removal of the Earth, as the Epicycle \ would cause (especially if its Semidiameter should not be above [23] l| of the Earths Semidiameter,) would scarce be sensible (if at all) to the remoter Planets; yet as to the nearer it might. Now, though what Galilaeo answers to a like Objection391 in his Hypothesis' (that its possible there may be some small difference, which Astronomers have not yet been so accurate, as to observe) might here perhaps serve the turne; Yet my answer is much otherwise; to wit, that such difference hath been observed, and hath very much puzzeled Astronomers to give an account of. About which you will find Mr. Horrocks, (in some

12 (this . . . done:) del. for publication 17 one main del.\ reason 21 especially (1) of the Moon, (2) some of them; alt. Oldenburg 25 the (1) Moon (2) nearer, alt. Oldenburg 29 turne (1) . \\ Yet my (2) ; Yet my alt. Oldenburg 391

answers to a like Objection: see GALILEO, Dialogo, 448-9; Opere VII, 480. 220

104. WALLIS to BOYLE, 25 April/[5 May] 1666 of his Letters392, whereof I did formerly, upon the Command of the Royal Society, make an Extract) was very much perplexed; and was seen, for want of other releif, to have recourse to somewhat like Kepler's amicable Fibres393, which did according to the several positions of the Moon, accelerate or retard the Moon's motion; which amicable Fibres he had no affection to at all (as there appears) if he could any other wayes give account of those little inequalities; and would much rather (I doubt not) have imbraced this Notion of the Common Center of Gravity, to Salve the Phaenomenon, had it come to his minde, or been suggested to him. And you find, that other Astronomers have been seen to bring in (some upon one Supposition, some upon another,) some kind of Menstrual Aequation, to solve the inequalities of the Moon's motion, according to her Synodical Revolution, or different Aspects (of New-moon, Full-Moon, &c.) beside what concerns her own Periodical motion. For which, this consideration of the common Center of Gravity of the Earth and Moon, is so proper a remedy, (Especially if it shall be found [24] precisely to answer those| Phaenomena, which I have not yet Examined, but am very apt to beleive,) that it is so far from being, with me, an Objection against it, that it is one of the reasons, which make me inclinable to introduce it. I must, before I leave this, adde one Consideration more, (which when I have done, I shall save you of further trouble for this time;) That if we shall upon these Considerations, thinke it reasonable, thus to consider the Common Center of Gravity of the Earth and Moon; it may as well be thought reasonable, that a like Consideration should be had of Jupiter and his four Satellites, which according to the Complication of their Several motions, will somewhat change the position of Jupiter, as to that common center of Gravity of all these Bodyes; which yet, because of their

2 was (1) fain, (2) seen, alt. Oldenburg 4 to (1) these (2) the 19 the |main del. reasons 21-22 (which . . . time;) del. for publication 392 in some of his Letters: i.e. Horrox's letters to William Crabtree, which were later published by Wallis in his edition of Horrox's Opera posthuma, Oxford 1673. See WALLIS-OLDENBURG 6/[16].IV.1664 and WALLIS-OLDENBURG 21.IX/[1.X].1664. 393 Kepler's amicable Fibres: see KEPLER, Astronomia nova, Prague 1609, 176-7 (Gesammelte Werke III, 246).

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Smallnesse, may chance to be so little, as that, at this distance, the change of his apparent place may not be discernable. And what is said of Jupiter, is in the like manner to be understood of Saturne and his Satelles, discovered394 by Hugenius: For all these Satellites are to their Principalls, as so many Moons to the Earth. And I do very well remember, in the Letters forecited, Mr. Horrocks expresseth some such little inequalities in Saturne's motion, of which he could not imagine what account to give, as if (to use his Expressions) this crabbed old Saturn had despised his Youth. Which, for ought I know, might well enough have been accounted for, if at that time the Satelles of Saturne had been discovered, and that Mr. Horrocks had thought of such a motion as the Common Center of Gravity of Saturn and his Companion, to be considerable, as to the guiding of his motion. You have now, in obedience to your Commands, an account of my [25] thoughts, as to this matter, though yet immature and unpolished: What use you will please to make of them, I shall leave to your Prudence. Desiring onely, that before you part with them, you will at lest secure me a Copy; this rude draught being indeed all I have of it; and I was loath to keep it so long by me as to transcribe it, having at present scarce so much time as to read it over before it goes. Which as it ows its birth to your desires, so it is to be disposed of at your Commands, as is Sir

Your honours Very humble Servant John Wallis

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3 his (1) Satellites (2) Satelles, |lately del.\ discovered alt. Oldenburg 7 not add. Oldenburg 10 the (1) Satellites (2) Satelles 12 his (1) Companions (2) Companion 16-25 Prudence, etc. alt. Oldenburg 394

discovered: see HUYGENS-WALLIS [3J/13.VI.1655 and HUYGENS-CONSTANTIJN HUYGENS [27.XJ/6.XI.1655; HUYGENS, CEuvres completes I, 362-3. 222

105. OLDENBURG to WALLIS, 5/[15] May 1666

105.

HENRY OLDENBURG to WALLIS 5/[15] May 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 7/[17].V.1666. In this letter Oldenburg sought permission to publish Wallis's hypothesis of the tides, as contained in WALLIS-BOYLE 25.IV/[5.V].1666. Furthermore he requested that Wallis further illustrate the question of the earth's moving in an epicycle.

106.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] May 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 19, 4 pp. (our source). Pages 2 and 3 contain the letter itself, p. 1 the address, p. 4 originally remained blank. The figures in the middle of p. 3 have been cut out by Oldenburg and pasted in w1. Above the cut Oldenburg has noted: 'The figures here cut out, and inserted in the letter book.' At top right: 'entered LB. 1. 308', and on p. 4 in Oldenburg's hand: 'Extract of (1) Dr Wallis's (2) a letter of Dr Wallis to M. Oldenburg, relating to his Theory concerning the Flux and reflux of the sea.' Postmark on p. 1: 'MA/9'.—printed: OLDENBURG, Correspondence III, 107-9. w1 Copy of letter sent with original figures inserted: LONDON Royal Society Letter Book Original 1, pp. 308-9 (figures pasted on p. 308) (our source for the figures). w 2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 264-5. Reply to: OLDENBURG-WALLIS 5/[15].V.1666.

Oxford. May 7. 1666.

[3]

Sir,

Yours395 of May 5. I received this morning. Your opinion of what I sent396, 395

Yours: i.e. OLDENBURG-WALLIS 5/[15].V.1666. what I sent: i.e. WALLIS-BOYLE 25!V/[5.V].1666, containing Wallis's hypothesis of the tides. This was published as 'An Essay of Dr. John Wallis, exhibiting his Hypothesis about the Flux and Reflux of the Sea' in Philosophical Transactions No. 16 (6 August 1666), 263-81. 396

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I fear may proceed as much out of affection as well as out of Judgement. As to the publishing of it I am not averse from it, if first it be approved by my Lord Brounker in particular, (whom I take for a good competent judge, if he have time to consider it,) and the Society. And then, if you think fit to do it for mee in your Transactions; you save mee the labour. Onely I must then beg some number of copies to dispose of to friends who will expect them from mee. For what you mention of the Earth making an Epicycle, if it do not appear sufficiently in the former papers, I think it will appear from these schemes. Supposing M the common Center of

gravity (of the Earth & moon as one Aggregate) to keep the line of the Ecliptick or Eccentrick Z£l; while the moon, in its menstruall motion, moves from A the full moon (when the Earth is between it & the Sun) by B, (7, D, to E the New moon (when the moon is between the Earth & the Sun,) the Earth must on the other side, move from a, by /?, 7, <5, to e; (else M, the common center of gravity, could not be, as it must be, in a line between the two bodies, whose common center it is.) And so, while the moon moves from the New at E (for we suppose the Sun to ly below Z£l & to be center of that circle of which Z£l is an arch, & near that Center) by F, G, H, to A the Full: The Earth must move from e, by I proceed (1) partly (2) as much 3 particular, (1) & (2) (whom 4 you (1) pleas breaks off (2) think 8 papers |about Tydes add. Oldenburg\ , I 9 appear |sufficiently del.\ from II motion, add. 16 two add. 18 of that |that del. ed. circle 19-225, 21 must move . . . Wallis. written at 90° to the left of first part of letter

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106. WALLIS to OLDENBURG, 7/[17] May 1666 [2] C, ?7, 0, to a. The difference from the former hypothesis is but this, that

supposeth the Earth allways in M; & this supposeth it a little beyond; & that M is not the place either of Earth or moon, but the common center of both; & therefore to describe an Epicicle about M as well as the Moon doth: like as in a balance, as the one end moves down the contrary moves upward on the same center. Or rather (in stead of Libra) to take the Statera, whose one Arm is 40 times as long as the other, (supposing the magnitude or weight of the Earth to be 40 times as much as that of the moon,) The two ends wil describe concentrick circles, but not Equall; the one as A B C D E F G H A, the other asa/3j6e^0a. And I make the Earth, notwithstanding to be the Primary Planet (& the moon the secondary) not as if the Earth made no Epicicle (as hath been supposed) but because it makes a less Epicicle than that of the moon; &, moreover, turns on its own Axis, (which the moon doth not,) & thereby gives motion to the moon; though, in the mean time, it be by the moons weight, thrust somewhat from its place. If you think fit, you may sometime move the Society about publishing Mr Horrocks's papers,397 which I did heretofore digest in order to it. My service to our friends as there shall be occasion, from your friend & servant John Wallis. [1] For my worthy friend Mr Henry Oldenburgh, at his house in the Palmal, near St James's London.

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2 it add.

4 both; (1) but (2) & 9 moon,) (1) Each end (2) The 10 e 7 r\ corr. ed. 39r

Mr Horrocks's papers: i.e. astronomical papers of Jeremiah Horrox. Cf. Wallis's letters to Oldenburg of 6/[16].IV.1664, 30.IV/[10.V].1664, and 21.IX/[1.X].1664. 225

107. WALLIS to OLDENBURG, 12/[22] May 1666

107. WALLIS to HENRY OLDENBURG Oxford, 12/[22] May 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 20, 4 pp. (our source). At top middle of p. 1 in Oldenburg's hand: 'Read before the Society May 16. 1666.' Beneath this: 'entered LB. 1. 312.' At top right of p. 1 Oldenburg has noted: 'An Account of Dr Wallis, sent (1) from Oxford (2) by way of letter to M. Oldenburg about Thunder and Lightning (a) . (b) , and the Effects thereof.'—printed: OLDENBURG, Correspondence III, 122-5. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 312-7. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 368-75. E First edition: Philosophical Transactions No. 13 (14 June 1666), 222-6 ('A Relation of an Accident by Thunder and Lightning, at Oxford') (partly). The letter was read at the meeting of the Royal Society on 16 May 1666; BIRCH, History of the Royal Society II, 89.

Oxford. May. 12. 1666. Sir

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I should scarce have given you so soon the trouble of another letter, were it not for an accident which happened here on Thursday last, May. 10. I had that afternoon about 4 of the clock heard it Thunder at some distance. About 5 of the clock the Thunder coming nearer to us it began to rain, & soon after (the rain withall increasing) the Thunder grew very loud & frequent & with long rattling claps, (though not altogether so great as I have some other times heard;) & the lightening with flashes very bright (notwithstanding the clear day light) & very frequent (when at the fastest, scarce a full minute between one flash & another; many times not so much, but a second flash before the thunder of the former was heard:) The thunder, for the most part, began to be heard about 8 or 10 second minutes after the flash (as I observed for a great part of the time by my minute-watch;) but once or twice, I observed it to follow (in a manner) immediately upon it, as it were in the same moment; & the lightening extreme red & fiery. I do not use to be much apprehensive of thunder & lightening, but I was at this time (I know not well why) 1-5 The 10th in the afternoon about 4 of the clock I heard alt. Oldenburg 6 began (1) a (2) to 7 (the rain withall increasing) add. 226

107. WALLIS to OLDENBURG, 12/[22] May 1666 very apprehensive (more then ordinary) of mischief to be done by it; for it seemed to mee to be very low & near us (which made me so particular as to observe the distance of the flash by the noise) & very frequent, & bright (so that had it been by night as it was by day it would have been very terrible,) and (though I kept within doores) yet I sensibly discerned a stinking sulphureous smell in the air. About 7 of the clock it ended. Before which time I had news brought mee of a sad accident upon the water at Medly398 (about a mile or somewhat more distant from hence.) Two scholars of Wadham College, being alone in a boat (without a waterman,) having newly thrust off from shoar, at Medly, to come homewards, standing near the head of the boat, were presently with a stroke of thunder or lightening, both struck off out of the boat into the water: the one of them stark dead, in whom though presently taken out of the water (having been by relation scarce a minute in it) there was not discerned any appearance of life sense or motion: the other was stuck fast in the mudde (with his feet downwards & his upper parts above water) like a post, & not able [2] to help himself out; but, beside a present stonying &| numnesse, had no other hurt: but was for the present so disturbed in his senses as that he knew not how hee came there out of the boat, nor could remember either thunder or lightening that did effect it: & was very feeble & faint upon it: which (though presently putt into a warme bed) hee had not throughly recovered by the next night, (& whether since he have or no, I know not.) Others in another boat about 10 or 20 yards from these (as by their descriptions I estimate) felt a disturbance & shaking in their boat, & one of them had his chair stroke from under him & thrown upon him; but had

3 flash. |by add. Oldenburg\ the noise) & very (1) th(r) breaks off (2) frequent, 7 accident (1) at (2) upon 8 somewhat (1) d breaks off (2) more 9 in (1) (—} (2) a boat |(without a waterman,) add.\ having 11 standing near the head of the boat, add. 16 his (1) he breaks off (2) upper parts 16 & (1) it was some (2) not 18 present (1) d breaks off (2) so 21 which (1) hee (2) (though 23 Others in (1) a boat (2) another 24 fe one (1) other (2) of them had |his add. chair 398

Medly: i.e. Medley in the north-west of Oxford between Fiddler's Island and Port Meadow. 227

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no hurt. These immediately made up to the others, & (some leaping into the water to them) presently drew them either into the boat or on shoar, yet none of them saw these two fall into the water (not looking that way) but heard one of them cry out for help presently upon the stroke: & smelt a very strange stinking smell in the ayr; which, when I asked him (that told it mee) what kind of stink? hee sayd, like such a smell as is upon the striking of flints together. Hee that was dead, (when by putting into a warm bed, & rubbing, & putting strong waters into his mouth, &c, no life could be brought into him,) was early the next morning brought to town; where (amongst multitudes of others who came to see it) Dr Willis399, Dr Mellington400, Dr Lower401, & myself, with some others, went to view the corps. Where we found no wound at all in the skin. The face & neck swart & black, but not more then might be ordinary by the settling of the bloud. On the right side of the neck was a little blackish spot about an inch long, & about a quarter of an inch broad at the broadest; & was as if it had been seared with a hot iron; & (as I remember) one somewhat bigger on the left side of the neck, below the eare. Streight down the breast, but towards the left side of it, was a large place about threquarters of a foot in length, & about two inches in breadth, (in some places more, in some lesse,) which was burnt & hard, like leather burnt with the fire, of a deep blackish red colour, not much unlike the scorched skin of a roasted pig: And on the fore part of the left shoulder such another spot about as big as a shilling, (but that in the neck was blacker & seemed more seared.) From the top of the right shoulder, sloping downwards towards that place in his brest, was a narrow line of the like scorched skin; as if somewhat had come in there at the neck & had run down to the breast & there spread broader. The buttons of his doublet were most of them off; which somej [3] 4 stroke: (1) Hee that was killed (2) & 6 such (1) as (2) a 10 Dr Willis, (1) my breaks off (2) Dr Mellington, |Dr Lower, add] & 15 fe (1) not part (2) about 23 seemed (1) to (2) more 24 in his (1) p breaks off (2) brest, 26 come (1) in |there add] at the (2) in 399

Dr Willis: i.e. Thomas Willis (1621-75). See WALLis-BoYLE 10/[20].IX.1663. Dr Mellington: i.e. Sir Thomas Millington (1628-1704), physician, M.D. at Oxford in 1659. He later succeeded Thomas Willis as Sedleian professor of natural philosophy. DNB. 401 Dr Lower: i.e. Richard Lower (1631-91). See WALLIS-BOYLE 25.IV/[5.V].1666. 400

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107. WALLIS to OLDENBURG, 12/[22] May 1666 thought might have been torn off with the blast getting in at the neck & then bursting its way out. But I can say nothing peremptory of this because I could not upon the view satisfy myself whether they were thus torn off or might have been wanting before; the greatest presumption was (to mee) that (besides 4 or 5 buttons wanting towards the bottome of the breast) there were about half a dosen together clear off from the bottom of the Collar downwards, & I do not remember that the rest of the buttons did seeme to be near worn out; but allmost new. The Collar of his doublet just over the forepart of the left shoulder was quite broken asunder, cloth & stiffening, streight downward as if cutt or chopped asunder but with a blunt tool; onely the inward linnen or fustian lining of it was whole; by which, & by the view of the ragged edges, it seemed manifest to mee that it was by a stroke inwards (from without) not outwards from within. His hat was strangely torn, not just on the crown but on the side of the head & on the brim; on the side of it was a great hole more then to put-in ones fist through it; some part of it being quite gone or struck away, & from thence divers gashes every way as if torn or cutt with a dull tool, & some of those of a good length allmost quite to the edges of the brim; &, beside these, one or two gashes more which did not communicate with that hole in the side. This allso I judged to bee by a stroke inwards, not so much from the view of the edges of these gashes (from which there was scarce any judgement to be made either way) but because the lining was not torn; onely ript off from the edge of the hat (where it was sewed on) on that side where the hole was made. But his hat being not found upon his head, but at some distance from him, it did not appear against what part of the head that hole was made. Upon the rest of his clothes I do not know of any further effect, nor did wee smell any sulphureous

1 in add.

6 together add. 8 be add. 10 stiffening, |as add. and del. streight |downwaxd add.\ as if cutt (1) with a (2) or 14 of (1) it (2) the had corr. ed. 18 length add. 20 This allso (1) seemed (2) I judged 21 of these gashes add. 23 the edge of add. 24-26 But his hat . . . was made. add. on left margin at 90°

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107. WALLIS to OLDENBURG, 12/[22] May 1666

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sent about them, which might be partly because it was now a good while after the time, & partly by reason of their being presently drenched in the water into which he fell. The night following (that is, on Friday night) Dr Willis, Dr Millington, Dr Lower, & myself with some chirurgions (beside a multitude of others) were present at the opening of the head, to see if any thing could be there discerned; but there appeared no sign of contusion, the brain full & in good order, the nerves hole & sound, the vessels of it were pretty full of bloud, but nothing was by any of them discerned to be| at all amisse. But it was by candle light, & they had not time to make very nice observation of it (the body being to be buryed by & by) & the crowd of people was a further hinderance: But if any thing had been considerably out of order to the view, it would surely have been by some of them discovered. Some of them thought they discerned a small crack or fissure in the skul, & some who helld it while it was sawing off sayd they felt it jarring in their hand, & there seemed to the eye to bee something like it, but it was so small as that by candle light we could not agree it certainly so to bee; perhaps by day light that & others might have been discerned. Some of the hair on the right temples was manifestly sindged or burnt, & the lower part of that ear blacker then the parts about it, but soft, & it might be onely the settling of the bloud. The upper part of the left shoulder and that side of the neck were allso somewhat blacker than the rest of the body, but whether it were by the blow which broke the collar, (& scorched the round red spot thereupon,) or onely by settling of the bloud there, I cannot say. Yet I think it might very well bee, that both on the head, & on this side of the neck there might be a very great blow & a contusion upon it (& seemes to have been so, by the tearing of the hat, & breaking the collar, if not allso cracking of the skull,) & yet no sign of such contusion or bruise, because, dying so immediately, there was not time for the bloud to gather to the part & stagnate there,

1 sent add. 3 into which he fell add. 4 Dr Millington, (1) & (2) Dr 7 sound, (1) nor it was (2) the vessels of it were 10 nice (1) of (2) observation 14 off (1) pr breaks off (2) sayd 19 it, add. 21 shoulder (1) was (2) and that side of the neck were allso 24 of add.

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[4]

108. OLDENBURG to WALLIS, IT/[27] May 1666 (which in bruises is the cause of blackness,) & it was but as if such a blow had been given on a body newly dead; which doth not use to cause such a symptome of a bruise, after the bloud ceaseth to circulate. Having done with the head, they opened the breast; & found that burning to reach quite through the skin, which was in those scorched places hard & horny, & shrunk up, so as it was not so thick as the soft skin about it; but no appearance of any thing deeper than the skin, the muscles not at all disordered or discoloured, (perhaps, upon the reason that was but now sayd, of the head neck & shoulder.) Having then taken off the sternon, the lungs & heart appeared all well, & well coloured without any disorder. This is the summe of what was observed; Onely that the whole body, was, by night, very much swelled more than in the morning; & smelt very strong & offensively: which might bee by the hotness of the weather, & by the heat of the place occasioned by the multitude of people: For it was a little close room, & seldome so few as fourty people in it most part of the day. I shall adde no more, Sir, but that I am

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your very humble servant, John Wallis. For Mr Henry Oldenburgh Esquire, Secretary to the Royall Society, London.

108.

HENRY OLDENBURG to WALLIS 17/[27] May 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 19/[29].V.1666. This letter apparently contained objections against Wallis's hypothesis of the tides, as raised at the meeting of the Royal Society on 16 May by Sir Paul Neile and by several

1 if add.

3 the bloud (1) leaves (2) ceaseth to 5 scorched add. 8 now add.

19-20 For . . . London, written on left margin at 90°

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109. WALLIS to OLDENBURG, 19/[29] May 1666 members; cf. BIRCH, History of the Royal Society II, 89 and Oldenburg's note entitled Objections Against Dr Wallis's Hypothesis of Tides [VI?.1666] (especially the first two objections).

109. WALLIS to HENRY OLDENBURG Oxford, 19/[29] May 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 21, 4 pp. (our source). At top right of p. 4 Oldenburg has noted at 90° to address: 'An Extract of a Letter of Dr Wallis to Mr Oldenburg, answering some Objections made at the Society to his Hypothesis concerning the Flux and Reflux of the Sea.' Beneath this: 'Read May 23. 1666. Entered (1) R (2) LB. 1. 320 Phil. Trans. 16.' Above address in Oldenburg's hand: 'To mention Vossius about Tides.' Postmark on p. 4: 'MA/21'.—printed: OLDENBURG, Correspondence III, 136-8. wl Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 320-2. w2 Copy of w 1 : LONDON Royal Society Letter Book Copy 1, pp. 378-80. Reply to: OLDENBURG-WALLIS 17/[27].V.1666. The letter was read at the meeting of the Royal Society on 23 May and provoked once more a discussion of Wallis's hypothesis of the tides. An account of this discussion was communicated to Wallis by Oldenburg in his letters of 29.V/[8.VI].1666 and of 31.V/[10.VI].1666. Wallis later revised the present letter together with those to Oldenburg of 2/[12].VI.1666 and of 8/[18].VI.1666 in the form of a letter dated 18/[28].VII.1666, which replied to the objections to his hypothesis, first raised at the meetings of the Royal Society on 16 and 23 May and subsequently collected by Oldenburg in his note Objections against Dr. Wallis's Hypothesis of Tides. Wallis's composite letter was printed in Philosophical Transactions No. 16 (6 August 1666), 281-9 ('An Appendix, written by way of Letter to the Publisher; Being an Answer to some Objections, made by Several Persons, to the precedent Discourse'). Cf. WALLIS-OLDENBURG 18/[28].VII. 1666.

Oxford May. 19. 1666. Sir,

5

I received yours402 of May 17. And am very well contented that exceptions be made against my hypothesis concerning the Tydes; being proposed but as a conjecture to be examined, & upon that examination rectified if there be occasion, or rejected if it will not hold water. To the former; how two 402

yours: i.e. OLDENBURG-WALLis 17/[27].V.1666. 232

109. WALLIS to OLDENBURG, 19/[29] May 1666 bodyes that have no ty can have one common center of gravity; that is (for so I understand the objection) can act or be acted in the same manner as if they were one, or were connected: I shal onely answere; that it is harder to tell How they have, than, That they have it. That the Loadstone & Iron have somewhat equivalent to a Tye, though wee see it not, yet by the effect we know. And it would be easy to shew that two loadstones at once applied in different positions to the same Needle, at some convenient distance, will draw it, not to point directly at either, but at some point between both; which point is, as to those two, the common center of attraction; & it is the same as if some one Loadstone were in that point. Yet have these two Loadstones no connection or tye, though a common center of virtue, according to which they act. And, as to the present case; how the Earth & Moon are connected, I will not undertake to shew (nor is it necessary to my purpose,) but that there is somewhat that doth connect them (as much as what connects the Loadstone & Iron) is past doubt, to those who allow them to be carried about by the Sun as one aggregate or body whose parts keep a respective position to one another; like as Jupiter with his 4 Satellites, & Saturn with his one. Some ty there is that makes those Satellites attend their Lords, & move in a body, though wee do not see the tye, or hear the word of command. And [2] so here. The other Objection; that the Annual High-tydes are at the two Aequinoxes; if they adde allso, & then onely: will, if it hold true, ease mee of half my work. For it is then easyly answered, that it depends upon the Obliquity of the Zodiack; the parts of the Aequinox answering to equall parts of the Zodiack, being near the Solstitial points greatest, & near the Aequinoctial points lest of all. But beside this Annual vicissitude of the Aequinoxes, (which my hypothesis doth allow & assert;) I beleeve it will be found there is another Annual vicissitude answering to the Apogaeum

7 at once applied add. 7 to (1) a (2) the same Needle, (a) will make (&) at 8 not (1) di breaks off (2) to 13 the (1) S(a) breaks off (2) Earth 15 doth add. 17 one aggregate or add. 18 with his (1) 4 Sa breaks off (2) 4 Satellites 20 body, (1) to (2) though 24 my (1) trouble (2) work.

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109. WALLIS to OLDENBURG, 19/[29] May 1666

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& Perigaeum. And that the greatest tydes of all will be found to be upon a result of these two cooperating. Which (as the Inequality of natural days, depending on these same causes) will light near the times I mention. To the contrary observation; that it doth not hold so at Chatham, & in the Thames as is observed in Rumny Marsh403: I must at present avrexeii/, & refer to a melius inquirendum. If the same persons, after this notice, shall find that upon new observations needfully taken, they do not find the Spring tides in February & November, as high as those in March & September, I shall then think the objection very considerable. But I am certain that I have many times found very high tydes in November, at London, as well as in Rumny. And the time is not yet so far past, but that it may be remembred (by yourself, & those who were then in London,) whether in last November; when the Tydes were so High at Dover, & Marget404, & so on-wards to Rumny Marsh, as to do in some of those places much hurt, (& much more in Holland;) whether, I say, there were not allso at London (on the Thames) very high Tydes. | But a good diary [3] of the hight & time both of high-water & low-water, for a year or two together, even at Chatham, or Greenwich; but rather at some place in the open seas, as at the Point in Cornwall, or the West parts of Ireland; would do more to the resolving of this point; than any verball discourse without it. But inough of this. Pray let my humble service & thanks be presented to my Lord Brouncker, who is pleased to give himself the trouble of particular perusing that piece. I am Sir

Yours &c. J. W.

25

1 be found |be found del. to 3 depending one these corr. ed. 4 hold (1) in (2) so 7 taken, (1) they (2) they 8 in (1) Ma breaks off (2) February 10 times (1) at London (2) found 15 say, (1) there (2) there 18 rather (1) upon (2) at 19 seas, (1) on (2) as (a) on (b) at 21 my (1) ser breaks off (2) humble 403 404

Rumny Marsh: i.e. Romney Marsh. Marget: i.e. Margate.

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110. WALLIS to OLDENBURG, 24 May/[3 June] 1666 As for Petr. Bles.405 I think after another week to send you his Tractate Quales sunt~ which will be about as big as both the other, that is, about a Quire of paper. That being a book belonging to a Private College406, not the publike library, I have obtained the favour to have it home to my own lodgings, & so can take my own time without being confined to the library time & place. What the treatise is that your letter mentions de Remediis Remediorum, I do not know; I remember the name of the book; but have forgot, of what Author, & so know not where to look for it, that I may compare it with this inituled de Remediis conversorum. As to the reconoissance you mention, I shal set no price upon it, but let them value it as they think fit, or you for them. [4]

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These For Mr Henry Oldenburgh, at his house in the Pallmall, near

St James's

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London.

110. WALLIS to HENRY OLDENBURG Oxford, 24 May/[3 June] 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 22, 4 pp. (p. 4 blank) (our source). Paper damage at bottom of p. 2. On p. 1 notes in Oldenburg's hand written across address: 'past Creed Church in Leadenhall Street a Tinman. Hugenbottom. In Leadenhall Street Mr Stone a Turner, a Candlestick with Scruer to move (up) and downe with a hallow tin-plate, and something for an umbrella, and a ball for oyle to move every way. In Angel-alley Mr Peadon a Clock-maker (for oyle bals, |for lamp del. ', and: 'NB. To inquire in litle Brittan about a book of P. Blesensis mentioned in this letter.' Postmark on p. 1: 'MA/25'.—printed: OLDENBURG, Correspondence III, 142-3.

2 6 6 7

, that is, about a Quire of paper add. library (1) time & (2) time that (1) he (2) your of the (1) (—} (2) book; add. 405 406

Petr. Bles.: i.e. Petrus Blesensis. See WALLIS-OLDENBURG 3/[13].IV.1666. Private College: i.e. Balliol College. See WALLIS-OLDENBURG 23.IV/[3.V].1666.

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110. WALLIS to OLDENBURG, 24 May/[3 June] 1666 Simultaneously with this letter Wallis sent a packet containing transcripts of Petrus Blesensis's tracts Quales suni and Epistola aurea de silentio servando by Moor's coach to London.

Oxford May. 24. 1666.

[3]

Sir,

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I have here sent you the tract of Petrus Blesensis, intituled Quales suntwritten by him to the King of England Hen. 2.407 who was allso Duke of Normandy & Aquitain; Being a complaint against the Bishops of Aquitain (Sanctoniensis408 & Lemovicensis409) for mis-government; seeking to the King for redresse. It is a smart piece, & well penned, for a Treatise of that age. I have made the more hast with it, by reason that I obtained the favor to have the book home to my own lodgings: And had transcribed a good part of it when I sent you my last. Of the two smal fragments in Merton College; that contra usum legum, is of Petrus Blesensis; but is allready printed; as being the latter part of his Epist. 26. as it is printed in 4°. The other, Epistola Aurea de silentio servando &c. I have transcribed, & annexed to this Quales sunt, in the last leaf of it. But whether it be of Petrus Blesensis, or no; I am not certain. There is another large treatise, 5 or 6 times as big as this; of which I find two Copies; one in the Bodlyan Library, & another in Merton College. Divided into two parts; each part in six Bookes; & those bookes into chapters. Written to Richard410 then Bishop of London, by Petrus Blesensis Archdeacon of the same Church: Intituled Remediarium Conversorum. Both copies are ancient; & as I may guesse by the hand, about the time of Edw. I.411 or Edw. 2.412 And I think there is no great question to be made but that they be his. But being so large, I dare not undertake to transcribe it myself. If they who [2] 4 Hen. 2. add. 6 Lemovicensis) (1) as very {—} (2) for mis-government; (a) & (6) seeking 7 that age (1) ; fe (2) . I (a) had (b) have 10 last. (1) The (2) Of the 20 Intituled (1) Remedium (2) Remediarium 22 there is (1) {—) (2) no 407

Hen. 2.: i.e. Henry II (1133-89), King of England (1154-89).

408

Sanctoniensis: i.e. Santoniensis, the southern French province Saintogne. Lemovicensis: i.e. the southern French province Limousin. 410 Richard: i.e. Richard de Belmeis II, bishop of London (consecrated 1152). 411 Edw. 1.: Edward I (1239-1307), King of England (1272-1307). 412 Edw. 2.: Edward II (1284-1327), King of England (1307-27). 409

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110. WALLIS to OLDENBURG, 24 May/[3 June] 1666 are concerned would yet have it done; I will yet seek further if I can find out any to undertake it. One or two Masters of Arts I have found out, who are pretty well acquainted with Old hands: But whether they will be persuaded to undertake the transcribing of it, I much fear. (I have given a brief account of it (to this purpose) in my Monition, praefixed to this piece I now send.) Your last letter413 representing them as somewhat indifferent as to transcribing of it: makes me not further to presse it here till I have further order. The Dialogue between the King & the Abbot414; was, sure inough, in the Lumlyan library415; & I think it is as sure that that library is since translated to Sir Rob. Cotton's416 (now Sir John Cotton's417.) So that I beleeve it may be there found. And (in London,) it will be more easy to find transcribers. As to that of the life of Henry. 2. I am not sure but that it may be printed amongst those Scriptores Brittanm"418 set out by Mr Bee419 a Bookseller in Little Brittain; or hee may possibly give you direction where or of whom to inquire after it. Or it may be, Mr (William) Sumner420 of Canterbury (who sett out the Saxon dictionary421, & is a great inqu(irer) into writings of English Antiquity) may give you an account of that, or other wor(ks of) Blesensis, if you

2 I have (1) somew breaks off (2) |found out, add.\ who 3 whether (1) it (2) they 4 I (1) have not (2) much 8 I (have add] further order. (1) I (2) The 8 Abbot; (1) fe (2) was, 12 life of (1) Edw. (2) Henry. 2. 16 be add. 413

letter: presumably OLDENBURG-WALLIS 17/[27].V.1666. Dialogue . . . Abbot: i.e. Petrus Blesensis's tract Dialogue inter Regum Henricum Secundum et Abbaten Bonavallis. 415 Lumleian library: i.e. the library of John, first Baron Lumley. 416 Sir Rob. Cotton's: i.e. Sir Robert Bruce Cotton. 417 Sir John Cotton's: i.e. Sir John Cotton (1621-1701), grandson of Sir Robert Bruce Cotton (1571-1631), DNB. 418 Scriptores Britanni: i.e. Roger TWYSDEN, Historiae Anglicanae Scriptores X, London 1652. 419 Mr Bee: i.e. Cornelius Bee (1636-72), London bookseller in Little Britain. 420 Sumner: i.e. William Somner (1598-1669), Canterbury antiquary, DNB. 421 Saxon dictionary: i.e. SOMNER, Dictionarium Saxonico-Latino-Anglicum, voces, phrasesque Anglo-Saxonicas, ... cum Latino, et Anglica vocum interpretatione complectens, Oxford 1659. 414

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111. OLDENBURG to WALLIS, 29 May/[8 June] 1666 write to him. I doubt that in Oxford we shall scarce find any (other) then what I have allready told you of. I am Sir

yours &c J. W.

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The packet comes by Mr Moors Coach; which will be at Saracens head without New-gate; where, if it be not brought you, I pray inquire for it.

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W

These For Mr Henry Oldenburgh, at his house in the Palmal, near St James's London.

111.

HENRY OLDENBURG to WALLIS [London], 29 May/[8 June] 1666 Transmission:

C Part draft of missing letter sent: LONDON Royal Society Early Letters O2, No. 177, 1 p. (our source). On verso notes in Oldenburg's hand, apparently connected with the meeting of the Royal Society on 23 May 1666. The note referring to Wallis reads as follows: 'Dr Wallis to be desired, that, if he hath any Observations about the Satellites of Jupiter, concerning the Periods of their revolutions, he would communicate them.'— printed: OLDENBURG, Correspondence III, 144. Date: Mentioned in WALLis-OLDENBURG 2/[12].VI.1666. Answered by: WALLIS-OLDENBURG 2/[12].VI.1666. Probably in this letter, but possibly also in OLDENBURG-WALLis 31.V/[10.VI].1666, Oldenburg provided an account of three objections which were raised against Wallis's hypothesis of the tides at the meeting of the Royal Society on 23 May 1666 (cf. BIRCH, History of the Royal Society II, 93). He also reported on Hooke's pendulum experiment, which was also performed on that occasion and which examined the compound pendulous motion of two connected bodies (ibid., 92 and OLDENBURG-BOYLE 8/[18].VII. 1666, OLDENBURG, Correspondence III, 154). In addition, he probably informed Wallis of the request made by the Society that Hooke and he should collaborate with the Earl of Sandwich in Spain in making observations in order to determine the geographical bearing of Madrid from London (cf. BIRCH, History of the Royal Society 238

111. OLDENBURG to WALLIS, 29 May/[8 June] 1666 II, 90). Finally, Oldenburg recommended to Wallis Isaac Vossius's work on the motion of wind and sea De motu marram et ventorum liber, The Hague 1663. In this or in his following letter, Oldenburg probably took up the issue of the transcripts of Petrus Blesensis, indicating to Wallis for the first time that the request for these had come from friends of his in Paris and that they were pleased with the work he had done (cf. JUSTEL-OLDENBURG [16J/26.V.1666).

To Dr Wallis. Some did offer to the consideration of the Society, 1. Whether it were consistent with this Hypoth. that the highest and lowest Menstrual Tides fall, not at the Full and New moon, but 3. dayes after422: and that in severall parts of the East-indies, the Highest and Lowest Tides happen in quadratures423. 2. Whether, when the full and Change of the moon happen in the Perigeum (when the moon moves swiftest) and in the Apogeum (when she moves slowest) it would not be strange, that the Spring and Neap-tides should fall out both at both those times of full and Change424.

3 higest corr. ed.

7-8 2. (1) Whether it be not strange, that the Spring and Neap-tides should be both at Full and Change, that is, at both such times, when the moon is either furthest from, or nearest to the Earth, and moves either swiftest or slowest. And, whether it not rather be highest water at one of those times, and lowest at the other. (2) Whether, (a) the full and Change of the moon hapning (b) when the full and Change of the moon happen in the (ao) Apo breaks off (bb) Perigeum |and Apogeum del] (when the moon moves swiftest |and slowest del.\ ) and in add. the Apogeum (when (aoa) it (666) she 9 that |both del] the 10 out |both add] at 422

Whether .. . after: This corresponds to the fifth objection in Oldenburg's note entitled Objections Against Dr. Wallis's Hypothesis of Tides [VI?. 1666], which was raised by Brouncker. 423 that . . . quadratures: i.e. the fourth objection in Oldenburg's note Objections Against Dr. Wallis's Hypothesis of Tides [VI?.1666], raised by Ent. 424 Whether . . . Change: This objection was raised by Goddard (cf. BIRCH, History of the Royal Society II, 93). It corresponds to the third objection in Oldenburg's note Objections Against Dr. Wallis's Hypothesis of Tides [VI?.1666].

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113. WALLIS to OLDENBURG, 2/[12] June 1666

112.

HENRY OLDENBURG to WALLIS 31 May/[10 June] 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS-OLDENBURG 2/[12].VI.1666. Answered by: WALLIS-OLDENBURG 2/[12].VI.1666 and WALLIS-OLDENBURG 8/[18].VI. 1666. This is the second of two letters replied to by Wallis on 2/[12].VI.1666, which dealt with the discussion of Wallis's hypothesis of the tides at the meeting of the Royal Society on 23 May 1666 as well as with the issue of the transcripts of Petrus Blesensis. Cf. OLDENBURG-WALLIS 29.V/[8.VI].1666 for more details of the contents of both letters and for a draft, the substance of which Oldenburg probably passed on to Wallis in the first letter. Further topics of this letter were the publication of Wallis's hypothesis of the tides as it was exposed in WALLIS-BOYLE 25.IV/[5.V].1666 and the impending Anglo-Dutch naval battle in the Channel.

113. WALLIS to HENRY OLDENBURG Oxford, 2/[12] June 1666 Transmission:

W Letter sent (with few minor alterations by Oldenburg): LONDON Royal Society Early Letters Wl, No. 23, 4 pp. (our source). At top of p. 1 beneath date: 'read June 20. 66. Enterd LB. 1. 329.' At top right of p. 4 Oldenburg has noted at 90° to address: 'Extract of a Letter of Dr Wallis to M. Oldenburg about Tides.' Beneath address in Oldenburg's hand: 'Ace. 4. jun.' Postmark on p. 4: 'JU/4'.—printed: OLDENBURG, Correspondence III, 147-50. w1 Copy of letter sent with original figure inserted: LONDON Royal Society Letter Book Original 1, pp. 329-31 (figure between p. 330 and 331) (our source for the figure). w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 388-92. Reply to: OLDENBURG-WALLIS 29.V/[8.VI].1666 and OLDENBURG-WALLIS 31.V/[10. Vlj.1666. The letter was read at the meeting of the Royal Society on 20 June 1666, together with Wallis's following letter of 8/[18].VI.1666; BIRCH, History of the Royal Society II, 98. The figure originally contained in the letter sent has been inserted in the Letter Book Original. 240

113. WALLIS to OLDENBURG, 2/[12] June 1666 Wallis later revised the present letter together with his letters to Oldenburg of 19/[29].V.1666 and of 8/[18].VL1666 in the form of a letter dated 18/[28].VII. 1666, in reply to the objections against his hypothesis of the tides raised at the meetings of the Royal Society on 16 and 23 May. This composite letter was published in Philosophical Transactions No. 16 (6 August 1666), 281-9. Cf. WALLIS-OLDENBURG 18/[28].VII. 1666.

(W)

Oxford. June. 2. 1666. Sir,

To yours425 of May 29, & 31, (which I am at once to answere) I shall not give you the trouble of any long Reply. What you mention of Isaac Vossius426, de motu maris et ventorum427, I have not yet seen; but shal inquire after it. To the particular varieties of Tides in severall places of the World: I cannot pretend to give a satisfactory account, for want of a full history of Tides. Because, (as is intimated in what I wrote in the Generall,) that the various positions of Chanells, Bays, Promontories, Gulfs, Shallows, Currents, Trade-winds, &c, must needs make an innumerable variety of accidents in particular places, of which no satisfactory account is to be given (though the generall hypothesis should prove never so true) without a due consideration of all those: Which is a task too great for mee to undertake, being so ill furnished with materials for it. To the particulars428 you mention, I may yet say something. 1. As to that of the Spring-tides falling out 2 or 3 days after the New and Full moons; I would attempt an answere, if I were certain, whether it be so in the open Seas, or onely in our Channells: For the answeres will not be the same, in both cases. If onely in our Channells, where the Tides find a large in-draught; but not in the open Seas: wee must seek the reason of it, from the particular position of these seas. If it be so in the wide open 13 true) (1) with (2) without 18 an add. 22 of it, (1) in the pa breaks off (2) from 425

yours: i.e. OLDENBURG-WALRUS 29.V/[8.VI].1666 and OLDENBURG-WALLIS 31.V/[10.VI].1666. 426 Vossius: i.e. Isaac Vossius (1618-89), Dutch polymath. 427 de motu . . . ventorum: i.e. VOSSIUS, De motu marram et ventorum liber, The Hague 1663, which ascribed to the sun an influence in causing the tides. Cf. Wallis's subsequent letter to Oldenburg of 8/[18].VL1666, in which he examined Vossius's tract. 428 particulars: cf. OLDENBURG-WALLIS 29.V/[8.VI].1666. 241

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Sea; then must wee seek a reason of it, from the generall Hypothesis. And till I know the matter of fact; I know not which to offer at: lest, while I attempt to salve one Appearance, I should fall foul of the other. To that of some places in the East-Indies; where, it is sayd, the Highest & Lowest tides happen in the Quarters. If it be so; I suppose the reason of it must be given from the particular situation of those places; whether Bays, Channells, Currents &c: to which because I know not the particulars; I know not how to attempt a satisfactory] Answere. [2] To that of Dr G.429 How it should bee that, if New-moons make the High-Tides (upon accelerating the Earths compound motion,) the Full moons (when it is slowest) should not make the lowest Tides; but that there should be High-Tydes at both times: The answere (if observed) is given in the First Papers430: Because the Effect is indifferently to follow, either upon a sudden Acceleration, or a sudden Retardation. Like as when a loose thing lyes on a moving Body: if the Body be thrust suddenly forward more then before, the loose body is cast back, or rather left behind, (not having yet obtained an equal impetus to that of the body on which it lyes:) But if stopped or notably retarded in its motion; the loose incumbent body is thrown forward, (by its formerly contracted impetus, not yet qualified or accommodated to the present stop or slowness of the Body on which it lyes.) Now both of these happening; the one at the New-moon, the other at the Full-moon; causeth high Tides at both. The Experiment431 you mention, is very proper to explain the Hypothesis (if I sufficiently understand it by your description.) And certainly,

1 Sea add. 3 to (1) salve (2) salve 4 To that of (1) the E breaks off (2) some 5 happen (1) the (2) in 5 the (1) reas breaks off (2) reason 6 particular (1) si breaks off (2) situation of (a) the (b) those 9 Dr Goddard. How alt. Oldenburg 10 High-Tides ( (1) by (2) upon accelerating the Earths (a) moti breaks off (b) compound motion,) the Full moons (ao) th breaks off (bb) (when 14 upon (1) ac breaks off (2) a 15 moving (1) Body (2) Body: if the Body be thrust (a) for breaks off (b) suddenly 17 yet (1) obtained (2) obtained 429

Dr G.: i.e. Jonathan Goddard. First Papers: i.e. in WALLIS-BOYLE 25.IV/[5.V].1666. 431 Experiment: i.e. Hooke's pendulum experiment carried out at the meeting of the 430

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113. WALLIS to OLDENBURG, 2/[12] June 1666 when wee see a long stick thrown in the Air (especially when one end is heavyer then the other) to turn itself so as that end which at first was forward conies to be behind; the regular line of motion, will be that which is described by the middle point or center of gravity; (about which both the ends do describe circles;) not by either of the two ends, whose motion is compounded of, that of the whole or its center of gravity, & that of their own Epicicle about it. And I cannot but acknowledge myself obliged to the Royal Society, & the members of it; who are pleased so far to relish my Conjectures; as not onely to give them the hearing; but allso to give [3] themselves the trouble of contriving Experiments to illustrate it.| But that which will give the clearest evidence to it (if it can be observed) will be from celestial Observation. As for instance432, supposing the Sun at O, the Earths place in its annual Orb at T: and Mars (in opposition to the Sunne, or near it) at 6; from whence Mars should appear in the Zodiack at 7, and would at Full moon be seene there to be, the moon being at C & the Earth at c: (and the like at the New moon;) But if the moon be in the first quarter at A, & the Earth at o; Mars would be seen not at 7 but at a; too slow: & when the moon at 5, the earth at b, Mars at /3; yet to slow: till at the full moon, the moon at C, the Earth at c, Mars will be seen at 7, its true place, as if the Earth were at T. But then, after the Full; the moon at D, the Earth at d; Mars will be seen,

1 (especially (1) to turn (2) when 3 to (1) bell breaks off (2) be behind 4 both add. 8 Royal Society, (1) that they (2) fe 9 but allso add. 13 the Sun at S, the alt. Oldenburg 13 the (1) Earth (2) Earths place 13 (in ... near it) add. 14 at Af; from whence alt. Oldenburg 14 Mars (1) we be (2) should appear (a) amongst the (6) in 15 Full moon (1) be seene (2) seem (3) be seene 16 New moon;) (1) but at (2) But Royal Society on 23 May 1666, which examined the compound pendulous motion of two connected bodies; cf. BIRCH, History of the Royal Society II, 92 and OLDENBURGBOYLE 8/[18].VIL1666, OLDENBURG, Correspondence III, 154. 432 As for instance: cf. the figure below.

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not at 7, but at <5; too forward: And yet more, when the moon (at the last quarter) is at E, the Earth at e: & Mars seen at s. If therefore Mars, (when in Opposition to the Sun) be found (all other allowances being made) somewhat too backward before the Full, & somewhat too forward after the Full-moon (& most of all at the Quadratures;) it will be the best confirmation of the Hypothesis. (The like might be fitted to Mars, in other positions, mutatis mutandis; & so for the other Planets.) But, if the Earths Epicycle should be so small as not to make any observable difference in Mars (& the superior Planets) yet possibly in Venus, or Mercury, or in the Moon, or its own place it may be discernable. But this is to be rather a matter of future Observation, than of present discourse. As to the motion you make concerning what my Lord Sandwich433 doth suggest; I should be very willing to comply with it, if wee had any Instruments in order, for Observation; or if we can yet get them in order time inough: I purpose to confer with Dr Wren about it, upon the first opportunity. As to what concerns your friends434 at Paris, you may name me to them if you think fit: I am glad they are so well pleased with what is done435. Before I proceed to more, I suppose you will receive further directions from them. Mean while I rest

1 forward: (1) An (2) And 2 quarter) |is add] at (1) D (2) E, 2 seen add. 4 made add. 5 it will be (1) a good (2) the best 7 for the (1) superior (2) other 8 as add. 10 in the (1) (same) (2) Moon 11 future add. 14 Instruments |here del.\ in 14 get |them add] in order (1) qu breaks off (2) time 18 you |think add] fit: |And del] I 433

my Lord Sandwich: i.e. Edward Montagu, first Earl of Sandwich (1625-72), admiral and general at sea, resident in Madrid 1666-8 as ambassador extraordinary, DNB. 434 your friends: i.e. Henri Justel. 435 pleased with what is done: i.e. with the collection of information on manuscripts of Petrus Blesensis in English libraries; cf. JUSTEL-OLDENBURG [16J/26.V.1666, OLDENBURG, Correspondence III, 132-4, 132.

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113. WALLIS to OLDENBURG, 2/[12] June 1666 Yours &c. J. Wallis. [4]

For my very worthy friend, Mr Henry Oldenburgh, at his house in the Palmal, near St James's. London.

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(w 1 , original figure)

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114.

WALLIS to HENRY OLDENBURG Oxford, 8/[18] June 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 24, 4 pp. (our source). At top right of p. 1: 'read June (1) 26: (2) 20: 66. enter'd LB. 1. 332'. At top left of p. 4 Oldenburg lias noted at 90° to address: 'Extract of Dr Wallis's letter to M. Oldenburg about M. Vossius his Hypothesis of the Flux and Reflux of the Sea, together with some further considerations upon his owne.' Beneath address: 'Rec. jun. 11. 66.' Postmar on p. 4: 'JU/{11}'.—printed: OLDENBURG, Correspondence III, 157-9. w Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 332-4. Reply to: OLDENBURG-WALLis 31.V/[10.VI].1666. This letter was read at the meeting of the Royal Society on 20 June 1666, together with Wallis's foregoing letter of 2/[12].VI.1666; BIRCH, History of the Royal Society II, 98. Wallis later revised the present letter, together with his letters to Oldenburg of 19/[29].V.1666 and of 2/[12].VI.1666, in the form of a letter dated 18/[28].VII. 1666, which replied to the objections against his hypothesis of the tides raised at the meetings of the Royal Society on 16 and 23 May. This composite letter was published in Philosophical Transactions No. 16 (6 August 1666), 281-9. See WALLIS-OLDENBURG 18/[28].VII.1666.

Oxford June. 8. 1666.

Sir,

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I have, since my last436 to you, gotten the book you mention437 of Isaac Vossius, de motu marium et ventorum; & perused it. And thank you for your recommendation of it. And though, I do not concur with him in his hypothesis; That all the great motions of the Seas &c, should arise onely from so small a warming of the Water, as to raise it (where most of all) not a foot in perpendicular (as in his 12th chapter): Nor, that there is any other connexion, between the Moons motion, & the Tydes menstrual

7 it (1) (wlier breaks off (2) (where 8 (as in his 12th chapter) add. 436

my last: i.e. WALLIS-OLDENBURG 2/[12].VI.1666. the book you mention: i.e. VOSSIUS, De motu marium et ventorum liber, The Hague 1663, which Oldenburg had recommended to Wallis in either his letter of 29.V/[8.VI].1666 or that of 31.V/[10.VI].1666. 437

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114. WALLIS to OLDENBURG, 8/[18] June 1666 period, than a casual Synchronism (as chapt. 16 & 18) Beside many other things in his Philosophy which I cannot allow: Yet I am well pleased with what is hystoricall in it, of the matter of fact; especially if I may be secure that hee is therein accurate & candide, not wresting the Phaenomena to his hypothesis. But I find nothing in it that doth induce mee to vary from my hypothesis. For granting his hystoricalls to be all true; the account of the constant current of the Sea West-ward, & of the constant Eastern blasts, &c, within the Tropick; is much more plausibly (&, I suppose, truly) rendred by Galileo438, long since, from the Earths diurnall motion, (which, near the Aequator, describing a larger circle than nearer the Poles, makes the current more conspicuous & swift; &, consequently, the Eddy or recurrent motion nearer the Poles, where this is more remisse;) than can easyly be rendred from so small a Tumor as hee supposeth. Not to adde; that his account of the progressive motion which he fansyeth to follow upon this Tumefaction; & by acceleration to grow to so great an height near the shore (as chap. 13 & 14th); is a motion which seems to mee too extravagant to be salved by any laws of Staticks. And that of the Moons motion onely Synchronizing with the Tydes, casually; without any Physical connexion: I can very hardly assent to. For it can hardly bee imagined, that any such constant Synchronism should be in nature; but where, either the one is the cause of the other; or both depend upon some common cause: which latter I take to be the case here. And where wee see so fair a foundation for a Physicall connexion; I am not prone to asscribe it merely to an independent Synchronism. In summe; His hystory [2] doth| well inough agree with my hypothesis: & I think the Phaenomena

1-2 (as chapt. . . . cannot allow add. 5 vary (1) my (2) from 8 (&, I suppose, truly) add. 10 (which, (1) at (2) near 12 nearer the Poles add. 14 he (1) fo breaks off (2) fansyeth |to follow add. upon 16 (as chap. 13 & 14th) add. 16 seems (1) so (2) to mee too 19 hardly (1) (—) (2) assent 21 but add. 22 cause add. 438

rendred by Galileo: cf. GALILEO, Dialogo sopra i due massimi sistemi del mondo, Florence 1632, dialogo quarto, esp. 419-21; Open VII, 452-4.

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much better salved by mine than his. But I find him generally (as well as others) to talk of the High Tydes in tempore Novilunii et Plenilunii at the times of New moon & Full moon; not, (as wee find it in our seas,) 2 or 3 days after. But, perhaps, hee means it onely in a more lax sense, about that time. And the like for that of the Aequinoctials. And if, (what you sayd was objected) that it be some days after, (as I know it useth to be in our narrow seas;) do hold allso in the wide Ocean: I can admitt it well inough; though, because I was not sure of it, I durst not build upon it. For the truth is, the Flux and Reflux of water in a vessell, upon the jogging of it; though it follow thereupon; yet is for the most part discernable sometime after: for there must, upon that jog, be some time for motion before the accumulation can have made a Tyde. And I do not know but that we must allow it in all the Periods. For as the Menstrual high Tide, is not till some days after the New-moon & full moon (at lest in some places;) so is the diurnall high water, about as many hours after the moons coming to south (I mean, at Sea; for in Chanells it varies to all hours, according as they are nearer or further from the open Sea:) And the Annual High Tydes, of November & February; somewhat later than (what I conjecture to be the causes of it) the greatest inequality of the Naturall days, happening in January, & October. But, as I sayd, I do not build upon this, (though I may admitt it:) because I would be first sure in point of Fact. And, for this reason, I did, in my papers (as I remember) onely hint, a suspicion; that (though mariners use so to speak) the High-water may be not (just) when the moon is south; nor the Spring-tides, (just,) at New & Full moon: But, about those times. And, since it hath been the custome hitherto, to speake of these periods, with that laxness of expression: I thought I was not obliged in my conjectural

5 And if, (1) as you said it was objected, (2) (what you said was objected) 7 narrow add. 7 wide add. 11 for (1) the (2) there 12 for motion add. 12 And I (1) think (2) do not know but that we (a) may (6) must allow it in (ao) all (66) all 14 some (1) time (2) days 17 nearer (1) of (2) or 23 that (1) the (2) (though 24 may add.

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[3] hypothesis (while it is yet but a Candidate,) to| speak more nicely. If the Hypothesis, for the main of it, be judged rationall: the niceties of it, are to be adjusted, in time, from particular observation. But (as I sayd in my last) the most certain criteria, will be from Astronomicall Observations: which, if they shall be found to favour it, will (to those who are intelligent in these affairs) put it as much out of doubt, as they do the goodness of the Copernican Hypothesis, before the Ptolemaick. The particular considerations you mention in your last439, concerning the printing it speedyly440: I concur with you in; if it be thought fit to be done at all. For being now made so publike as to be openly discoursed of in the Society; it is not to be supposed but that news of it will quickly go further: & give others occasion to deliver their thoughts of it. And therefore, if that noble Lord, our President441, approve of it; & you so think fit: you shall have my consent to use it as you please. I thank you for your informations about M. Cassini's papers442; &; of our own naval affairs443; &c resting

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your friend & servant John Wallis.

[4] These For my worthy friend, Mr Henry Oldenburgh, at his house

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4 certain (1) cha breaks off (2) criteria 6 doubt, (1) if (2) as 6 of the (1) Ptolemaick (2) Copernican 9 in; add. 10 as to be (1) commonly (2) openly 15 informations (1) of (2) about 16 affairs; (1) which have since received a further con breaks off (2) fee 439

your last: i.e. OLDENBURG-WALLIS 31.V/[10.VI].1666. printing it speedyly: i.e. the publication of Wallis's hypothesis of the tides as set out in WALLIS-BOYLE 25.IV/[5.V].1666. It was printed in Philosophical Transactions No. 16 (6 August 1666). 441 President: i.e. William Brouncker. 442 M. Cassini's papers: presumably Giovanni Domenico CASSINI, Martis circa axcm proprium revolubilis observationes Bononiae ... habitae, Bologna 1666, which had been sent to Oldenburg from Paris; cf. OLDENBURG-BOYLE 8/[18].VI.1666, OLDENBURG, Correspondence III, 155. 443 our own naval affairs: probably concerning the Anglo-Dutch naval battle in the Channel, which took place in the days after Oldenburg's letter of 31.V/[10.VI].1666. 440

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115. OLDENBURG: Objections, [June? 1666] in the Palmal, near St James's London.

115. HENRY OLDENBURG: Objections against Dr Wallis's Hypothesis of Tides [June? 1666] Transmission:

C Note: LONDON Royal Society Early letters Wl, between No. 20 and No. 21, 1 p. (verso blank). This note contains the five objections against Wallis's hypothesis of the tides, raised at the meetings of the Royal Society on 16 and 23 May 1666, as well as the names of the fellows, who had made them; cf. BIRCH, History of the Royal Society II, 89 (objections 1 and 2) and 93 (objections 3 to 5). Oldenburg apparently had informed Wallis of the first two in his letter of 17/[27].V.1666, and of the other three in one of his letters of 29.V/[8.VI].1666 and 31.V/[10.VI].1666. The present note was possibly occasioned by the design to publish a reworked version of Wallis's replies to the objections in his letters to Oldenburg of 19/[29].V. 1666, 2/[12].VI.1666, and 8/[18].VL1666. This revised version, written as letter dated 18/[28] July 1666, takes up literally the formulations of Oldenburg's note, which accordingly must have reached Wallis by one of Oldenburg's now missing letters or have been communicated to him during his stay in London around 20 June 1666.

Objections Against Dr Wallis's Hypothesis of Tides.

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1. That it appears not, how two Bodies, that have no tye, can have one common Center of Gravity. Sir P. Neile444. 2. That at Chatam, and in the Thames, the Annuall Spring-tides happen about the Aequinoxes, not, as Dr Wallis affirms to have been 10 observed elsewhere about the beginning of February and November. Severall. 8 Spring- |and Neap- del.\ tides 10 elsewhere add. Wallis 444

Neile: i.e. Paul Neile. 250

116. WALLIS to OLDENBURG, 18/[28] July 1666 3. That, Supposing the Earth and moon to move about a compound Center of Gravity, if the highest Tides be at a New Moon, when the Earth is farthest from, and the moon nearest to the Sun; and the Tides abate, as the Earth approaches nearer, till she come into the supposed Circle of her Annuall Motion; it may be demanded, Why they doe not abate, as the Earth corns still nearer to the Sun within the said Circle and so, why we have not one spring, and one Neap-Tide in every Course of the moon. Dr Godd.445 4. That in some parts of the Eastindies the Highest Tides are in the Quadratures of the moon. Sir G. Ent446. 5. That those High Tides happen not with us just at the Full and Change, but about 3. dayes after. Lord Brounker.

116.

WALLIS to HENRY OLDENBURG Oxford, 18/[28] July 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 25, 6 pp. (our source). The text lias been slightly amended and corrected by Oldenburg for publication. Together with place, date, and salutation, the last paragraph and signature have been deleted. At top of p. 1 Oldenburg has written: 'An APPENDIX, written by way of letter to the Publisher. Being an Answer to some Objections, made by severall Persons, (1) upon their p breaks off (2) to the precedent Discourse.' On p. 6 (originally blank): 'an abstract of foregoing Letters for the Phil. Trans. Not to be entered.'—printed: OLDENBURG, Correspondence III, 184-91. E1 First edition of letter sent (including Oldenburg's corrections and an alteration according to Wallis's instructions given in WALLIS-OLDENBURG 2/[12].VIII.1666): Philosophical Transactions No. 16 (6 August 1666), 281-9 ('An APPENDIX, written by way of Letter to the Publisher; Being an Answer to some Objections, made by several Persons, to the precedent Discourse') (our source for the figure). w Latin translation of E1: LONDON Royal Society MS. 368, No. 1, pp. 30-44.

I 3. (1) That, supposing (2) Supposing (3) That, Supposing 9 some parts of add. 9 Highest |and Lowest del] Tides II High | and Low del. Tides 445

Dr Godd.: i.e. Jonathan Goddard. See WALLIS-OLDENBURG 30!V/[10.V].1664. Sir G. Ent: i.e. Sir George Ent (1604-80), physician, fellow of the Royal Society since 1660. 446

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116. WALLIS to OLDENBURG, 18/[28] July 1666 E2 First edition in Latin (based on a translation of El different from w): POTTER, TlvOajopas /j.eTe^ipvxo';, sive Theses Quadragesimales in Scholis Oxonii publicis pro forma habitae ... Quibus ... adjecta est Cl. D. D. Wallisii ... Dissertatio Epist. ad D. Boyle De Fluxu & refluxu Maris, Leiden 1684, 171-90. E3 First edition of w: WALLIS, Opera mathematica II, 750-4 (De Aestu Maris, Epistola secunda. (Antecedentis Appendix, & simul edita:) Qua quorundam Objectionibus Respondetur. Ad D. Oldenburgium, mense Julio, 1666 scripta). This letter, intended for publication, is a reworked version of Wallis's letters to Oldenburg of 19/[29].V.1666 and of 2/[12].VI.1666, in which he had replied to the objections raised against his hypothesis of the tides by members of the Royal Society. In reworking these letters, Wallis referred literally to the formulations used by Oldenburg in his note Objections against Dr Wallis's Hypothesis of Tides, which presumably was written for this purpose. Wallis also included his discussion on Vossius's De motu marium, taken from his letter to Oldenburg of 8/[18].VL1666, together with his views on Gassendi's Syntagma philosophicum. Wallis sent a Latin translation (w) of this letter (and of his letters to Boyle of 25.IV/[4.V].1666 and to Oldenburg of 24.VII/[3.VIII].1666, which likewise were published in Philosophical Transactions No. 16) to Oldenburg in February 1667/8; see OLDENBURG-WALLIS 24.XII.1667/[3.I.1668], WALLIS-OLDENBURG 1/[11].II.1667/8, and WALLIS-OLDENBURG 8/[18].II. 1667/8.

(W) Oxford July 18. 1666.

Sir, I received yours; & am447 very well contented, that Objections be made 5 against my Hypothesis concerning Tydes: being proposed but as a conjecture to be examined; &, upon that Examination, rectifyed, if there be occasion; or rejected, if it will not hold water. 1. To the first objection of those you mention; That it appears not how two Bodies, that have no tye, can have one common center of Gravity10 that is (for so I understand the intendment of the objection) can act or be acted in the same manner, as if they were connected: I shall onely answere, that it is harder to shew How they have, than That they have it. That the Loadstone & Iron have somewhat equivalent to a Tye; though we see it not, yet by the Effects wee know. And it would be easy to shew, that two 15 Load-stones, at once applyed, in different positions, to the same Needle, II were (1) (—) (2) connected 44r

l received yours; & am . . . : The introduction and the replies to the first two objections are nearly literally taken from WALLIS-OLDENBURG 19/[29].V.1666. 252

116. WALLIS to OLDENBURG, 18/[28] July 1666 at some convenient distance; will draw it, not to point directly to either of them, but to some point between both; which point is, as to those two, the common center of Attraction; & it is the same, as if some one Load-stone were in that point. Yet have these two Load-stones no connection or tye, though a common center of Virtue, according to which they jointly act. And as to the present case, How the Earth & Moon are connected; I will not now undertake to shew (nor is it necessary to my purpose;) but, That there is somewhat, that doth connect them, (as much as what connects the Loadstone, & the Iron, which it draws,) is past doubt to those, who allow them to be carryed about by the Sun, as one Aggregate or Body, whose parts keep a respective position to one another: Like as Jupiter with his four Satellites, & Saturn with his one. Some Tye there is, that makes those Satellites attend their Lords, & move in a Body; though wee do not see that Tye, nor Hear the words of Command. And so here. 2. To the 2d objection; That, at Chatham & in the Thames, the Annuall Spring-tydes, happen about the Aequinoxes; not (as this Hypothesis doth suppose elsewhere to have been observed) about the beginning of February & November. If their meaning be, that Annuall High Tydes, doe then happen, & then onely: If this prove true, it will ease mee of half my work. For it is then easyly answered, that it depends upon the Obliquity of the Zodiack; the parts of the Aequinoctial answering to equal parts of the Zodiack, being near the Solstitial points greatest, & near the Aequinoctial points lest of all. But beside this Annuall Vicissitude of the Aequinoxes, not to say, of the 4 Cardinall Points, (which my Hypothesis doth allow & assert;) I beleeve it will be found, that there is another annual vicissitude answering to the Suns Apogaeum and Perigaeum. And that the greatest Tydes of all, will be found to be upon a result of these two causes cooperating: which (as doth the Inequality of Natural days,

4 in that |one del.\ point 5 Virtue, (1) by (2) according to which they (a) act (6) jointly 7 now add. 16 Spring-tydes, \& Neap-tydes, del. happen 17 observed) (1) about (2) about 19 then (1) & (2) happen, 24 not to say, of the 4 Cardinall Points, add. 28 causes add. 28 doth add.

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depending on these same causes) will light nearer the times, I mention. To what is sayd to be observed at Chatham & in the Thames, contrary to that I allege as observed in Rumney Marsh: I must at present o:7rexetz/> & refer to a melius inquirendum. If those, who object this contrary observation, shall, after this notice, find, upon new Observations needfully taken, that the Spring-tydes in February & November, are not so high, as those in March & September; I shall then think the Objection very considerable. But I do very well remember, that I have seen in November, very high Tydes at London, as well as in Rumney Marsh. And, the time is not yet so far past, but that it may be remembered (by yourself or others then in London) whether in November last, when the Tydes were so high at Dover, at Deal, at Margate, & all along the Coast from thence to RumnyMarsh, as to do in some of those places much hurt, (and, in Holland, much more;) whether, I say, therej were not allso, at the same time, at London, [2] (upon the Thames) very high Tydes. But a good Diary of the Height & Time both of High-water, & Low-water, for a year or two together, even at Chatham, or Greenwich; but rather at some place in the open Sea; as at the Lands-end in Cornwall, or on the West parts of Ireland, or at St Hellens, or the Barmodas, &c: would do more to the resolving of this point, than any verbal discourse without it. 3. To the 3d Objection; That supposing the Earth & Moon to move about a common center of gravity: if that the Highest tides be at the Newmoon, when the Moon being nearest to the Sun, the Earth is farthest from it, & its compound motion at the swiftest; & that the Tydes abate as the Earth approacheth nearer, till it comes into the supposed Circle of her An-

1 depending (1) one (2) on 1 To (1) the contrary Observation which (2) what 5 find, (1) that (2) upon 10 but that (1) you may remember (2) it may be remembered 12 at Deal, add. 12 & (1) so (2) all along the Coast |from thence add] to 14 at the same time, add. 14 at London, (1) very h breaks off (2) (upon 18 or at St Hellens, or the Barmodas, add. 22 that add. 23 when the (1) Earth is farthest fr breaks off (2) Moon (a) is nearest to the Sun, & the Earth farthest (b) being nearest to the Sun, the Earth is farthest 24 motion (1) (in) (2) at

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116. WALLIS to OLDENBURG, 18/[28] July 1666 nual Motion: It may be demanded, Why do they not still abate as the Earth comes yet nearer to the Sun & the swiftness of its compound motion still slackens? And so, why have wee not Springtides at the New-moon (when the motion is swiftest;) & Neap-tydes at Full-moon; (when the motion is slowest;) but Spring-tides at both? The Answere (if observed) is allready given in my Hypothesis itself. Because the Effect is indifferently to follow, either upon a suddain Acceleration, or a suddain Retardation. (Like as a loose thing, lying on a moving body; If the Body be thrust suddainly forward; that loose thing is cast back, or rather left behind, not having yet obtained an equal impetus with that of the Body, on which it lyes; But if stopped, or notably retarded; that loose incumbent is thrown forward, by its formerly contracted impetus not yet qualified or accommodated to the slowness of the Body, on which it lyes.) Now both of these happening, the one at the New-moon, the other at the Full-moon, do cause High-Tydes at both.448 4. To the 4th Objection; that the Highest Tydes are not at all places, about the New-moon & Full-moon; & particularly, that, in some places of the East-Indies, the Highest Tydes are at the Quadratures: I must first answere in general; That as to the particular varieties of Tydes in severall parts of the World, I cannot pretend to give a satisfactory account, for want of a competent History of Tydes, &c. Because (as is intimated in what I wrote in the generall) the various positions of Channells, Bays, Promontories, Gulfs, Shallows, Currents, Trade-winds, &c. must needs

1 It may be demanded, add. 2 yet add. 2 the swiftness of add. 3 why (1) wee have (2) have wee not (a) a Springtide at . . . & a Neap-tyde Springtides at ... & Neap-tydes 3 (when the motion is swiftest;) add. 4 Full-moon; (1) but Sprin breaks off (2) (when 6 in my | first Papers which propose the del. Hypothesis 8 moving (1) board, (2) body; 11 incumbent jthing del. \ is thrown 16 places, (1) at (2) about 18 first (1) pr breaks off (2) answere 21 of a (1) full (2) competent History of Tydes (a) . (b) , fee. 448

(b)

3. To the 3d objection . . . High-Tydes at both: cf. WALLIS-OLDENBURG 2/[12]. VI.1666 ('To that of Dr G. . . . ')•

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make an innumerable variety of Accidents in particular places, of which no satisfactory account is to be given from the generall Hypothesis (though never so true) without a due consideration of all those. Which is a task too great for mee to undertake, being so ill furnished with materialls for it. And then as to the particular instance of some places in the East-Indies] where the Highest Tydes are at the quadratures: I suppose, it may be chiefly intended of those about Camboia, & Pegu. At which places, beside that that they are situate at the in-most parts of Vast Bays, or Gulfs, (as they are called;) have allso vast In-draughts of some hundred miles within land, which when the Tydes are out, do ly (in a manner) quite dry: And may therefore very well be supposed to participate the Effect of the Menstrual Tydes many days after the cause of them happens in the open Sea, upon a like ground as in Straights & narrow chanells the Diurnall Tydes happen some hours later than in the Ocean. And a like account must be given of particular accidents in other places, from the particular situation of those places, as Bays, Chanells, Currents, &c.449 5. To the 5th Objection, That the Spring-Tydes happen not, with us, just at the Full & Change, but two or three days after: I should with the more confidence attempt an Answere, were I certain, whether it be so in the open Seas, or onely in our Channells. For the Answeres will not be the same in both cases. If onely in our Chanells, where the Tydes find a large in-draught; but not in the Open Seas: wee must seek the reason of

4 furnished (1) for (2) with. 5 of (1) those (2) some 6 where the (1) High-ty breaks off (2) Highest 8 of (1) a Vast Bay or Gulf, (which they call the Golf of Bengale,) (2) Vast Bays, or Gulfs, (as they are called;) |they add. Oldenburg] have 9 allso (1) a vast In-draught (2) vast In-draughts 9 miles (1) in (2) within land, which (a) at (b) when 10 do (1) ly by (2) ly 11 therefore (1) upon (2) very 12 Tydes (1) at (2) in the ope breaks off (3) some (4) many 15 places, (1) different fro breaks off (2) from 16 places, (1) whether (2) as 19 were I (1) sur breaks off (2) certain, 449

4. To the 4th Objection ... Currents, &x.: This is a significantly extended version of the corresponding passage in WALLIS-OLDENBURG 2/[12].VI.1666 ('To that of some places . . . ' ) . 256

116. WALLIS to OLDENBURG, 18/[28] July 1666 [3] it from| the particular position of these places. But if it be so generally in the wide Open Seas; wee must then seek a reason of it from the generall Hypothesis. And, till I know the matter of Fact, I know not well, which to offer at; lest while I attempt to salve one, I should fall foul of the other.450 I know that Mariners use to speake of Spring Tydes at the New & Full of the Moon; though I have still had a suspition that it might be some days after, as well in the open Seas as in our narrower Channells; (& therefore I have chosen to say, in my papers, About the New & Full, rather than At the new & Full; & even when I do say At] I intend it in that laxer sense in which I suppose the Mariners are to be understood, for Near that Time:) Of which suspition you will find some intimations even in my first Papers. But this though I can admit; yet, because I was not sure of it, I durst not build upon it. The truth is, the Flux & Reflux of Water in a Vessel, by reason of the jogging of it; though it follow thereupon; yet is, for the most part, discernable some time after. For there must, upon that jog, be some time for Motion, before the Accumulation can have made a Tide. And I do not know but that wee must allow it in all the Periods. For as the Menstrual High Tyde, is not (at lest with us) till some Days after the Full & Change; so is the Diurnall High water, about as many Hours after the Moons coming to South; (I mean, At Sea: for in Chanels it varies to all Hours, according as they are nearer or further from the open Sea:) And the Annual High-Tydes of November & February; somewhat later than (what I conjecture to be from the same causes) the greatest Inaequalities of the Natural Days, happening in January & October. But this though I can admit, yet (till I am sure of the matter of Fact) I do not build upon. And since it hath hitherto been the custome, to speak with

6 a (1) sp breaks off (2) suspition 8 to (1) speak ab breaks off (2) say, 8 than add. 10 sense (1) at (2) in 11 intimations (1) in m breaks off (2) even 12 yet, (1) I (2) because 14 Vessel, (1) upon (2) by reason of 23 conjecture to (1) the cause of it (2) be 25 not (1) or (2) build 450

5. To the 5th Objection ... foul of the other: cf. WALLIS-OLDENBURG 2/[12].VI.1666 ('To the particulars . . . ' ) . The following discussion is not to be found in any of the earlier letters. 257

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that laxness of expression; assigning the Times of New-moon, Full-moon, & Quadratures, with the Moons coming to South; for, what is near those times: I did not think myself obliged in my conjectural! Hypothesis, (while it is yet but a Candidate) to speak more nicely. If the Hypothesis for the main of it be found Rationall; the Niceties of it are to be adjusted, in time, from particular Observation. Having thus given you some Answere to the Objections you signify to have been made to my Hypothesis; & that in the same order your Paper451 presents them to mee: I shall next give you some account of the Two Books which you advised mee to consult, so far as seemes necessary to this busynesse: Which, upon your intimation, I have since perused; though before I had not. And first, as to that of Isaac Vossius, De motu marram et ventorum; Though I do not concurr with him in his Hypothesis; That all the great Motions of the Seas &c, should arise onely from so small a warming of the water as to raise it (where most of all) not a Foot in perpendicular, (as in his 12th Chapter;) Or that there is no other connexion between the Moons motion, & the Tydes menstrual period, than a casual Synchronism (which seems to be the doctrine of his 16th & 18th chapters;) Beside many other things in his Philosophy, which I cannot allow: Yet I am well enough pleased with what is Historical in it, of the matter of Fact: Especially if I may be secure, that he is therein accurate & candide, not wresting the Phaenomena to his own purpose. But I find nothing in it, which doth induce me to vary from my Hypothesis. For, granting his Historicals to be all true; the account of the constant Current of the Sea Westward, & of the constant Eastern Blasts, &c, within the Tropicks- is much more plausibly, & (I suppose) truly, rendered by Galilaeo long since, from the Earths Diurnall motion; (which, near the Aequator describing a greater Circle, than nearer the Poles, makes the Current to be there [4] more conspicuous & swift; &, consequently, the Eddy, or recurrent motion,

1 that (1) car breaks off (2) laxness 2 South; (1) for the times (2) for, 7 some (1) of (2) Answere to the Objections you signify to (a) be (6) have been made |by severall persons add. Oldenburg\ to 23 to his (1) Hypothesis. (2) own purpose. 24 my |own del.\ Hypothesis. 451

your Paper: i.e. Oldenburg's note entitled Objections against Dr Wallis's Hypothesis of Tides [VI7.1666]. 258

116. WALLIS to OLDENBURG, 18/[28] July 1666 nearer the Poles, where this is more remisse:) than can easyly be rendered by so small a Tumor, as hee supposeth. Not to adde; that his account of the Progressive motion, which he fansieth to follow upon this Tumefaction, and by Acceleration to grow to so great a height near the Shoar, (as in Chap. 13. & 14.) is a notion; which seems to mee too extravagant to be salved by any laws of Staticks. And that, of the moons motion onely Synchronising with the Tydes, casually, without any Physicall connexion; I can very hardly assent to. For it can hardly be imagined, that any such constant Synchronism should be in nature; but where, either the one is the cause of the other, or both depend upon some common cause. And where we see so fair a foundation for a Physical connexion; I am not prone, to ascribe it to an Independent Synchronism. In summe; His History doth well enough agree with my Hypothesis; & I think, the Phaenomena much better salved by mine, than his.452 And then as to Gassendus, in his second Tome De Philosophia Epicuri, where hee speaks of De Aestu Mans453: I find him, after the relating of many other Opinions concerning the cause of it, inclining to that of Ga/«7aeo454, ascribing it to the Acceleration & Retardation of the Earths motion, compounded of the Annual & Diurnall: And moreover attempting to give an account of the Menstrual Periods from the Earths carrying the Moon about itself, as Jupiter doth his Satellites; which together with

13 Phaenomena |are add. Oldenburg] much 15 in his discourse De Aestu Maris; I find him E1 (according to Wallis's instructions) 16 hee discourses De Aestu Maris: alt. Oldenburg 16 I find (1) , after many other Opinions related (2) him, after the relating of many other Opinions 18 Accelation corr. Oldenburg 20 Periods |, (which Galilaeo doth not, as in my Hypothesis I have intimated,) add. and del.\ from 452

And first, as to that of Isaac Vossius ... than his: cf. WALLIS-OLDENBURG 8/[18].VL1666. 453 where hee speaks of De Aestu Maris: i.e. GASSENDI, Syntagma philosophicum, pars 2 ('Physica'), sectio 3, membrum prius ('De rebus terrenis inanimis'), liber 1 ('De globo ipso telluris'), caput 4 ('De aestu, seu fluxu fe refluxu maris'); Opera omnia (1658), II, 27-33. Cf. WALLIS-OLDENBURG 2/[12].VIIL1666. 454 that of Galilaeo: i.e. GALILEO, Dialogo sopra i due massimi sistemi del mondo, Florence 1632, 'dialogo quarto'. 259

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them is carried about by the Sunne, as one Aggregate. (And that the Earth with its moon is to be supposed in like manner to be carried about by the Sun, as one Aggregate; cannot be reasonably doubted, by those who intertain the Copernican Hypothesis, & do allow the same of Jupiter & his Satellites.) But though he would thus have the Earth & Moon looked upon as two parts of the same Moved Aggregate; yet he doth still suppose (as Galileo had done before him,) that the line of the Mean Motion of this Aggregate, (or, as he calls it, motus aequabilis ac veluti medius,) is discribed by the Center of the Earth, (about which center he supposeth both its own revolution to be made, and an Epicycle described by the Moons motion;) not by another Point, distinct from the centers of both, about which, as the common center of gravity, as well that of the Earth, as that of the Moon, are to describe several Epicycles. And, for that Reason, fails of giving any clear account of this Menstrual Period. (And in like manner, he proposeth the consideration as well of the Earths Aphelium & Perihelium, as of the Aequinoctial & Solstitial Points, in order to the finding a Reason of the Annual Vicissitudes; but doth not fix upon any thing, in which himself can Acquiesce: And therefore, leaves it in medio as he found it.) It had been more agreeable to the Laws of Staticks, if he had, (as I do,) so considered the Earth & Moon as two parts of the same moveable, (not so, as he doth, aliam in Centra et sequentem praecise revolutionem axis, aliam remotius ac velut in circumferentia, but,) so, as to make nei-

1 by add.

1 Aggregate (1) : fc so the Earth with its Moon. (2) . (And 2 in like manner add. 5 But (1) forasmuch as (2) though 7 the Mean add. 8 (or, as he ... medius,) add. 9 he supposeth add. 10 made, (1) & the (2) and |an add.\ Epicycle 11 distinct (1) from (2) from the centers of both, (a) which is (2) about which, as 13 are to add. 14 Menstrual (1) motion (2) Period. 15 of the (1) Earths A breaks off (2) Earths 18 leaves |it add.\ in 20 , (as I do,) add. 22 (not (1) as (2) so 260

116. WALLIS to OLDENBURG, 18/[28] July 1666 ther of them the Center, but both out of it, describing Epicycles about it: like as, when a long stick thrown in the Ayr, whose one end is heavyer than the other, is whirled about, so as that the End, which did first fly foremost, becomes hindmost; the proper line of motion of this whole body is not that, which is described by either end, but that, which is described by a middle point between them; about which point each end, in whirling, describes an Epicycle. And indeed, in the present case, it is not the Epicycle described by the Moon, but that, described by the Earth, which gives the Menstrual vicissitudes of motion to the water; which would, as to this, be the same, if the Earth so move, whether there were any moon to move or not; nor would the Moons motion, supposing the Earth to hold-on its [5] own course, any whit concern the motion of the Water. | But now, (after all our Physical, or Staticall Considerations,) the clearest Evidence for this Hypothesis (if it can be had) will bee from Celestiall Observations. As for instance; Supposing the Sun at 5; the Earths place in its Annual Orb at T; and Mars (in opposition to the Sun, or near it) at M; From whence Mars should appear in the Zodiack at 7, & will at Full-moon be seen there to be; the Moon being at C & the Earth at c: (and the like at the New moon.) But if the Moon be in the First quarter at A, & the Earth at o; Mars will be seen not at 7, but at a; too slow: And when the Moon is at B, & the Earth at 6, Mars will be seen at /?; yet too slow: till at the Full moon, the Moon at (7, the Earth at c, Mars will be seen at 7, its true place, as if the Earth were at T. But then, after the

2 stick (1) & (2) thrown 3 which (1) was fi breaks off (2) did 4 the (1) rep breaks off (2) proper 4 of this (1) Aggregate (2) whole 6 in whirling, add. 8 Moon, (1) (which hath (2) but 9 water; |& del. which 10 moon (1) or not (2) to 13 Physical, (1) fe (2) or 14 for this Hypothesis add. 14 (if it (1) could (2) can be had) (a) would (b) will 17 & (1) would (2) will 20 Mars (1) would (2) will 21 Mars (1) would (2) will 22 Mars (1) would (2) will

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Full, the Moon at D, the Earth at d; Mars will be seen, not at 7, but at <5; too forward: & yet more, when the Moon (at the last Quarter) is at E, the Earth at e, & Mars seen at e. If therefore Mars (when in opposition to the Sun) be found (all other allowances being made) somewhat too backward before the Full-moon, & somewhat too forward after the Full-moon, (and most of all, at the Quadratures:) it will be the best confirmation of the Hypothesis. (The like may be fitted to Mars in other positions, mutatis mutandis; & so for the other Planets.)455 But this proof, is of like nature as that of the Parallaxis of the Earths Annual Orb to prove the Copernican Hypothesis. If it can be observed; it proves the Affirmative: but if it cannot be observed; it doth not convince the Negative, but onely proves that the Semidiameter of the Earths Epicycle is so small as not to make any discernable Parallax. And indeed, I doubt, that will bee the Issue. For the Semidiameter of this Epicycle, being little more than the Semidiameter of the Earth itself, or about 11 thereof (as is conjectured, in the Hypothesis, from the Magnitudes & Distances of the Earth & Moon compared;) and there having not as yet been observed any discernable Parallax of Mars, even in his nearest position to the Earth; it is very suspicious, that here it may prove so too. And whether any of the other Planets will be more favourable in this point, I cannot say. And thus you have an Abstract at once, of what I have in several letters given you in answere to the objections which at several times you have represented. Nor shall I further inlarge at present, that I be not over tedious. 1 Mars (1) would (2) will 2 at E, |fe del] the 5 somewhat |to corr. ed.\ (1) backward (2) forward 7 (The like (1) make (2) may 8 for (1) ot breaks off (2) the 9 of (1) the (2) like 9 of the (1) Gr breaks off (2) Earths 15 than (1) that of (2) the 15 itself, |or about l| thereof add] (as (1) wee have (2) is 17 not add. 455

But now, (after . . . for the other Planets.): cf. WALLIS-OLDENBURG 2/[12].VI.1666 ('But that which will give . . . ' ) . See below for the corresponding figure printed in E1, a manuscript source of which has survived neither with WALLIS-OLDENBURG 2/[12].VL1666 nor with the present letter. 262

117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 Yours &c. John Wallis. (E\ figure)

117. WALLIS to [HENRY OLDENBURG] Oxford, 24 July/[3 August] 1666 Transmission:

E First edition of missing letter sent: Philosophical Transactions No. 16 (6 August 1666), 289-94 ('Animadversions of Dr. Wallis, upon Mr Hobs's late Book, De Principiis & Ratiocinatione Geometrarum.') (our source).—reprinted: OLDENBURG, Correspondence III, 193-8. w Latin translation of E: LONDON Royal Society MS. 368, No. 2, pp. 45-54. Date and place of the letter, but not its addressee are mentioned in Philosophical Transactions: 'These [i.e. Animadversions] were communicated by way of Letter, written in Oxford, July 24. 1666. to an Acquaintance of the Author . . . ' (p. 289). However, it seems

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117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 to be clear from WALLIS-OLDENBURG 2/[12].VIII.1666, that it was addressed to Oldenburg. According to Wallis's instructions given in WALLIS-OLDENBURG 2/[12].VIIL1666, the printed version of the letter was enlarged by one paragraph (see critical apparatus). Wallis sent a Latin translation (w) of this letter (and of his letters to Boyle of 25.IV/[5.V].1666 and to Oldenburg of 24.VII/[3.VIII].1666, which likewise were published in Philosophical Transactions No. 16) to Oldenburg in February 1667/8; see OLDENBURG-WALLIS 24.XII.1667/[3.I.1668], WALLIS-OLDENBURG l/[ll].II.1667/8, and WALLIS-OLDENBURG 8/[18].II. 1667/8.

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Since I saw you last, I have read over Mr. Hobs's Book456 Contra Geometras (or De Principiis & Ratiocinations Geometrarum) which you then shewed me. A New Book of Old matter: Containing but a Repetition of what he had before told us, more than once; and which hath been Answered long agoe. In which, though there be Faults enough to offer ample mat ter for [290] a large Confutation: yet I am scarce inclined to believe, that any will bestow so much pains upon it. For, if that be true, which (in his Preface) he saith457 of himself, Aut solus insanio Ego, aut solus non insanio; it would either be Needless, or to no Purpose. For by his own confession, All others, if they be not mad themselves, ought to think Him so: And therefore, as to Them, a Confutation would be needless; who, its like, are well enough satisfied already: at least out of danger of being seduced. And, as to himself, it would be to no purpose. For, if He be the Mad man, it is not to be hoped that he will be convinced by Reason: Or, if All We be so; we are in no capacity to attempt it. But there is yet another Reason, why I think it not to need a Confutation. Because what is in it, hath been sufficiently confuted already; (and, so Effectually; as that he professeth himself not to Hope, that This Age is like to give sentence for him; what ever Nondum imbuta Posteritas may do.) Nor doth there appear any Reason, why he should again Repeat it, unless he can hope, That, what was at first False, may by oft Repeating, become True. I shall therefore, instead of a large Answer, onely give you a brief Account, what is in it; &, where it hath been already Answered. 456 Book: i.e. HOBBES, De principiis et ratiocinations Geometrarum ubi ostenditur incertitudinem falsitatemque non minorem inesse scriptis eorum, quam scriptis physicorum et ethicorum. Contra fastum professorum Geometriae, London 1666; HOBBES, Opera philosophica IV, 385-465. 45r saith: i.e. HOBBES, De principiis et ratiocinatione Geometrarum, sig. A3Y.

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117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 The chief of what he hath to say, in his first 10 Chapters, against Euclids Definitions, amounts but to this, That he thinks, Euclide ought to have allowed his Point some Bigness; his Line, some Breadth; and his Surface, some Thickness. But where in his Dialogues4"58, pag. 151, 152. he solemnly undertakes to Demonstrate it; (for it is there, his 41th Proposition:) his Demonstration amounts to no more but this; That, unless a Line be allowed some Latitude; it is not possible that his Quadratures can be True. For finding himself reduced to these inconveniences; 1. That his Geometrical Constructions, would not consist with Arithmetical calculations, nor with what Archimedes and others have long since demonstrated: 2. That the Arch of a Circle must be allowed to be sometimes Shorter than its Chord, and sometimes longer than its Tangent: 3. That the same Straight Line must be allowed, at one place onely to Touch, and at another place to Cut the same Circle: (with others of like nature;) He findes it necessary, that these things may not seem Absurd, to allow his Lines some Breadth, (that [291] so, as he speaks, While a Straight Line with its Out-side doth at one place] Touch the Circle, it may with its In-side at another place Cut it, &c.) But I should sooner take this to be a Confutation of His Quadratures, than a Demonstration of the Breadth of a (Mathematical) Line. Of which, see my Hobbius Heauton-timorumenus4159, from pag. 114. to p. 119. And what he now Adds, being to this purpose; That though Euclid's Ytrmeiov, which we translate, a Point, be not indeed Nomen Quanti; yet cannot this be actually represented by any thing, but what will have some Magnitude; nor can a Painter, no not Apelles460 himself, draw a Line so small, but that it will have some Breadth; nor can Thread be spun so Fine, but that it will have some Bigness; (pag. 2, 3, 19, 21.) is nothing to the Business; for Euclide doth not speak either of such Points, or of such Lines. He should rather have considered of his own Expedient, pag. 11. That, when one of his (broad) Lines, passing through one of his (great) Points, is supposed to cut another Line proposed, into two equal parts; we are to understand, the Middle of the breadth of that Line, passing through the middle of that Point, to distinguish the Line given into two equal 458 Dialogues: i.e. HOBBES, Examinatio et emendatio mathematicae hodiernae, London 1660. 459 Hobbius Heauton-timorumenus: i.e. WALLIS, Hobbius Heauton-timorumenos, Oxford 1662. *60ApeUes: i.e. Apelles of Kolophon (2nd half of 4th century BC), Greek painter.

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parts. And he should then have considered further, that Euclide, by a Line, means no more than what Mr. Hobs would call the middle of the breadth of his; and Euclide's Point, is but the Middle of Mr. Hobs's. And then, for the same reason, that Mr. Hobs's Middle must be said to have no Magnitude; (For else, not the whole Middle, but the Middle of the Middle, will be in the Middle: And, the Whole will not be equal to its Two Halves; but Bigger than Both, by so much as the Middle comes to:) Euclide !s Lines must as well be said to have no Breadth; and his Points no Bigness. What he saith here, Chap. 8. & 19. (and in his fifth Dial. p. 105. &c.) [292] concerning the Angle of Contact; amounts but to thus much, That, by the Angle of Contact, he doth not mean either what Euclide calls an Angle, or any thing of that kind; (and therefore says nothing to the purpose of what was in controversie between Clavius^1 and Peletarius^2, when he says, that An Angle of Contact hath some magnitude:) But, that by the

9-10 Bigness. || In like manner, When Euclide and others do make the Terme or End of a Line, a Point: If this Point have Parts or Greatness, then not the Point, but the Outer-Half of this Point ends the Line, (for, that the Inner-Half of that Point is not at the End, is manifest, because the Outer-Half is beyond it:) And again, if that Outer Half have Parts also; not this, but the Outer part of it, and again the Outer part of that Outer part, (and so in infinitum.) So that, as long as Any thing of Line remains, we are not yet at the End: And consequently, if we must have passed the whole Length, before we be at the End; then that End (or Punctum terminans) has nothing of Length; (for, when the whole Length is past, there is nothing of it left. And if Mr. Hobs tells us (as pag. 3.) that thisj End is not Punctum, but only Signum (which he does allow non esse nomen Quanti) even this will serve our turn well enough. Euclid's Sr/^etoi/, which some Interpreters render by Signum, others have thought fit (with Tally) to call Punctum,: But if Mr. Hobs like not that name, we will not contend about it. Let it be Punctum,, or let it be Signum (or, if he please, he may call it Vexillum.) But then he is to remember, that this is only a Controversie in Grammar, not in Mathematicks: And his Book should have been intituled Contra Grammaticos, not, Contra Geometras. Nor is it Euclide, but Cicero, that is concern'd, in rendring the Greek S?7/i«oz/, by the Latine Punctum, not by Mr. Hobs's Signum. The Mathematician is equally content with either word. || What inserted in E (according to Wallis's instruction in WALLIS-OLDENBURG 2/[12].VIII.1666; 461

Clavius: i.e. Christoph Clavius (1538-1612), Jesuit priest and teacher of mathematics in the Jesuit College at Rome. 462 Peletarius: i.e. Jacques Peletier (1517-82), French mathematician and physician. He became embroiled in a lengthy controversy with Clavius over the angle of contact. 266

[292]

117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 Angle of Contact, he understands the Crookedness of the Arch] and in saying, the Angle of Contact hath some magnitude, his meaning is, that the Arch of a Circle hath some crookedness, or, is a crooked line: and that, of equal Arches, That is the more crooked, whose chord is shortest: which I think none will deny; (for who ever doubted, but that a circular Arch is crooked! or, that of such Arches, equal in length, That is the more crooked, whose ends by bowing are brought nearest together?) But, why the Crookedness of an Arch, should be called an Angle of Contact; I know no other reason, but, because Mr. Hobs loves to call that Chalk, which others call Cheese. Of this see my Hobbius Heauton-timorumenus, from pag. 88 to p. 100. What he saith here of Rations or Proportions, and their Calculus; for 8. Chapters together, (Chap. 11. &c,) is but the same for substance, what he had formerly said in his 4th. Dialogue, and elsewhere. To which you may see a full Answer, in my Hobbius Heauton-tim. from pag. 49. to p. 88. which I need not here repeat. Onely (as a Specimen of Mr. Hobs's Candour, in Falsifications) you may by the way observe, how he deals with a Demonstration of Mr. Rook's^3, in confutation of Mr. Hobs's Duplication of the Cube: Which when he had repeated, pag. 43. He doth then (that it might seem absurd) [293] change those words, aequales] quatuor cubis DV; (pag. 43. line 33.) into these (p. 44. /. 5.) aequalia quatuor Lineis, nempe quadruplus Rectae DV: And would thence perswade you, that Mr. Rook had assigned a Solide, equal to a Line. But Mr. Rook's Demonstration was clear enough for Mr. Hobse's Comment. Nor do I know any Mathematician (unless you take Mr. Hobs to be one) who thinks that a Line multiplyed by a Number will make a Square; (what ever Mr. Hobs is pleased to teach us.) But, That a Number multiplyed by a Number, may make a Square Number; and, That a Line drawn into a Line may make a square Figure, Mr. Hobs (if he were, what he would be thought to be) might have known before now. Or, (if he had not before known it) he might have learned, (by what I shew him upon a like occasion, in my Hob. Heaut. pag. 142. 143. 144.) How to understand that Language, without an Absurdity. Just in the same manner he doth, in the next page, deal with Clavius. For having given us his words, pag. 45 1. 3. 4. Dico hanc Lineam Perpendicularem extra circulum cadere (because neither intra Circulum, 463

Mr. Rook's: i.e. Lawrence Rooke (1622-62), astronomer, professor of astronomy (from 1652) and professor of geometry (from 1657) at Gresham College in London, DNB. 267

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nor in Peripheria;) He doth, when he would shew an errour, first make one, by falsifying his words, line 15. where instead of Lineam Perpendicularem, he substitutes Punctum A. As if Euclide or Clavius had denyed the Point A. (the utmost point of the Radius,) to be in the Circumference: Or, as if Mr. Hobs, by proving the Point A, to be in the Circumference, had thereby proved, that the Perpendicular Tangent AE had also lyen in the Circumference of the Circle. But this is a Trade, which Mr. Hobs doth drive so often, as if he were as well faulty in his Morals, as in his Mathematicks. The Quadratures of a Circle, which here he gives us, Chap. 20. 21. 23. is one of those Twelve of his, which in my Hobbius Heauton-timorumenus (from pag. 104. to pag. 119) are already confuted: And is the Ninth in order (as I there rank them) which is particularly considered, pag. 106. 107. 108. I call it One, because he takes it so to be; though it might as well be called Two. For, as there, so here, it consisteth of Two branches, which are Both False; and each overthrow the other. For if the Arch of a Quadrant be equal to the Aggregate of the Semidiameter and of the Tangent of 30. Degrees, (as he would Here have it, in Chap. 20. and There, in the close of Prop. 27;) Then is it not equal to that Line, whose Square is equal to Ten squares of the Semiradius, (as, There, he would have it, in Prop. 28. and, Here, in Chap. 23.) And if it be equal to This, then not to That. For This, and That, are not equal: As I then demonstrated; and need not now repeat it. The grand Fault of his Demonstration (Chap. 20.) wherewith he would now New-vamp his old False quadrature; lyes in those words Page 49. line 30, 31. Quod Impossibile est nisi ba transeat per c. which is no impossibility at all. For though he first bid us draw the Line Re, and afterwards the Line Rd: Yet, Because he hath no where proved (nor is it true) that these two are the same Line; (that is, that the point d lyes in the Line Re, or that Re passeth through d:) His proving that Rd cuts off from ab a Line equal to the line of Be, doth not prove, that ab passeth through c: For this it may well do, though ab lye under c, (vid. in case d lye beyond the line Re, that is, further from A:) or though it lye above c, (vid. in case d be nearer, than Re, to the point A.) And therefore, unless he first prove (which he cannot do) that Ad (a sixth part of AD) doth just reach to the line Re and no further; he onely proves| that a sixth [294] part of ab is equal to the Line of Be. But, whether it lye above it or below it, or (as Mr. Hobs would have it) just upon it; this argument doth not 268

117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 conclude. (And therefore Hugenius's assertion464, which Mr. Hobs, Chap. 21. would have give way to this Demonstration, doth, notwithstanding this, remain safe enough.) His demonstration of Chap. 23. (where he would prove, that the aggregate of the Radius and of the Tangent of 30. Degrees is equal to a Line, whose square is equal to 10 Squares of the Semiradius;) is confuted not onely by me, (in the place forecited; where this is proved to be impossible;) but by himself also, in this same Chap. pag. 59 (where he proves sufficiently and doth confesse, that this demonstration, and the 47. Prop. of the first of Euclide, cannot be both true.) But, (which is worst of all;) whether Euclid's Proposition be False or True, his demonstration must needs be False. For he is in this Dilemma: If that Proposition be True, his demonstration is False, for he grants that they cannot be both True, page 59 line 21. 22. And again, if that Proposition be False, his Demonstration is so too; for This depends upon That, page 55. line 22. and therefore must fall with it. But the Fault is obvious in His Demonstration (not in Euclid's Proposition:) The grand Fault of it (though there are more) lyes in those words, page 56. line 26. Erit ergo MO minus quam MR. Where, instead of minus, he should have said majus. And when he hath mended that Error; he will find, that the major in page 56. line penult, will very well agree with majorem in page 57. line I (where the Printer hath already mended the Fault to his hand) and then the Falsum ergo will vanish. His Section of an Angle in ratione data', Chap. 22. hath no other foundation, than his supposed Quadrature of Chap. 20. And therefore, that being false; this must fall with it. It is just the same with that of his 6. Dialogue. Prop. 46. which (besides that it wants a foundation) how absurd it is, I have already shewed; in my Hobbius Heauton-timor. page 119. 120. His Appendix465, wherein he undertakes to shew a Method of finding any number of mean Proportionals, between two Lines given: Depends upon the supposed Truth of his 22. Chapter; about Dividing an Arch in any proportion given: (As himself professeth: and as is evident by the Construction; which supposeth such a Section.) And therefore, that failing, this falls with it. 464

Hugenius's assertion: see HUYGENS, De circuli magnitudine inventa, Leiden 1654, theor. IX prop. IX; (Euvres completes XII, 136-9. 465 Appendix: i.e. De mediis proportionalibus in genere; HOBBES, Opera philosophica IV, 466-84.

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And yet this is otherwise faulty, though that should be supposed True. For, In the first Demonstration; page 67. line 12. Producta Lf incidet in I; is not proved; nor doth it follow from his Quoniam igitur. In the second Demonstration; page 68. line 34. 35. Recta Lf incidit in x; is not proved; nor doth it follow from his Quare. In his third Demonstration; page 71: line 7. Producta YP transibit per M; is said gratis', nor is any proof offered for it. And so this whole structure falls to the ground. And withall, the Prop. 47. El. I doth still stand fast (which he tells us, page 59, 61, 78. must have Fallen, if his Demonstration had stood:) And so, Geometry and Arithmetick do still agree, which (he tells us, page 78: line 10.) had otherwise been at odds. And this (though much might have been said,) is as much as need to be said against that Piece.

118.

HENRY OLDENBURG to WALLIS 26 July/[5 August] 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 2/[12].VIII. 1666. In this letter, as in the one following it, Oldenburg related political news concerning the Second Anglo-Dutch Sea War and discussed the printing of Wallis's three contributions to the Philosophical Transactions for August. Cf. WALLis-BoYLE 25.IV/[5.V].1666, WALLIS-OLDENBURG 18/[28].VII.1666, and WALLIS-OLDENBURG 24.VII/[3.VIII].1666.

119.

HENRY OLDENBURG to WALLIS 31 July/[10 August] 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 2/[12].VIII. 1666.

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120. WALLIS to OLDENBURG, 2/[12] August 1666 In this letter, as in OLDENBURG-WALLIS 26.VII/[5.VIII]. 1666, Oldenburg related political news and discussed the printing of Wallis's contributions to the Philosophical Transactions for August.

120.

WALLIS to HENRY OLDENBURG Oxford, 2/[12] August 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 26, 4 pp. (our source). Postmark on p. 4: 'AU/{—}'.—printed: OLDENBURG, Correspondence III, 203-5. Reply to: OLDENBURG-WALLIS 26.VII/[5.VIII].1666 and OLDENBURG-WALLIS 31.VII/ [10. VIII].1666. The present letter is concerned chiefly with the forthcoming publication of WALLISBOYLE 25. IV/[5.V].1666, WALLIS-OLDENBURG 18/[28].VII.1666, and WALLIS-OLDENBURG 24.VII/[3.VIII].1666 in the Philosophical Transactions. Oldenburg carried out all of Wallis's instructions in time for the issue of August.

Oxford. Aug. 2. 1666.

Sir, I thank you for your two Letters, of Thursday, & Tuesday last466. And for your Intelligence in both. Which I looked upon as the most rationall of any letters that came hither, (at either of those times,) that I mett with. I hear not of that of De Ruyter467 by any Letter but yours. But wee had some suspitious conjectures of his death before, as well because of their suddain flight; as from a circumstance that some bodyes letters related; that the Dutch Admirall, were observed to take-in their Flag. Of which wee could make no other conjecture, but that it was an intimation to the Fleet, that they were thence forth to take their Orders from the 3 Letters, jthat del] of 7 some (1) suspitions (2) suspitious conjectures 7 death (1) by (2) before 466

your two Letters, ... last: i.e. OLDENBURG-WALLIS 26.VII/[5.VIII].1666 and OLDENBURG-WALLIS 31.VII/[10.VIII].1666. 467 De Ruyter: i.e. Michiel Adriaanszoon de Ruyter (1607-76), Dutch admiral, who fought in the Anglo-Dutch 'St James's Day Fight' on 25 and 26 July 1666. Oldenburg was misinformed in believing that the admiral had been killed in this battle. 271

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next superiour officer. But even of this wee are no more certain than of his death, if it bee so. But wee wonder here, that in the Narrative there is no notice taken of what Sir Jeremy Smith's468 Squadron did: which makes us doubt, that the Elizabeth's News of his Firing 4 of the Dutch Ships, comes to nothing. As to the businesse of Mr Hobbs469; If it be not yet too late: I think it much better that it should come into the book of this Month; than stay till September. For it will seem too short a piece, to bee so long about: being, indeed, but little more than a days work; scarce a day & a half. I am apt to think; that the whole, both of the Tides470, & this, will come into less then 4 sheets; perhaps 3, or 3^. In my Hypothesis of Tides471; I forgot to blott out those few lines, (about Romney Marsh,) in the former place; which come again (for substance) towards the End: & therefore might be spared in the former place. In my Letter of it472: Where Gassendus is cited; I have (according to my Book) cited what he hath De Aestate, as in his Philosophia Epicurea which I now find; in the last Edition of his Works to be removed into his Physicks473. Therefore let that Citation be onely De Aestu Man's, without referring to either Book: &| so the Reader may seek it in whether of the [2]

1 superiour (1) sup breaks off (2) officer 8 will (1) (—} (2) {—) (3) seem 12 Tides; (1) is (2) I 15 In my (1) letter paper (2) Letter 16 cited (1) him as (2) what 17 to be (1) in his Physica (2) removed 18 Marls (1) {—} (2) without 468

Smith's: i.e. Sir Jeremy Smyth, commander of the 'Blue Squadron' in the battle on 25 and 26 July 1666. 469 businesse of Mr Hobbs: i.e. the publication of WALLIS-OLDENBURG 24.VII/ [3.VIII].1666, on the topic of Hobbes's De principiis et ratiocinatione geometrarum, in Philosophical Transactions. 470 both of the Tides: i.e. WALLIS-BOYLE 25.IV/[5.V].1666 and WALLIS-OLDENBURG 18/[28].VIL1666. 471 In my Hypothesis of the Tides . . . : cf. WALLIS-BOYLE 25.IV/[5.V].1666. 472 In my Letter of it ...: cf. WALLIS-OLDENBURG 18/[28].VII.1666. 473 what he hath De Aestate ... into his Physicks: i.e. GASSENDI, Syntagma philosophicum, pars 2 ('Physica'), sectio 3, membrum prius ('De rebus terrenis inanimis'), liber 1 ('De globo ipso telluris'), caput 4 ('De aestu, seu fluxu & refluxu maris'); Opera, omnia (1658), II, 27-33. Cf. GASSENDI, Philosophiae Epicuri syntagma; Opera omnia (1658), III, 1-94. 272

120. WALLIS to OLDENBURG, 2/[12] August 1666 two hee please: Unless you should think fit to cite Both; which may be done in the margin. In that against Mr Hobs474, (where I speak of a Points having no Greatness, because else the whole would contain somewhat more then the two halves, viz. so much as is the Middle.) You may adde to this purpose, if you please. In like manner; When Euclide & others, do make the Term or End of a Line, a Point. If this Point have Parts, or Greatnesse: then not the Point, but, the Outer Half of this Point ends the Line, (for, that the Inner Half of that Point is not at the End, is manifest; because the Outer half is beyond it.) And again, if that Outer Half have Parts allso; not this, but the Outer Part of it; & again the Outer part of that Outer Part; (and so in infinitum.) So that, as long as Anything of Line remains, wee are not yet at the End. And, consequently, if we must have passed the Whole Length before we be at the End; then that End (or Punctum Terminans) hath nothing of Length; (for when the Whole Length is past, there is nothing of it left.) And, if Mr Hobs, tell us, (as pag. 3.) That this End is not Punctum; but onely Signum (which he doth allow non esse nomen Quanti:) even This will serve our turn well inough. Euclide's ST//IO.OZ/, which some Interpreters render by Signum, others have thought fit (with Tw%475) to call Punctum: But if

1 two (1) you (2) hee I Both (1) . (2) ; which may be done in the margin. 7 Euclide (1) fee (2) & others, 8 or Greatnesse: add. 10 Half add. I I because the Outer half is beyond it. add. 13 Part; (1) fee. (2) (and 17 (or Punctum Terminans) add. 18 Whole \Length add] is 19 left.) (1) So that the Very End, (a) or (&) that is the Punctum Terminans, (2) And therefore no Magnitude. (3) And, if 19 (as pag. 3.) add. 20 is (1) onely Signum (2) not 474 475

In that against Mr Hobs . . . : cf. WALLIS-OLDENBURG 24.VII/[3.VIII].1666. Tully: i.e. Cicero.

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120. WALLIS to OLDENBURG, 2/[12] August 1666 Mr Hobs like not that Name] wee will not contend about it; let it be Punctum, or let it be Signum, (or, if he please, he may call it Vexillum.) But then hee is to remember, that this is onely a controversy, in Grammar', not, in Mathematicks: And his Book should have been intituled Contra Grammaticos; not, Contra Geometras. Nor is it Euclide, but Cicero, that is concerned, in Rendering the Greek Sr/^etoi/, by the Latine Punctum, not by Mr Hobs's Signum. The Mathematician is equally content with either Word.

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If it be too late to put that confutation into the Transactions of this month: Yet you may at lest give an Advertisement in the end; that you had Received a Letter from mee containing] a confutation of Mr Hobsjs [3] book Contra Geometras mentioned in your last Months Transactions. Which Letter you would here have entered; but that it would have made the account of this Months Transactions too much exceed the Proportion of other Month's; & therefore you have thought fit to reserve it to be inserted in the next. I have no more to adde at present, but that I am. yours &c. J. W.

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I shall call upon Dr Wren about that Book476. But I doubt hee will scarce be persuaded to make Observations477. For I find him no way propense to it. And, without him, I know not well how to undertake it, being destitute of Instruments for it. 25

[4]

These For Mr Henry Oldenburgh, at 1 like (1) it not; (2) not that Name; wee will not contend |with him del] about (a) the Name; (b) it; let it be Punctum, or (ao) let (66) Sign breaks off (cc) let 10 put (1) this (2) that (a) Letter (6) confutation 12 Received (1) (—) (2) a Letter 13 your add. 13 Transactions. (1) Showing (2) Which 476

that Book: not identified. Observations: This refers to the determination of the geographical bearing of Madrid from London conducted in collaboration with the Earl of Sandwich; cf. WALLISOLDENBURG 2/[12].VI.1666. 477

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121. JOHN COLLINS to WALLIS [London], 2/[12] August [1666] Transmission:

C Draft of (missing) letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 186r-186v (our source).—printed: RIGAUD, Correspondence of Scientific Men II, 463-5. Answered by: WALLIS-COLLINS 7/[17].VIII.1666.

Reverend Sir Upon the expectation of your arrivall here I have hitherto forborne to send you my booke of accompts478, which is so excentrique to your studies as I thought it unworthy your acceptance, but notwithstanding have sent it herewith. That Pasted quadrant was of Mr. Markes479 fixing, but now he is more carefull. As to my Navigation480 it is altogeather unworthy the view of a Geometer or that Character you are pleased to affoard it. I am informed you have had the view of an Algebraick treatise of Bartholinus481 intituled Dioristice482. In regard it is not to be had, my humble request is that you would vouchsafe me the title of it exactly with the time and 6 so (1) meane (2) excentrique 7 acceptance, (1) which notwithstanding is herewith sent you. (2) but notwithstanding have sent it herewith. 9 carefull (1) , and my Navigation is (2) . As to my Navigation it is 9 unworthy (1) of that (2) the view of a Geometer or that 478

booke of accompts: i.e. COLLINS, An Introduction to Merchants Accounts, London 1653. 479 Markes: i.e. John Marke (fl. 1665-79). 480 Navigation: i.e. COLLINS, The Mariners Plain Scale New Plain'd, London 1659. 481 Bartholinus: i.e. Erasmus Bartholin (Rasmus Berthelsen) (1625-98), Danish mathematician and physician. 482 Dioristice: i.e. BARTHOLIN, Dioristice sen aequationum determinations duabus methodis propositae, Copenhagen 1663. 275

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place of its Edition as also your censure of the Bookes following, if they have passed your view. Josephi Hebraei Bibliotheca mathematica483 8 Francofurti Andraei Alexandri484 Mathemologium485 fo. Sanclari486 Apologia pro Archimede et Euclide487. I never saw any of them, but the two latter are in the Marquese of Dorchesters488 Library. May I presume a little further I would likewise entreate your information concerning the Manuscript489 of Galileos in your Library, whether it be his mechanick Problemes which Mr Salusburie490, whilst living, complained he could not obtaine. And when your Occasions shall next invite you hither, if not to troublesome, another Request would be to have a sight of the thin folio of Valerius491 intituled Subtilium indagationum lib. primus492 to compare with an Approach some of our Gagers make use of to Compute the Segment of an upright Cone betweene two parralell Plaines whereof one passeth through the Axis which is as followeth: As the Diameter of the Base + parralell distance of those plaines To the Semidiameter of the Base 7 presume (1) so farre (2) a, little further 10 not (1) go (2) obtaine. 14 Gagers (1) for (2) make 14 of (1) a (2) an upright 483

Josephus Hebraeus Bibliotheca mathematica: not identified. Alexandri: i.e. Andreas Alexander (b. c.1475), professor of mathematics at the University of Leipzig 1502-4. 485 Mathemologium: i.e. Andreas ALEXANDER, Mathemalogium prime partis Andree alexandri Ratisbonensis mathematici super novam et ve.te.ram logicam Aristotelis, Leipzig 1504. 486 Sanclari: i.e. George Sinclair (d. 1696), professor of philosophy at the University of Glasgow between 1654 and 1666 and again from 1689 until 1691, when he was appointed professor of mathematics at the same institution. DNB. 487 Apologia pro Archimede et Euclide: not identified. 488 Marquese of Dorchesters: i.e. Henry Pierrepont (1606-80), first Marquis of Dorchester (created 1645) and second Earl of Kingston, fellow of the Royal Society, DNB. 489 Manuscript: cf. WALLIS-COLLINS 7/[17].VIII. 1666. 490 Salusburie: i.e. Thomas Salusbury, who edited Mathematical collections and translations, 2 vols., London 1661-5. 491 Valerius: Luca Valerio (1552-1618), Italian mathematician and natural philosopher, member of the Accademia dei Lincei until his expulsion in 1616 for publically teaching the doctrine of Copernicus. 492 Subtilium . . . primus: i.e. VALERIO, Subtilium indagationum liber primus, sen quadratura circuli et aliorum curvilineorum, Rome 1632. 484

276

121. COLLINS to WALLIS, 2/[12] August [1666] So Segment of the Circumscribing Cilinder betweene those plaines To the Segment of the Cone. In the annexed figure, if poi be A through the axis, rp a Semidiameter

of the Base, uo plane through Axis, ae the other Plaine; re : rp :: so Segment Cylinder ua To Segment Cone oneuo. And for the Second Segment of a Sphere to shun the use of the Area of a Segment of a Circle some use the Proportion following: Let a Sphere be cut by 3 parralell Plaines, one passing through the

Center as /c, the other two above it, as be, ad, and let a fourth Plaine [186V] from d to o be erect to the former, | the Approach used is: As the Soliditie of the Centrall Zone cadfc

4 4 6 8 9 9

plane through. Axis add. ae the other (1) parralell distance of the Plaines it is (2) Plaine; Segment of (1) th breaks off (2) a Plaines, |where del] one Center (1) , the other two on (a) one side (6) the same (2) as Plaine (1) as do (2) from d to o

277

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123. WALLIS to COLLINS, 7/[17] August 1666

5

Is to the Solid on the halfe Segment dfod So is the Soliditie of badeb, (which represents a Zone excluding the Center) To the second Segment dend. With what safety and how farre these Approaches may be used I have not abilities sufficient to iudge, vouchsafe not to be offended that I trouble your iudicious eyes therewith, and account me one of those that are desirous to obtaine the aspect of your favour and to manifest himselfe Sir

Your much obliged Servitor J. Collins

10

Aug 2

122.

HENRY OLDENBURG to WALLIS 4/[14] August 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 11/[21].VIII. 1666. This letter apparently contained political news and informed Wallis of the despatch to him of Hevelius's Descriptio Cometae anno 1665.

123.

WALLIS to JOHN COLLINS Oxford, 7/[17] August 1666 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 187r-187v (our source). On left margin of f. 187V at 90° in Collins's hand: 'from Dr Wallis A second Segment Computed'.—printed: RIGAUD, Correspondence of Scientific Men II, 465-7. Reply to: COLLINS-WALLIS 2/[12].VIIL1666. 1 to the (1) hal breaks off (2) Solid 6 not (1) Geom breaks off abilities

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123. WALLIS to COLLINS, 7/[17] August 1666 Oxford Aug. 7. 1666.

Sir, I thank you for yours493 of the 2d instant, & your Book494 of Merchants Accounts which you were pleased to send with it. Which I have perused, & find very full & satisfactory as to what it undertakes. I am sorry I am not able to give you a better account in the particulars of your letter. The Books you mention, whether I have seen all of them, or not, I cannot tell; (for I have so ill a memory at Names, that I can undertake but little for it:) But, if I do not much mis-remember, I saw some of them, if not all, in my Lord Brounkers hands, more then a year ago. When I did onely slightly look them over, having then no more time then onely so to do. I do not remember, that in those I looked on, I did take notice of the contents of them as very considerable for any thing of new discoveryes; but rather illustrating, or putting into some new form, things before known: Which made mee less concerned as to inquiring farther after them. That of Bartholinus's Dioristice495, I do not remember that I have seen at all: But he is like inough to perform well what he undertakes; being formerly a Scholar of Schotens496, & well understanding his way of (as he calls it, Cartesian,) Geometry. I think the other three were amongst those my Lord Brounker shewed mee; & perhaps that of Bartholine's allso. The manuscript of Galilee497, which you saw with mee; is not his Mechanick Problems498; but rather an Introduction to them: Which (that 14 new add. 15 concerned (1) in (2) as to 22 them: (1) Cont breaks off (2) Which 493

yours: i.e. CoLLiNS-WALLis 2/[12].VIIL1666. Book: i.e. COLLINS, An Introduction to Merchants Accounts, London 1653. 495 Dioristice: i.e. BARTHOLIN, Dioristice sen aequationum determinationes duabus methodis propositae, Copenhagen 1663. 496 Schotens: i.e. Frans van Schooten (1615-60). Bartholin edited Schooten's elementary lectures in Geometry on the cartesian method: SCHOOTEN, Principia matheseos universalis seu introductio ad Geometriae methodum R. des Cartes, Leiden 1651. He also translated two pieces by Debaune into Latin for inclusion in Schooten's second edition of Descartes's Geometria (1659/61). 497 manuscript of Galilee: i.e. the Oxford manuscript of Galileo's Traitato delle Meccaniche (Opere II, 155-90), OXFORD Bodleian Library MS. Savile 45 ('Dell' utilita che si traggano della Mechanica fe de' suoi Instromenti'). 498 Mechanick Problems: This treatise on mechanical problems mentioned at the end of Galileo's Trattato delle Meeeaniehe is now lost. See DRABKIN and DRAKE (ed.), Galileo Galilei on motion and on mechanics, 136, 182. 494

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123. WALLIS to COLLINS, 7/[17] August 1666

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I might the better give you an account of it) I have since read over. It's subject is About the Utility of Mechanick Engines, wherein it consists: which hee shews to be chiefly in this, that they do exchange Strength for Time; performing the same in so much a longer Time, as it is done with less Strength. In shewing which, hee explains the Balance, the Leaver, the Wheel, the Pully, the Screw, (&, particularly, Archimedes's Waterscrew,) & the force of a Stroke. Referring, (in the close,) to his Mechanick Problems, which (he says) should be annexed to this discourse. That499 of Valerius, (if I forget it not,) I intend to bring with mee next time I come to London, that you may see it as you desire. When it will bee, I cannot well tell. I passed through London not long since, both as I went into Kent, & as I came back again: But I made so short a stay both times, &; lodged so far out of town when I was near it, that I could not (as I desired) get so much spare time as to make you a visit. Which therefore (with some other things I would not willingly have omitted) I was then forced to let alone. The Methode (you mention) of Approach, in the Second-Segment of a Sphere; may perhaps come pretty near the matter, so as to serve a Gagers occasions (where a preciseness is not attended;) But it allways gives the segment] en(^i bigger than indeed it is. For it is manifest, to the first [187V] view; that the segment fod doth decrease towards
I have formerly (according to the method I shewed you) computed 3 be add. 13 & lodged . . . near it, add. 499

That: i.e. VALERIC, Subtilium indagationum liber primus, sen quadratura circuli et aliorum curvilineorum, Rom 1632. 280

124. WALLIS to OLDENBURG, 11/[21] August 1666 such a segment as nlmp; supposing the Spheres semidiameter 1,00000; and od = 0, 75000; on = 0,50000. (that is, R = 1. on = \. od = f.) And found the content; 2,93040 proxime. (that is, more then 2, 93039, & less than 2,93040.) The whole operation is too large to insert. But you may (at your leisure) computing it according to the Methode you mention; find how near that comes to the truth: & thereby make an estimate of the goodness of it. The Post-hour approaching, allows mee not time this morning to do it. Nor can I give you any estimate of your other rule of Approach for the Segment of cone between two parallel plains perpendicular to the Base: Which will require a longer time to examine. What I sayd of the fixing of the pasted Quadrant; was only an Intimation which might bee of use to Mr Marks500 in order to rendring them more exact. Nor shall I now adde more, than that I am Your Friend to serve you John Wallis.

124.

WALLIS to HENRY OLDENBURG Oxford, 11/[21] August 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 27, 4 pp. (p. 3 blank) (our source). At top right of p. 1 beneath date: 'read Aug: 15. enter'd LB. 1. 346.' At top left of p. 4 Oldenburg has noted at 90° to address: 'From Dr Wallis to M. Oldenburg concerning some Barometrical Observations'. Postmark on p. 4: 'AU/13'.— printed: OLDENBURG, Correspondence III, 207-8. wl Copy of letter sent: LONDON Royal Society Letter Book Original 1, p. 346. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 408-9. Reply to: OLDENBURG-WALLIS 4/[14].VIII.1666. Answered by: OLDENBURG-WALLIS mid-VIII.1666.

1 such add. 1 1,00000; |and add.\ od = 0,75000; 6 estimate (1) how (2) of 13 use |to add.\ Mr Marks 500

Marks: i.e. John Maxke (fl. 1665-79). 281

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124. WALLIS to OLDENBURG, 11/[21] August 1666 The present letter was read at the meeting of the Royal Society on 15 August 1666; cf. BIRCH, History of the Royal Society II, 109.

Oxford Aug. 11. 1666 Sir

According to yours of Saturday last501, I have made inquiry at both our Carriers for Hevelius's book502, which you say was sent by the Coach 5 the day before: But I cannot hear of it at either of them. I was loth to give you the trouble of making inquiry concerning it presently; supposing it might have been left in the Ware-house, & would come down by this Weeks Wagons: But neither can I yet hear of it. Which puts me upon the necessity of giving you the trouble of making some inquiry after it: 10 And, by your next, to let me know, by which of the Coaches it was sent, (whether by Moor at the Saracens head without New-gate, or by Bartlet at the Oxford-Arms in War wick-lane;) & to whom it was delivered. Wee have nothing of News from hence to send you, in requitall of that wherewith you are pleased to oblige mee. Onely that Dr Fell (the Dean 15 of Christ-church) is now our Vice-Chancellour503: And that God hath yet been pleased to preserve this Place from the Infection. Unless you shall think this worth your noting; that whereas I have been wont to observe constantly in my Baroscope; that the Sun-shining did make the Quicksilver to rise; (wherein the Observations of others have concurred with 20 mine:) I have this summer, especially in this hott time in June, July, & August (save that for the end of June & beginning of July, I can give no account, as having been from home;) I have observed the Quicksilver constantly to rise in| the night; & to fall in the day notwithstanding the [2] extreme hot sun-shine: I was thinking at first, it might bee from the winds 25 beginning to stir more in the mornings & the day time, than in the still night: But that hath been many times so indiscernable; & the other so constant, that I can hardly think it is wholly to be attributed to that. You may inquire whether others of your correspondents, who observe the 4 say add.

21 (save (1) f breaks off (2) that for the end of (a) July (b) June 501

yours of Saturday last: i.e. OLDENBURG-WALLIS 4/[14].VIIL1666. Hevelius's book: i.e. HEVELIUS, Descriptio Cometae anno aerae Christ. MDCLXV . . . cui addita est Mantissa prodromi cometici, Danzig 1666. 503 Dr Fell . . . Vice-Chancellour: John Fell (q.v.) was elected vice-chancellor of the University of Oxford on 3 August 1666, DNB. 502

282

125. OLDENBURG to WALLIS, mid-August 1666 Ayrs weight, have taken notice of the same with them that I have done here. I am glad to hear of Mr Boyles being better of his Aguish distemper. I thought indeed hee had been at Leez504; otherwise I should not have so much neglected inquiring after him. Pray present my very humble service to him; from

5

Sir

Your humble servant John Wallis. [4] For my very worthy Friend

Henry Oldenburgh Esquire, at his house in the Palmal near St James's London.

10

125.

HENRY OLDENBURG to WALLIS mid-August 1666 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 18/[28].VIII. 1666. Reply to: WALLIS-OLDENBURG 11/[21].VIII.1666. This letter apparently contained political news and Oldenburg's report on his efforts to discover the whereabouts of the copy of Hevelius's Descriptio cometae anno aerae Christ. MDCLXV which he had earlier dispatched to Oxford. Another copy of the book was sent to Wallis in September. See BIRCH, History of the Royal Society II, 114.

1 with them add. 3 so much add. 504Leez:

i.e. Lees Priory in Essex, the home of Boyle's sister Maxy. 283

126. WALLIS to OLDENBURG, 18/[28] August 1666

126.

WALLIS to HENRY OLDENBURG Oxford, 18/[28] August 1666 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 28, 4pp. (our source). At top right of p. 1 beneath date: 'read. Aug. 22. entered LB. 1. 347'. At top left of p. 4 Oldenburg has noted at 90° to address: 'From Dr Wallis to Mr Oldenburg about |the del] Observations of Tydes from the beginning of Sept. to the end of November.' Postmark on p. 4: 'AU/20'.—printed: OLDENBURG, Correspondence III, 212-14. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, p. 347. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 409-10. Reply to: OLDENBURG-WALLis mid-VIII. 1666. Part of the present letter was read at the meeting of the Royal Society on 22 August 1666; see BIRCH, History of the Royal Society II, 111. The passages concerning tides and Wallis's proposal for further observations were reproduced in Philosophical Transactions No. 17 (9 September 1666), 297-8 ('Some Inquiries and Directions concerning Tides, proposed by Dr. Wallis, for the proving or disproving of his lately publish't Discourse concerning them.').

Oxon Aug. 18. 1666.

Sir, I thank you for the Copies of this Months Transactions505; which I have received from Bartlet (viz. a parcell containing 20 of them.) I am sorry the other Book506 (which is not yet come to hand) hath given you so much 5 trouble: as I perceive it hath done. I should advise, for the future, rather to send by Moor; who is, of the two Carriers, much the more carefull. I forgot to tell you in my last507, that Dr Millington508 had from Dr Wren, that Book509 you sent for, to bring up to you; who tells mee upon his 10 return from London that hee hath left it with Dr Goddard for you; from whom I hope you have at length received it. One mis-adventure in the coming of it; hath been, that while I was last at London, Dr Wren had left it at my house to bee sent: which upon my return I knew not of, till 505

this Months Transactions: i.e. Philosophical Transactions No. 16 (6 August 1666). the other Book: i.e. Hevelius's Descriptio cometae anno aerae Christ. MDCLXV. 507 my last: i.e. WALLis-OLDENBURG 11/[21].VIII.1666. 508 Dr Millington: i.e. Thomas Millington. 509 that Book: not identified. Cf. WALLIS-OLDENBURG 2/[12].VIII.1666. 506

284

126. WALLIS to OLDENBURG, 18/[28] August 1666 upon your letter510 I called upon him for it again. Hee supposing it had been sent; & I not knowing it had been left here. But had it come to me before my coming to London, I had then brought it with mee. And so soon as I knew of it, I sent it the next morning by Dr Millington. When you write to Mons. Hevelius511, pray return him my thanks for the Book hee sent mee; (which you need not take notice of to him, but that it came safe inough to my hands; as perhaps it yet may: If not; I would not have you more solicitous about it. I shal buy mee one when there are copies come over to be sold.) And you may send him as from mee one of these Transactions: who perhaps will not be displeased with the new hypothesis. And the like to Mons. Hugens, when you write to him. I could wish, that somebody, would at London, or Greenwich, or rather nearer to Sea (but any where, rather then not at all,) make particular observation of all the Spring tides (New Moon & Full moon) between this & the end of November; & take account of the Hour, & of the Perpendicular height. [2] That wee may see whether those in| September, or those in November bee highest, and if of the Ebbs too, it were not amisse. (It may be easyly observed by a mark made upon any standing Post in the Water; & by any Waterman, or other understanding person who dwells by the Water-side.) And I think it would not be unacceptable to the Society to see how well Observation will agree or disagree with my Hypothesis.I thank you for the news you sent512, about the successe of Sir Rob. Holmes513: which 1 your (1) late (2) letter 2 come |to me add] before (1) I (2) my 7-9 If not; I would . . . mee one when (1) they (2) there are copies come over to be sold. add. above line and on left margin 9 may add.

12 or rather (1) (low) breaks off (2) nearer 17 , and if of the Ebbs too, it were not amisse add. 20 well add. 510

your letter: i.e. OLDENBURG-WALLS 26.VII/[5.VIII]. 1666 or OLDENBURG-WALLIS 31.VII/[10.VIII].1666. 511 When you write to Mons. Hevelius . . . : cf. OLDENBURG-HEVELIUS 29.VIII/ [8.IX].1666 (OLDENBURG, Correspondence III, 215-18, 215). Hevelius had meanwhile independently dispatched further copies of his Descriptio cometae; cf. HEVELIUSOLDENBURG 5/[15].IX.1666 (ibid., 224-5). One of these copies was sent to Wallis in September; see BIRCH, History of the Royal Society II, 114. 512 you sent: presumably in OLDENBURG-WALLIS mid-VIII.1666. 513 Sir Rob. Holmes: i.e. Sir Robert Holmes (1622-92), governor of the Isle of Wight and admiral of the navy, who destroyed ships and stores of the islands Vlieland and 285

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126. WALLIS to OLDENBURG, 18/[28] August 1666

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comes confirmed to us from all hands. I wish the issue of it may bee a good peace; which well established would bee much more acceptable then the News of Desolations, though of our Enemies; though this, as the case stands, be good News too. It seems you hear no more news about Petrus Blesensis514: whether the last papers were received; &, how liked. I shall be glad to hear, what satisfaction Hevelius's New Book doth give, to our Astronomers, concerning the points in dispute between him & Auzout515. You may please to inquire allso, whether when the Tydes be highest, the Ebbes be lowest, et contra: Or rather (which suites best with my Hypothesis) when the Tydes are highest (both in the Annual & Menstrual Periods) the Low-Waters be not allso at Highest; And, at Nepe-tides, the Ebbes allso very low. You will excuse mee that I do thus, without methode, write things confusedly as they come to mind. I am, Sir, Your affectionate friend & servant John Wallis.

15

zv

25

I should expect that the spring-tides now coming, & those at the beginning of September, should not be so high as those at the middle of September; & then lower again at the beginning of October; & after that higher at the middle of October, & higher yet about the beginning of November, (at the usual times of Springtides after the New & Full.)| [3] You may please in the copies you send to Hevelius, & Hugenius, (& in what more you think fit,) to amend some of these slips at the Presse which I observed in reading. (The first is the most material, because it disturbs the sense, both by leaving out a word, & by missing the Point: the rest are, most of them not considerable; unlesse that of Line for

3 ; though, this, as the case stands, be (1) good (2) good News too add. 5 : whether the last papers were received; fe, how liked add. 20 at the (1) middle (2) beginning 23 to amend add. 25 out add. 26 Point: (1) ) (2) the Terschelling on 8 August 1666, DNB. 514 Petrus Blesensis: see WALLIS-OLDENBURG 3/[13].IV.1666. 515 in dispute between him & Auzout: This refers to the lasting controversy between Hevelius and Auzout over their observations of the comet of 1664. 286

127. WALLIS to [OLDENBURG?], 23 October/[2 November] 1666 Sine, twice.) Pag. 269. line 27. Read; motion o/B about the Center; G is allso. p. 272. 1. 2. dele may. p. 273. 1. 2. read, of the Two Bodies, p. 274. 1. 13. the Annual, p. 277. 1. 24. Naturall Days. p. 280. 1. 34. fain (instead of seen) p. 281. 1. 16. of his. 1. 27. a Notion, p. 292. 1. 33. with what. p. 293. 1. 4. inough without. 1. 43. to the Sine of. p. 294. 1. 1. to the Sine of. 1. 44. Arithmetick. If you do not from yourself (& that it bee not now too late,) I desire you would from mee present one of them to our President516 (with my service, & thanks for his pains about it;) & to Mr Boyle, (whose it is:) &, if you think good, to Dr Wilkins517, Sir Robert Moray, Sir Paul Neile, or whom you think fit. And, if you have opportunity, to Capt. Titus518.

[4] These For my worthy Friend, Mr Henry Oldenburgh, at his house in the Palmal, near St James's London.

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127. WALLIS to [HENRY OLDENBURG ?] Ashford, 23 October/[2 November] 1666 Transmission:

Manuscript missing. Existence and date: Recorded in BIRCH, History of the Royal Society II, 119. This letter, which was read at the meeting of the Society on 31 October 1666, contained an account 'of the great hight of the tides about Hythe and Romney-marsh, two or three days after the last new moon of the said month of October; which the doctor looked

1 , twice add. 3 p. 274. 1. 13. the Annual, add. 516

President: i.e. William Brouncker. Dr Wilkins: i.e. John Wilkins, q.v. 518 Capt. Titus: i.e. Silas Titus, q.v. During the Anglo-Dutch war, Titus was captain of a company in the lord-admiral's regiment of foot. 517

287

130. FELL to WALLIS, 18/[28] November [1666] upon as agreeing with his hypothesis, though the people in the country imputed it very much to the great winds'. Cf. OLDENBURG, Correspondence III, 275. On his return from Kent, Wallis gave a personal account of his observations at the meeting of the Society on 28 November; BIRCH, History of the Royal Society II, 129.

128. JOHN FELL to WALLIS October/November 1666 (i) Transmission:

Manuscipt missing. Existence and date: Referred to in FELL-WALLIS 18/[28].XI.1666. As mentioned in FELL-WALLIS 18/[28].XL1666, Fell had written to Wallis twice beforehand on the topic of the University's latest dispute with the city of Oxford. It is probable that he sent this and the following letter to Ashford where Wallis was staying at the time, following the death of his brother Henry.

129. JOHN FELL to WALLIS October/November 1666 (ii) Transmission:

Manuscipt missing. Existence and date: Referred to in FELL-WALLIS 18/[28].XL1666.

130. JOHN FELL to WALLIS [Oxford], 18/[28] November [1666] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 82r-83v (f. 82V and 83r blank). On f. 83V in Wallis's hand: 'Nov. 18.', and in another hand: 'ut videtur 1667 Dr fell.' Above address in unknown hand: 'Mrs Letchfelde be at her Motheres Loginds to speake with you'. Postmark on f. 83V: 'NO/19'.

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130. FELL to WALLIS, 18/[28] November [1666] Evidently, Wallis was neither in Oxford nor in London at the time the present letter was written. Contrary to the endorsement, the year was clearly 1666, when Wallis is known to have spent at least part of October in his home town of Ashford (see WALLISOLDENBURG 23.X/[2.XI].1666) and then to have stayed a number of weeks in London before returning to Oxford. At the meeting of the Royal Society on 28 November he gave an account of his recent observations of the tides in Kent (see BIRCH, History of the Royal Society II, 129). 18 November was a Sunday; it would have been timed to arrive in London on Tuesday, coinciding with Wallis's arrival in the city. Wallis's lodgings in Drury Lane were outside the area affected by the Great Fire.

Sir, I have wrote twice to you519 concerning our quarrel with the town, & the need we have of your assistance. Mrs Wallis sends me word that you will be in London on Teusday. I pray as soon as you can call upon Dr Jenkins who lodges at the bishop of Rochesters520, at the Chancery in Westminster. I am solicitous in this matter, as knowing that much of our future quiet will depend upon the successe of this contest. I therefore recommend it to your care & remain

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Sir: Your Assured friend J. Fell.

Nov: 18 [83V] For the reverend Dr Wallis at Mr Stagges house at the signe of the Stagge in little Drury-lane neer the Bell back gate there D.D.

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wrote twice to you: not found. bishop of Rochesters: i.e. John Dolben (1625-86), canon of Christ Church 1660, D.D. 1660, dean of Westminster (1663-83), bishop of Rochester 1666, archbishop of York 1683, DNB. 520

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131. FELL to WALLIS, 5/[15] December [1666?]

131.

JOHN FELL to WALLIS [Oxford], 5/[15] December [1666?] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 84r-85v (f. 84V and 85r blank). On f. 85V in unknown hand: 'Dec. 5. (1667)'. Postmark on f. 85V: 'DE/7'.

Sir

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I had wrote to you ere this time, but that I was told you purposd to be here, as upon teusday last. Be pleasd to pursue your intendment of seeing my lord Chancellor521, for I sent him word, that you would wait upon him with an account of our affair. Mr. Crooks522 confident assertion that the matter in Controversy was determined against us in the late Kings time, did a little stratle me; till I considerd, that men thought themselves at large to say any thing for the cause they defend: & observed that both our Statutes is plain as to our Government of the market, & King Charles his Charter523 so particular therein that tis impossible to find more expressive words. I am told, I know not how truly, that the towns last Charter which they obtained since the king came in, is Cloyd with a Salvo of all our Privileges & rights. You will as you have opportunity signify to my lord by how many vexatious ways the town are usd to drive strangers from buying & selling here. When you return I shall in several particulars desire your assistance, at present I wish you health, & good success in your affairs & remain Your Assured friend, & faithfull servant J. Fell.

Dec: 5 20

For The reverend Dr Wallis

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lord Chancellor: i.e. Edward Hyde. Crooks: possibly Unton Croke (fl. 1658), B.C.L. at Oxford (1649), barrister of the Inner Temple (1653), high sheriff of Oxfordshire (1658). 523 King Charles his Charter: i.e. the 'great charter' of Charles I of 3 March 1636; cf. Wallis's Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.H]. 1667/8. 522

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132. OLDENBURG to WALLIS, c.8/[18] January 1666/7 at Mr Stagges house at the signe of the Stagge in little Drury-lane near the Bell back gate, these D.D.

132. HENRY OLDENBURG to WALLIS c.8/[18] January 1666/7 Transmission: Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 19/[29].I. 1666/7.

133. HENRY OLDENBURG to WALLIS 15/[25] January 1666/7 Transmission: Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 19/[29].I. 1666/7.

134. WALLIS to HENRY OLDENBURG Oxford, 19/[29] January 1666/7 Transmission: W Letter sent: LONDON Royal Society Early Letters Wl, No. 29, 4 pp. (p. 3 blank) (our source). At top left of p. 1: 'read Jan: 23: 66-7. entered LB. 1. 375.' On p. 4 Oldenburg has noted beneath, address at 180°: 'Where he got the distances of the 12th, |p. 87 in mantissa add] twice after Febr. 4. he observed not with a sextant p. (1) 16. (2) 17. sub finem.'—printed: OLDENBURG, Correspondence III, 312-14. w 1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 375-6. w'2 Copy of wl: LONDON Royal Society Letter Book Copy 1, pp. 443-6.

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134. WALLIS to OLDENBURG, 19/[29] January 1666/7 Reply to: OLDENBURG-WALLIS c.8/[18].I.1666/7 and OLDENBURG-WALLIS 15/[25].I. 1666/7. The letter was read at the meeting of the Royal Society on 23 January 1667; see BIRCH, History of the Royal Society II, 143.

Oxford Jan. 19. 1666./7. Sir,

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Of all the friends from whom I receive Letters, there are none who's letters seem less a trouble than those from you: so little need have you to apologize for your favours as if I could look upon them as other than so. They come allways so well fraught not onely with kindnesse but with so good ingredients that the oftener they come they are the more wellcome. I received both that524 of Jan. 15. & that525 of the week before: & am onely troubled that I am not able to give so proportionall a return to them as they do deserve. Truth is, I had been so long from home, & find my hands now so full of Businesse, that I have hardly time to read books or give account of them. Hevelius book526 hath had ill fortune: For beside that I hear no news of the first Copy527 that miscarryed between London & Oxford; the other Copy528 was allso gone once & I gave it for lost: being possessed by the Theeves who plundered us of other things. But, it seemes, bookes not being for their turn; they had but pulled it in peeces (being in quires) & thrown some in one place & some in another; where after they were gone I picked it up again in the dark; & found, when I came to my Inne, that I had lost none of it.

3 friends (1) the (2) from 8 fe |and del.\ am 14 gone (1) {once,} (2) once, 18 again in the dark; add.\ & found, (1) I had (2) when 524

525 526

that: i.e. OLDENBURG-WALLIS 15/[25].1.1666/7.

that: i.e. OLDENBURG-WALLIS c.8/[18]. 1.1666/7.

Hevelius book: i.e. Hevelius's Descriptio cometae . . . cui addita est Mantissa.

527

first Copy: This had been dispatched by Oldenburg in August 1666. Cf. WALLISOLDENBURG 11/[21].VIII.1666. 528 other Copy: i.e. one of the copies sent to London by Hevelius about August 1666, which was passed on to Wallis in September; cf. HEVELIUS-OLDENBURG 5/[15].IX.1666 (OLDENBURG, Correspondence III, 224-5) and BIRCH, History of the Royal Society II, 114. 292

134. WALLIS to OLDENBURG, 19/[29] January 1666/7 I have yet had no time since I came home from London529, to peruse it with that care I would do; (&, before, I had it not:) But, upon what I find upon a cursory reading, hee seemes to mee to have taken a vast deal of pains, & used much accurateness. I must trust his ingenuity for the truth of his Calculation (for it were too long a task to examine it all over:) But supposing it true (as I have no reason to doubt;) & that the result answeres the account hee had formerly given us of the place & motion, I see not why wee should disbeleeve him in matter of Fact. For I must needs assent to him for the preferring the Observations by Quadrants, sextants, & such other instruments (for places & distances) much before those of the Telescope; though this allso be in its kind an Exellent Instrument, where those can not be used. The great controversy Between him & Mons. Auzout, (whether the Comet came to the first or second Star of Aries,) I know not how to reconcile. I have no reason to suspect that either of them would willingly falsify an Observation: And yet how both can be salved, without allowing two Phaenomena, I cannot tell. That there should be two, seemes somewhat odde; yet is not impossible: for we see, presently after, yet another (undoubtedly) near the same place (viz. the latter Comet.) But this Observation being waved by Hevelius, I see noe reason why all the rest should not bee admitted. And what shall become of this doubtfull appearance must for ought I see, be left [2] undetermined. | Concerning the Tydes; I am well pleased with the care Mr Powle530 hath used in getting observations531 upon the Severn (or branches of that

1 to peruse Hevelius's his Mantissa with alt. Oldenburg 2 care (1) I (2) I 2 But, by what alt. Oldenburg 3 upon (1) ap breaks off (b) a 18 after, (1) a third (2) yet 19 Observation (1) Hevelius, waving, (2) being waved by Hevelius, 20 admitted. (1) (having this immedia) breaks off (2) And 529

home from London: Following his stay in Kent, Wallis had spent a number of weeks in London, evidently attending to University affairs. He had been present at the meeting of the Royal Society on 28 November (cf. WALLIS-OLDENBURG ? 23.X/[2.XI].1666) and probably also at other meetings in December. 530 Powle: i.e. Henry Powle (16307-92), lawyer and politician, fellow of the Royal Society. On retirement from parliament, made master of the rolls in 1690. DNB. 531 observations: Powle had been asked by Oldenburg for assistance in obtaining further observations of the tides in order to verify Wallis's hypothesis (cf. WALLIS-OLDENBURG

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Arm of the Sea which comes up that way.) I find that the times of the High Tydes (by his narrative) in those Seas, are not at the two Aequinoxes, but rather (as their Seamen call it) at St Davids Stream, & Michaelmas stream; the one about as long before the Vernal, as the other is after the Autumnall Aequinox. Like as, on the Coasts of Kent it happens, where the High Tydes are at Candlemas & All hollantide, at an equall distance the one before the other after, the respective Aequinox. But in those on the Coast of Kent (which is much further distant from the open Sea) it is at a greater distance, before & after, than on the West of England, which is nearer the great Ocean. So that the other cause, which together with that of the Aequinoxes, doth operate in determining the time of the Annuall High Tydes; doth it seemes, not operate so strongly near the Main Ocean, as in the In-land channell. The observation of the difference between the Day-Tydes & the Night-Tydes, that the one is higher in summer, the other in winter: I am very well pleased with. And I looke on it as a good confirmation of my Hypothesis: for, according to it, it ought so to bee. And, I suspect, (as consonant to the Hypothesis,) that it must bee so as to the Springtydes of New-Moon to Full Moon: viz. that the one is higher at one time of the year; the other at the contrary. Which is consonant to what our Seamen in Kent observe; that they have not two High streams together: (but when that of the New-moon is very high, at the Fullmoon following it is otherwise.) Which is that which in some of my former letters I have recommended to bee observed, (as consonant to my hypothesis,) though I had not then heard any intimation that it was observed by any. To the businesse of Petrus Blesensis532; I have formerly made what 3 fe (1) St (2) Michaelmas 5 Coasts of (1) the (2) Kent 10 is is del] nearer 12 Annuall High (1) streames (2) Tydes 17 And (1) the like (2) , I 25 then jheard add.\ any intimation that |it add] was (1) seen (2) observed 18/[28].VIIL1666, OLDENBURG-POWLE 12/[22].IX.1666; OLDENBURG, Correspondence III, 228-9, and POWLE-OLDENBURG late IX.1666; ibid., 235-6). He gave an account of these observations at the meeting of the Royal Society on 12 December 1666. Powle also prepared a paper, which was read at the meeting on 19 December; BIRCH, History of the Royal Society II, 133-4. Wallis probably attended (at least) one of these meetings; cf. WALLIS-OLDENBURG 7/[17].III.1667/8 (ii). 532 businesse of Petrus Blesensis: see WALLIS-OLDENBURG 3/[13].IV.1666.

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135. OLDENBURG to WALLIS, 24 January/[3 February] 1666/7 inquiry I could well concerning his pieces to be found here; but can hear of none more than what I have formerly mentioned: amongst which those two pieces; the life of Henry the 3d, & the Dialogue with the Abbat, are none. I directed you formerly533 to Sir Joh. Cotton's library, where one of them is more likely to be found. His Remediarium Conversorum, is for the main subject of it, Morall or Theologicall: But doubtlesse (as in most of his other pieces) it will be found intermingled with many other things that will concern the affairs of the Church in those times. But I have not yet read so much of it as that I can say it on my own Knowledge. I have again looked on it; & find it to be about twice as much as both the other peeces put together: That is, about 4 quires written as that was. And may be transcribed if they please. At the rate (I suppose) which you mentioned. I shall adde onely that I am Sir

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your affectionate friend & servant J. Wallis. [4] For Mr Henry Oldenburgh, in the Pallmall near St James's London.

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135. HENRY OLDENBURG to WALLIS 24 January/[3 February] 1666/7 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 31.I/[10.II]. 1666/7.

5 is (1) most (2) more 533

directed you formerly: i.e. in WALLis-OLDENBURG 24.V/[3.VI].1666. 295

136. WALLIS to OLDENBURG, 31 January/[10 February] 1666/7

136. WALLIS to HENRY OLDENBURG Oxford, 31 January/[10 February] 1666/7 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 30, 2 pp. (our source). At top of p. 1: 'read Feb: 6: 1666-7 Entered LB. 1. 380.' On p. 2 Oldenburg lias noted to the left of address at 90°: 'The Society will (1) per breaks off (2) endev breaks off (3) persuade (4) endeavor to persuade those Astron. that have made observ. to publish them as soon as conven. { }, which will give light'. Postmark on p. 2: 'FE/1'.— printed: OLDENBURG, Correspondence III, 330-1. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 380-1. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 449-51. Reply to: OLDENBURG-WALLIS 24!/[3!I]. 1666/7. The present letter was read at the meeting of the Royal Society on 6 February 1667, at which it was ordered 'that the president and the lord bishop of Exeter be desired to invite these persons, to whom, besides Dr. Wallis, the perusal and examination of Mr. Hevelius's book had been committed, that they would meet with them at a convenient time, to consider of this account of Dr. Wallis, and that additions or alterations were fit to be made therein, in order to send an answer to Mr. Hevelius'. See BIRCH, History of the Royal Society II, 147.

Jan. 31. 1666/7. Oxford. Sir,

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According to yours534 of Jan. 24. I have again cast an eye on Mons. Hevelius two bookes of the late Comet (his Prodromus535, & Mantissa536.) I do not see but that his observations deserve credit inough as far as the 12th or 14th of February. So far they do not differ so much from the place of the Comet observed from others, but that the Error may, with as much probability at lest, be cast on their part as on his. Especially his Instruments being much better; & himself a diligent &; long experienced 5 the (1) 14th (2) 12th 534

yours: i.e. OLDENBURG-WALLIS 24.I/[3.II].1666/7. Prodromus: i.e. HEVELIUS, Prodromus cometicus, quo historia, cometae anno 1664 exorti cursum, faciesque, diversas capitis ac caudae accurate delineatas complectens, nee non dissertatio, de cometarum omnium motu, generations, variisque phaenomenis, exhibitur, Danzig 1665. 536 Mantissa: i.e. HEVELIUS, Descriptio Cometae ... cui addita est Mantissa prodromi cometici, Danzig 1666. 535

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136. WALLIS to OLDENBURG, 31 January/[10 February] 1666/7 observer. But his Observation of Febr. 18. I much doubt of; & you see, hee is content to wave it. I am much inclined to be of the opinion which our own observations at home, (of my Lord Brounker our Noble President, Mr Hook, & others,) seem to confirm. (They being disinterested persons, & having no temptation to be biassed.) And do therefore incline to beleeve that it passed by the Second star of Aries, not (as Hevelius would have it) by the First. I should be more confirmed in that opinion if Mr Hookes Observations & other at London, do (as I think they do) trace the comets way consonantly thither; & were continued day by day without any considerable intermission. If Mons. Hevelius did, notwithstanding, observe any thing of that nature near the first of Aries on the 18th of Febr: I must take that to be rather somewhat else than this Comet. And, though none else have observed such a second Phaenomenon but himself at that place, (& it be therefore the more doubtfull,) yet since no body else did then & there look for it, & the thing in itself be nothing impossible; I had rather avrexetz', than determine any thing concerning it. It was a little unhappy, that hee stumbled on that Observation; because, I am apt to think, he was thereby diverted from seeking after the Comet in its true place; which otherwise hee might in likelyhood have discerned, as others did. I am sorry Mr Hook hath not, all this while, drawn up his Observations of that (& the other Comet, which followed in the ensuing April, 1665.) into some methode, (with a scheme fitted thereunto,) & published them. Which I wish he would yet do. For I look on those, as much more considerable than divers of those published by others; & mentioned in Hevelius his Mantissa. Hee would thereby both do himself right, & the Society. As to the business of Tydes; As Mr Powle hath observed a different season of the Annual High-Tydes, on the Severn, from those on the Coast of Kent: So I could wish, wee had alike account about other parts; as on the Coasts of Sussex, Hamshire, Devonshire, Cornwall &c: And of Essex, Norfolk, Lincolnshire, &c, & so on towards Scotland. For I suppose there may be such seasons observed by the Seamen, or Fishermen of those Coasts; & that they may at several places fall at several times: & that

7 if ( 1 ) 1 (2) Mr Hookes 12 than (1) the (2) this Comet, (a) Which then (&) And, 21 in (1) the foil breaks off (2) the 29 about |the del. other

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137. COLLINS to WALLIS, [January? 1666/7] comparing of them may give us further light. But this at present from Sir

Your very humble servant, John Wallis. 5

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The other lost copy of Hevelius his last book537 is at last come to hand; which now I find was here left for mee in my absence: But I knew not of it till this week. I shall send you one of them back again; with the first convenience. These For Mr Henry Oldenburg at his house in the Pall-mall, near St James's, London.

[2]

137. JOHN COLLINS to WALLIS [London, January? 1666/7] Transmission:

C Draft of (missing) letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 192r-193v (f. 192r and 193V blank) (our source).—printed: RIGAUD, Correspondence of Scientific Men II, 468-70. Answered by: WALLIS-COLLINS 17.1666/7.

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Reverend Sir

[192V]

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Mr Pitts a Stationer in little Brittaine (of whome I can give no other then a good Character) understanding from Mr Branker539 (a late fellow 17 late add. 53r

last book: i.e. Hevelius's Descriptio cometae. Pitts: i.e. Moses Pitt (fl. 1654-96), London publisher, author, and bookseller, estab lishe d at the 'White Hart' in Little Britain and at the 'Angel' in St Paul's Churchyard, DNB. 539 Branker: i.e. Thomas Brancker, q.v. 538

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137. COLLINS to WALLIS, [January? 1666/7] of Exceter Colledge who now lives with the Lord Brereton540 in Cheshire) that you had prepared sundry Tracts541 for the Presse, is very desirous to undertake any of them, and to treate with you to that purpose promising they shall come out on very good Paper like Schootens Miscellanies542, and the Cutts to be suitable thereto, which may be done by Mr Marke543, and indeed, passing by Morning, and Night, and at Noone too if need require I should affoard my endeavour to have it Carefully corrected. The said Mr Pitts hath sent & presents you with as much of Dr Pells delayed Booke544 as is yet Printed, and intends to send the rest when finished. We are very sorry to heare of your Losses by Robberie; if money were so lost, it might have been prevented by Paying it here on the account of Sir Thomas Pennystone545, former of the Excise, and receiving it there of him. I have inclosed sent a note546 of some Bookes lately come forth and hope to procure some of them ere long. Talkes there are as if some about Hamburgh547 had discovered a Series of the Ratio of Polygons inscribed and Circumscribed about a Circle which would be of great use in making

4 very add. 6-7 and indeed, . . . corrected, add. 8 & presents add. 8 delayed add. 12 former of the Excise, add. 14 long (1) wherewith to (2) . Talkes 14 if (1) the Germans (2) some 15 of (1) figures (2) Polygons 540

Brereton: i.e. William Brereton, q.v. Tracts: i.e. OUGHTRED, Clavis mathematicae. The editio quarto, was eventually reprinted in Oxford in 1667. However, it would appear that Collins was not aware at this time which tracts were concerned. 542 Miscellanies: i.e. SCHOOTEN, Exercitationum mathematicorum liber V. continens sectiones triginta miscellaneas, Leiden 1657. 543 Marke: i.e. John Marke. 544 Booke: i.e. RAHN, An Introduction to Algebra, translated out of the High Dutch into English by T. Brancker . . . Much altered and augmented by D[r]. Pfell]., London 1668. 545 Pennystone: not identified. 546 note: not found. 547 some about Hamburgh: possibly Bertrand de La Coste (fl. 1663-76), French artillery officer in service with Friedrich Wilhelm, Elector of Brandenburg and later (until 1683) resident in Hamburg. He published various tracts on the quadrature of the circle, including Klarer Bewiis van het Quadraet des Circels, 1666. Cf. PAISEN-OLDENBURG 1/[11].1.1668/9; OLDENBURG, Correspondence V, 293-6 and LA COSTE-OLDENBURG 30.I/[10.II]. 1668/9; ibid., 366-8. 541

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the Sine or tangent of any arke in the quadrant on Demaund, but as farre as I know the ranke remaines to be Compleated, and I presume it cannot be expected so well from any other as your selfe. This easily presents its selfe, Triangle Square Pentagon Hexagon Inscribed 1 2 3 Circums. 4 4 4 4 but the Continuance and Compleating of the ranke will beget admiration.

If my Presumption offend not, I would adde that there might be many sorts of Conchoid Lines. As let ABC be a Parabola whose base AC is indefinitely extended, let this Parabola slide along ACD drawing the ruler HP after it which alwaies lyes over the Point P and H, and where the Parabolicall line and this line or ruler HP intersect, let there be Points made in a Curve which will be Conchoidall or Asymptotick, of which there may be innumerable kinds. Now such a Curve being proposed, how to find the figure and the Pole is not yet handled. But I have a Suspition that the Meridian line of Mercators Sea Chart may be made by some such kind of Curve. For let AB be radius, BT a line of Tangent, and supposing the first degree of Mercators Meridian Line equall to one degree of tangent with every 10 degrees of Meridian on B as Center, let every 10 degrees of 3 expected (1) better (2) so well 4 This (1) pre breaks off (2) easily 11 let (1) a ruler (2) this 12 and H add. 14 of which (1) kind (2) there 15 Now (1) how (2) such 15 how add. 20 on B as Center add. 300

138. WALLIS to COLLINS, January? 1666/7

[193r] Secant be Intersected, and a Curve passing through those Intersections will resemble a Conchoid line. Vouchsafe to excuse the trouble I put on your Patience, if you should chance to consider these things, and make some usefull Discovery therein, I doubt not but in time you will make it redound to the Publique benefitt of learning. I remaine

138. WALLIS to JOHN COLLINS January? 1666/7 Transmission:

Manuscript missing. Existence and date: This letter is a reply to COLLINS-WALLIS 17.1666/7 and is mentioned in and answered by COLLINS-WALLIS 2/[12].II.1666/7. It was therefore probably written in January 1666/7. Wallis apparently explained that he was intending to have Oughtred's Clavis reprinted and probably stipulated the conditions under which this was to take place. Collins passed the letter to the publisher Moses Pitt.

139. JOHN COLLINS to WALLIS [London], 2/[12] February 1666/7 Transmission:

C Draft of (missing) letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 188r-189v (f. 189r enclosure, f. 189V blank) (our source). On right margin

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139. COLLINS to WALLIS, 2/[12] February 1666/7 of f. 189V at 90° in Collins's hand: 'To Dr Wallis'.—printed: RlGAUD, Correspondence of Scientific Men II, 470-3. Reply to: WALLis-CoLLiNS 17.1666/7. Answered by: WALLIS-COLLINS 5/[15].II. 1666/7. Enclosure: Note on a general method for solving equations.

February 2d 1666/7. Reverend Sir

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I received yours548 and communicated it to Mr Pitts549 who very much honours your advice and thankes you for it, but if there be any other that is willing to bargain for the said Impression, he is not desirous to interpose for these reasons. 1. He is engaged in the Drs Booke550 already, but cheifly the impression is double the Number that ought to have been of a Mathematicall booke, the best whereof though sure of Sale are but slow. Mr Briggs Arithmetica Logarithmica551 being to numerous an Impression has been tendred about the streets for Is Qd each. The like I say of Mr Barrowes Euclid552. Mr Sutton553 and myselfe, as Mr Marke554 well knowes, have bought diverse of them at 1s a booke in quires.

4 and thankes you for it add. 5 that is (1) about the s breaks off (2) bargaining (3) willing to bargain 5 not (1) willing (2) desirous 7 He is engaged in (1) a treatise of the like nature al breaks off (2) the Drs Booke already, but cheifly add. 8 is | above del] double 9 whereof (1) are (2) though 10 have been corr. ed. 548

yours: i.e. WALLIS-COLLINS 17.1666/7. Pitts: i.e. Moses Pitt. See COLLINS-WALLIS 17.1666/7. 550 Drs Booke: i.e. RAHN, An Introduction to Algebra, translated out of the High Dutch into English by T. Brancker . . . Much altered and augmented by D[r]. Pfell]., London 1668. 551 Arithmetica Logarithmica: i.e. BRIGGS, Arithmetica Logarithmica, London 1624. 552 Euclid: i.e. BARROW, Euclidis elementorum libri XV breviter demonstrate,, Cambridge 1655. 553 Sutton: i.e. Henry Sutton. 554 Marke: i.e. John Marke. 549

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139. COLLINS to WALLIS, 2/[12] February 1666/7 2. There are sundry Tracts of Algebra expected, first from beyond Sea, as that of Chaveau555, one promised, the manuscript whereof is by Hudden556 ab ovo ad montem557, an intire posthumous treatise of this nature left by Tacquett558, which Meursius559 of Antwerpe intends to print, 2 Volumes in folio of Reinaldinus560 in the presse at Florence, Fermatius561, lastly the 3d volume of Des Chartes562 is said to containe all of his that is tolerable and mentioned by Borellus563 in his life. At home Mr Kersy564 hath a laborious treatise565 in English ready

1 2. (1) Chaveaus Algebra (2) There are 1 first add. 2 the manuscript whereof is add. 5 Reinaldinus (1) at Florence (2) in the 5 Florence, \(1) diverse (2) Fermatius, add. lastly 555

Chaveau: i.e. Jean-Baptiste Chaveau, professor of mathematics in Paris. See DESCARTES-MERSENNE [18J/28.1.1640/1; MERSENNE, Correspondance X, 434-46; VERDUS-MERSENNE end 1641-43?; ibid., X, 833-5; PELL-CAVENDISH 8/[18].II. 1644/5; ibid., XIII, 373-5, and CAVENDISH-PELL [23.VII]/2.VIII.1648; ibid., XVI, 462-5. 556 Hudden: i.e. Johann Hudde (1628-1704). Collins refers probably to the major work which Hudde had planned to publish under the title De natura, reductions, determinations, resolutione atque inventione aequationum. Hudde mentions this work in his Epistola prima de reductions aequationum, (Geometria a Renato Des Cartes, ed. Schooten, Amsterdam 1659, I, 402). The studies on which it was to be based were apparently contained in a notebook which disappeared after coming into the hands of a nephew following his death. See HOFMANN, 'Mutmafiungen iiber ein derzeit verschollenes Manuskript von Hudde', Nova Acta Leopoldina N. F. 30 (1965), Nr. 173, 459-66 (HOFMANN, Ausgewahlte Schriften I, 425-32). 55r ab ovo ad montem: 'ab ovo usque ad mala', HORACE, Satirae, 1, 3, 6. 558 Tacquett: i.e. TACQUET, Opera mathematica . . . demonstrata et propugnata a Simone Laurentio Veterani, Antwerp 1669. 559 Meursius: i.e. Jacob Meurs, printer in Antwerp. 560 Reinaldinus: i.e. Carlo Renaldini (1615-98); RENALDINI, Ars analytica mathematum, 3 vols., Florence 1665-9. The reference is probably to the volumes 2 and 3. 561 Fermatius: possibly Diophanti Alexandrini arithmeticorum libri sex . . . Accessit doctrinae analyticae inventum novum collectum ex variis ejusdem D. de Fermat epistolis, Toulouse 1670. 562 Des Chartes: i.e. DESCARTES, Lettres, ed. Clerselier, vol. 3, Paris 1667. 563 Borellus: i.e. Giovanni Alfonso Borelli (1608-79), Italian mathematician and physiologist. 564 Kersy: i.e. John Kersey (1616-77), surveyor and teacher of mathematics in London, DNB. 565 treatise: i.e. KERSEY, The Elements of that Mathematical Art commonly called Algebra, 4 vols., London 1673. 303

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for the presse which is promised Mr Stephens566 who will undertake it when Paper is more reasonable. The said Mr Kersy hath made Notes on the Clavis567 and to say the truth doth not admire any thing in it save what concernes the 10 and succeeding Elements of Euclid. Mr Sunning an aged Minister neare Nuneaton in Warwickshire hath commented on the Clavis, which he left with Mr Leyburne568 to be printed, but one Mr Anderson569 a knowing Weaver told Mr Sunning that the Clavis its selfe and his Comment thereon were immethodicall, and the Precepts for educing the rootes of an affected Aequation maime and insufficient. Mr Pitts upon Supposition as above is notwithstanding willing to deale for the Impression, provided there be an Engagement that it shall not be reprinted till the Impression be sold570, and because it is already common that he may have liberty to increase it with such Comments or Explications as he shall be advised (by yourselfe instar omnium if willing) by his freinds to annex to it; if also there be not another Clavis571 or Introduction ready for the presse prepared by Mr Gilbert Clarke572, 1 presse (1) and (2) which 8 a most add. and del.\ knowing 9 Comment (1) thereto (2) thereon were |altogeather del.\ immethodicall, 12 Supposition (1) that others are not abo breaks off (2) as above 14 because ... common add. 16-17 shall (1) thinke fitt (2) be advised . . . his freinds 17 to it; (1) and fur breaks off (2) if also 566

Stephens: i.e. Robert Stephens, London bookseller. Clavis: OUGHTRED, Arithmeticae in numeris et speciebus institutio, quae . . . totius mathematicae clavis est, London 1631. Published in the second edition (and thereafter) as Clavis mathematicae, London 1648. 568 Leyburne: probably the mathematical practitioner William Leybourne (16261716), whose brother Robert Leybourne (fl. 1645) was a printer at Monkswell Street, Cripplegate. In early years, they had worked together. 569 Anderson: i.e. Robert Anderson (fl. 1661, ob. 1710), silk-weaver and mathematical practitioner. 5ro provided . .. sold: Already the second Latin edition of Oughtred's Clavis mathematicae had been held back by the printer, for fear of not being able to sell it profitably. See HAAK-MERSENNE 6/[16].VIII. 1647; MERSENNE, Correspondance XV, 353-6. 571 another Clavis: i.e. CLARK, Oughtredus explicatus, sive commentarius in clavem mathematicam Oughtredi, London 1682. 572 Clarke: i.e. Gilbert Clark (or Clerke) (1626-97?), mathematics tutor and maker of dials, DNB. 567

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139. COLLINS to WALLIS, 2/[12] February 1666/7 (which we shall endeavour to know) who lives with Sir lustinian Isum573 whithin 7 miles of Northampton, who in the preface of a treatise concerning some Experiments in Philosophy intimates he wrote a Comment on the Clavis which lay long in the hands of a Printer by whome he was abused, meaning Leyburne. Mr. Cocker574 our famous Englich Graver and Writer, now a Schoolemaster at Northampton, about halfe a yeare since directed a freind of his that came up, to see me, who intimated that Mr Clarke finding his Notes obstructed, had prepared either a new Clavis or a [188V] method of Aequations and further enlargements] upon Des Chartes. Mr Dary575 thinkes he hath of late so mended his Precepts for the obtaining the rootes of an Aequation that they shall undoubtedly give the quality of every assumed figure in the roote, whether to great or too little, and render the least roote in the Aequation or Discover the impossibility thereof; and the like he can doe for the greatest, which I hope by some other opportunity to transcribe and send them to be submitted to your iudicious censure. As concerning the Booke of Dr Pells Scholar576 I thinke the Dr did little concerne himselfe in it till the Introduction was past, and to speake plainly I account that Introduction much worse then Principia

1 (which . . . know) add. 2 in (1) a treatise about (2) the 6 now . . . Northampton, add. 8 Clavis (1) with (2) or 9 enlargements (1) of (2) upon 10 mended (1) the (2) his 11 rootes (1) that (2) of 11-13 the quality of every [assumed add] figure . . . and render add. 15 to (1) send an breaks off (2) transcribe 16 censure (1) and examined a breaks off (2) . As 181(1) thinke (2) account 18 Introduction (1) unworthy not to be (2) much 573

Isum: i.e. Sir Justinian Isham (1611-75), M.P. for Northamptonshire (elected 1661), fellow of the Royal Society, DNB. 574 Cocker: i.e. Edward Cocker (1631-75), teacher of writing and arithmetic, and engraver in London, author of numerous books on arithmetic, DNB. 575 Dary: i.e. Michael Dary (1613-79), tobacco-cutter and self-taught mathematician, author of The General Doctrine of Equation, London 1664. 576 Scholar: i.e. Johann Heinrich Rahn (1622-76). 305

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Matheseos universails577. I have not been yet concerned in the correcting of any part of that Booke, and have observed many errors therein. I perceive you have had the happinesse to light upon the same manner of effecting Cubique Aequations in your letter578 to the Lord Brouncker that I have heard the Dr sometimes mention, to wit, by Waving indented Lines to crosse a right line, by which Waving lines I presume he meant the Diameters of the Parabolasters when the ordinates are inclined, and in regard the Dr is not likely in this treatise to publish something else I have heard him discourse of, I thinke fitt to suggest it. 1. That any Aequation being proposed he can by his generall Doctrine of Limits determine how many of its rootes are possible. 2. I have seen 2 long Scroles or tables of Numbers relating to a Biquadrate Aequation having all the powers extant; in one the Resolvend or homogeneum Comparationis hath been kept constant, and the Coefficient of any one tearme at a time altered successively in an Arithmeticall Progression, till one or more of the rootes have lost their possibility, and the said table hath given the rootes very neare. The other keeping the Coefficients constant hath given the rootes all along when the Resolvend hath been an Arithmeticall Progression; and such tables he saith are not very hard to make in relation to any Aequation and that he disuseth the generall Method.

I not (1) ye breaks off (2) been 4 Aequations (1) that I (2) in 5 Dr (1) often talke of (2) sometimes mention 6 which (1) Cur breaks off (2) Waving II Limits (1) de breaks off (2) vel (3) determine 12 or tables of Numbers add. 13 having . . . extant add. 15 tearme |at a time add.\ altered (successively add] in 16 one (1) of (2) or 19 he (1) hath (2) saith 20 and that . . . Method add. 5rr

Principia Matheseos universalis: i.e. SCHOOTEN, Principia matheseos universalis, sen introductio ad Geometriae methodum R. des Cartes, ed. Erasmus Bartholin, Leiden 1651. 578 letter: i.e. WALLIS-BROUNCKER 5/[15].XIL1656. 306

140. COLLINS to WALLIS, 2/[12] February 1666/7, enclosure I presume these things have been considered by your selfe and therefore I crave pardon for my impertinencies, and returne my utmost thankes for your manifold favours, Subscribing my selfe Your most affectionate humble Servitor

140.

JOHN COLLINS to WALLIS 2/[12] February 1666/7, enclosure: Note on a general method for solving equations Transmission:

C Draft of (missing) note sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 189r (our source).—printed: RIGAUD, Correspondence of Scientific Men II, 473-4. Enclosure to: COLLINS-WALLIS 2/[12].II.1666/7. The present note, apparently based at least in part on a transcript made by Collins, takes up one of the central themes discussed in the letter.

For the effecting of all manner of Aequations in Lines I have heard it thus in generall Described.

By the Doctrine of Limits he assumes a limited Line AB and finds a point above it from whence for each paire of Rootes issues a Curve 2 and (1) togeather with Mr Pitts returne our utmost thank breaks off (2) returne my 4 humble add. 7 (1) But of (2) By 7 AB (1) of a de breaks off (2) and 8 point (1) under (2) above 307

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PCD

AC

rhit

AiL

„,-,,-, which crossing the right line at C, £7, £>, F, gives

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for Negative rootes, and if it be an Aequation AF of an odde degree as Cubick, the Point F might fall beyond B shewing an impossibility of another roote, and if a Curve doe not touch at all, 2 rootes loose their Possibility, and all these may be one Continued indented Curve, and then me thinkes they might be the Diameters of the Parabolasters at the end of Dr Wallis his Arithmetick579. What lesse might be expected from Dr Pell whose Idea of Mathematicks580 is extant in Duries581 reformed Schoole582 printed in 1651, where Page 42 he saith according to his Prescribed Method he can deduce not only all that ever is to be found in our Antecessors writings, and whatsoever they may seem to have thought on but also all the Math. Inventions, Theoremes, Problemes and Precepts, that it is possible for the working wits of our Successors to light upon, and Page 45, to determine all the Number of Problemes that can be conceived concerning any thing Propounded, and for Excise in these methods my Lord Brereton583 informes me hath wrote a quire of Paper on this question. affirmative rootes and

5

for

1 PEF add. 1 right line \AB del. at C, \ ( 1 ) and D, (2) E, D, F, add] gives 1 AE add. 1 for an affirmative (1) roote (2) rootes corr. ed. 2 AF add. 2 for | a del] Negative rootes 2 it be (1) a Cube (2) an 3 degree (1) th breaks off (2) as 4 and if (1) they do not touch (2) a 9 Schoole (1) Page (2) printed 579

Arithmetick: i.e. WALLIS, Arithmetica infinitorum. Idea of Matheniaticks: i.e. PELL, An Idea of Mathematicks, [London 1650]. 581 Duries: i.e. John Dury (1596-1680), protestant divine, mainly active on the continent in efforts to achieve Christian unity, DNB. 582 reformed Schoole: i.e. DURY, The Reformed-School: And the Reformed LibrarieKeeper . . . Whe.re.unto is added I. An Idea of Mathematicks. II. The Description of one of the chiefest Libraries which is in Germanie, erected and ordered by one of the most Learned Princes in Europe, London 1651. 583 Brereton: i.e. William Brereton, q.v. 580

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141. WALLIS to JOHN COLLINS Oxford, 5/[15] February 1666/7 Transmission: W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 190r-190v (f. 190V blank). On left margin off. 190V at 90° in Collins's hand: 'Dr Wallis his Letter commending the Clavis'.—printed: RIGAUD, Correspondence of Scientific Men II, 474-6. Reply to: COLLINS-WALLS 2/[12].II.1666/7. Answered by: CoLLiNS-WALLis c.lO/[20].II.1666/7.

Oxford. Febr. 5. 1666/7

Sir, In answere to yours584 of the second instant, concerning the proposition I made about Mr Pitts his dealing for the Impression of Mr Oughtreds Clavis585, I have not much more to say than what I expressed allready. It was onely a motion of my own, which Mrs Lichneld586, who hath the Impression to sell, knew not of. And I took my rise onely from the overture your letter587 mentioned from Mr Pits of his willingness to print Mathematick books. And I thought it seasonable inough to inform him of this, which is allready printed; which (in the scarcity of Books since the fire) might be vending, while others are but in preparing. Whether the number be too great, or the Book not so vendible, the Bookseller (who understands his trade) is a more competent Judge then I: But for the goodness of the Book in itself; it is that (I confesse) which I look upon as a very good Book, & which doth, in as little room, deliver as much of the fundamentall & usefull part of Geometry (as well as of Arithmetick or Algebra) as any book I know: And why it should not be now Acceptable, I do not see. It is true, that, as in other things, so in Mathematicks, Fashions will dayly alter; & that which Mr Oughtred designed by Great Letters, may be now by others be designed by small; But a Mathematician will with 16 as ( 1 ) 1 (2) any 19 dayly add. 884

yours: i.e. COLLINS-WALLIS 2/[12].II.1666/7. proposition . . . Clavis: presumably in WALLis-CoLLiNS 17.1666/7. 586 Lichfield: i.e. Ann Lichfield (d. 1672). 58r your letter: i.e. COLLINS-WALLIS 17.1666/7. 585

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the same ease & advantage understand Ac, & a3, or aaa. Nor will Euclide or Archimedes cease to be Classick Authors, & in request: though some of their considerable Propositions, be, by Mr Oughtred & others, delivered now in a more advantageous way according to mens present apprehensions. And the like I judge of Mr Oughtreds Clavis, which I look upon (as those pieces588 of Vieta who first went in that way) as lasting Books, & Classick Authors in their kind. To which notwithstanding, every day may make new Additions. But this is beside the busyness; (For a Book-seller, is to judge that a good book which sells well; what ever wee Students judge of them.) If Mr Pitts think it convenient for him to deal for the Impression as it is; & let mee know on what terms hee is willing to take it: Mrs Lichfield is my neighbour & I can easyly propose it to her, & give him her Answere. As to the particulars hee mentions: I think it reasonable that shee should ingage not to put forth another Impression till hee may have a reasonable time for the putting off of this: (How many years that is to be judged at, Booksellers can better judge than I.) And, as for adding any thing to it, to be sold & bound with it; I think shee hath no reason to be against it, provided it be not so as to become a prejudice to her coppy. But, I confesse, as to my own judgement, I am not for making the book Bigger; because it is contrary to the design of it; being intended for a Manuall or Contract: whereas Comments, by inlarging it, do rather destroy it. I should rather advise those who would publish any thing that way, to do it as a work of their own, than as a Comment on this. For if Mr Oughtred had intended it to be large; hee could with more ease have made it much bigger than it is. But it was by him intended, in a smal Epitome, to give the substance of what is by others delivered in larger Volumes. I told you in my last, what price shee expects for it as I have formerly understood from her, (viz. 40' for the Impression; which is about 9g the book) if they agree about the price; I beleeve other things will be easyly inough accommodated. If hee will have anything else adjoyned to it: the paper & print of that is to be further agreed on. 7 day (1) mak breaks off (2) may 11 on add. 12 it: (1) if hee let mee know (2) Mrs 14 till (1) his (2) hee 16 that (1) will (2) is 28 her (1) . If (2) , (viz. 588

pieces: i.e. ViETE, In artem analyticam isagoge, Tours 1591.

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142. COLLINS to WALLIS, [c. 10/[20] February 1666/7] I thank you for your advertisement of Bookes expected. Resting in hast your loving Friend, John Wallis. Mrs Lichfields daughter589 is now in London, & lodgeth at Mrs Stevens's590 house in Castle yard in Holborn, over against Mag-py Court: Mr Pits, if he please, may speak to her about it. Or let mee know his mind here; which he please.

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142. JOHN COLLINS to WALLIS [London, c.lO/[20] February 1666/7] Transmission:

C Draft of (missing) letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 194r-195v (f. 194r blank) (our source). At top of f. 195r and 195V calculations in Collins's hand.—printed: RIGAUD, Correspondence of Scientific Men II, 482-5. Reply to: WALLis-CoLLiNS 5/[15].II.1666/7. [194V] Reverend Sir

I received yours591 of the 5th instant in answer to mine592 of the 2d preceding, since the Writing whereof Mr Pitts, having conference with Mr Thompson593, a Bookseller who lived in Pauls Church yard and now in little Brittaine, was by him informed that he was importuned since the fire

12 a Bookseller . . . Brittaine add. 13 he was (1) in treaty (a) with M breaks off (b) since the fire (2) importuned since the fire to treate 589

daughter: i.e. Margaret Lichfield. Stevens's: not identified. 591 yours: i.e. WALLIS-COLLINS 5/[15].II.1666/7. 592 mine: i.e. COLLINS-WALLIS 2/[12].II.1666/7. 593 Thompson: i.e. Samuel Thompson. 590

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to treate with Mrs Leichfeild594 for the Impression595, who offered it for 36', and the said Mr Thompson would not give above 32' for it, the rather because it is printed upon worse Paper then the former Impression; he having one of them here did also shew it. I prevailed with them to meete being neare togeather, and Mr Pitts declared his unwillingnesse to interpose without the consent of Mr Thompson, who offered and desired him to take what part or share thereof he would. I cannot prevaile with them to bid more for it; they say it is not a booke so much inquired for here, as in the Universities, and they both Doubt it will not sell without a Comment, and Mr Thompson sayes he was long posessed of Mr Clarkes596 Comment, who would freely have imparted it to any one to print, and presumes he may have it againe if he request it, and affirmes it is very large and will make above 20 sheetes, and this they agree upon, if they may have liberty to Print that Comment here by it selfe, and to sell it apart, or bound up with the Booke as they see convenient, and that the booke be not reprinted so long as they have 100 Bookes left, they will then give 32 and no more for the Impression. This is not different from what

5 Mr Pitts (1) being unwilling (2) declared his unwillingnesse to interpose (a) of his (b) without 6 offered (1) them (2) and desired him to take 10 sayes (1) Mr Clarke left his Comment (2) he 11 to print, (1) he sayth (2) and presumes he may have it againe if he request it, and (a) allead breaks off (b) affirmes 13 make (1)21 (2) above 14 here add. 15 as they see convenient add. 17 Impression. (1) There is (a) nothi breaks off (b) not more in all then (2) This is not different from what 594

Leichfeild: i.e. Ann Lichfield (d. 1672). Impression: i.e. the third edition of Oughtred's Clavis mathematicae, printed by Leonard Lichfield in Oxford in 1652. 596 Clarkes: i.e. Gilbert Clark. 595

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142. COLLINS to WALLIS, [c. 10/[20] February 1666/7] Mr Thompson saith himselfe formerly Designed, and doth thinke the widdow597 will be inclinable to, forasmuch as her Daughter598 was lately with him about it. Now as to the booke it selfe, Dr Croone599 and Mr Colwall600 can attest that the late Mr Foster601 of Gresham College seldome heard it mentioned but tooke occasion to utter his Dislike of it, (and Dr Croone [195r] hath formerly said as much to Mr Thompson). By reason whereof in anno 1649 I asked Mr Foster what Authors he would advise unto; he replyed that the Algebra of Scheubelius602 (out of which Mr Bunning603 hath

3 about it. (1) That purpose (2) Now as to the Booke it selfe I alwaies esteemed it a good Compendium but know withall as (3) Now as to the booke it selfe, 4 the late add. 4 College (1) never (2) seldome 6-314, 6 Mr Thompson). \(1) The said Mr Foster advised shewing his dislike (2) I once occasionally (3) Upon occasion (4) By reason whereof in anno 1649 I asked (ao) him (66) Mr Foster . . . in one volume. add.\ I have 8 the Algebra of add. 8 Mr Bunning (1) tooke (a) his (b) some (2) hath taken some 597

widdow: i.e. Ann Lichfield. Daughter: i.e. Margaret Lichfield. 599 Croone: i.e. William Croone (or Croune) (1633-84), physician, professor of rhetoric at Gresham College (1659-70?). In 1670 he was appointed successor to Sir Charles Scarborough in Company of Surgeons. Fellow of the Royal College of Physicians and fellow of the Royal Society. DNB. 600 Colwall: i.e. Daniel Colwall (d. 1690), wealthy London citizen and patron of the sciences; fellow of the Royal Society (from 1661 onwards), DNB. 601 Foster: i.e. Samuel Foster. In 1646 or 1647, Wallis received from Foster a theorem entitled De triangulo sphaerico, which he afterwards published in his Mechanica, part 2, Oxford 1670, ch. 5, prop. 24; Opera mathematica I, 869. 602 Scheubelius: i.e. Johann Scheubel (or Scheybl) (1494-1570), professor of mathematics in the University of Tubingen; SCHEUBEL, Algebrae compendiosa facilisque descriptio, qua depromuntur magna Arithmeticus miracula, Paris 1551. 603 Bunning: church minister in Warwickshire. See CoLLiNS-WALLis 2/[12].II.1666/7. 598

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taken some of his Notes), Stifelius604, Clavius605, Dybuadius606, Stevin607 did fully handle the Surds, and Euclids irrationall lines, that Harriott608, Herigon609, Deschartes610 & Ghetaldus611, sufficiently the Specious and Exegetick part, not mentioning Vieta or Mr Oughtred612, whose workes might then be scarce and not so large as now, Vieta613 not being then in one volume. | I have likewise seen other small good Compendiums, as [194V]

1 Dybuadius, add. 3 Ghetaldus, (1) aboundantly (2) sufficiently the Specious (a) part (6) and 4 Mr Oughtred, jmuch of add. and del] whose workes (1) I should be very sorry the world should want (2) seeme to be tal breaks off (3) in the former Authors (4) might 6 small add. 604

Stifelius: i.e. Michael Stifel (1487-1567), reformed theologian (Lutheran) and mathematician; STIFEL, Arithmetic/! Integra, Nuremburg 1544. 605 Clavius: i.e. Christoph Clavius (1537-1612); CLAVIUS, Algebra, Rome 1608. 606 Dybuadius: i.e. Christoffer Dybvad (15787-1622), born in Copenhagen, physician and canon in Lund. He died in Kallundborg, where he had earlier been imprisoned for writing against the aristocracy and the government; DYBVAD, In geometriam Euclidis prioribus ex elementorum libris comprehensam demonstratio linealis, Leiden 1603; In arithmeticam irrationalium Euclidis decimo Elementorum libra comprehensam demonstratio linealis & numeralis, Arnhem 1605; In arithmeticam rationalium Euclidis septimo, octavo & nono elementorum libris comprehensam demonstratio, Arnhem 1605; Problema de arcuum descriptions ex triangulorum apicibus, Copenhagen 1606. 607 Stevin: i.e. Simon Stevin (1548-1620), Flemish mathematician and engineer; STEVIN, L'arithmetique contenant les computations des nombres arithmetiques ou vulgaires; aussi I'algebre ..., Leiden 1585; Appendice algebraique, Leiden 1594. 608 Harriot: i.e. Thomas Harriot (1560-1621), mathematical practitioner, adviser to Walter Raleigh and later to the Earl of Northumberland; HARRIOT, Artis analyticae praxis, London 1631 (a selection of his writings published posthumously by W. Warner). 609 Herigon: i.e. Pierre Herigone, pseudonym of Clement Cyriaque de Mangin (c.1570c.1642), physician and teacher of mathematics in Paris; CYRIAQUE DE MANGIN, Cursus mathematicus nova, brevi et clara methodo demonstrata, 5 vols., Paris 1634-7; Supplementum cursus mathematici, continens . . . L'isagoge de I'algebre . . ., Paris 1642. 610 Deschartes: i.e. DESCARTES, La geometrie, Leiden 1637; Geometria, ed. Fr. v. Schooten, Leiden 1649; 2 vols., Amsterdam 1659/61. 611 Ghetaldus: i.e. Marino Ghetaldi (c.1566-1627), Italian mathematician; GHETALDI, De resolutione & compositions mathematica, libri quinque, Rome 1630. 612 Oughtred: By the time of the third Latin edition of Clavis mathematicae (Oxford 1652), numerous chapters had been enlarged and seven appendices added. 613 Vieta: i.e. VIETE, Opera mathematica, in unum Volumen congesta, ac recognita, ed. Schooten, Leiden 1646.

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142. COLLINS to WALLIS, [c. 10/[20] February 1666/7] Vaulezards614 & Humes615 Epitome of Vieta, Durett616, Beaugrand617, Henrioone618, Lantz619, Minher620, Cardanus621 in 8°, Leotauds623 Geometria Practica, which is rather a small tract of Conicks and the Problematick part of Speculative Geometry, to which might be added some I have not seen, as Andreas Arzets624 Clavis Mathematicae, in English 1 & Humes add. 1 Durett, add. 2 Henrioone, add. 2 in 8°, |Gerrards622 Inventione Novelle, del. Leotauds 4 added (1) divers others (2) some 5 Mathematicae (1) . Now whereas Anderson the weaver hath misliked it (2) , in English . . . Balaam, 614 Vaulezards: i.e. Jean Louis de Vaulezard (fl. 1630); VAULEZARD, Introduction en I'art analitic ou nouvelle algebre de Viete, Paris 1630. 615 Humes: i.e. James Hume (fl. 1636), Scottish mathematician. He was a member of the circle in Paris around his compatriot Alexander Anderson; HUME, Algebre de Viete, Paris 1636. 616 Durett: i.e. Nicolas Duret (17th century); DURET, L'algebre, effections geometriques et partie de I'exegetique nombreuse de Viete, Paris 1644. 6lr Beaugrand: i.e. Jean Beaugrand (1595-1640), student of Viete and sometime mathematician to Gaston, Duke of Orleans. Later a refugee in the Netherlands, he claimed that Descartes' work was copied from Harriot; BEAUGRAND, Ars analytica, Paris 1630. A late edition of VIETE, In artem analyticam isagoge (Paris 1631) includes notes by Beaugrand. 618 Henrioone: i.e. Denis Henrion, pseudonym of Clement Cyriaque de Mangin (see above); CYRIAQUE DE MANGIN, Collection, ou, Recue.il de divers traitez mathematiques, Paris 1621. 619 Lantz: i.e. Johannes Lantz (1564-1638); LANTZ, Institutionum arithmeticarum libri quatuor cum, appendice fractionum astronomicarum, editio secunda, Munich 1619. The first edition was published in 1616. 620 Minher: i.e. Valentin Mennher de Kempten (fl. 1550), German-Dutch arithmetician; MENNHER, Pratique briesve pour cyfrer et tenir Livres de compte, Antwerp 1550; revised edition, Arithmetique seconds, Antwerp 1556; revised edition, Pratique pour brievement apprendre a ciffrer, & tenir livre de Comptes, avec la Regie de Coss, & Geometric, Antwerp 1565. 621 Cardanus: i.e. Geronimo Cardano (1501-76); CARDANO, Practica arithmeticae singularis, Milan 1539; Artis magnae sive de regulis algebraicis liber unus, Nuremburg 1545. 622 Gerrards: i.e. Albert Girard (1595-1632); Invention nouvelle en I'algebre, Amsterdam 1629. 623 Leotauds: i.e. Vincent Leotaud, q.v.; LEOTAUD, Elementa geometriae practicae, Dole 1631. 624 Arzets: i.e. Andreas Arzet (1604-75), born in Konstanz, Jesuit and teacher of mathematics and rhetoric; ARZET, Clavis mathematica, Innsbruck 1634.

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Billingslys625 Idea626, Gibsons627 Syntaxis, Balaam628, in Dutch, which I understand not, Sybrant Hantz629, Frans vander Huips630, &c. All these are in 8° or 12°. In 4° in 1661 came out Gerrard Kinckhuysens631 Algebra inbound after his Analytick Conicks632, which I presume to be an excellent Introduction. He was the first author of the Probleme633 of the 3 Sticks in Des Chartes and hath published in his Conicks his owne Solution634 thereof. I shall mention one more in high Dutch I have not seen, [1951 Academia Algebra635. Now concerning Mr Anderson636 the Weaver, a reserved person, I

1 which I understand not, add. 2 Huips, (1) in high Dutch the (2) fee. All these are (a) small (b) in Sin (1) 1660 (2) 1661 4 inbound . . . Conicks add. 5 first add. 9 a reserved person, add. 625

Billingslys: probably Henry Billingsley (fl. 1551-1606), first English translator of Euclid; BILLINGSLEY, The Elements of Geometry of the most auneient philosopher Euclide of Megara, London 1570. 626 Idea: possibly John Dee's mathematical preface to Billingsley's translation of the Elements. 62r Gibsons: i.e. Thomas Gibson (fl. 1655); GIBSON, Syntaxis mathematica: or, A construction of the harder problemes of Geometry: with so much of the conicks as is therefore requisite, and other more ordinary and usefull propositions intermixed: and tables to several purposes, London 1655. 628 Balaam: i.e. Barlaam (12907-1348), monk of the order of Holy Basilius, bishop of Geraci, Naples; BARLAAM, Logistiea, sive arithmeticae algebraicae libri VI, Strasbourg 1572. 629 Sybrant Hantz: i.e. Sybrandt Hansz Cardinael (1578-1647), mennonite Christian and arithmetician in Amsterdam; CARDINAEL, Arithmetiea; ofte, Reecken-konst, 4 vols., Amsterdam 1640-54; Tractatus geometricus, Amsterdam 1617. 630 Huips: i.e. Frans van der Huips (fl. 1654), examiner of maritime pilots; HUIPS, Algebra, Amsterdam 1654. 631 Kinckhuysens: i.e. Gerard Kinckhuysen (d. 1661); KINCKHUYSEN, Algebra ofte stelkonst, Haarlem 1661. 632 Conicks: i.e. KINCKHUYSEN, De grondt der meetkonst ofte een korte verklaringe der kegelsneeden, Haarlem 1660. 633 Probleme: see SCHOOTEN, Additamentum, in: DESCARTES, Geometria, ed. Schooten, I, Leiden 1659, 369-89. This problem had first been published by Jacob van Wassenaer in Den on-wissen wiskonstenaer, Leiden 1640. 634 Solution: see KINCKHUYSEN, De grondt der meetkonst, 70-91. 635 Academia Algebra: i.e. FAULHABER, Academia algebrae, Ulm 1631. 636 Anderson: i.e. Robert Anderson.

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142. COLLINS to WALLIS, [c. 10/[20] February 1666/7] never had any Papers or so much as a Theoreme from him. Mr Leake637 accounts him very able in Algebra and Solid Geometry, and one if he may be beleived that hath exalted the 15 Problemes of Vieta of Tactions of Circles638 to Tactions of Spheres, who affirmed above 4 yeares since that he found the Second Segments of a Sphaere or Sphaeroid by Inscription of a Parralellipipedon whose sides produced cut of 12 second Segments. I have mett with nothing of such a tendency except in Andersons639 1. Decad of Math. Exercises, at Paris 1619, where he proposes to find a Parral. with a Square Base equall to a given Solid that is lesse then a Cube inscriptible in a Sphere, and finds two such Parralellipipeds the lesser greater Base one height then the Cube, and ol a greater lesser other V [195 ] and these being equall, their plaines| produced cutt of 12 Second Segments of the one equall to 12 of the other. I want abilities to drive it home, and Doubt whether any person living hath done it either before or so well as your selfe, but hope my suggesting of what others (if they have done) keepe to themselves may be no Offence to you who will impart it. In my Letter640 about the Meridian Line I forgott to mention a Curve that may be supposed to passe through every 10th degree of Secant, if crossed with the Extents of the length of every 10th degree of arch, the Center for the

2 him (1) a Gyant in (2) very able in Algebra and 8 at Paris 1619, add. 8 proposes (1) & Solid (2) to 9 lesse (1) as (2) then 11 the |given del] Cube, 15 done add. 16 keepe |to add. themselves 16 who will impart it add. 18 if [supposed to be del] crossed 63r

Leake: i.e. John Leake (fl. 1650-86), mathematical practitioner and teacher of mathematics in London, editor (with George Serle) of Elements of Geometry, London 1661. 638 fifteen problems of Vieta of Tactions of Circles: Collins presumably refers to VIETE, Apollonius gallus; Opera mathematica, ed. Schooten, 325-46. 639 Andersons: i.e. Alexander Anderson (1582-1619?), mathematician, native of Aberdeen, taught mathematics in Paris in early 17th century, where he was in the circle of Viete and Hume; ANDERSON, Exercitationum mathematicarum decas prime, Paris 1619. 640 Letter: i.e. COLLINS-WALLIS 17.1666/7.

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143. WALLIS to OLDENBURG, 12/[22] February 1666/7 Compasses being the Extreamity of the Radius, which may be of some use in the Plaine Chart. The Querie is whether it is yet considered or treated of by Geometers. Mr Pitts nor my selfe have no reason to thinke your Motion unseasonable at such a time as this is, but returne you hearty 5 thankes for your trouble in it, as also for your hint that somewhat else might be Preparing whilst the Clavis is selling; any thing of yours will be most acceptable to the Learned, and to the Stationer also, whome I hope and beleive will deale candidly, and put your Workes as they deserve into a more Splendid Garbe then those already extant. May your endeavours 10 therefore have good successe and your selfe happinesse, so wisheth Your obliged thankfull Servant Sir Samuells Cryptography641

143. WALLIS to HENRY OLDENBURG Oxford, 12/[22] February 1666/7 Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 31, 4 pp. (our source). On p. 3 (otherwise blank) three drawings probably concerning Hevelius's Selenographia (one apparently being a rough sketch of Hevelius's lunar map, the others concerning stereographical projection), but without clear connection to the letter itself (see apparatus). On p. 4 Oldenburg has noted beneath the address: 'Of Febr. 12. 1666/7'.—printed: OLDENBURG, Correspondence III, 341-2.

1 which (1) will (2) may 2 use (1) wh breaks off (2) in 2 it is add. 5 your trouble in add. 7 I (1) doubt not but will (2) hope |and beleive add.\ will 8 as they deserve add. 641

Sir Samuells Cryptography: i.e. MORLAND, A New Method of Cryptography, humbly presented to the most serene Majesty of Charles the II., [London 1666]; Samuel Morland (1625-95).

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143. WALLIS to OLDENBURG, 12/[22] February 1666/7 Oxford Febr. 12. 1666./7. Sir

I told you last week642, that the other Copy of Hevelius his Book643 being come to hand, I intended to send your last back again; which I do this day by Moor's Wagon, at whose Warehouse I hope it will bee safe for you. I find upon the putting it up, that it hath received some dammage in one or two of the sheets; by the incivility of those who had once the possession of it, & would have kept it, if they had thought it worth stealing: But, as it is, it is better than lost. I might indeed have sent you back the other Copy which is a little fairer; but that is allready bound up with some other peeces, & so not convenient to bee sent, & therefore I must desire you to accept of this as it is, rescued from those who had made a prey of it. As to the contents of it; the truth is I know not what to say. Hee speakes so fair for two Phaenomena, that though I am not forward to grant it, yet I am not able to answere some arguments so as to satisfy myself. But I should be glad to see the series of observations made by our own members, from which I should expect more satisfaction than from those of Auzout, whose Observations are but very lame as to the desiding the doubt. For though I doubt not at all, but that there did a comet pass by the second of Aries; yet I am not sure but there might another pass by the first allso, as Hevelius would have it. And therefore (as I hinted in my last) I could yet wish, that some publike account were given of our English Observations, by Mr Hook, or Dr Wren, or both: especially th th [2] if there be any considerable ones, between the 12 & 18 of February.) I have with it, adjoyned another book644, tyed to it, which I must desire your favour to deliver it to the Count Ubaldini645; it being a Manuscript which hee is to restore to Sir Robert Moray. Which I the rather send with yours, not onely for the convenience of it; that one trouble may do both, but because it hath once scaped the same danger of being lost with it. I have even now been asked, whether there bee not some Copies of 17 members, (1) wli breaks off (2) from 642

told you last week: Wallis probably refers to his letter of 31 January. Book: i.e. HEVELIUS, Descriptio Cometae, Danzig 1666. 644 another book: not identified. 645 Ubaldini: i.e. Carlo Ubaldini, Count of Montefeltri, Tuscan nobleman and protestant proselyte. Proposed by Oldenburg, he was elected fellow of the Royal Society on 21 November 1667. See OLDENBURG-BOYLE 10/[20].VIII. 1665; OLDENBURG, Correspondence II, 457. 643

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143. WALLIS to OLDENBURG, 12/[22] February 1666/7

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Hevelius his Selenography646 sent over to be sold on his account. I think I was told about a year agoe, that there were. If so, I can help to a chapman for one of them. And, it may bee, for the rest of his pieces all that are yet extant. I write it now while it is fresh in my mind, lest otherwise I should forget it. I have nothing of news from hence, that is worth writing: & shall therefore adde no more but that I am Your affectionate friend & servant John Wallis.

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These For Mr Henry Oldenburgh, my honoured friend; at his house in the Palmal near St James's London. together with a parcell, containing two Bookes, tyed together. 9 [On p. 3, probably in Wallis's hand:]

5 worth (1) (yet) (2) writing 646

Selenography: i.e. HEVELIUS, Selenographia: sive, lunae descriptio: atque accurata, tarn macularum ejus, quam motuum diversorum, aliarumque omnium vicissitudinum, phasiumque, telescopii ope de.pre.he.nsarum, delineatio, Danzig 1647.

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144. FELL to WALLIS, 19 February/[l March] [1666/7?]

144. JOHN FELL to WALLIS [Oxford], 19 February/[l March] [1666/7?] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 95r-96v (f. 95V and 96r blank). Top third of f. 96 is missing. At top of f. 95V endorsement in unknown hand: 'febr. 19'. The present letter evidently reflects Fell's efforts to restore discipline to the University at large after he had been installed as vice-chancellor in August 1666. The problem of prostitution was a particular concern at this time, making it likely that this letter written at the beginning of 1666/7. See WOOD, Life and Times II, 82-3, 95-6.

Sir

I am enformed by two witness upon Oath, of a Woman who commonly entertains Scholars & others; & was seen in the Act of lewdnesse with a Scholar: She pretends to be a Married Woman, & hir pretended husband is party to this practice. I pray enform me in a line or two, whither notwithstanding marriage, or the pretence of it I may proceed against the woman, as against a Common Whore; or what course is to be taken. I am Sir

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Your Assured friend J. Fell.

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Feb: 19.

[96V] For The reverend Dr Wallis these D.D.

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146. WALLIS to OLDENBURG, 21/[31] March 1666/7

145.

HENRY OLDENBURG to WALLIS 19/[29] March 1666/7 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 21/[31].III. 1666/7. Oldenburg informed Wallis of observations of the tides made by Samuel Colepresse and reported to him in a letter (COLEPRESSE-OLDENBURG 6/[16].III. 1666/7; OLDENBURG, Correspondence III, 355-7). He apparently also reported on French experiments on the transmitting of blood from one animal to another, confirming those performed by Richard Lower, and enclosed a theory of motion by an unnamed author. The author was in fact William Neile, as made clear by Oldenburg's endorsement on WALLISOLDENBURG 21/[31].III.1666/7.

146.

WALLIS to HENRY OLDENBURG Oxford, 21/[31] March 1666/7 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 32, 4 pp. (p. 3 blank) (our source). At top right of p. 1 beneath date: 'Entered L.B. 1. 410.' On p. 4 Oldenburg has noted beneath address: 'From Dr Wallis to Mr Oldenburg About Tides, and Mr Neiles propositions of motion.' Postmark on p. 4: 'MA/22'.—printed: OLDENBURG, Correspondence III, 372-4. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 1, pp. 410-1. w2 Copy of w1: LONDON Royal Society Letter Book Copy 1, pp. 484-5. Reply to: OLDENBURG-WALLIS 19/[29].III. 1666/7.

Oxford March 21. 1666./7.

Sir, I thank you for yours647 of the 19th instant, which came to my hands this morning: & for the severall particulars in it testifying your kindness 64r

yours: i.e. OLDENBURG-WALLis 19/[29].III.1666/7. 322

146. WALLIS to OLDENBURG, 21/[31] March 1666/7 for me. Mr Colepresse648, it seemes, hath observed649, at Plimouth, the Tydes in February, consonant to my Hypothesis; which I am not sorry for: and I am pretty confident, that the more observations are made of this kind, the more will that Hypothesis be confirmed. I do expect, that Observations should not, at all places, just answer the same time for the High Tydes: But I am apt to think, that the Annuall Highest Tydes are to be expected, before the Vernall Aequinox, & after the Autumnal: But whether at one Moneth, or two Moneths distance, or some intermediate time, I think is very subject to variation, according to the divers situation of places. For since that it doth not depend upon a single cause, as on the Zodiacks Obliquity onely, or onely on the Suns Apogaeum & Perigaeum; but on a complication of both; (as doth that of the Inaequality of Natural days;) It is not strange that the one or the other's operation should be more or lesse predominant, according to the different position of places on the Globe, & the nearness or remotenesse from the open Sea; & other the like adventitious causes, which may check or promote the efficacy of the one or the other of those principal causes, & so cast the time nearer to this or that of those different seasons which those causes would respectively design without the complication of the other. The Hypothesis in Philosophy650, which you were pleased to send, (without telling me, Who's it is,) is, as to many parts of it, rationall inough; though I will not promise to comply with it in all. The main 6 Annuall (1) High (2) Highest 8 at add. Oldenburg

10 as (1) of (2) on 11 18 18 20 22

or (1) only (2) onely different add. would (1) singly (2) respectively design (a) of the (b) without were add. to (1) bee (2) comply

648

Mr Colepresse: i.e. Samuel Colepresse (d. 1669), correspondent of Oldenburg at Plymouth. 649 hath observed: see COLEPRESSE-OLDENBURG 6/[16].III. 1666/7 (OLDENBURG, Correspondence III, 355-7, 356), of which Oldenburg clearly had informed Wallis in his letter of 19/[29].III.1666/7. 650 Hypothesis in Philosophy: As appears from Oldenburg's endorsement on this letter, this was presumably a preliminary version of William Neile's Hypothesis of Motion (OLDENBURG, Correspondence V, 519-24), which was discussed in the correspondence between Oldenburg and Neile not before December 1668 and read at the meeting of the Royal Society on 29 April 1669 (BIRCH, History of the Royal Society II, 361).

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146. WALLIS to OLDENBURG, 21/[31] March 1666/7

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thing in it; That a Body at Rest, will so continue till some positive cause put it in motion; And, a Body in motion, will continue in motion, & with the same Celerity, and the same way, till some positive cause do either stop or alter it: I take to bee very true. And I think I have sufficiently demonstrated it from its proper cause; in some papers of myne651 which Mr Boyle was pleased to have transcribed for him some years ago. I think it is, a notion taken for granted by Galilaeo, Toricellio, & others of later times; but I do not know that any of them have gone about to prove it: but rather (as the proposer of what you now send) take it for a Postulate. I shall not at present give you any further trouble concerning it; nor that other particular you mention of Dr Lowers652 Operation of transmitting Bloud from one Animal to another, confirmed by the French operation653 in imitation of his: Onely I could wish that| those of our own Nation, P] were a little more forward than I find them generally to bee (especially the most considerable) in timely publishing their own Discoveries, & not let strangers reape the glory of what those amongst ourselves are the Authors. But the Post is going, & I have no time (for fear of loosing him) to say more than that I am Your affectionate friend & servant.

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These For my honoured friend, Mr Henry Oldenburgh, at his house in the Palmal, near 2 will (1) so continue (2) continue in motion, 3 and the same way, add. 5 myne |of del. which 7 others (1) later than (2) of 8 have (1) given (2) gone 11 Dr Lowers (1) experiment (2) Operation 13 Nation, (1) with (2) were 651

some papers of myne: not identified. Dr Lowers: i.e. Richard Lower. 653 French operation: i.e. the successfull repetition of Lower's experiment, an account of which Oldenburg had received from France and read at the meeting of the Royal Society on 21 March 1666/7; see BIRCH, History of the Royal Society II, 161. Possibly this experiment had been performed by the physiologist Jean-Baptiste Denis (c.16401704). 652

324

147. WALLIS to CLENDON, 5/[15] May 1667 St James's London.

147. WALLIS to JOHN CLENDON Oxford, 5/[15] May 1667 Transmission:

W Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 21r (verso blank). At the foot in Ubaldini's hand: 'May the 6 th : 1667. Received of Tho: Stagg five poundes the rentes of this bill by me. Ubaldini &x.'

Oxford May 5. 1667. Cousin Clendon, I sent you a while agoe, a Bill desiring you to pay to Count Ubaldini654, the summe of Five pounds on my account, (which I suppose is done;) this is to desire you, with your first conveniency, to pay him another Five pounds on my account. About 10 days hence I think to be in London. Mean while resting Your loving Uncle John Wallis. For Mr John Clendon at his lodgings in Fig-tree Court in the Inner Temple; or at Mr Stag's house in little Drewry lane over against the BellTavern.

148. HENRY OLDENBURG to ROBERT BOYLE London, 17/[27] September 1667 Transmission:

C Letter sent: LONDON Royal Society Early Letters OB, No. 63, 6 pp. (our source). Postmark on p. 6: 'SE/17'.—printed: OLDENBURG, Correspondence III, 476-8; BOYLE, Correspondence III, 335-9. 654

Ubaldini: i.e. Carlo Ubaldini, Count of Montefeltri.

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148. OLDENBURG to BOYLE, 17/[27] September 1667 E1 BOYLE, Works (1744), V, 365-6. E2 BOYLE, Works (1772), VI, 239-41. The present letter, Oldenburg's third to Boyle following his release after two months' imprisonment in the Tower of London, was apparently also intended for Wallis, as the postscript makes clear.

London Sept. 17. 1667.

Sir,

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Yesterday, when I receaved the favor of yours656, I receaved also the first forrain letter657, since my enlargement; which putts me in good hope of reestablishing my former philosophicall commerce. By it I perceave, the Parisian Virtuosi658 employ themselves much in Anatomicall disquisitions. They have lately made dissections of some Fishes, and of a Lyon; the account whereoff I look for in print, by the first oportunity. At Caen659 they have dissected the eye of an owl; which also I exspect the particulars off. Monsr Steno660, I hear has publisht a Treatise661 de Motu Musculorum; and Godartius^'2 an Addition to his Book De Jnsecfe663; another Dutch 1 [Postscript beneath date at 18CP:] Dr Sherley655 is now with me, and your humble servant. 655

Dr Sherley: i.e. Thomas Sherley or Shirley (1638-78), physician, DNB. yours: i.e. a now missing letter of Boyle written on or shortly before 15/[25].IX.1667. 65r forrain letter: i.e. a now missing letter presumably of Henri Justel. 658 Parisian Virtuosi: i.e. the group around the anatomist Claude Perrault (1613-88). 659 At Caen: i.e. at the Academic de Physique at Caen, founded in 1662 by PierreDaniel Huet (1630-1721). Oldenburg received an account of the dissection experiment at Caen not later than in December; cf. OLDENBURG-BOYLE 17/[27].XII.1667 (OLDENBURG, Correspondence IV, 58-61, 59-60; BOYLE, Correspondence III, 379-81, 381). 660 Steno: i.e. Niels Stensen (1638-86), Danish anatomist. 661 Treatise: i.e. STENSEN, Elementorum myologiae specimen, sen musculi descriptio geometriea. Cui accedunt cam's carchariae dissectum caput et dissectus piscis ex canum ge.ne.re. . . ., Florence 1667. 662 Godartius: i.e. Jan Goedart (1620-68). 663 Book de Insectis: i.e. GOEDART, Metamorphosis naturalis, ofte historische beschryvinghe van den oirspronck, aerd, eygenschappen ende vreemde veranderinghen der wormen, rupsen, maeden, vliegen, witjens, byen, molten en dierghelijcke dierckens meer, niet uyt eenighe boecken, maer alleenelijck door eygen ervarenheydt uytgevonden, beschreven, ende na de konst afgeteyckent, Middelburgh [1662]; also published in Latin translation: Metamorphosis et historic, naturalis insectorum, Middelburgh [1662]. The second part was published in 1667 in Dutch as well as in Latin translation. 656

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148. OLDENBURG to BOYLE, 17/[27] September 1667 man664 being also at work upon the same subject. I know not, whether, during my confinement, you had, or heard of the Tetras Anatomicarum Epistolarum Malpighii (Siculi) et Fracassati (Pisani) de Lingua et Cerebro- together with an Exercitation De Omenfo665. It is in print, and written for by me, to be sent over. I pray, Sir, if you can conveniently, order the injecting into the veines of a Dog some Oile of Sulphur, and see, what will be the consequence of it, and [2] observe particularly, whether the dogg will| feed with an extraordinary avidity upon it. I wish also, that an injection might be made with oyle of Tartar; and, in case the dog dye (as 'tis like he will) that he may be opened, and his bloud well observed. The R. Society hath not met these 2. months; but I hope, they will shortly meet again. I know not, what deadnes there is upon the members of it. I intend to urge, God willing, that those, who have hitherto appear'd to be cordiall for the advancement of its dessein, may combine together with a new zeal, and try to inspire the like concern and fervor into others, that have been coole enough hitherto, to the end, that so noble and usefull an institution may not fall to the ground, in an age, so full of knowledge and curiosity, and enjoying so considerable advantages to improve and increase the same. Sir, how declining soever the words of your last letter seeme to be, as to the publishing of those good things, you are known to have yet in store, I am confident you cannot obtaine of your generosity the hiding of the residue of your Talent. And I shall be very glad to see what you have retrived of your Treatise of Heat and Flame666, and, if you will allow me, to communicate the List of those Experiments to the publike, to [3] provoke others | to make tryals accordingly. I humbly thank you for the

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months; (1) as well (2) but Sir, (1) however (2) how of (1) your (2) the List of those add. publike, (1) for (2) to

664

another Dutch man: presumably Jan Swammerdam (1637-80). Tetras ... Omento: i.e. Tetras anatomiearum epistolarum de lingua et cerebro clariss. Marcelli Malpighii ac Caroli Fracassati quibus anonymi accessit exercitatio de omento, pinguedine et adiposis ductibus, Bologna 1665. 666 Treatise of Heat and Flame: Boyle's dialogue of heat and flame was never published and is now lost. 665

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copy of the augmented Treatise of Formes667, you intend me; And I am persuaded, the Society would take it very ill, if you should passe them by, without presenting them with an Exemplar of the same. I intend one for D. Beale668, as soon as any number of them comes hither; he being one of the most inquisitive men after all your productions. Mr Colepresses papers669 to me will certainly be yet found in your Scrittore670, since I cannot find them, upon the search thorough all my papers: And I remember, that though I mentiond them once, and you were ready to returne them me, yet I stopt you, saying, I had not yet occasion for them. For civill affairs; we look every day for the arrival of the Dutch Ambassadors: And I wonder, they were not here sooner, to prevent Monsr Ruvigny671, who is said to endeavor a stricter allyance between Us and France. 'Tis discoursed, that the United Provinces have furnisht great sums of money for the defence of the Spanish provinces; for which they are to receive, by way of security, Anwerp and some others of the Spanish Townes. Which, if true, how it will serve to keep the peace between France and Holland, I understand not. My Parisian friend672 hath these particulars; 1. Le Roy est maitre d'une bonne partie de la Flandre,

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augmented Treatise of Formes: i.e. the second edition (Oxford 1667) of Boyle's The origine of formes and qualities, according to the corpuscular philosophy (Oxford 1666), which was augmented by a discourse of subordinate forms. 668 D. Beale: i.e. John Beale (1603?-c.l683), D.D., rector of Yeovil in Somerset, interested in agricultural and horticultural matters, elected fellow of the Royal Society in 1663, DNB. 669 Mr Colepresses papers: i.e. Colepresse's observations on the tin mines of Devonshire and Cornwall, mentioned in COLEPRESSE-OLDENBURG 26.VII/[5.VIII]. 1667 (OLDENBURG, Correspondence III, 457-60, 459) and later read to the Royal Society in the meeting on 7 November 1667 (BIRCH, History of the Royal Society II, 208). 670 Scrittore: i.e. escritoire. 671 Monsr Ruvigny: i.e. Henri de Massue, Marquis de Ruvigny et Raineval (1618-89), French general, Huguenot, ambassador at the English court, later resident at Greenwich. 672 Parisian friend: i.e. Henri Justel. 328

148. OLDENBURG to BOYLE, 17/[27] September 1667 et en estat de pousser ses conquestes bien plus avant. 2. Monsr de Touraine673 fortifie Deynse674, et a des troupes a Alost675 pour incommoder les ennemis et ruiner leur quartiers.| Cependant, 1'armee subsistera en Flanders des contributions, et le Roy ne sera pas oblige de faire de grandes despenses. 3. Les Princes du Rhin sont dans nos interests. 4. La Suede doit envoyer Monsr de Dona676 pour travailler a la paix: le pape s'en mesle aussi; mais avec tout cela on ne croit pas, qu'elle se face. 5. Le siege de Candie677 continue tousiours: Ceux de la ville se defendent vaillament. II y a quatre mois, qu'ils resistent au Grand Visir, qui avoit pres de 60000. homines. 6. Le G. Seigneur678 veut, que la Pologne rompe avec le Moscovite, et qu'elle laisse les Cosaques en paix, qui sont soubs sa protection. Cela obligera I'Emperour d'avoir des troupes sur la frontiere.

[4]

The Duke of Yorks679 son was baptised Sunday last, and named Edgar. This maketh many look back upon that brave Saxon King, Ed
Monsr de Touraine: i.e. Henri de La Tour d'Auvergne, Vicomte de Turenne (161175), French, marshal. 674 Deynse: i.e. Deinze, southwest of Ghent. 675 Alost: i.e. Alost (Aalst), between Ghent and Brussels. 676 Monsr de Dona: i.e. Kristoffer Delficus Dohna (1628-68), Swedish diplomat, sent as ambassador to Holland and England. 677 Candie: i.e. Heraklion on Crete, which was besieged by the Turkish Grand Vizier Achmed Koprulii (1635-76) from May 1667 to 1669. 678 G. Seigneur: i.e. Sultan Mohammed IV (1641-92). 679 Duke of Yorks: i.e. James, Duke of York (1633-1701), King James II of England (1685-8). His son Edgar, who was baptized on 15 September 1667, died in 1669. 680 Edgar: i.e. Edgar (944-75), King of England (959-75).

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149. WALLIS to JENKINS, 26 October/[5 November] 1667 because he had so deceitfully usurped from him the faire Elfrida, to whom he was sent by Edgar to view her celebrated beauty, and to engage her for his nuptiall bed, if the thing agreed with fame. I leave you, to think the rest, remaining Sir Your very humble servant H. O.

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My humble service to Dr Wallis, I pray, to whom the communica- [5] tion both of the Philos. and civill part of this letter will perhaps not be unacceptable. I doe not find by your last letter, that such an attempt of Transfusion, for which I suggested Examples in my former681, will be undertaken by any body there. [6]

For his Noble friend Robert Boyle Esquire At the house of Mr Crosse an Apothecary in Oxford. 149.

WALLIS to LLEWELYN JENKINS Oxford, 26 October/[5 November] 1667 Transmission:

W Letter sent: KEW The National Archives PRO SP 29/221, No. 41, 4 pp. (pp. 2 and 3 blank). On p. 4 to the left of address at 90° in unknown hand: 'send by bartlett at the gray hospice in holborn by 8 a clock.', and: 'Dr Wallis touching University charters in Line's Case. 26. oct. 67.' Anwered by: JENKINS-WALLIS 31.X/[10.XI].1667. The background to the present and subsequent letters are disputes between the city and the University of Oxford which arose after Fish Lyne, an attorney, was caught in an alehouse by University proctors on their night walk. Lyne slighted the vice-chancellor's court by not obeying a summons to attend it and afterwards pleaded himself free from 8 letters corr. 681

my former: i.e. OLDENBURG-BOYLE 12/[22].IX.1667 (OLDENBURG, Correspondence III, 473-75, 474; BOYLE, Correspondence III, 333-5, 333). 330

149. WALLIS to JENKINS, 26 October/[5 November] 1667 the University's jurisdiction at the Court of Common Pleas. This raised doubts about the fundamental rights and privileges of the University which the city thereafter sought to exploit.

Oxon. Octob. 26. 1667. Sir,

I have seen yours to the Vicechancellor682. Farendons case683 Mr. Hopkins684 hath at large, it wants but comparing. Chaces685 case686 is in the Printed year books. Wendovers case I think I have notes of, but not at large. Blake & Hunton, where the Privilege is allowed after imparlance, I have allso. Slone against Medeford, I think I have notes of, but am not certain. The French Petition687 you desire, I cannot get time inough, unless possibly by the Post to morrow morning. The writings under seal marked. N 20. N 12. N 21. are all by authority of Parliament; & Hopkins hath the originalls with him. Which are all material. So is that, H 5. and N 3.688 which hee hath with the writ in pursuance of it (of which Hopkins hath a copy,) viz. 27 Apr. 48 Edw. 3. And the charter of 29 Ed. 3. marked N 5.689 where (upon a submission) the power of taking malefactors is taken from the town & given to the Universitie: with the writ in pursuance of it (of which Hopkins hath a copy) 28 July 29. Edw. 3.690 whereby all writs or commissions formerly

13 which hee hath add. 14 copy,) (1) the other (2) viz. 27 Apr. 48 Edw. 3. 15 (upon a submission) add. 682

Vicechancellor: i.e. John Fell, q.v. Farendons case: cf. Wallis's Notes on Martivall's Case 1667? and his Notes relating to the allowance of the claim in the Exchequer. 684 Hopkins: i.e. William Hopkins, q.v. 685 Chaces: probably Thomas Chace (d. 1449), master of Balliol College (1412-28), chancellor of the University of Oxford (1426). 686 Chaces case: cf. Wallis's Notes relating to the allowance of the claim in the Exchequer. 687 French Petition: cf. WALLIS-JENKINS 2/[12].XL1667 and Wallis's Notes relating to the allowance of the claim in the Exchequer. 688 N 3: i.e. the Charter of 12 April 10 Ed. 3; see Wallis's Notes relating to the allowance of the claim in the Exchequer. 689 N 5: i.e. the Charter of 27 June 29. Edw. 3; see WALLIS-JENKINS 2/[12].XL1667. 690 28 July 29. Edw. 3: i.e. the writ directed to the mayor and bailiffs in pursuance of the charter of 27 July 29. Edw. 3. See WALLIS-JENKINS 2/[12].XL1667. 683

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151. WALLIS to JENKINS, 27 October/[6 November] 1667 granted to the town to that purpose, are revoked. So that the University must have that power, or else it is no where. All these Hopkins hath with him, & divers others to the same purpose. I can say no more (lest the carrier be gone) save that I am Your humble servant J. Wallis.

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[4]

These For the Right Worshipfull Dr Luellin Jenkins, Judge of the Court of Admiralty, at Exeter-house London.

150. LLEWELYN JENKINS to WALLIS 26 October/[5 November] 1667 Transmission:

Manuscript missing. Existence: Referred to in and answered by WALLIS-JENKINS 27.X/[6.XI].1667.

151. WALLIS to LLEWELYN JENKINS Oxford, 27 October/[6 November] 1667 Transmission:

E First edition of (missing) letter sent: WYNNE, Life of Sir Leoline Jenkins, vol. II, 656-7. Reply to: JENKINS-WALLS 26.X/[5.XI].1667. Answered by: JENKINS-WALLIS 31.X/[10.XI].1667. The present letter evidently contained a copy of at least part of the charter 22 Edw. III.

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151. WALLIS to JENKINS, 27 October/[6 November] 1667 Oxon, Octob. 27. 1667. Sir,

I thank you for the judicious Hints you was pleased to impart to me in your Letter691 of yesterday, they have been very serviceable to me in my Searches, and are a fresh and convincing Proof to us, how much your Thoughts and Inclinations are for the Service of the University. But I do not find that there was any Composition made between the University and Town, Anno 3 Ed. III. as your Letter692 to Mr. Vice Chancellor693 intimates: But An. 22 Ed. III. there was some such Composition, of which I have found the Original in French, and have here sent you a Copy of it, which I can attest to be true, having written and examined it myself. That which therein doth most concern our present Business, is the Acknowledgment of the Town, that the Chancellor hath Cognisance of Townsmen, as well as Scholars, and ex officio, as well as at the Suit of the Party, and particularly in Disturbances of the Peace, and in Matters which concern the University, as an University; in which Cases they are bound to appear in Person, and not by a Proctor; in other Cases indeed they may constitute a Proctor if they please, save where the Chancellor, for some just Reasons, shall see Cause to require a personal Appearance. But upon what Occasion this Composition was made, I cannot tell, nor do I yet find the Petition you mention, of the Townsmen to the University. I remember somewhat to that Purpose, in a Business concerning the Mercate, as Picage and Stallage, &c. But I take it to be about Hen. VI. his Time; and I believe the like was about 26 Ed. I. for there was then a Conflict between the University and Town, and afterwards a Composition. For Presidents of Privilege in the Exchequer I have two of Cambridge, one of J. Pooley, upon an Information about Card-making, ex libra ordination. Scacc. de term. Hill. 7. Feb. 24. Eliz. The other of J. Barker, upon an Information about selling of Wines above the Value dismissed upon a Privy Seal. Hill. 28 Eliz. rot. 125. ex parte Remem. Reg. in Sec. and yet the Barons of the Exchequer are not more particularly mentioned in| [657] their Charter than in ours, as may be seen in Jo. Lilly1?, Case, Hill. 1656. in Ban. Com. where the University's Plea was allowed after imparlance. This in hast from yours, &c. John Wallis. 691

Letter: i.e. JENKINS-WALLIS 26.X/[5.XI].1667. Letter: evidently the letter also referred to in WALLIS-JENKINS 26.X/[5.XI].1667. 693 Vice Chancellor: i.e. John Fell, q.v. 692

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152. WALLIS to JENKINS, 31 October/[10 November] 1667

152. WALLIS to LLEWELYN JENKINS Oxford, 31 October/flO November] 1667 Transmission: E First edition of (missing) letter sent: WYNNE, Life of Sir Leoline Jenkins, vol. II, 655-6. Answered by: JENKINS-WALLIS 5/[15].XL1667.

Oxon, Octob. 31, 1667. Sir,

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The Cognisance of all personal Actions whatsoever, (Maim and Felony only excepted) where a Scholar, or Scholar's Servant is a Party, hath as well by Prescription as by Grant, belonged to the Chancellor of the University of Oxford: and this hath been thought so necessary for the well Government of the University, that at least from K. John downwards, it hath been granted and confirmed by almost every King and Queen that hath reigned; and still as any new Objections or Evasions did arise, new Clauses were inserted to obviate or explain them. As in K. Henry III. Time, in a Concurrence of Privilege (the Constable of the Castle of Oxon pretending an Exemption of the King's Jews, as under the King's special Protection) it was first found by Verdict, and afterwards solemnly adjudged, that the University's Privilege was to have the Preference; and afterwards the same was inserted by particular Words in other Charters. In Ed. I a Scruple was made, because the Privilege was not claim'd before imparlance, and thereupon a special Writ was directed to the Justices of the King's Bench, to allow the Privilege notwithstanding; and in following Charters was inserted the Clause, tarn in causis jam pendentibus quam postea inchoandis. In the Case of Prohibitions, it is particularly granted, that the Chancellor should proceed non obstantibus prohibit: regiis, by Charters of Hen. III. Ed. I.II.III. and so downwards; and the like for other Writs whatever from Superior Courts, the Chancellor is not to be molested in the Exercise of his Jurisdiction, but to proceed notwithstanding; and many such Proceedings have been allowed, as you may find by the Presidents and Instances hereunto annexed. As to the Objection, that no Privilege is to be allowed against the King, it is provided against by that Clause, etiamsi tangat nos haeredes vel successores nostros vel ministros 334

152. WALLIS to JENKINS, 31 October/[10 November] 1667 nostros quoscunque in the Charters of Hen. IV. and Hen. VIII, &c. which hath been so strictly observed formerly, that the Chancellor hath been allowed his Claim of Cognisance in his own Case, (as in Dr. CTmres694, in the Reign of Hen. VI.) upon an Indictment in K. B. for an Assault on a Marshal of the Court, executing a Writ of that Court (it seems) on a privileged Person. And by the Charters of Ric. II. Hen. IV. Hen. VIII. &c. all Judges, Justices, and Ministers whatever, are required not to intermeddle; and in case they should intermeddle, yet to surcease upon the Chancellor's Monition, or Intimation. When it was a Question, whether Grants in general Terms, without a particular Enumeration, would be conclusive; to obviate this, in the Charters of Ric. II. Hen. IV. Hen. VIII, &c. are very large Enumerations, both of Causes, Judges, and Courts, closed with general comprehensive Words, and that Expressions in general Terms should be interpreted as valid, as if more particularly expressed and recited, and that the Words and Clauses should be interpreted most strongly in Favour of the University: And least all these Privileges might be thought not grantable by Charter, or the sole Authority of the King, they are by Act of Parliament, 13 Eliz. not only confirmed, but Enacted to be as valid in Law to all Intents and Purposes, as if the said Charters were verbatim recited in the said Act; and the Substance of those Privileges, tho' not in so full Words, have been by former Parliaments granted, as in the Times of Ed. II, &c. and unless these Rights be allowed us, the Government of the University cannot subsist. The Vice Chancellor's Officers cannot arrest a Person, or execute a Writ, but they are presently sued for it in the Courts above; and though perhaps it be but a Matter of 40s. it shall cost the University as many Pounds to defend their Jurisdiction toties quoties. The Towns People always countenancing all vexatious Actions of this kind, and frequently bearing the Charges of them. Especially since my Lord Chief Baron Hales695 refused to allow Mr. Wilkins696 his Privilege in the Exchequer. In Confidence of which President, they now lay their Actions in the Exchequer; of which kind there are several there now depending. Particularly that of Pn'nce697 against Litchfield^98, and

694

Chares: probably a misspelling for Chace. See WALLIS-JENKINS 26.X/[5.XI].1667. Hales: i.e. Sir Matthew Hale (1609-76), lord chief baron of the Exchequer (1660). 696 Wilkins: i.e. Timothy Wilkins. See DILLINGHAM-WALLIS 27.X/[6.XI].1662. 697 Prince: i.e. Daniel Prince, Serjeant of the city of Oxford. 698 Litchfield: i.e. Solodell Lichfield (d. 1671). 695

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Powise9g by quo Minus, for arresting Prince upon the Vice Chancellor's Warrant, and imprisoning him till he put in Bail to answer. And Fish Lyne supported by the Town at this Time, molesting the Vice Chancellor and Proctors, and questioning their Jurisdiction against him for Tipling and Noctivaga|tion, and many other Instances of like kind. So that if such [656] vexatious Proceedings receive Countenance above, it will be impossible to preserve either the Jurisdiction, or good Government of this Place, in any tolerable Order. It having already cost the University, within these few Years since his Majesty's Return, some hundreds of Pounds, to defend themselves against these vexatious Actions. And for preventing of which, the Care of our Princes hath been in former Times so great, as to fence our Jurisdiction against it by all Means imaginable. It being of that great Concern, that the Governours of the University, who are intrusted with the Education of the Youth of the whole Nation, might be in a Capacity to keep this Place in Quietness and Order, and to prevent Wickedness and Debauchery. And that which is aimed at by Prince is such a notorious and publick Injury, as we hope no reasonable Person, or Court of Justice, can think fit to countenance it. The Townsmen making it their Business to ingross all Trade, that they may sell at excessive Rates; nor will suffer any Foreigner (no, nor Citizen, if not free of their pretended Corporation of Mercers) so much as to sell in open Market, any Commodities which they call Mercery Wares, which they extend from Linen and Woollen Cloaths, Stuffs and Silks, even to Mouse Traps and dark Lanthorns. And whosoever shall sell thereof, they vex by Threatnings, Suits, Arrests, Informations, Indictments, and all the vexatious Ways they can imagine. And for arresting one of their Serjeants for throwing down a Foreigner's Wares, and spoiling them, for being exposed in open Market to sell, is the Cause of Prince's Action; which we hope will appear to be so unreasonable, as not to find any Countenance or Allowance. And though in one Case the Chief Baron did refuse the Plea of Privilege, because then we had not Presidents ready to show, that it had been allowed in the Exchequer against the King's Debtor, as they call it; yet we hope, since our Charters are so full, and such Privileges have been formerly allowed, when his Lordship sees so just a Claim derived to us, by such Numbers of Royal Grants and Confirmations, (though it must be confessed they have not met with the same Success at all Times, or in all Places) that he will think fit for the future to allow them to us. This is somewhat, Sir, 699

Powis: not identified.

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153. JENKINS to WALLIS, 31 October/[10 November] 1667 of what may be properly urged on our Behalf, why Prince should not be countenanced in such vexatious Proceedings; the rest, I doubt not, wi be full supplied by yourself. I am, &c. John Wallis.

153. LLEWELYN JENKINS to WALLIS [London], 31 October/flO November] 1667 Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 76r-77v. On f. 77V in Wallis's hand: 'Ultimo oct. 1667 Dr L. Jenkins'. Abbreviations 'Qu.' for 'Question' and 'wh.' for 'whether' have been silently expanded for reason of better legibility. Reply to: WALLIS-JENKINS 26.X/[5.XI].1667 and WALLIS-JENKINS 27.X/[6.XI].1667. Answered by: WALLIS-JENKINS 2/[12].XL1667.

Ult. Oct. 67.

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I have yours700 by Bartlett, & that other701 by the Poste; for both which I thank you. Yesterday's putt off, & this daye's hearing produces no more than that we are to begin again. Line is to bring either an alias writt of Priviledge or (as some would have it & as I find the Rule of Court is to be) a Habeas Corpus & we are to Return the Cause as we first intended. This we must provide for, against next Terme. The Court & our Councell are pleasd to phansie, the Question to be this; whether the university can make a By-Law to subject the Townsmen (a Distinct Body) to this mulch? & they cannott be taken off from these Termes which have a visible Byasse in them to our Disadvantage. For we doe not exact this mulch from them but as they are the same Aggregate Body with our selves; I mean under no other Governors but ours as to the Peace & the night-watch. I have the return that you perfected (after my little hammering) now before me. It has the materialls and the forme of what we are to Answer to, with 10 fe as I find the Rule of Court is to be add. 16 from them add. roo

yours: i.e. WALLIS-JENKINS 26.X/[5.XI].1667. that other: i.e. WALLIS-JENKINS 27.X/[6.XI].1667.

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some little changes, which must be left to the Clerks & the Councell here. Onely in order to fortifie our selves as much as we can, I must crave leave to offer to your Consideration & search; 1. whether there be any thing in your Charters or other Muniment Collections or writings that doe Enunciative or obiter declare the University to a Corporation time out of mind? For that must be our Basis. That being prov'd or granted, I think the Power of making By-Lawes (at least for our own Body) cannott be denyed us. 2. what time the University had the Custody of the peace and the punishment of malefactors granted to it with Command to the Town to be Ayding & Intending the Chancellor which Command | I hope 761 infers an Exclusion of the Town from Medling in their own Right; & if so; the Power of the watch was in the University or no where. 3. whether There be any salvo's in the Town-Charters, or Restrictions upon them, as to our Liberties; and whither in any of the Compositions between you & Them it be not confessd declar'd or imply'd that the Government of the night, and the Custody of the Peace belongs to the University. If you meet with any thing argumentative in these 3 Questions we shall offer it to Councell that it may be either putt into the Body of our Return, or else kept in store for the Argument to Justine our Return. For we are, (we must be supposd) to doe all that we doe by a By Law. Therefore we must make good 1. that we had Power to make one, which our being a Corporation implyes. 2. that we have Right to bring the Townsmen under it; which we must make out by shewing that we have been allwaies entrusted with the Observation of the peace &c. I pray by the way, satisfie your self whither there be any thing in the origine or Extent of that Power which the steward exercises that can make for us in this busines. Sly that our By Law is reasonable; which the stat. that speaks the Penalty to be 40s in X° Hen. 6th as also the many Disguises of our scholars makes very reasonable. When I doe hear from you I shall wait upon the Clark Mr Robinson that we intend to employ to modell our Return before we bring it to our Councell or offer it to the Court. I have taken order to search for the two Cambr. Precedents, according to your Directions. All that I have 4 or other Muniment Collections or writings add. 8 2 . (1) when th breaks off (2) what 11 infers (1) & (2) an 22 to add. ed. 28 in X° Hen. 6th add. 29 upon (1) you (2) the

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154. WALLIS to HOPKINS, 1/[11] November 1667 brought me from the Exchequer Pleas, is that Dr Pierce702 of Cambr. mo [77r] pleadedj Q- Eliz. Arch, to one Pie that sued him upon a Bond 10 Jacobi the plaintive demurrd, the Defendent joyns in Demurrer, affirming his Plea to be good: but no further Proceedings. I take order to take out the french Petition. I wish I had your Inspeximus of 6to Hen. 5th Rot. 8 or else that of Edw. 4. that I might cast my Eye over them before I come hence. I am

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Your most humble & assured servant L. Jenkins [77V] For the Reverend Dr Wallis D. D. this.

154. WALLIS to WILLIAM HOPKINS Oxford, I/[11] November 1667 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-JENKINS 2/[12].XL1667. Enclosed with this letter were copies of the University charters mentioned in WALLISJENKINS 2/[12].XI.1667.

155. WALLIS to LLEWELYN JENKINS Oxford, 2/[12] November 1667 Transmission:

W Letter sent: KEW The National Archives PRO SP 29/222, No. 31, 4 pp. On p. 4 to the right of address in Jenkins's hand: 'Dr Wallis touching the Allowance of the 702

Pierce: possibly Edwaxd Peirce, Trinity Hall, Cambridge. LL.B. 1635, LL.D. 1639.

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155. WALLIS to JENKINS, 2/[12] November 1667 University Priviledges in the Common Pleas & the Exchequer. 2 Nov. 67.' Postmark on p. 4: 'NO/4'. Reply to: JENKINS-WALLIS 31.X/[10.XI].1667. Answered by: JENKINS-WALLIS 5/[15].XL1667. Enclosure: Wallis's Extracts from records on the rights and privileges of the Universities of Oxford and Cambridge.

Oxford Novemb. 2. 1667.

Sir, In answere to your Quere's703; you may please to take these things which occur at present. 5

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1. To that, whether the University be a Corporation by prescription. I have this to offer for the Affirmative. 1. That it is evident by all our charters: For if the University were not a Corporation; they were not capable of a grant; nor could injoy any hereditaments for perpetuity: Whereas charters have been granted &; privileges injoyed & allowed without controll. 2. Our charters are granted to the Chancellor, Masters, & Scholars; or Chancellor & Scholars, &c; and their Successors: Now where there is no Corporation, there can be no Successors. So that the very grants of such charters, implyes as a corporation, at lest soe long: which will be inough to prove the Prescription. 3. Wee are owned as such a body capable of succession; not onely in charters; but the Parliament itself, at lest as high as 18° Edw. 3. where there is a large decision in Parliament of the differences between the University & Town: (revised amongst our Charters in that of King Charles704.) 4. In my Lord Cooks Jurisdiction of Courts (which is, I think, the second of his Institutes,)705 where he speakes706 of the Courts of the Universities: he says expressely that the Universities

9 been add. ed. 10 Chancellor, |the del. Masters 14 owned (1) s breaks off (2) as 20 that (1) they w breaks off (2) Corpor breaks off (3) the 703

your Quere's: see JENKINS-WALLIS 31.X/[10.XI].1667. that of King Charles: i.e. the 'great charter' of Charles I of 3 March 1636; cf. the Extracts from the University Archives enclosed in WALLIS-JENKINS 28.1/[7.II].1667/8. 705 Lord Cooks Jurisdiction . . . second of his Institutes,): The jurisdiction of courts is in fact dealt with in COKE, The fourth part of the institutes of the laws of England: concerning the jurisdiction of the courts, London 1648. 706 speakes: see COKE, The fourth part, 227. 704

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155. WALLIS to JENKINS, 2/[12] November 1667 were Ancient Corporations by prescription' (even before that of the Act of Parliament 13° Eliz.) 2. To that, How long time the University have had the custody of the peace granted them with order to the Town to be ayding &c. I say 1. this wee claim by prescription allso; for the first rise of it doth not appear. But 2. ever since the 14th of K. John the Mayor, Bailifs & 50 Townsmen, have been bound to take an Oath to the University, to maintain & defend their privileges. 3. there be divers writs extant, or copies of them, some to the Sherifs of Oxford & Berks, some to the Mayor & Bailifs of Oxon, in the times of Hen. 3. Edw. 1. Edw. 2 Edw. 3. &c. requiring them to be aydant & assistant to the Chancellor & his deputies; not to suffer any wrong to be done them; &, if any be done, to give them assistance for their redress. 4. divers such writs in particular to the same persons requiring them to apprehend & imprison & keep prisoners malefactors &c at the chancellors command; & that as well within their liberties as without when he or the Proctors shall require it (divers of such writs Mr Hopkins707 hath at London with him.) 5. in divers writs directed to the Chancellor (and particularly one of 24 Nov. 12 Edw. 2d. by the King & Parliament) hee is sayd to have custodiam pacis tanquam minister noster &c. (Mr Hopkins hath a copy, but we have not the original.) & some others we have to the same purpose 6. In Letters Patents to the Chancellor Proctors & Masters, they are commanded (17 Jun. 4 Ric. 2) upon forfeiture of all they can forfeit, to make such as Ordinances as that the peace 2 [In left margin in Wallis's hand:] desunt.

3 long (1) of (2) time 3 have had the custody add. 6 the Mayor, Bailifs fc 50 (1) citi breaks off (2) Townsmen, add. 7 have (1) take breaks off (2) been 8 3. | and del. there 10 fee. (1) directin breaks off (2) requiring 12 to (1) do (2) be 14 prisoners (1) such (2) malefactors 18 by (1) Parliament (2) the 19 to (1) be custo breaks off (2) have 21 In (1) a Writ of (2) Letters 22 Ric. 2) (1) as they (2) upon ror

Hopkins: i.e. William Hopkins, q.v.

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be unviolablyl preserved: and, if need be cum posse totius villae to make [2] resistance to disturbers. 7. The first institution of the 4 Aldermen & 8 assistance, is in a grant to the University (not to the towne) in tranquillitatem Universitatis, for the arresting malefactors, vagabundos de node & for keeping the Assise &c: & so were indeed University Officers, for those ends; & were from their first institution to take an Oath to that purpose: as is evident by the Charter of Hen. 3.708 (whereby they are constituted;) recited with the rest in K. Charles Charter. 8. It is expressely resolved in Parliament 18 Edw. 3. that for miscarriages herein both they & the Bailifs are responsible to the Chancellor; etiam de balivis dominationi regis cognoscant qui aliter se gerant in officio suo quam facere debent (see it in K. Charles Charter.) 9. When they abused this power against the University, which they were to act for the University, it was (upon a submission of their whole jurisdiction, power, privileges, &c, which were thereupon Capta in manum Regis, taken wholly from them & given to the University by the Charter of 27 June 29°. Edw. 3. (see it in K. Charles Charter.) & they required nullatenus se intromittere; sed humiliter pareant et intendant: & the like in a charter of 32 Edw. 3. 10. In pursuance of that charter of 27 Jun. 29 Edw. 3. a writ is directed to the Mayor & Bailifs &c ; dated 28 July. 29 Edw. 3. that, whereas a former to them directed of 3 Octob. 27. Edw. 3. authorizing them to take malefactors, disturbers of the peace, noctivagators &c; it is thereby declared that he had now given that power to the University; & declares that Commission & all others to that purpose to be voyd & requires them to be brought into chancery by a day assigned to be cancelled: The copy of that Commission, & of the Revocation Mr Hopkins hath; the original is not in our custody, but

I need be (1) to (make) (2) cum posse totius villae to make II facere add. 16 27 June add. 18 et (1) obedeant (2) intendant 19 of (1) Ed breaks off (2) 29 Edw. 3. (3) 27 19 Bailifs fee; (1) that whereas they had before obtained (2) dated 22 fee add. 25 copy of (1) the (2) that 25 fe |of add.\ the Revocation 708

Charter of the charter.

Hen. 3: Wallis has neglected to supply the complete reference to

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[3]

the record I suppose may be found in the tower, by the date. 11. In all commissions for the Peace, there was wont to be a reservation of the University Jurisdiction; ita tamen ut vos A. B. C. &c neque aliquis vestrum in eis quae ad jurisdictionem cancellarii Universitatis spectant nullatenus vos intromittatis; or to that purpose. And this continued; till the time of Hen. the 8. his charter, wherein that clause being put in, universally, as to all commissions, it was left out in the particulars as needless: The copies of many such commissions I have; & can direct where they are to be found in the Tower. | 12. In the French composition 22° Edw. 2. (which I sent you,) it is agreed & acknowledged that the Chancellor hath cognizance in cases that concern the peace, & the body of the University as a body: And somewhat in the Composition of 37 Hen. 6. & that in Parliament 18 Edw. 1. (the latter of those in King Charles Charter; a copy of the former, is subjoined to that Copy of the Town charters which Mr Hopkins hath.) 3. To your 3d Quaere; whether any salvo of the University Priviledges in the Town charter: I answ. there is: & particularly in that which is their principall, in 1° Edw. 3. that salvo is twice or thrice inserted. Or, if it had not, yet at the time of our grant 29° Edw. 3. after the great conflict, the Town Charters were under their own seal, submitted to the Kings pleasure, & were actually seized in manum regis; & it is questionable whether they were ever validly restored. A copy of the town charters, together with a copy of K. Charles his Charter I sent up yesterday709 to Mr Hopkins; & hee hath allso with him all or most the particulars above mentioned: with a note of them, which gives a short account of the contents of each. I think you may do very well to take some occasion of acquainting yourself with Mr Crouch710, who is one of the Burgesses for Cambrige, &

I , by the date add. 9 (which I sent you,) add. II cases (1) of (2) that 13-14 (the latter . . . Mr Hopkins hath.) add. 19 the (1) m breaks off (2) Town Charters were (a) by (6) under 21 validly add. 23 Mr add. 709

sent up yesterday: i.e. with WALLIS-HOPKINS 1/[11].XI.1667. °Crouch: presumably Thomas Crouch (1610-79), fellow of King's College and later of Trinity Hall, Cambridge. Member of parliament for the University 1660-79. rl

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155. WALLIS to JENKINS, 2/[12] November 1667

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understands the affairs of that university very well: & in this businesse which is of common concernment to both, I doubt not but he may do us very good service. To your question; about Erles his case; the debt was contracted out of the University: but not out of the extent of our cognisance: for the words of K. Hen. 8. his Charter711, say expressely: whether within the sayd University or City, or the four hundreads adjacent, or within the Counties of Oxford or Berks, aut ubicunque infra regnum nostrum Angliae. The Charters of 1° Hen. 5. (enrolled Ter. Hill. 6° Hen. 5. Rot. 80.) & that of 1° Edw. 4. (enrolled Ter. Hil. 12° Edw. 4. Rot. 13. ex parte Remoratoris Regis,) with all the particulars recited in those charters, (though possibly in somewhat a different order,) are contained in K. Charles his great charter: (a copy of which I sent Mr Hopkins yesterday;) to which I have put a table in the beginning denoting the pages where every charter is to be found. If in those enrollments be contained any thing more than those charters (with the branches of them) I have no notice thereof: Nor of the occasion on which those enrollments were made. And if, there, be any considerable case resolved for us; it may be worth taking out so much, or a breviat of it, without copying the Charters themselves, of which we have the Originalls. The Origine of our Stewards power; is the charter of 7° Hen. 4. Revised by K. Charles. The businesse of a Habeas Corpus, wil be but another years delay; & wee shall bee but where wee are: For the form of that being; whereas the body of F. L.712 is in your custody, (ut dicitur,) these are to require you to have the corpus et causae captionis et detentionis: wee can make no other return, but that hee was neither captus nor detentus. (For he was not arrested, but onely cited.) And so no cause of either. I have sent you the transcript of two cases in the Exchequer allowed 2do(l) you (2) us 5 our (1) jurisd breaks off (2) cognisance 10 ex parte Remoratoris Regis, add. 23 but add. 24 (ut dicitur,) add. 24 you add. 28-345, 6 I have sent . . . sent you copies, at 9(f in left margin 711

K. Hen. 8. his Charter: i.e. the charter of Henry VIII of 1 April 1523 ('Wolsey's charter'); cf. the Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II]1667/8. rl2 F. L.: i.e. Fish Lyne. 344

155. WALLIS to JENKINS, 2/[12] November 1667

[4]

to Cambridge, according as I had them from Cambridge. And what I have of Howtons case 12° Edw. 3. I think it might be worth searching the Rolls at large (if you have not done it) of which the book of the Exchequer is but a breviat. And consider, whether there be not any way open for us now to obtain such a writ as that of Qu. Eliz: or that of Edw. 3. of which here I have sent you copies. | I think, if you had shewed Judge Brown713, that Charter of 12 Apr. 10 Edw. 3. (marked N. 3.) of which Mr Hopkins hath the originall (marked in the margent on the right hand in two places with a +) and the writ which refers to it, of 27 Apr. 48 Edw. 3. it might have satisfied him as to the businesse of Contempt: especially if you shewed him allso Martivells case714 22 Edw. 1. This, I hope, may serve for answere to that of yours to mee Oct. 31. &; (so much as I remember of) that to Mr Vice-chancellor715. from Sir

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Your very humble servant John Wallis. These For my honoured Freind Dr LLuellin Jenkins at his lodgings in Exeterhouse, in the Strand, London.

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I as 1(1) have (2) had 5 a add. 713

Brown: i.e. Sir Samuel Browne (d. 1668), serjeant-at-law (1648) and justice of the common pleas (1660). 714 Martivells case: cf. Wallis's Notes on Martivell's case [1667?]. 715 Vice-chancellor: i.e. John Fell, q.v.

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156. WALLIS to JENKINS, 2/[12] November 1667, enclosure

156.

WALLIS to LLEWELYN JENKINS Oxford, 2/[12] November 1667, enclosure: Extracts from Records on the Rights and Privileges of the Universities of Oxford and Cambridge Transmission:

W Notes sent: KEW The National Archives PRO SP 29/222, No. 311, 4 pp. (p. 4 blank). Enclosure to: WALLIS-JENKINS 2/[12].XI.1667.

That John Pooley, a privileged man, shal not be impleaded in the Exchequer. But before the Chancellor of the University.

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For as much as it appeareth that John Pooley of the Town of Cambridge is one of the priviledged persons within the University of Cambridge: It is therefore ordered that he shall not plead to any Information exhibited in this Court against him for Cards by the procurement of Mr Bowes & Mr Bedingfield: But shall be sued before the Vicechancellor of the University of Cambridge if that shal seem good to the said Plaintifs. Ex libro Ordinationum Scaccarii de Termino Sancti Hilarii, videlicet Sabbato septimo die Febr. Anno 21° Reginae Elizabethae; Ex parte Remembranciae Reginae. Processus coram Baronibus Scaccarii contra Johannem Boxter Universitatis Bedellum in causa informationis pro retalliatione vini excessive pretio contra formam Statuti.

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Memorandum quod Hugo Bonnell &c. Inter recorda de Termino Sancti Hilarii anno 28° Reginae Elizabethae Rot. 125. ex parte Remembranciae Reginae in Scaccario. A Writ of Privy Seal to the Court of the Exchequer for surcease from Process in the cause of John Boxter & others, touching an Information for selling of wine at unlawful prices. 346

156. WALLIS to JENKINS, 2/[12] November 1667, enclosure Elizabeth by the grace of God Queen of England France & Ireland Defender of the Faith &c. To the Treasurer Chancellor & Barons of our Exchequer and our Atterny General Greeting. Whereas wee are given to understand that one Hugh Bonnel, a person given to unquietnesse and of an ill will to our University of Cambridge about three years since exhibited several Informations in our sayd Court of Exchequer against one John Boxter & others the Vinteners inhabiting in our Town of Cambridge; alleging by the same that they had sold & uttered wines by retail, as by the Quart & Pynt, above the rates and prices limited & appointed by the laws 6 statutes of this our Realm; & thereby had forfeited divers summes of mony. Whereupon the liberties & privileges of our sayd University in the sayd Town of Cambrige by us & our noble progenitors long time sithence to them granted are by means thereof drawn into question; by which, as it seemeth, the pricing of all manner of wines to be sold within our sayd Town, ought to be ordered by the Chancellor & other Officers of the sayd University, as by the pleadings to the sayd Informations remaining of Record in our sayd Court of Exchequer more at large doth & may appear. For that the like Actions or Suits have never been had or moved to the impeachment of the sayd liberties & Privileges in our time or in any of the Reigns of our sayd Progenitors till now: And that we tendering the good & quiet estate of our sayd University as graciously as any of our sayd Progenitors, for the special & natural favour wee bear unto learning: Do think it most meet & convenient to take both themselves & their causes wholly into our defense and direction. Know ye therefore that wee [2] not minding the sayd| John Boxter & other the vinteners either now or hereafter inhabiting in our sayd Town of Cambridge or any of them, to be vexed sued troubled or disquieted in body goods lands or Tenements, for selling or for uttering by retail any wines in our sayd Town above the rates & prices limited or appointed by our Laws or Statutes: So that the sayd Boxter & other the vinteners there, be by the Chancellor Masters & Scholars of the sayd University licensed to sell wines, & the prices rated & set down by the Chancellor & other the Officers of our sayd Universitie, according to their claim, by the ancient customes usage & charters: Do will 24 [In left margin in Wallis's hand:] forte, protection. 7 in add. 17 our add. 27 sued (1) molested (2) troubled 30 the | sayd del. Chancellor 347

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156. WALLIS to JENKINS, 2/[12] November 1667, enclosure & command you our sayd Treasurer Chancellor Barons & Aterny general, & every of you, utterly to surcease for ever of & from all & all manner of process suits, quarels, proceedings, demands, judgements, & executions, sued moved or made by the sayd Hugh Bonnell, or any other, for us him 5 or any of themselves, in our name; or hereafter to be sued moved or made for us or any of them in our behalf, in for or concerning the premisses; Any law statute use or course of our sayd Exchequer, or any ambiguity doubt question matter or cause whatsoever you or any of you to the contrary moving notwithstanding. And further our Will & pleasure is, that 10 upon sight hereof you do forthwith dismiss the sayd Boxter, & other the sayd Vinteners from & out of our sayd Court of Exchequer touching the sayd Informations & Causes aforesayd. And to the intent the Students scholars & Professors of learning in our sayd University of Cambridge may the better attend their studies, & the Vinteners now inhabiting our 15 sayd Town of Cambridge, or which at any time hereafter shal be resident or abiding there, may be the more quiet without any further molestation; Our will & pleasure is that you cause the tenour hereof to be entered of record in our sayd Court of Exchequer for your better remembrance & warrant hereafter, for the accomplishment of our princely mind & inten20 tion towards that our sayd University in this behalf. And these our letters shal be your sufficient warrant & discharge in this behalf. Given under our Privy seal at our Mannour of Greenwich the 26th day of April in the 30th year of our Reign. William Parker. Concordat cum Originali Brevi sub privato Sigillo remanente in Cu25 stodia. Remembrancia ex parte Reginae in Scaccario.| [3] Breve Regis, quid cancellarius habeat allocationem Chartae suae coram Justitiariis Regis de cognitione Placiti, ubi altera pars est scholar is. 30

Rex dilectis et fidelibus suis Gulfredo le Scrope et sociis suis Justitiariis ad placita coram Rege tenendum assignatam Salutem. Cum per literas nostras patentes concesserimus pro nobis et heredibus nostris tarn pro tranquillitate et quiete Universitatis Oxoniae, quam pro securiori ibidem 27 [In left margin in Wallis's hand:] Edw. 3. 8 doubt (1) matter question (2) question matter 20 University in (1) that (2) this 348

156. WALLIS to JENKINS, 2/[12] November 1667, enclosure pacis nostrae conservatione, quod Cancellarius ejusdem Universitatis qui pro tempore fuerit, habeat cognitiones transgressionum quarumcunque tarn in suburbiis extra muros dictae villae quam infra muros eiusdem villae factarum, ubi Clericus fuerit una partium, exceptis placitis de morte hominis et mahemio: Et quod transeuntes per villain Oxoniae respondeant coram Cancellario praedicto de contractibus et transgressionibus factis Scholaribus tarn in suburbiis extra muros villae praedictae quam infra villain eandem et non de contractibus et transgressionibus forinsecis extra villain et suburbia praedicta factis, prout in literis nostris praedictis plenius continentur; et vobis mandaverimus quod Cancellarium Universitatis praedictae, libertate praedicta coram vobis uti et gaudere permitteretis juxta tenorem literarum nostrarum praedictarum; et quod ipsum Cancellarium contra tenorem earundem, non molestaretis in aliquo seu gravetis. Ac jam ex parte dicti Cancellarii acceperimus, quod licet ipse per Thomam de Bradwell Attornatum suum libertatem praedictam in quodain placito quod pendet coram vobis per brevem nostrum inter Johannem le Blake de Todington et Editliam uxorem ejus, et Adam de Houghton de Oxonia de quadam transgressione eisdem Johanni et Editha per praedictum Adam apud Oxoniam (ut dicitur) illata, pro eo quod idem Adam unus Clericorum Universitatis praedictae (ut dicitur) existit, calumniaverit, et eandem libertatem praefato Cancellario petierit allocari, ut idem Cancellarium cognitionem placiti praedicti habere possit juxta formam concessionis nostrae praedictae sibi inde factae. Vos nihilominus pro eo quod dictus Cancellarius libertatem praedictam die quo dictum breve coram vobis returnatum fuerit seu aliis diebus sequentibus non calumniaverit, nee chartam nostram sibi de libertate hujusmodi concessam tune coram vobis exhibuit, libertatem illam praefato Cancellario allocare distulistis, super quo supplicavit sibi per nos remedium adhiberi. Nos nolentes praedicto Cancellario super libertate praedicta ex causa hujusmodi taliter praejudicari, Vobis mandamus, quod eo non obstante quod libertas praedicta per praedictum Cancellarium die quo praedictum brevem coram vobis returnatum fecit, seu aliis diebus sequentibus calumniata non extitit, nee idem Cancellarius chartam nostram praedictam et inde sic concessam coram vobis tune exhibuit, ut est dictum; praefatum Cancellarium libertate ilia coram vobis uti et gaudere permittatis juxta tenorem 15 de add. 17 Adam de (1) Howton (2) Houghton 27 tune (1) vobis (2) coram 34 ut (1) dicitur (2) est dictum; |praefatum add.\ Cancellarium

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literarum nostrarum praedictarum, ipsum contra tenorem earundem non molestantes in aliquo seu gravantes. Teste meipso apud Westmonasterium, decimo die Febr. anno regni duodecimo. Per ipsum Regem. Ex Rot. claus. de Anno duodecimo Regis Edw. 3. parte prima membr. 18. in Turre Londoniae. In thesaurario Scaccarii, in libro Abbreviationum placitorum ab anno 22° (an 12°) Edw. 3. usque ad annum ejusdem 44.

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fol. 98. Placita coram Domino Rege apud Westmonasterium de termino Sancti Hilarii Anno regni Regis Edw. 3. post conquestum 12°. Oxoniae 33. Cognitio placiti transgressionis ubi scholaris est unus seu altera pars, allocetur Cancellario Universitatis ibidem, juxta chartam Regis Johannis. Et Brevem Regis mittitur hie Justitiariis, quod dicto Cancellario allocetur dicta libertas, non obstante quod non clamavit dictam libertatem super returnum placiti, sed permisit placitum hie placitari.

157. WALLIS: Notes on Martivell's case [1667?] Transmission:

W Notes: KEW The National Archives PRO SP 29/441, No. 34, Ip. On verso in Jenkins's hand: 'Severall Cases of our Proceedings in the University - non obstantibus Prohibitionibus Regis, by Dr Wallis Custod. Archivorum Universitatis.' Evidently, Wallis sent these notes to London to support the University's case in the Court of Common Pleas, suggesting a date in late 1667.

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In the Treasury of the Exchequer, scil. In libro Abbreviationum de Placitis coram Domino Rege habitis. Ubi P. 103. 104. In Placitis scil. de Termino Michaelis in Banco Regio A° Edw. primi 22°. Oxon. Rot. 162. 3 regni (1) {—} (2) duodecimo. 350

157. WALLIS: Notes on Martivell's case, [1667?] Rogerus de Codesford implacitat Magistrum Rogerum de Martivallibus, pro eo quod ipsum traxit in Curia Christianitatis pro terris in laico feodo ipsius Rogeri contra Prohibitionem Domini Regis in Oxon: Allegando quod cum quidam Willelmus Banker et Robertus de Kirketon traxissent ipsum Rogerum de Codesford in Placitum in Curia Christianitatis coram praedicto Rogero Cancellario Oxon. in Aula ipsius Magistri, in praesentia Archidiaconi Rosensis et aliorum; et postquam ipse Rogerus porrexisset dicto Magistro Rogero prohibitionem Regis, ne praedictum placitum teneret &c. praedictus Magister, spreta Prohibitione Regis nihilominus tennit praedictum placitum in Curia Christianitatis &c. quousque ab ingressa Ecclesiae fuit suspensus &c. Magister dicit, quod die et anno &c fuit Cancellarius Universitatis Oxon. et quod praedicti Willelmus et Robertus fuerant scholares ejusdem Universitatis Oxon, qui coram eo conquesti fuerunt, quod cum praedictus Rogerus de Codesford, eis, praedictas domos ad inhabitandum per unum annum dimisisset, idem Rogerus de Codesford conventionem inde postea eis non tenuit &c. Cujus quidem contractus cognitio ad ipsum Cancellarium spectabat &c. ratione libertatum Scholaribus ejusdem Universitatis a Regibus Angliae concessus et hactenus usitatus, et maxime per chartam Domini Regis nunc, quam profert in haec verba. Edwardus Dei gratia &c. Omnibus ad quos praesentes literae pervenerint salutem. Sciatis quod de gratia nostra speciali, concessimus scholaribus Universitatis Oxon, quod de omnibus actionibus personalibus possunt convenire Burgenses et alios laicos municipii Oxon. coram Cancellario ejusdem Universitatis; et quod per prohibitionem nostram super hoc nequaquam impediantur. In cujus rei Testimonium has literas nostras fieri fecimus patentes pro nostrae libito voluntatis duraturas. T. meipso apud Westmonasterium 30 Octobr. a° Regni nostri 3°. Unde idem Cancellarius ratione libertatis praedictae bene advocat, quod ipse placitum praedictum tenuit, ut in actione personali. Et qui a praedicto Rogero de Codesford cognovit, quod praedicti Willelmus et Robertus, non implacitaverunt de domibus praedictis ipsum Rogerum de Codesford clamando feodum seu liberum tenerentur, nisi tantum inhabi27 [In left margin in Wallis's hand:] This is made perpetual by a charter of K. Edw. 3. confirmed by aftercharter & Act of Parliament. 6 Rogero coram del.\ Cancellaxio 21 ad add. 30 qui (1) praedictus Rogerus (2) a praedictus Rogerus corr. ed.

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158. JENKINS to WALLIS, 5/[15] November [1667] tationem unius anni, sicut praedictum est; per quod bene licebat praedicto Cancellario praedictum placitum tenere ut in minutis contractibus &c. Consideratum est, quod praedictus Cancellarius sine die, et Rogerus de Codesford in misericordia. See a like proceeding In the year book of Edw. 3. De anno 40° Edw. 3. Term. Pasch. fol. 17. 6. printed 1565. And the same abbridged in Brooks abbrigement. Tit. Praescription. f. 145. §8. Attachement sur prohibition. And Tit. Jurisdiction, §4. fol. 49. And Faringdons Case 2° H. 4. Vice Chancellor Indicted for disarming a Mareschall of the Kings to execute a writ of that Court (it seems) on a scholar. The Court grants the Cognisance of the case to the University: & dismisseth it their own Court.

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158. LLEWELYN JENKINS to WALLIS [London], 5/[15] November [1667] Transmission: C Letter sent: OXFORD University Archives WPj/16/1, f. 80r-81v (f. 80V and 81r blank; on 81r arithmetic calculation in pencil in Wallis's hand). On f. 81V beneath address in unknown hand: '5 Nov (1667)'. Postmark on f. 81V: 'NO/5'. Reply to: WALLis-jENKiNS 31.X/[10.XI].1667 and WALLIS-JENKINS 2/[12].XI.1667.

5. Nov.

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Sir

I have two of yours716 for which I can never sufficiently thank you. I now 10 [In left margin in Wallis's hand:] In the treasury of the Exchequer. Rot. Placit. coram Regem. Term. Trin. An. 2. H. 4. Rot. 13°. 10 Vice Chancellor add. 11 Kings (1) & (a) executed (b) executin breaks off (2) to execute 716

two of yours: i.e. WALLIS-JENKINS 31.X/[10.XI].1667 and WALLIS-JENKINS 2/[12].XL1667. 352

159. WALLIS to JENKINS, 11/[21] November 1667 plainly see my mistake. For I did not know that the Restriction about forrein Tracts was taken off by H. 8. There shall not a word of what you have written fall to the Ground if I can help it: and I have directed a search for the pleads you mention. I would offer to your consideration one thing more, as to Noctivagators I mean, to search the old Registers of the v. chancellors courts in your Archives, for proceedings and judgements in such misdemeanors. Mr Bate717 has calld upon me some dayes since; and I will not fayle to have an Eye to his Busines here, as also to give you, Sir, and him an accompt of what paves as soon as proceedings are had upon the Commission dispatched at Cant. I am

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Sir

your most humble & assured servant L. Jenkins

[8F] For the Reverend Dr Wallis Doctor in Divinity, Publiqu. Professor at his House nere the Scholes in Oxford

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WALLIS to LLEWELYN JENKINS 11/[21] November 1667 Transmission: Manuscript missing. Existence and date: Mentioned in and answered by JENKINS-WALLIS 16/[26].XI.1667.

5 old add. 717

Bate: not identified.

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160. WALLIS to OLDENBURG, 16/[26] November 1667

160.

WALLIS to HENRY OLDENBURG Oxford, 16/[26] November 1667 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 33, 4pp. (p. 3 originally blank) (our source). At top middle of p. 1: 'read Nov: 28: 67. entered LB. 2. 98.' On p. 4 Oldenburg has noted to the right of address at 90°: 'From Dr Wallis to M. Oldenburg concerning a Rule, whereby (for any year assigned) the Cycles of the Sun and Moon, and the Roman indiction being given, to find the Number of the Julian period.', and beneath address: 'Rec. Nov. 18. 67.' On p. 3 calculations in unknown hand (see apparatus). Postmark on p. 4: 'NO/{—}'.—printed: OLDENBURG, Correspondence III, 598-601. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 98-101. w2 Copy of w1: LONDON Royal Society Letter Book Copy 2, pp. 116-18. The letter was read at the meeting of the Royal Society on 28 November 1667, where it was ordered, 'that the doctor should be desired to send his demonstration, and to consider, whether this rule could not be made more easy and less operose'; BIRCH, History of the Royal Society II, 218. The calculation on page 3 gives the number of the Julian period for the values of the cycle of the sun and the moon and of the indiction given by Wallis in the present letter. It is based on the rule of the Jesuit mathematician Jacques de Billy (1602-79), a short account of which was published without demonstration in Journal des Scavans No. 36 (6 Sepember 1666), 430 ('Extrait d'une lettre du P. de Billy de la Comp. de Jesus, du 22. Aoust a Dijon'). A summary is also printed in Philosophical Transactions No. 18 (22 October 1666), 324 ('A Problem For finding the Year of the Julian Period by a new and very easie Method'). As Wallis reports in De periodo Juliana, published as part of Exercitationes ires (London 1678), he had been occupied with the question of calculating the number of the Julian period since 1652. The rule stated at the end of De periodo Juliana (68-9) largely agrees with that given in the present letter.

Oxford Novemb. 16. 1667. Sir,

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Having heard lately that some enquiry718 hath been made, about an expedite Rule, whereby (for any year assigned) the Cycle of the Sun, the Cycle of the Moon, & of the Roman Indiction, being given; to find the Number of the Julian Period: I thought it not amisse to take this occasion of

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enquiry: not ascertained.

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160. WALLIS to OLDENBURG, 16/[26] November 1667 renewing our intercourse (which hath been for some time intermitted719) by transcribing somewhat of what I wrote, about two years since, to that purpose. Dato; Quotus sit annus aliquis, Cycli Solaris, Cycli Lunaris, atque Indictionum: Quotus sit ille annus Periodi Julianae, Invenire. I. Exposito Numero Cycli Solaris, (vel hujus excessus supra 19, si hunc numerum excedat,) Addantur continue 9, (abjectis semper 19, quoties ultra hunc numerum excurritur,) donee occurrat expositus Numerus Cycli Lunaris. II. Per numerum locorum hujus seriei, uno minus; multiplicentur 28: Et Facto, addatur expositus Cycli Solaris numerus. III. Ex numero sic invento; abjiciantur, quoties fieri potest, 15: (vel, dividatur numerus ille per 15:) Residuoque (vel ipsi 15, si residuus sit 0;) Addantur continue 7, (abjectis semper 15, quoties ultra hunc numerum excurritur,) donee occurrat expositus Cycli Indictionum numerus. IV. Per numerum locorum hujus seriei, uno minus; multiplicentur 532: Et Facto, addatur Numerus secundo praecepto repertus. Et habetur Periodi Julianae numerus quaesitus. Exemplum.

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Anno Domini, 1665. Cyclus Solis, 22. Lunae, 13. Indictionum, 3. Cycl. O. 22 (= 19 + 3.) I. 3.12.2.11.1.10.19.9.18.8.17.7.16.6.15.5.14.4.13. Cycl. 3 . II. 28 [1.8.15.7.14.6.13.5.12.4.11.3. Cycl. Indict: 25

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Est itaque 719

Period. Julian.

for some time intermitted: Wallis's last surviving letter immediately preceding the present one is that of 21/[31].III. 1666/7.

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160. WALLIS to OLDENBURG, 16/[26] November 1667 I could give you several other Rules of the like nature (with the Demonstrations of them) if it be desired. But, one being sufficient, I shall give you at present no farther trouble, from Sir,

Your affectionate friend & humble servant, John Wallis.

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After I had written this letter, I thought it convenient to give the Rule allso in English; with the Example a little more explained: which is done on the other side.j [2] The Number of the Cycle of the Sun, the Cycle of the Moon, & of the Indictions, (for any year,) being given: To find the Number of the Julian Periode, for the same year.

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I. To the Number of the Suns Cycle proposed, (or, if it be greater than 19, to its excesse above 19,) make continual Addition of 9, (still casting away 19, so oft as it comes to exceed this number,) till you meet with that of the Moons Cycle proposed. II. By the Number of Additions so made, (which is less by 1, than the number of places, in that rank of numbers,) Multiply 28: And, to the Product, adde the Number of the Suns Cycle proposed. III. From the Number thus found, cast away 15 as oft as may be; (or divide that Number by 15, noting the Remainder:) And, to the Remainder (or to 15, if there be no remainder) make continual Addition of 7, (still casting away 15, so oft as it exceeds this number,) till you meet with that of the Indiction proposed. IV. By the Number of Additions here made, (or the Number of places in this rank of numbers, wanting 1,) Multiply 532: And, to the Product, Adde the Number found by the second direction. And so you have the Number of the Julian Periode, desired.

23 (or to ... remainder) add.

356

160. WALLIS to OLDENBURG, 16/[26] November 1667 Example. In the year of our Lord 1665. The Cycle of

the Sun, the Moon, the Indiction

22. 13. 3.

Cycl: O, 22 = 19 + 3. I. 3.12.2.11.1.10.19.9.18.8.17.7.16.6.15.5.14.4.13. Cvcl: 3 . [UU5.7.14.6.13.5.12.4.11.3. Cycl. Indict:

5

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15

So is

the Julian Period.

The example particularly explained. I. The number of the Suns Cycle for the year of our Lord 1665, (in which I first contrived this rule,) being 22; that is, more than 19, by 3: To this Excess, making continuall Additions of 9, & casting away 15 so often as it comes to exceed this number; (till I meet with 13, the Number of the Moons Cycle;) I have this rank of numbers (made by 18 continual additions,) consisting of 19 places. (And it is as many as in any case can happen.) 3.12.2.11.1.10.19.9.18.8.17.7.16.6.15.5.14.4.13. II. By 18, (the Number of Additions here made; or the number of places wanting one,) I multiply 28: And, to the Product 504, I adde 22, (the number of the Suns Cycle proposed,) which makes 526. III. This number 526, divided by 15, gives the Quotient 35, (which

12 IV. add. ed.

18 year (1) 1665 (2) of 20 continuall (1) d breaks off (2) Additions 22 of the (1) Suns Cycle (2) Moons 23 as (1) great a number (2) many

357

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160. WALLIS to OLDENBURG, 16/[26] November 1667 here is of no other use than to tell us how often 15 may be subducted from 526,) and the Remainder 1: To which remainder, making continuall Additions of 7, & casting away 15 so oft as it comes to exceed this number; I have this series, (made up by 11 continual Additions,) consisting of 12 5 places. (And more than of 15 places, it cannot be in any case.) 1.8.15.7.14.6.13.5.12.4.11.3. IV. By 11, (the number of places, wanting 1,) I multiply 532: And, to the Product 5852, I adde 526, (the number found in the second direction,) which makes 6378. And this is the Number of the Julian Period for the 10 year proposed.

15

These For my honoured Friend Henry Oldenburgh Esquire at his house in the Old Pal-mal, near St James's London.

[4]

10 [On page 3 in unknown hand:]

Us (1) no (2) of 2 from (1) it,) (2) 526,)

358

22 4845 9690 9690 106590 54600 20748 7980) 181938 (22 15960 22338 15960 6378

13 3 4200 6916 4200 20748 12600 54600

161. JENKINS to WALLIS, 16/[26] November 1667

161. LLEWELYN JENKINS to WALLIS [London], 16/[26] November 1667 Transmission:

C Letter sent: OXFORD University Archives WPj/16/1, f. 78r-79v. On f. 79r (originally blank) algebraic calculations by Wallis in pencil. On f. 79V beneath address in Wallis's hand: '16 Nov 1667'. Postmark on f. 79V: 'NO/16'. Reply to: WALLIS-JENKINS 11/[21].XL1667.

16. Nov. 67. Sir

yours720 of the 11th came not till yesterday morning to Hand. What you write concerning Mr Bate's Busines I shall retain & make use of in due time and Place. Sanity of mind in those Cases is not made out by the Affirmations or Beliefe of a stander by. So weighty an Act as that of Testing it, must be done in a series of sane words & Actions, going before & following after: at least a sedate mind & the Animus testandi must be clear & indisputable. I'le speak with Mr Crowch721 about Boxters' Case & other Things that may ayd us. I come from communicating your Line to Mr S. Holloway722. He is positive this will never come to a Jury: nay, that this is a certain way to declin it. Our Remedy likewith to recover the Mulch will be (if we cash him) entire unto us, notwithstanding the fictions: & he says our return must not vary any thing in matter or forme from that in painter's Case723: onely we must deduce our Power of making byLaws more at full. So that we must Cuivis in sua Arte. I subscribe & so does Sir G. Sweit724 to your opinion that the watch-maker must sue the Serjeants in the v. chancellor's Court. We shall see how they'l support their by-Law against your charter. Phe. Taylors of Ipswichs Case 9 Crowch about add. 720

yours: i.e. WALLIS-JENKINS 11/[21].XL1667. Crowch: i.e. Thomas Crouch. 722 Holloway: i.e. Charles Holloway (d. 1679), serjeant-at-law. 723 painter's Case: cf. Wallis's Notes relating to the allowance of the claim in the Exchequer. 724 Sweit: i.e. Sir Giles Sweit (Sweet) (15867-1672), regius professor of civil law at Oxford 1661-72. Jenkins had been appointed deputy professor of civil law to him in 1662. 721

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162. OLDENBURG to WALLIS, 26 November/[6 December] 1667

5

11. 1. Cooks Repts does not seem to me to make for them, if our Charter be home for the Ministry of scholars, notwithstanding they may have it affirm'd by| the Judges of Assize. But let us have a Care that the man be Bona fide priviledg'd; & within the meaning as well as the Line of our charters. I would gladly see the Libell before it be given in against him; & that article of the Priviledge & Act of Parliament affirming it layd home. I am

[781

Sir

Your most humble & assured servant L. J.

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15

[791

For the Reverend Dr Wallis D. D. publiqu. Professor at his House nere the Scholes in the University of Oxford.

162. HENRY OLDENBURG to WALLIS 26 November/[6 December] 1667 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 30.XI/[10.XII]. 1667. Oldenburg communicated to Wallis the method devised by Sluse for solving biquadratic equations geometrically, which he had received in SLUSE-OLDENBURG [14]/24.XL 1667; OLDENBURG, Correspondence III, 594-8. Oldenburg had indicated his intention of doing so in OLDENBURG-BOYLE 25.XI/[5.XII].1667; OLDENBURG, Correspondence III, 61015. Cf. BIRCH, History of the Royal Society II, 217-18. Oldenburg presumably also informed Wallis of the discussion of his rule for finding the number of the Julian period at the meeting of the Royal Society on 28 November 1667 (cf. BIRCH, History of the Royal Society II, 218) and referred to de Billy's rule printed in Philosophical Transactions No. 18 (22 October 1666), 324. 360

163. WALLIS to OLDENBURG, 30 November/flO December] 1667

163. WALLIS to HENRY OLDENBURG Oxford, 30 November/[10 December] 1667 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 34, 4 pp. (our source). At top of p. 1 Oldenburg has noted: 'A Letter from Dr Wallis to M. Oldenburg Containing his Demonstration of the Method, formerly sent by him (see p. 80.) for finding the Number of the Julian Period.' At top right beneath date: 'Enter'd LB. 2. 107'. On p. 4 beneath address in Oldenburg's hand: 'Received Dec. 2. 67.', and to the right of address at 90°, again in Oldenburg's hand: 'Dr Wallis's letter.' Postmark on p. 4: 'DE/2'.— printed: OLDENBURG, Correspondence III, 626-9. w1 Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 107-10. w 2 Copy of w1: LONDON Royal Society Letter Book Copy 2, pp. 127-31. Reply to: OLDENBURG-WALLis 26.XI/[6.XII].1667. Answered by: OLDENBURG-WALLIS 24.XII.1667/[3.I.1668].

Oxford Novemb. 30. 1667. Sir,

In answere to yours725 of Nov. 26. (without further preface,) As to what you mention of the Methode sent from Paris726, for rinding the number of the Julian Period, mentioned (as you say) in the Transactions N°. 18. I have not yet seen it, &, so, can say nothing to it. As to the Demonstration of that methode which I sent you in my last727, (which with my humble service, you may please to let my Lord Brounker see, whom I hope by this time I may congratulate as President for the following year728;) you have it thus: (applyed to the particular instance there given, which may,

7 methode | which add. \ I 9 congratulate (1) fe (2)as 10 it (1) br breaks off (2) briefly (3) thus 725

yours: i.e. OLDENBURG-WALLIS 26.XI/[6.XII].1667. Methode sent from Paris: i.e. de Billy's rule printed in Journal des Scavans No. 36 (6 September 1666), 430, a short account of which was printed in Philosophical Transactions No. 18 (22 October 1666), 324 ('A Problem For finding the Year of the Julian Period by a new and very easie Method'). 727 my last: i.e. WALLIS-OLDENBURG 16/[26].XI.1667. 728 President for the following year: Brouncker was re-elected president of the Royal Society at the meeting on 30 November 1667; BIRCH, History of the Royal Society II, 220. 726

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163. WALLIS to OLDENBURG, 30 November/[10 December] 1667

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mutatis mutandis, be as well fitted to any other.) 1. The Number of the Suns Cycle, being 22. It must bee in the 22d year either of the first, second, third, or some other revolution of that period. That is, in an arithmeticall progression, beginning at 22, & increasing by 28. 2. The 22d year of the first Revolution (being allso the 22d year of the Julian Period,) the Cycle of the Moone (because of 22 — 19 + 3) is 3, not 13, as in our case: wee are therefore to seek in what other Revolution this may first happen. 3. To find this; because of 28 = 19 + 9; I consider that for every Revolution of the Suns Cycle, is added to that of the Moons number (beside one whole revolution hereof,) the number 9; (still casting away 19 as there is occasion.) And therefore, the first time 22 comes in the Suns cycle, there being 3 in the Moons; the second time 22 comes in the Suns, there must be (in the Moones) 12 (that is 3 + 9;) the third time, 2: (that is, 12 + 9 — 19.) & so on-ward; According to my first Precept, Cycl. O, 22 (= 19 + 3) 3.12.2.11.1.10.19.9.18.8.17.7.16.6.15.5.14.4.13. Cycl: 2> (which rank of additions is so easyly made, as that after two or three numbers, it runs on as fast as you can write it; & the same order of 19 places, is in all cases the same, onely beginning at severall places; & so being once done serves for all:) 4. Now because I do not here meet with 13, till the 19th place; I conclude that 13 in the Moons Cycle, doth not concur with 22 in the Suns, till the 19th Revolution; that is, not till after 18 intire revolutions (or 28 x 18 = 504 years,) & 22 yeares onward of the 19th Revolution. For which Reason, in my second Precept, I bid multiply 28 by 18, & then adde 22: which makes, 526; which is therefore the first year of the Julian

I mutandis, (1) fit (2) be 3 year (1) of (2) either 7 the (1) Number (2) Cycle 73, (1) (which is in our case is 13.) fe (2) not 13, as in our case: II number add. 13 And (1) therefore (2) therefore 14 the (1) next (2) second 15 the (1) moo breaks off (2) Moones 25 after add. 271(1) ad breaks off (2) bid

362

163. WALLIS to OLDENBURG, 30 November/flO December] 1667 Period, when these two Numbers (O 22, S> 13,) can meet. And it is the number of the Dionysian Period (compounded of the Suns & Moons cycle) in which it allways so meets. 5. Now if, at the same time, the Indiction had been 3 (as is proposed) that year had answered the Case: But, dividing 526 by 15, I find (by the [2] Remainder) the Indiction then to be 1. And therefore I am to seek for| some other such conjunction of O 22, S> 13, which may concur with the Indiction 3. 6. Now because this doth not come again till after 532 years (which is the Revolution of the Dionysian Period, compounded of the two Cycles of Sun, & moon; that is 532 — 28 x 18,) which contains 35 intire revolutions of the Cycle of Indictions (because of 15 x 35 = 525) & 7 years onward: therefore, the next time these meet, the Indiction will be promoted by 7. 7. To find therefore, in what Revolution of the Dionisian period, the number 526 (in which O 22, & 3) 13, do meet) will meet 3 for the Indiction; To the Number 1, (which I find to agree with 526 in the first Dionysian Period,) I adde continually 7, (still casting away 15 as there is occasion,) And so I find, that to 526 in the second revolution, answeres 8; in the third, 15; & so onward. For which Reason I bid, in my Third precept, to make this continuall addition of 7 &c; till I come to 3. 1.8.15.7.14.6.13.5.12.4.11.3. Cycl: Indict. (which is so easyly continued as you see, & being once continued to 15 places, serves for all cases, onely beginning as there is occasion, at several parts of it.) 8. Now because I do not here find 3, till the 12th place; I conclude that the Indiction 3, doth not concur with the Dionysian 526, (wherein O 22, & 3 13, do allways & onely meet,) till the 12th Revolution of the Dionysian period: Therefore according to my 4th Precept; for 11 intire

1-3 And it is ... so meets, add. 7 may (1) be accomodate to (2) concur with 9 come add. 12 onward: |And del] therefore 13 be (1) forward (2) promoted 15 526 ( (1) when (2) in which 18 that (1) with (2) to 24-25 of it.) (1) Whe breaks off (2) \\ 8. Now 27 fe onely add. 28 period: (1) There (2) Therefore

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163. WALLIS to OLDENBURG, 30 November/[10 December] 1667

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revolutions, I multiply 532 (the Revolution of that period) by 11; & to the product 5852 (= 532 x 11) I adde 526 (the number in the 12th period current,) which gives the number 6378, the number current in the Julian period. So that the whole operation (the numbers given being O 22, S> 13, Indict. 3.) is but this Cycl. O, 22 (= 19 + 3.) 3.12.2.11.1.10.19.9.18.8.17.7.16.6.15.5.14.4.13. Cycl. 5) . 28 x 18 = 504 +22 11.8.15.7.14.6.13.5.12.4.11.3. Indict. Per: Dionys: 526 532 x 11 = 5852 +526 6378 Period. Jul. Where though I have been fain, in many words to set down the Notion, the operation is short & easy. And I know not well how it can be made more facile. The like methods (mutatis mutandis) may be assigned; in what order soever you place the three Cycles: whether you begin with the Sunnes, & then the Moones, & lastly the Indictions, (as here:) Or first the Suns, then the Indictions, then the Moons: Or first the Moons, then the Suns, then the Indictions: Or first the Moons, then the Indictions, then the Suns: Or, first the Indictions, then the Suns, then the Moons: Or (lastly) first the Indictions, then the Moons, then the Suns. But of this sufficient. | [3] The Construction of Slusius729, for resolving a Biquadratick Equation by a Parabola & a Circle; I have examined: & find it to effect what it undertakes. The methode from Paris730, for depressing a Biquadratick Equation,

18 order (1) the (2) soever 27 I have (1) & (2) examined 27 it (1) affirmed. (2) undertakes. 729

Construction of Slusius: see SLUSE-OLDENBURG [14J/24.XI.1667; OLDENBURG, Correspondence III, 594-6, 594-5. 730 methode from Paris: possibly the Regula nova Deprimendi aequationes quatuor Dimensionum ad Sum gradum (OLDENBURG, Correspondence III, 433-4), which had been sent to Oldenburg from Paris in summer 1667. 364

164. OLDENBURG to WALLIS, 10/[20] December 1667 to a Cubick: I have not yet considered, sufficiently to give you an account of it by this Post. Resting Yours &c. John Wallis. [4]

These For my honoured freind Mr Henry Oldenburg; at his house in the Old Pelmell near St James's London.

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164. HENRY OLDENBURG to WALLIS London, 10/[20] December 1667 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 25r-26v (f. 26r blank) (our source). Postmark on f. 26V: 'DE/10'.—printed: OLDENBURG, Correspondence IV, 23-4. Answered by: WALLIS-OLDENBURG 13/[23].XIL1667.

London Decemb. 10. 67. Sir,

In stead of giving you by this MyLord Brounkers thoughts upon your last (which I could not yet doe, he being still absent731) I shall acquaint you with two particulars, you will, I think, be well pleased with: One is, the notice of a Book732, lately publisht at Paris, entitled,

731

he being still absent: Brouncker was at the time staying in Chatham. His absence from London was also the reason why Oldenburg had not written a week earlier. Cf. OLDENBURG-BOYLE 3/[13].XII.1667; OLDENBURG, Correspondence IV, 4-7, 7. 732 Book: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667. After receiving a copy in March 1668, Wallis commented on this book in WALLISOLDENBURG 30.III/[9.IV].1668.

365

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164. OLDENBURG to WALLIS, 10/[20] December 1667

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Francisci Du Laurens Specimina Mathematica, duobus libris comprehensa, quorum Primus, Syntheticus, agit de Genuinis Matheseos principiis in genere, in specie autem de veris Geometriae Elementis hucusque nondum traditis. Secundus, Analyticus, de methodo Compositionis atque Resolutionis fuse disserit, et multa nova complectitur, quae subtilissimam Analyseos artem mirum in modum promovent. I am promised733 to have this book sent me by the first opportunity, together with two others, 1. De vi Percussionis734, Borelli Itali. 2. De motu Musculorum735, Stenonis. The other particular is a Probleme, sent from the Author736 of this Book, which he recommends not for the difficulty of its Solution, but for the Inferences, that may be thence made, of some very fine proprieties of the Circle,| by him affirmed not to have been discovered hitherto. [251 Problema.

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Numero datis, circuli cbd radio eg, et inscripta bd extra circulum porrecta ad libitum, ita tamen, ut ad ductam cf acta parallela by, circuli circumferentiam secet in puncto y; Verum (non autem vero proximum, quod quidem per Sinuum Canones facile fieri potest) valorem exhibere lineae by.

6 fuse (1) diffe breaks off (2) disserit, 9 1. add.

12 Solution, (1) as (2) but for 14 by him add. 733

I am promised: evidently by Justel; see JusTEL-OLDENBURG [6J/16.XI.1667 (OLDENBURG, Correspondence III, 577-9, 578). Although Justel had dispatched two copies of this book not later than December 1667 (cf. JUSTEL-OLDENBURG [25.XII.1667J/4.I.1668; OLDENBURG, Correspondence IV, 84-5), Oldenburg did not receive them until February 1668; cf. OLDENBURG-SLUSE 26.II/[5.III].1667/8 (OLDENBURG, Correspondence IV, 209-10, 209). 734 De vi Percussionis: i.e. BORELLI, De vi percussionis liber, Bologna 1667. 735 De motu Musculorum: i.e. STENSEN, Elementorum myologiae specimen, sen musculi descriptio geometrica, Florence 1667. 736 Author: i.e. Francois Dulaurens. Oldenburg read this problem at the meeting of the Royal Society on 12 December 1667, where the problem 'was ordered to be communicated for solution to Mr. Collins'; BIRCH, History of the Royal Society II, 226. 366

164. OLDENBURG to WALLIS, 10/[20] December 1667

I pray, Sir, if you have leasure, consider it, and what is intimated above of the proprieties of the Circle to be drawn from thence. I am now printing, in the Transactions of this month, Mr John Collin's method and demonstration737 for finding the Julian Period, which a pretty while agoe he hath been ready with, and mentioned at the Society with applause, altogether differing of yours, I think. At the latter end of this, or the beginning of the next week, the book will be printed off; and your judgement, upon that part especially, is also desired by Sir Your faithfull humble servant H. O.

[26V] For his much honor'd friend Dr John Wallis Savilian Professor of Geometry in Oxford

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9 part (1) is (2) especially 73r

Collin's method and demonstration: This method was based on de Billy's rule pubslihed in Journal des Scavans No. 36 (6 September 1666), an account of which was printed in Philosophical Transactions No. 18 (22 October 1666); see WALLISOLDENBURG 16/[26].XL1667. Collins's method and its demonstration were printed in Philosophical Transactions No. 30 (9 December 1667), 568-75 ('A Method For finding the Number of the Julian Period for any Year assign'd, the Number of the Cycle of the Sun, the Cycle of the Moon, and of the Indictions for the same year, being given; together with the Demonstration of that Method'). 367

165. WALLIS to OLDENBURG, 13/[23] December 1667

165.

WALLIS to HENRY OLDENBURG Oxford, 13/[23] December 1667 Transmission:

W1 Draft of letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 28r-28v. On f. 28V three additional figures, not inserted in the letter sent. W2 Letter sent (fair copy of draft, of which some passages and longer marginal note are omitted): LONDON Royal Society Early Letters Wl, No. 35, 2 pp. (our source). At top of p. 2 above address at 90° note by Oldenburg (apparently arising from a meeting with Brouncker and written in preparation of Oldenburg's reply): 'The sense is That DF being ad libitum, is to be chosen of such a (./) length, (2) quantity, that BY may be rational. || As to Collins, he makes it a rational quantity in this instance, but hath not given a rule for it, for A (a) may (b) or GF may (ao) be (66) prove irrational by his methode. 11 Give always such a progression of numbers, that thereby lengthening the corde the parallel-line BY, may prove a rational quantity.' Beneath address in Oldenburg's hand: 'Rec. Dec. 16. 1667.' Postmark on p. 2: 'DE/16'.—printed: OLDENBURG, Correspondence IV, 36-8 (Latin original); 38-9 (English translation). W3 Enlarged and elaborated version of the mathematical section of the letter: OXFORD Bodleian Library MS. Add. D. 105, f. 27r-27v (our source). E First edition of letter sent: WALLIS, Opera mathematica II, 597-8. Reply to: OLDENBURG-WALLis 10/[20].XII.1667. Answered by: OLDENBURG-WALLIS 24.XII.1667/[3.I.1668]. The text of W3 represents a revised and extended version of the solution to Dulaurens's problem, as it was contained in the letter to Oldenburg. It takes up the issue raised by Brouncker to which Oldenburg refers in his reply and which he also notes in the endorsement. It can be assumed that Wallis wrote most if not all of this revised version after having received that letter. Cf. Wallis's Solution to Dulaurens's Problem 8/[18].II. 1667/8, which also discussed the case of rationality of BY taking up the reasoning of W3'.

(W)2 Oxford Decemb. 13. 1667.

Sir, The Books mentioned in yours738 of Dec. 10. I shall be willing inough to 738

yours: i.e. OLDENBURG-WALLIS 10/[20].XIL1667. 368

165. WALLIS to OLDENBURG, 13/[23] December 1667 see. Especially if the first739 of them do answere the title to which it doth pretend. Problema vero quod spectat, ibidem his verbis propositum, Problema. Numero datis, Circuli CBD radio CG, et inscripta BD, extra circulum porrecta ad libitum, ita tamen, ut ad ductam CF acta parallela BY, circuli circumferentiam secet in puncto Y; Verum (non autem vero proximum, quod quidem per Sinuum Canones facile fieri potest,) valorem exhibere lineae BY.

Hoc est (si ego ipsius mentem satis assequor;) Datis magnitudine (numeris designata) tribus hisce rectis; Nempe, Circuli BDG, turn radio CG, turn inscripta BD, turn hujus (ad libitum protractae) continuatione DF; cui occurrat in F, recta a Centro ducta CF; et huic parallela intelligatur inscripta BY: Quaeritur hujusce BY verus valor, numeris exhibendus. Quod sic solvo. Dico; Datis numeris, r = CG, 26 = BD, et / = DF: Erit Quadratum semissis rectae BY. 7 circuli (1) peripheriam (2) circuniferentiani 17 rectae BY. Hujusque radix quadratica r39

first: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667.

369

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165. WALLIS to OLDENBURG, 13/[23] December 1667 Dernonstratio.

Sint, CG = CD = CY = r.

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BE = ED = \BD = b.

DF = f .

BA = AY = \BY = CH = y.

Erunt EF = b + f.

BF = 26 + /.

EFq = b2 + 2bf + f2. BFq = 462 + 4&/ + f2. 2 CEq = CDq - EDq = r - b2. CFq = CEq + EFq = r2 + 2bf + f2. Item, in similibus triangulis, CFE, BFH, CF . CE :: BF . BH = CA. Adeoque CFq . CEq :: BFq . BHq = CAq. Hoc est r 2 + 2bf + f2 . r2 - b2 :: 462 + 46/ + f2

Ergo, 15

Hoc est, Ideoque

Quod erat demonstrandum.

17 Marginal note and additional figures in Wl: Nota. Si H sit citra C (puta, remotius ab F,) figura prima740 convenit. Si inter C et C?, vide figuram secundam (in versa 740

figura prima: i.e. a figure identical to the first figure in W2. The additional figures 2 and 3 in W1 are drawn on the reverse ('versa pagina'). 370

165. WALLIS to OLDENBURG, 13/[23] December 1667 Habes itaque Problema, eo saltern sensu quo (utut satis obscure traditum) intelligendum existimo, constructum et demonstratum. Assumo autem tres rectas, CG, BD, DF, datas esse: Quoniam, nisi hoc velit problema, aut saltern quod huic sit iaoSvvanov, poterit esse BY cujusvis longitudinis quae totam diametrum non excedit. Quaenam autem sint illae Circuli proprietates novae, hinc detegendae, quae non aliunde haberi possunt, nondum assequor. Vale.

5

Tuus Joh: Wallis. [2] These

For my honoured friend Mr Henry Oldenburg, at his house in the Pal-mal near St James's London.

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(W3)

Problema. Datis magnitudine (numeris designata) tribus hisce rectis; Nempe, Circuli BDG, radio CG, inscripta BD, hujusque (ad libitum protracta) continuatione DF; cui, in F, occurrat recta a Centre CF; et huic parallela

pagina.) Si in ipso C; erit BY — 0; ut in fig. 3. (quippe quae per B dianietro parallela ducitur, circuluni in B tangit.) ubique eadeni.

4 huic add. 19 Circuli (1) BY (2) BYD (3) BDG

371

20

165. WALLIS to OLDENBURG, 13/[23] December 1667 intelligatur inscripta BY: Quaeritur hujusce BY longitude, numeris item exhibenda.

Solutio. 5

Dico; Datis numeris r =CG,2b-BD, et f=DF erit Quadratum semissis rectae BY.

Dico porro; Si punctum F, quod ultra D supponitur, contingat citra D (ut in figuris quinque sequentibus,) erit Quadratum semissis BY. 7 quinque add.

372

165. WALLIS to OLDENBURG, 13/[23] December 1667

Demonstratio. Sint

CG = CB = r. BE = \BD = b. Datae. DF = ±f. (Nempe; +/, si F sit ultra D: -/, si citra;) 5A = i^y = y, Quaesita. Erunt. EF = b±f. EFq = b2 ± 2bf + f2. CEq = CBq - BEq = r2 - b2. CFq = EFq + CEq = r2 ± 2bf + f2. BF = 2b± f. BFq = 462 ± 46/ + f2. BHq = CAq = CBq - BAq = r2 - y2. Item (in similibus triangulis CFE, BFH,} CF. CE :: BF . BH. Et CFq.CEq :: BFq.BHq = CAq. Hoc est; r2 + f 2 ± 2bf .r2 -b2 :: 4fe2 + f 2 ± 46/ .

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Adeoque y

2

CJuod erat demonstrandum. Vel etiam, posito K ubi CA, BD, (productae si opus) se mutuo intersecant; propter similia Triangula BFH, RFC, CFE, KCE, KB A. Erit (in fig. prima, quod caeteris accommodabitur, mutatis ut opus erit + et -,) EF = b+ f .EC = :r2 -b2 :: EC.EK = et KB = BE-EK = b-

Et KF = FE + EK = b + f +

9-10 4&2 + /2±46f. (1) r2 -y2 (2) 15 KFC, add. 18 W («) Et

corr. ed.

373

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165. WALLIS to OLDENBURG, 13/[23] December 1667 Item KF

FC

KB=

seu y

BA=y=

[27^] Tandem petitur, 5

Ut BY sit rationalis. Respondeo.

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1. Si illud vellet problema ut primo propositum; nimis imperfecte propositum fuit; Quippe illud quod petitur faciendum, nempe verum valorem rectae BY exhibere, nihil tale insinuet. Dixisset igitur numero rationail exhibere- vel saltern numero exhibere] et non simpliciter exhibere. 2. Illud tamen sic solvo. Quo fiat 11 Cum sit (ut demonstratum est) rationalis (positis r, 6,ex hypothesi rationalibus) requiritur ut / rationalis

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(quo sit 2b2 — r2 + bf rationalis) ita sumatur ut sit etiam ^/ : r 2 + / 2 + 26/ : rationalis: Hoc est, ut sit r2 + / 2 + 2bf quadratus. Quod sic habetur. erit Cum sit b < r, adeoque r 2 +/ 2 +26/ 2bf :< r + f. Esto r + f — c, (adeoque c rational!, propter rationales r, /.) Hujusque propterea quadratus r 2 +/ 2 +c 2 +2/r—2cr—2c/ = r 2 +/ 2 +2fe/: Adeoque c2 + 2/r - 2cr - 2cf = 2bf: Et 2/r - 2bf - 2cf = 2cr- c2; adeoque / (adeoque rationalis c < r —6, nisi velimus ut

2r — 2b — 2c sit negativus.) Ponatur (quo fractionis Denominator fiat singularis) 2r — 26 — 2c — 2x; adeoque r — b — c — x, qui itaque est minor quam r — b, (et quidem negativus, si c > r — 6,) et r — 6 — x = c; et 25

propterea 2cr = 2r2 - 26r - 2xr, et c2 = r2 + 62 + a;2 - 2r6 - 2rx + 2bx, et 2cr - c2 = r2 - b2 - x2 - 2bx, et

/, (sumpto pro x, quovis rational! qui minor sit quam r — b.) Adeoque bf Adeoque et 2b2

12 est) (l)y= (2) \BY = 16 r2 + f2 + 2rf; (i) quadrat! r 2 + (g) erit 17 r + / — c, ( (i) sumpto (2) adeoque 18 propterea add. 19 = 2bf: (1) Et (a) 2rf (6) c2 - 2cr = 2&f + 2c/ - 2r/ (g) Et 21 sit (1) quantitas (2) negativus 23 si (1) c> 6 &reafc« off (2) Or-b 25 2cr - c2 = (i) {—} (2) r 2 - &2 - x2 - 2bx 374

166. OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] (propter •J : r2 + f2 + 2bf :=r + f-c

= f + b+ x =

\ erit

Hoc est, posito Radio r inscripta BD — 26, rationalibus; sumptoque x rational! quovis modo minor sit quam r—6; erit b— rationalis. Quod erat faciendum. Eademque facile accommodantur (mutatis mutandis) aliis casibus; puta cum F cadit intra circulum. Vel etiam, (hoc est / — DF) quantisi ita sumatur or, ut proveniat tas negativa; indicium est punctum F quod supponitur ultra D, cadere citra D- hoc est, BD secabit CG intra circulum; aut etiam ex parte C productum. Similiter, si pro y = BA, proveniat b — x negativa quantitas; indicio est BY non prorsum (ut in schemate principal!) sed retrorsum sumendum: Quod fieri potest si BD minor sit quam subfuerit negativa quantitas; tensa quadrantis; non secus. Item, si erit BY major quam BD, et centre propius jacebit. Similiter, si pro EK proveniat quantitas negativa; indicio est punctum K quod supponitur citra E, jacere ultra E. Item, si pro KB, proveniat negativa quantitas; erit K, quod supponitur ultra B versus D, revera citra B, in DB ex parte B producta. Atque alibi similiter. Quorum omnium determinationes non sunt difficiles: Nempe, qua ratione ponenda erit 6, vel sumenda x, ut hoc aut illud contingat.

166. HENRY OLDENBURG to WALLIS London, 24 December 1667/[3 January 1668] Transmission: C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 29r-30v (our source). At bottom of f. 30r part draft of Wallis's reply. Postmark on f. 30V: 'DE/24'.—printed:

1 erit (1) 2&2 + 4 breaks off (2) 2b2x - 4r2 breaks off (3) y = (4) 5 Quod erat faciendum, add. 7 proveniat (1) pro / = DF quant breaks off (2) corr. ed. (a) qu breaks off (b) (hoc est / (aa) ; (bb) = 10 pro (1) BA, proveniat (2) y = BA, proveniat corr. ed.

12 si (1) BY (2) BD 17 K, (1) citr breaks off (2) quod 18 similiter. (1) Quod (2) Quorum

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166. OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] OLDENBURG, Correspondence IV, 82-4. Reply to: WALLS-OLDENBURG 30.XI/[10.XII].1667 and WALLIS-OLDENBURG 13/[23]. XII. 1667. Answered by: WALLIS-OLDENBURG l/[ll].II.1667/8.

London Dec. 24. 67.

Sir,

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Since MyLord Brounker's returne I exhibited to him your two last741, concerning the Demonstration of the numbers for finding out the Julian Period, and that other of the Parisian Probleme. His Lordship is satisfied with the former; and as to the latter, he understands the Proposers meaning and demand to be in short this, That DF, being ad libitum, is to be chosen of such a quantity, that BY may be rational. A certain person742 here, makes it a rational quantity in this Instance, but hath not given a Rule for it, because, as MyLord Brounker observed in his paper, by his Method, GF may prove irrational. This I thought necessary to acquaint you with. Mr Boyle will doubtlesse give you part in the news, I send him743 by this post from abroad, and particularly of a considerable Experiment744, if not mis-represented, of Transfusion, practised very lately at Paris upon a Mad man in the presence of 8. able physitians there. | [29V] 745 Dr John Palmer , now Arch-deacon of Northampton, hath begun to correspond with me Philosophically. He presses746, that, since Tanger 5 other add. 7 demand (1) is, (2) to be in short this, 8 being |drawn del] ad libitum 741

your two last: i.e. WALLIS-OLDENBURG 30.XI/[10.XII].1667 and WALLIS-OLDENBURG 13/[23].XIL1667. 742 A certain person: probably Collins, to whom the problem was communicated by the Royal Society with the request for its solution; BIRCH, History of the Royal Society II, 226. 743 I send him: i.e. OLDENBURG-BOYLE 24.XII.1667/[3.I.1668] (OLDENBURG, Correspondence IV, 78-80), which was sent together with the present letter. 744 Experiment: i.e. the experiment on blood transfusion by Jean-Baptiste Denis, an account of which was contained in JuSTEL-OLDENBURG 18/[28].XII. 1667 (see OLDENBURG, Correspondence IV, 61). 745 Palmer: i.e. John Palmer (1612-79). 746 presses: see PALMER-OLDENBURG 12/[22].XII. 1667 (OLDENBURG, Correspondence IV, 34-5). 376

166. OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] is ours747, a Correspondence may be setled for Observations to be made of the Moons Meridian Altitude both there and at Edinburg, Aberdeen, or Catnes748, as well as at London, on the same days; to the end, that by taking a Chord of 17. 20. or 25. deg. (in stead of the semidiameter of the Earth) to subtend the parallactick angle, the distance of the S> might be solidly demonstrated. The like correspondence he would have held for observing the Eclipses of the Moon, with Smyrna, Aleppo or Bantam749 in the East, and with Bermuda, Barbados or Jamaica in the West, for the correcting of our Maps. We shall take this into Consideration. Since this, upon an information, I receaved, that this Learned man has a way of resolving all Equations and hard problems of Arithmetick by Regula Falsi, I have writ to him again750, and desired him to communicate it, if the information be true. Besides, I have engaged him, to procure for us from good Husbandmen the practice of Agriculture in Northamtonsh. and to take notice of all the Observables in that shire, for communication. Sir, I intend, if God vouchsafe me Life and health, to put the Transactions into Latin between this and Whitsontide, there being several particulars in it, communicated by able persons, and new, which ought to r [30 ] be kept from usurpation, by this way. You would | oblige me not a litle, and somewhat ease me, by taking yourself the pains, to turne your owne excellent Hypothesis about Tides751 into your owne Latin, as soon as conveniently you can. I am

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Sir

Your faithful humble servant H. Old. I have desired752 Mr Boyles favor, to see, whether the sub-bibliotheca14 from (1) a, good Husbandman (2) good Husbandmen 20 by add. 747

Tanger is ours: Tangier was occupied by England from 1662 to 1684. Catnes: i.e. Caithness. 749 Bantam: i.e. Bandung on Java. 750 writ to him again: i.e. OLDENBURG-PALMER 21/[31].XII. 1667 (OLDENBURG, Correspondence IV, 71-2). This request was also contained in the opening letter of the correspondence with Palmer, OLDENBURG-PALMER 3/[13].XII. 1667 (OLDENBURG, Correspondence IV, 3-4). 751 Hypothesis about Tides: i.e. WALLIS-BOYLE 25.IV/[5.V].1666, which had been printed in Philosophical Transactions No. 16 (6 August 1666). 752 I have desired: see OLDENBURG-BOYLE 24.XII.1667/[3.I.1668] (OLDENBURG, Cor748

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167. JENKINS to [WALLIS?], [January 1667/8?]

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rius of Oxford can give a good account and catalogue of the good books printed in England, these 10. or 12. years last: Monsr Carcavy having earnestly desired me753, to procure him such an one, in order to furnish the French kings Library with such Books accordingly. Pray, be pleas'd to Joyn with M. Boyle herein. A gentleman754 living in the Franche-Comte, that has been a great Traveller in the Levant, desires to know, what able men England has at present in the Oriental Tongues, especially in Arabick, Persian, Turkish. If at Oxford there be others, that ar skilld in them, besides Dr Pokock755 and Mr Hide756, I pray, signify it to me: And it may be, those two persons can give you and me notice of all others in the nation. For his much honor'd friend Dr John Wallis, Savilian Professor of Geometry in Oxford.

[30V]

167. LLEWELYN JENKINS to [WALLIS ?] [London, January 1667/8 ?] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 90r-90v (f. 90V blank). On f. 90r beneath, text at 180° calculation probably in Wallis's hand. On f. 90V in unknown hand: 'Dr Jenkins letter'. It would appear that the present request was directed to Wallis at the University archives in connection with the Fish Lyne case. Wallis possibly decided to make the transcript himself and sent it with his letter to Jenkins of 28 January 1667/8.

1 the |good add] books 3 one, (1) for (2) in order respondence IV, 78-80, 79-80). 753 Monsr Carcavy having earnestly desired me: via JuSTEL-OLDENBURG ll/ [21]?.XIL1667 (OLDENBURG, Correspondence IV, 29-31, 30). 754 gentleman: not identified. 755 Dr Pokock: i.e. Edward Pococke. 756 Mr Hide: i.e. Thomas Hyde. 378

168. WALLIS to JENKINS, 21/[31] January 1667/8 Thursday Morn. Sir

I perceive there is a pluries habeas Corpus, and an Attachment coming down against me. I must therefore beg the use of King Charles his great Patent757 this morning, that I may transcribe the whole. I build upon, & send it up to some Friends for

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I am

Your most humble servant L. Jenkins

168. WALLIS to LLEWELYN JENKINS 21/[31] January 1667/8 Transmission: W Letter sent: KEW The National Archives PRO SP 29/233, No. 4(ii), 2 pp. (p. 2 blank). Enclosure: Notes relating to the allowance of the claim in the Exchequer.

Jan. 21. 1667./8.

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Sir

Be pleased, in my breviat758; to mend one particular, in the reciting of Painter's: where instead of according to Dr Chaces case' put according to the case of 9 Hen. 6. (viz. Term. Mich. 9 Hen. 6. cas. 24.) which is not Dr Chace's759 case; but another; which allso is to our purpose, as well as the printed case of Dr Chase. For Dr. Jenkins.

Yours. J.W.

Dr chace's case is 8° Hen. 6. 757

King Charles his great Patent: i.e. the 'great charter' of Charles I of 3 March 1636; cf. Wallis's Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II].1667/8. 758 breviat: i.e. the enclosed Notes relating to the allowance of the claim in the Exchequer. 759 Chace's: see WALLIS-JENKINS 26.X/[5.XI].1667.

379

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169. WALLIS to JENKINS 21/[31] January 1667/8, enclosure

169.

WALLIS to JENKINS 21/[31] January 1667/8, enclosure: Notes relating to the allowance of the claim in the Exchequer Transmission:

W Paper sent: KEW The National Archives PRO SP 29/233, No. 4(i), Ip. Endorsed on verso: 'Instructions from Dr. Wallis, out of the Archives, touching Prince's the Townserjeant his suit against the Bedles in the Exchequer for executing the V. Chancellors warrant. 21. Jan. 67.' Enclosure to: WALLIS-JENKINS 21/[31].1.1667/8.

For the Allowance of our claim in the Exchequer in the Bedles case arrested for executing the Vicechancellors760 warrant. 5

That the Priviledge is to be allowed in Exchequer as well as other courts. There is the Precedent of Poolye's case for Cambridge. Which is expressely allowed. Boxters case: Which by a writ of Privy seal (which is of record in the Exchequer, ex parte rememoratoris Reginae) is superseded.

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To the objection, that they will not allow it in our case, where our jurisdiction is concerned. Wee may plead. The charter of 12 Apr. 10 Ed. 3. marked N 3. at the marks in the margin +. +. That the Prohibitio regis non curret &c. That the chancellor is not to be molested for false imprisonment by any writs from above.

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4 other (1) cl breaks off (2) courts 6 case [allowed del] for 7 seal add. 11 Wee (1) pr breaks off (2) may 760

Vicechancellors: i.e. John Fell, q.v. 380

169. WALLIS to JENKINS 21/[31] January 1667/8, enclosure A writ thereupon directed to the Chancellor &c. 27 Apr. 48. Edw. 3. (of which you have a Copy out of the Proctors Old Statute book writ about 250 years ago) Authorizing the Chancellor to proceed notwithstanding such writs.

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761

And a clause in K. Hen. 8 Charter to that purpose. Provided the Vicechancellor submitted to that way of Appeal provided in the University. Which end in chancery. The French composition 22 Edw. 3. (marked P. fase. XIII. 1.) That the Chancellor is to proceed in cases that concern the University as a University. Farendons case; indicted for an assault on the Marshals-man of the Kings bench executing the Kings writt. Farendon762 was Comissary or (as we now call it Vicechancellor) & is claimed by the Chancellor & allowed, in Hen. 4. time. You have the Record at large.

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Dr Chaces case, 9 Hen. 6.763 in the printed year-Book. (Where the jurisdiction was questioned to amerce townsmen for not paving the streets.) Hee claims cognizance of his own cause in his own court, (where that particular is of large debated by the judges) & it is allowed. Painters764 case: 7 King James. Where the question was of Jurisdiction; whether the Vicechancellor might amerce the Bailifs for Night-walking

4 the | Vice del] Chancellor 6-8 And a clause . . . in chancery, add. 11 as as University corr. 14 in (1) Edw. 4 (2) Hen. 4. 18 of (1) the cause (2) his 21 might (1) amit (2) amerce 761

K. Hen. 8 Charter: i.e. the charter of Henry VIII of 1 April 1523 ('Wolsey's charter'); cf. Wallis's Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II].1667/8. 762 Farendon: i.e. William Farington (d. 1420), chancellor's commissary at the University of Cambridge 1401. 763 Dr Chaces case, 9 Hen. 6.: cf. the correction in WALLIS-JENKINS 21/[31].I.1667/8. 764 Painters: i.e. Thomas Painter, bailiff; TYACKE (ed.), History of the University of Oxford IV, 113-14.

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170. WALLIS to JENKINS, 28 January/[7 February] 1667/8 who pretended the Kings business, as hues & cryes &c. And it was allowed in the Kings bench to be of the same nature with that of Dr Chace765. And Painter remanded till the Vicechancellor release him. 5

10

The market in particular, which is here the case, The whole power the Town hath in the market (at lest the cheif) depends on that clause in their charter of Edw. 3. Clericus Merkati nostri se non intromittet &c. Which is taken from them, (after the great conflict upon their submission) and granted to the University in the same words. Et quod Clericus Mercati se in presentia nostra vel hereditorum nostrorum de cetero non intromittat infra burgum ilium aut suburbia ejusdem de officio suo exercendo de aliqua se eundem burgum tangente vel suburbia. 10 Jan. 32 Edw. 3. You have the copy of the town charters by themselves. And of this in the University, in that of K. Charles766.

170. WALLIS to LLEWELYN JENKINS Oxford, 28 January/[7 February] 1667/8 Transmission: W1 Draft of letter sent: OXFORD University Archives WP7/16/1, f. 86r-89v (f. 86V-88V enclosure, f. 89r-89v blank). W2 Letter sent: KEW The National Archives PRO SP 29/233, No. Ill, 12 pp. (pp. 2 and 11 blank, pp. 3-10 enclosure) (our source). On p. 12 at 90° to address in Jenkins's hand: '16 Reasons against the Leasing of Felon's Goods to the Citty of oxford by Dr Wallis 28. Jan. 67.' Postmark on p. 12: 'JA/29'. Answered by: JENKINS-WALLIS 30.I/[9.II].1667/8. Enclosure: Extracts from the University Archives. The background to the present letter is evidently the attempt on the part of the city of Oxford to bring the conflict with the University to an end, fearing that custom and trade might be taken away from townsmen. After both sides failed to reach an agreement, it was decided that the vice-chancellor should proceed in the courts of justice against the city. See WOOD, Life and Times II, 128-9.

2 of the (1) Cha breaks off (2) same 3 remanded (1) to (2) till 765

that of Dr Chace: cf. the correction in WALLIS-JENKINS 21/[31].I.1667/8. that of K. Charles: i.e. the 'great charter' of Charles I of 3 March 1636; cf. Wallis's Extracts from the University Archives 766

382

170. WALLIS to JENKINS, 28 January/[7 February] 1667/8 Oxford Jan. 28 1667./8.

Sir, I suppose Mr Vicechancellor767 hath given you an account of Yesterday's Convocation; in which, hee gave a brief Narrative of what hath lately passed in order to a Treaty with the Town. Whereupon Delegates were appointed to consider of it, but not to conclude: (not cum nuda relatione ad domum, which the masters would not passe; but cum relatione ad domum et approbations ejusdem.) After which, he sent for mee at night, & desired mee to draw up somewhat and send to you, concerning those dangerous consequences which are apprehended in making a Lease to the Town, of Felons goods &c, with the Powers appurtenant thereunto. Which last night & this morning I have been doing, as the shortness of Time would permit. Which that it might be the more cleare to you; I have caused the Words of our charters which grant us those Rights & Powers, to be transcribed768; and then, what did, for the present, occurr to mee of danger in putting over those Powers to the Town. (Though I will not undertake to be so well skilled in the Law, as in so short a time to foresee all.) What use you are to make of it; I presume you have Instructions from Mr Vice-chancellour. I understand allso from him, that the Habeas Corpus for Fish Lme769 is come down; which I suppose will be sent up to you. To which it will be necessary (out of the Instructions I sent) to get a Return drawn up by the best Clarks you can find, to be ready to annex to the Writ: And this, the sooner the better. For we are not so far to trust to the Treaty on Foot, as to be surprised by being too secure. And I think it would be very Necessary that Justice Brown770 were, before hand, acquainted with the Particulars of the Return which I drew up, that he may have timely leisure to consider of them, (being yet strange to him,) which in so short a time as it will be pleading at the Bar, cannot be done.

13 permit. (1) & because it was blamed and not so likely to be ready, I may which that serve what use you are to make of it; I presume you have W1 17 as |in so short a time add] to foresee jthem del. all.) 767

Vicechancellor: i.e. John Fell, q.v. transcribed: i.e. the enclosed Extracts from the University Archives. r&9 Fish Line: i.e. Fish Lyne. See WALLIS-JENKINS 26.X/[5.XI].1667. 770 Brown: i.e. Sir Samuel Browne (d. 1668). 768

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure

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In the Exchequer; I hope care hath been taken that wee bee not surprised there, as having deserted our Plea for not pleading to it on the first Saturday in this Term, as was the order of last Term. If the suit there, be wholly withdrawn by consent; I think there is no danger of it. But if onely suspended; wee must be vigilant that wee be not surprised. I could wish allso that wee had a Copy (if not an Exemplification under Seal) of Dr Chace's case at the Kings Bench, in Hillary term 8° Hen. 6. out of the Records. For though, in the Printed Year-book, the pleadings favour us; yet it is not there expressely sayd, that the cause was dismissed. And the case is extremely pertinent to that now in the Exchequer. Sir, I hope you will excuse mee for the troubles I give you in these affairs, from Sir

Your humble servant, John Wallis.

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These For Dr LLuellin Jenkins Judge of the Court of Admiralty at Exeter house London.

[12]

171. WALLIS to LLEWELYN JENKINS Oxford, 28 January/[7 February] 1667/8, enclosure: Extracts from the University Archives Transmission:

W Draft of paper sent: OXFORD University Archives WPj/16/1, f. 86V-88V. w Paper sent (transcript in scribal hand with corrections and additions by Wallis): I hope (1) so much care . . . taken as (2) care ... that 6 Copy (1) of th breaks off (2) (if 7 at the Kings Bench, add. II mee |(I hope) del] for 384

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure KEW The National Archives PRO SP 29/233, No. 1111, 2 pp., and No. Ill II, 5 pp. (our source). Enclosure to: WALLIS-JENKINS 28.I/[7.II].1667/8.

The Words of King Henry the 8th his Charter771 concerning Felons Goods &c. 1° April. 14° Hen. 8.

[I, p. 11

At insuper de uberiore gratia nostra damus et concedimus praedictis cancellario et scholaribus dictae universitatis quaecunque annum, Diem, strepum, Vastum, Deodandum et Thesaurum Inventum, catalla felonum, utlagatorum, fugitivorum damnandorum felonum de se et felonum in exigendis positorum sive aliter delinquentium pro quo bona sive catalla sua amittere sive forisfacere debeant infra villain Oxoniae praedictam et suburbia ejusdem cujuslibet et quorumcunque ligeorum sive subditorum nostrorum. Nee non omnium bonorum et catallorum quorumcunque manuopere captorum, et quod bene liceat eisdem cancellario, commissario, deputato sive vicem gerenti et scholaribus et successoribus suis omnia praedicta [fines, amerciamenta, redemptiones exitus, forisfacturas,] Deodanda, Thesaurum inventum, annum, diem, vastum, catalla felonum, utlagatorum, fugitivorum, damnandorum, felonum de se, felonum in exigendis positorum et caetera praemissa quaecunque et qualitercunque forisfacta, sive forisfacienda et omnia quae ad nos, haeredes et successores nostros de dictis finibus, amerciamentis, exitibus, die, et Anno vasto, streppo, Deodandis, catallis felonum, utlagatorum, fugitivorum, damnandorum, felonum de se, et in exigendis positorum, sive aliter (ut praedicitur) forisfactis quoquo modo pertinere possit per ipsos et Ministros suos levare, colligere percipere et habere tarn per manus suas proprias, sive assignatorum suorum, sive Deputatorum suorum quam per manus vicecomitis, Escaetoris et aliorum Ofnciariorum nostrorum quorumcunque et seipsos in seisinam inde ponere absque calumpnia, impetitione vel impedimento nostro vel haeredum nostrorum Officiariorum vel Ministrorum nostrorum, vel Haeredum nostrorum praedictorum aut aliorum Justiciariorum,

1-3 The Words ... Hen. 8. add. Wallis 14 [fines . . . forisfacturas,] add. Wallis 19 dictis finibus, amerciamentis, exitibus, add. Wallis 771

King Henry the 8th his Charter: i.e. the charter of 1 April 1523 ('Wolsey's charter'). 385

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Escaetorum, Vicecomitum, Coronatorum, Majorum Ballivorum, Constabulariorum, seu Ministrorum nostrorum vel Haeredum nostrorum quorumcunque. The words of King Charls's Charter in 3° Marti 11° Car. I.772

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Insuper cum Dominus Henricus illius nominis Octavus, Antecessor noster per literas suas patentas praerecitatas praefatis cancellario et scholaribus dictae universitatis et successoribus suis (inter alia) dederit et concesserit quod ipsi quaecunque annum, diem, strepum, vastum, Deodandum, The10 saurum Inventum, catalla felonum utlagatorum, fugitivorum damnandorum, felonum de se, et felonum in exigendis positorum sive aliter delinquentium pro quo Bona sive catallaj sua amittere sive forisfacere deberent [I, P. 2] Infra villain Oxoniae et suburbia ejusdem cujuslibet et quorumcunque ligeorum sive subditorum suorum. Nee non omnium Bonorum et catallo15 rum quorumcunque manuopere captorum in proprios Universitatis usus per se vel Officiarios et Ministros suos colligerent et levarent ac omnia alia circa eadem bona et catalla felonum ac aliorum ibidem mentionatorum exequerentur et facerent, prout in praedicta Antecessoris nostri charta plenius continetur. Sciatis ulterius quoddam nos ex Ampliori et uberiori 20 gratia nostra special! ac ex certa scientia ac mero motu nostris pro nobis Haeredibus et Successoribus nostris iisdem cancellario, Magistris, et Scholaribus et successoribus suis in perpetuum damus et concedimus per praesentes, quoddam ipsi eadem omnia et singula Bona et catalla felonum aliorumque praementionatorum ac debita etiam Jura et credita sive per 25 scripta vel alio quovis modo ad ipsos pertinentia tarn infra praedictam Universitatem Oxoniensem et ipsius praecinctum, quam etiam Infra Civitatem Oxoniae et suburbia ejusdem accidentia sive emergentia habeant et teneant sibi et successoribus suis libere et quiete et pacifice absque compute seu aliquo alio proinde nobis Haeredibus et successoribus nostris 30 reddendo seu faciendo. Ita quod ubi et quandocunque iisdem Cancellario 2-3 quorumcunque. |Et Insuper volumus et per praesentes concedimus quod praedictus cancellarius et scholares ac omnes eorum servientes et cujuslibet eorum serviens aut eorum seu universitatis dictae Ministri firmarii et Tenentes sui, et eorum quilibet ubicunque fuerint exonerati et quieti sint de quibuscunque prisis chiminagiis captionibus carriagiorum Equorum Carractarum plaustrorum. del. Wallis] rr2

King Charls's Charter in 3° Marti 11° Car. 1.: i.e. the 'great charter' of Charles I of 3 March 1636. 386

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Magistris et scholaribus, vel Cancellarii locum tenente et successoribus suis certo vel probabiliter innotuerit aliquem vel aliquos crimen feloniae commisisse aut in suspitionem feloniae incidisse infra universitatem praedictam ejusve praecinctum vel infra Civitatem Oxoniae ejusve suburbia. Tune Cancellarius ejusdem universitatis vel ejus locum tenens per seipsum, vel ejusdem universitatis procuratores eorumve Deputatos officiarios vel Ministros suos hac in parte appunctuatos vel appunctuandos, in quamcunque domum aut alium locum ubi hujusmodi bona et catalla aut alia praementionata, Nee non scripta et chirographa eorundem felonum aut Delinquentium Debita jura et credita quovismodo tangentia sive concernentia esse, abscondi aut in quorumcunque manibus detineri contigerit libere et impune ingredi liceat et scrutinio sive examine super iisdem facto et habito eadem omnia sic inventa vel in posterum invenienda ut Cancellario Magistris et scholaribus Universitatis praedictae forisfacturae virtute et vigore praesentis istius nostrae concessionis in quorumcunque custodia fuerint vendicare, arrestare, capere, seisire et impune asportare et post eorum debitam convictionem in usum proprium ejusdem universitatis convertere. Resistentes autem et contradicentes tanquam eos qui Cancellario ejusve Commissario et procuratoribus aliisve officiariis et Ministris Universitatis praedictae in executione officiorum suorum Resistunt et contradicunt secundum leges et statuta Regni nostri, vel secundum Jura et consuetudines ipsius Universitatis debite castigare et punire. These are the Rights & Powers which are by the intended Lease for 200 years to be passed over to the Town, for a Noble a year. For which Concession, the Town offer by their Indenture (to which we are to be parties,) to acknowledge a Power in the Vice-chancellor & Proctors 1. to punish according to the Laws of the Kingdom such as they find Wandering after Ten of the clock at night in summer, or Nine of the clock in Winter, & before four in the morning. 2. To search Inns Taverns and notoriously suspected houses, and such as they find there Debauched, to punish according to the Lawful usages of the University. 3. Such as they shal find notoriously debauched, or resisting the Vice-chancellor or Proctors, they promise not to abbet or defend, but suffer them to be punished according [II, p. 1] to the present statutes of the University.

5 vel ejus . . . ejusdem universitatis add. Wallis 23-34 These axe ... the University, add. Wallis 26 1. add. 387

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Reasons against the passing away the Universitys Right in Felons Goods &c, with the Powers concerning the same by lease to the Citty of Oxford for 200 yeares at the Rent of a Noble a yeare. 5

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1. There is no priviledg or Right which the University hath which is more certaine, then what they have to these, (no not the power of holding convocations and graunting degrees) they being thinges which cannot passe from the crowne by prescription, (which is all the citty have to pretend) but by words of Expresse graunt which the University have. But the citty have not; Which is a thing so certaine and knowne, that no lawyer in England will doubt of it. See the Abbot of Strata Marcella's case. And former decisions for the University. 2. It is the whole Royalty for the university that is intended thus to passe. Which is a thing of that Nature, that No man who hath but the spirit of a Gentleman would part with it for a much greater value then what he makes of it. 3. It is such a Royalty as scarce any Corporation in England doth Enjoy, beside this University, (No not the Citty of Londont selfe.) For though divers Corporations may be found which have some of these thinges, yet very few, (if any) that have all. 4. It would be very unreasonable; when as K. Henry the Eight hath graunted us such a Right which to the Citty was Never graunted: & K. Charles by his Charter (obtained by the great care of the late Archbishop Laud, our then Chancellor) hath so much Enlarged it, perhaps beyond all president, that the University should now make no better use of this, which for theire sake was taken from the Kinges Almoner, then to passe it away to the Citty of Oxford for a Noble a yeare. 5. Whereas it is pretended, that it hath beene a matter of contention, and that wee Never made any thing of it. I say as to the first, that there is no power or priviledg, which wee have but is and hath beene soe. And if makeing it a matter of contention will wrest of it out of our hands. Wee must part with all; even that of holding convocations and makeing statutes for our owne Body. For the towne have not only made complaints in parliament and otherwise against us oftentimes, but even preferred

8 praesumption corr. Wallis 9 pretend) and corr. Wallis 25 precedent W 388

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Indictments against the Chancellor, Procter and Convocation for so doing; as holding unlawfull Conventicles, contrary to the laws of the land, & incurring the penalties of Praemunire or the like. Which occasioned the Charter of K. Henry the Fourth for the steward of the University who might hold pleas of such Indictments. At which Charter the Towne was so much offended. 6. The Matters of contention are not hereby avoided; But Rather increased. For, (beside that the Excercise of this power by the towne will sure create new causes of contention;) the other perticulars now actualy in contest as the power of the Market, and of priviledged men's trading, & theire arresting our Bedles for Executing the vicechancellors warrant, [II, p. 2] are not at all provided for; But in Effect deserted, even then when wee have spent above an hundred pound in defence of our Rights therein, and are fair ennough to have them decided for us. Which thinges breakeing off in the posture they now are, will seeme Rather to passe against us. So that all our Rights therein must bee deserted too, or else wee must contend for them at a greater disadvantage. And the power of the night watch, which is the only thing thought hereby to bee secured, will by this graunt if it passe, bee made more disputable then ever. Of which by and

by-

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7. As to that of our makeing nothing of it, wee must upon the same Reason part with all the Rest, For what hath the University (as to mony) made of the Maior and Citizens takeing theire yearly Oath of fealty to us: which yeilds us not a penny and that of theire offerring on Saint Scholastica's Day, which is But 42 pence to the University, (and 21 pence to the vicar of Saint Marys.) Why should wee not therefore for another Noble passe away this Right also? And the like for the Assises of Bread and Beere. And of weights and measures, and the Inspection of the Streets and pavements. Which have beene (and are) matters of contention and do Realy cost more then wee make of them, And the like of the clarkship of the Market (if at least it bee not to passe by this graunt) why should it

1 Proctors W 5 Indictments. (1) \\ The (2) At 12 when wee scarce spent an hundred corr. Wallis 13 have (1) fair inough to have them (2) spent W 18 thing (1) the pretended (2) thought W 19 passe, jwill del. bee 26 Why should . . . Right also? add. w, add. in margin W 389

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure not bee graunted away? Since it doth not yeild the University any thing? And doth not so much as pay the Universities charges to the dark of the Market. 8. The thing is not so inconsiderable as is pretended. For the University have Really Received very considerable summes on this account, 5 sometimes Threescore and Tenn or four score poundes at a time. And what ever at any time the University have actualy seised, the towne Never had the Confidence to sue for: But suffered us to injoy it as knowing they had no Right to set forth as evidence in the case; and have only by vio10 lence laid handes on what they can get. And though the University have not converted all these summes to theire owne proper use yet some part thereof they have so imployed, and part of it by way of charity, for the payment of Just debts, (which must otherwise have beene lost) and for Releife of widdowes and Orphants of such as have made away themselves, and for other charitable uses; According to the true contentment of such 15 forfeitures, which were otherwise to come to the Kinges Almoner for such uses. From whom they were taken to be given to us, and when heretofore there hath beene differences concerning such thinges It was not Betweene the University and Citty But Betweene the Kinges Almoner and the University, the Citty never interposing, for the Goods of Felons de se &c. 20 till since the Kinges Almoner did Relinquish them to the University. To whom they do belong if not to us. 9. If it were no more But a power Reserved in us for an equitable Releife of the widdowes and children of persons whose Estates are soe forfeited, it were at least a Christian work to doe so much of charity. For 25 though the law do make forfeit the Estates of such persons, yet there is equity that the Innocent widdowes and children should be considered.) [II, p. 3] And even the disposing of thinges thus forfeited belonging ordinerly to the Kinges Almoner it may be well presumed to be so Intended. Only that it should be left to his or the Kinges discretion how far and in what 30 way to do it. But if wee put it out of our handes into the Cittys they are not to expect the least Releife. But to be totaly Ruined (for it is cleare ennough by the Experience of theire practice what mercy is to be found at theire handes.) And though sometimes Rather then the persons should seek for shelter from the Universities Right, the Citty Baileifs are faine to 35

11 owne (1) use (2) proper use 15 contendment corr.

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure accept of some composition, (knowing that they have no title) wee shall soone see what use they will make of it; and how little soever it may be thought worth, while it is ours it will soone bee found a very considerable purchase to them. 10. If it were indeed nothing worth: yet were it better to keep it for nothing, then to sell it for Nothing, that so wee might at least transmit the Right to our successoures as it hath beene Derived to us, and if wee do not think it worth while to effect our Right therein yet let us not by our act preclude those who may come after us. Who certainly will deservedly blame this generation when they shall see so faire a Royalty (of which they shall then find the Town to make so much advantage) passed away by us for a Noble a yeare. Which will not by them be looked upon as a Rent, But Rather as a perpetual Reproach, for haveing passed it away at so cheap a Rate. Nor need wee feare our Not doing it will be a matter of charg or contention. For if wee do but tacitely permit the towne to usurp upon us without our passing it away by an Act of our owne, that contention will soe long cease. Nor will they Revive the contention so long as wee are willing to be silent. 11. If it be thought a thing soe little worth, wee might do well first to try whether the towne will be content to passe away theire No-Right at so cheap a Rate as wee are Ready to passe away our Right. And if they will I should advise for peace sake, that wee may be purchasers. If not, it is a signe they look upon it as better worth. 12. When as it hath beene hitherto thought a great oversight in our predecessours that when the Royalty of North-gate hundred was to be sold, the University suffered the towne to buy it, when they might themselves have purchased it: (Because thereby the towne Inlarged theire Jurisdiction to our prejudice:) Wee do hereby extend it not only to North-gate hundred and Holy-well, But to Saint Clements and the whole precincte

I no Title) yet when wee shall have given them a legall-title; wee shall soon see what they will make of it. And how W 9 Act (1) prejudice (2) preclude W II Town add. Wallis 14 it add. Wallis 15 but so silely permit corr. Wallis 19 (1) 11. Whereas it hath been hitherto thought a great oversight in our predecessors; yet when the Royalty of North-gate Hundred was to be sold. (2) 11. If it W

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure

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of the University. As far as Saint Bartholomews Bagly-wood, Botly, and Gofford Bridge, which Jurisdiction though, while in the Universities hand It hath beene moderatly used, yet when the towne shall have it, both wee and our Neighboures will soone find the Inconveniency. For the Rights and powers Appurtenant thereunto, which are now about to be passed away extend to the whole praecincts of the University, which extend some miles further then those of the Towne. 13. The clarkship of the market is hereby (in effect) graunted away. For that depending cheifly on forfeitures, (which the University officers do commonly give to the poor) All these fall into the Townes hands under those general words in our charters sen aliter delinquentium pro quo bona sua sive catalla amittere sen foris facere debent. Which will extend to these and to all other forfeitures whatsoever: suppose by praemunire or many other wayes beside Felonyes and Treasons. 14. The Right of the Night watch, which is the only thing thought hereby to be secured to us, is so far from being secured that is indeed given away, so far as it was wont to bee controverted. For the controversy betweene the University and the towne about the Night watch heretofore, was not, whether the Vicechancellore & proctors may walk and punish Noctivagatores. For that was alwayes unquestionable till just Now, But, whether they only and not the Major and Baileifs also. And even this hath airways hitherto been judged for us, as in the parliament 18° Edw: 3. and other parliaments; and at Lawe in Paynters case at the Kinges Bench 7° Jacobi; and at the Councill table, by the consent of lot bodyes 10° Jacobi: beside our charters Ed: 3. and many more. But this graunt doth evidently passe that away; alloweing to the towne (as our officers, though not thencforth under our correction) at lest an equal power therein (and

1 Bartlilomews corr. ed. 8 away ( 1 ) the profit of (2) . For that 9 For | the profit of add] that W 9 University (1) doth commonly (2) officers 10 commonly {—} to corr. Wallis 16 secured by us corr. Wallis 16 indeed taken or seeken away corr. Wallis 21 hath also corr. Wallis 26 alloweing (1) power therein (and in much more) (2) to 27 thenceforth add. W

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure in much more) under the pretence of searching for Felons, or for goods writings or evidences belonging to such; even in Colledges and scollers chambers and studies. And what is like to be the effect of such concurrent powers (when the proctors with theire company and the baileifs with theirs, shall walk at once) may be easily foreseene by what happened 29° Edw: 3. when upon a much lesse pretence of Authority the townes-men in warlike manner slew so many of the schollers and committed so great outrages for which all their Rights were seised into the Kinge's hands, and so much of theire former powers taken from them. And if the powers mentioned in King Charles Charter as hereunto appertaining, be but Real; it will soone be evident whether the towne be fit to be intruded with such powers; and whether Archbishop Laud would have taken care to have them so fully expressed had he thought that wee would have passed them to the towne and then be confronted with our owne charters. Who will not spare to stretch it to the utmost and turne our owne charters upon [II, p. 5] us. 15. So much of the Night watch as is pretended to be secured to us (that it may be lawfull for the Vicechancellor and procters to search and punish Noctivagatores) is not at all advanced by it. For if wee have the power without theire acknowledgment, wee need it not. If not: theire concession signifies Nothing: For it is not a thing in theire power to graunt. Nor will theire acknowledgment preclude any single person from bringing his Action to try our power so oft as they please: Nor will it be any plea in law against such an Action. No nor hinder them from countenancing, under hand, any who will soe molest us. Beside that it is so lamely expressed as that (if wee be parties to the Indenture) it will do us more hurt then good: Wee seeming thereby to accept of that as our whole Right which they there acknowledg. 16. No Act of Parliament of doubtfull consequence is usualy made more then a Probationer at first till Experience may discover what is not 1 in add. 2 even in Colleges & Scholars chambers fe Studies add. in margin W 4 (when the Proctors with their company, & the Bailifs with theirs shall walk at once,) add. W 4 company (1) when (2) and 8 all the rights corr. Wallis all their Rights were seized into the Kings hands, and add. in margin W 14 Who will not spare to stretch it to the utmost; & turn our own Charters upon us. add. W

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172. BROUNCKER: Solution, [December 1667/January 1667/8]

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at first foreseene. It would therefore be very imprudent for us, at the first dash, to passe away such a Right accompanied with such powers for 200 yeares: When it is not possible to foresee what mischeifes may follow upon it. These which I have mentioned being but some few of what an expert Lawyer might shew us. And all the consequences of Removing the ancient boundes of such powers as these, nothing but time can discover: when it will be part Remedy or Redresse.

172. WILLIAM BROUNCKER: Solution to Dulaurens's Problem [December 1667/January 1667/8] Transmission:

C Copy of note in Oldenburg's hand: LONDON Royal Society Classified Papers 24, No. 11, p. 1. This copy of Brouncker's solution to Dulaurens's problem, which had been read at the meeting of the Royal Society on 12 December 1667 (BIRCH, History of the Royal Society II, 226), is possibly identical with (a copy of) the note which was enclosed in OLDENBURG-BOYLE 28.1/[7.II].1667/8 (OLDENBURG, Correspondence IV, 120-3) and intended for Wallis; see BOYLE-OLDENBURG 1/[11].II.1667/8 (OLDENBURG, Correspondence IV, 139-40, 139).

Problema transmissum Parisiis, et lectum coram Societate R. die 10

Numero datis Circuli cbd radio eg, et inscripta bd extra circulum porrecta ad libitum, ita tamen ut ad ductam cf acta parallela by circuli circumferentiam secet in puncto y; Verum, non autem vero proximum (quod quidem per sinuum canones facile fieri potest) valorem exhibere lineae by.

1 for us add. W

5 might ( 1 ) discover (2) shew us 5 And | all add. \ the consequences W

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173. JENKINS to WALLIS, 30 January/[9 February] 1667/[1668]

Solutum a Nobilissimo Vice-comite Brouncker, ut sequitur.

CG = R BD = D sit x quaelibet quantitas rationalis, minor differentia Radii et Lineae datae. BY

DF =

173. LLEWELYN JENKINS to WALLIS [London], 30 January/[9 February] 1667/[1668] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 91r-91v. On f. 91V endorsement in unknown hand: '30 Jan. 1667', and postmark: 'JA/30'. Reply to: WALLIS-JENKINS 28.I/[7.II].1667/8.

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] Thursd. 30. Jan. 67 Sir

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I have your Line773, and your Reasons774 for which I humbly and heartily thank you. Wee'l provide as well as we can for the H. Corpus. & if Mr Just. Brown775 be accostable he shall not be unacquainted with what your line intimates: we shall not, I hope, be surprisd in the Exchequer & I will look after Dr Chace his Case. I beg Mr V: Chancellor776 that the Delegacie will please to think of some Persons more capable then I am to serve them. I shall not fayle to give them and their subdelegates the very best & uttmost of my service, in this & all other occasions. I shall make the best use I can of your Reasons, among our friends & Adversaries (for such also we have) here and shall remain Sir

Your most humble & thankfull servant L. J.

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[91V]

For the Reverend Dr Wallis D. D. publique Professor & Gustos Archivorum at his House neer the Scholes in Oxford.

174.

WALLIS to HENRY OLDENBURG Oxford, I/[11] February 1667/[1668] Transmission:

W1 Paxt draft of letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 30r (written on OLDENBURG-WALLIS 24.XII.1667/[3.I.1668]) (our source).—printed: OLDENBURG, Correspondence IV, 83. 773

Line: i.e. WALLIS-JENKINS 28!/[7!I].1667/8. Reasons: i.e. the 'Reasons against the passing away the Universitys Right in Felons Goods', which were enclosed in Wallis's letter of 28.1/[7.II]. 1667/8 (Extracts from the University Archives). 775 Brown: i.e. Sir Samuel Browne (d. 1668). 776 Mr V: Chancellor: i.e. John Fell, q.v. 774

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] W2 Letter sent: LONDON Royal Society Early Letters Wl, No. 36, 4 pp. (p. 4 blank) (our source). On p. 1 beneath address in Oldenburg's hand: 'Rec. Febr. 3.' Postmark on p. 1: 'FE/3'. — printed: OLDENBURG, Correspondence IV, 141-2. Reply to: OLDENBURG-WALLis 24.XII.1667/[3.I.1668]. Answered by: OLDENBURG-WALLIS 4/[14].II.1667/8.

(W1) Febr. 1. 1667. 777

The French Probleme if understood of rational! numbers, is to imperfectly expressed. For the demand being onely verum valorem exhibere lineae BY; If meant of rational numbers, it should have been numero ratio(na)le exhibere or at lest numero exhibere. B(ut) I find, my Lord Brouncker hath satisfied778 it in that sense: so (that) I need not spend further thoughts about it. J. Wallis.

(W*)

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Oxford Febr. 1. 1667.

[3]

Sir

Your last779 to niee I have observed, & accordingly have put your Transactions Numb. 16.780 into Latine. Which hath been finished some weekes, & when you please shall be sent. The French Probleme, which you say is to be understood of rational Numbers: was (if so) too imperfectly expressed. For the demand being 4 onely (1) of (2) verum 6 exhibere \exhibere del. or 16 of (1) whole (2) rational 777

French Probleme: i.e. the problem of Dulaurens contained in OLDENBURG-WALLIS 10/[20].XII.1667. 778 my Lord Brouncker hath satisfied: cf. BROUNCKER Solution to Dulaurens's Problem, XII.1667-I.1667/8. 779 Your last: i.e. OLDENBURG-WALLIS 24.XII.1667/[3.I.1668]. 780 Transactions Numb. 16.: i.e. Philosophical Transactions No. 16 (6 August 1666), which contained Wallis's hypothesis of the tides (WALLis-BoYLE 25.IV/[5.V].1666), the appendix to this hypothesis (WALLis-OLDENBURG 18/[28].VII.1666), and Wallis's review of Hobbes's De principiis et ratiocinatione Geometrarum (WALLIS-OLDENBURG 24.VII/[3.VIII].1666).

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] onely verum valorem exhibere lineae BY: if ment of rational numbers; it should have been nume.ro rationali exhibere, or at lest nume.ro exhibere. But since that exposition, I have spent no more thoughts on it; (choosing rather to imploy my first leisure hours on your other task.) But I find, by 5 yours781 lately to Mr Boyle, that My Lord Brounker hath satisfied it in that sense. The Correspondences, which you say are desired, I like very well of. As to the books lately printed, I presume Mr Boyle hath given you an account782 what is to be expected hence. For those skilled in Arabick; we 783 10 have here, (beside Dr Pocok,) Mr Clark one of our Bedles & Mr Hyde784 our Library keeper, (both heretofore imployd in the Biblia Polyglotta785,) Mr Huntington786, a fellow of Merton College; Mr Bernard787 a fellow of St Johns College, & now Procter; Mr Marsh788 a fellow of Exeter College; Mr Smith789 of Magdalene College; & some others. There is elsewhere, 5 that (1) Mr (2) My Lord 9 what is to be expected hence add. 10 Clark |one of our Bedles add] & Mr Hyde |our Library keeper add] , (both (1) sometim breaks off (2) heretofore 781

yours: i.e. OLDENBURG-BOYLE 28.1/[7.II]. 1667/8 (OLDENBURG, Correspondence IV, 120-3). This letter apparently contained Brouncker's solution of Dulaurens's problem, which was destined for and passed on to Wallis by Boyle; see BOYLE-OLDENBURG l/[ll].II.1667/8 (OLDENBURG, Correspondence IV, 139-40, 139), and BROUNCKER Solution to Dulaurens's Problem XII.1667-1.1667/8. 782 account: see BOYLE-OLDENBURG 29.XII.1667/[8.I.1668] (OLDENBURG, Correspondence IV, 93-5, 94-5). Boyle's attempts in Oxford to obtain a catalogue of books printed in England during the previous twelve years were not successful. Oldenburg had requested the catalogue for Carcavi. 783 Mr Clark: i.e. Samuel Clarke. 784 Mr Hyde: i.e. Thomas Hyde. 785 Biblia Polyglotta: i.e. S.S. Biblia Polyglotta. Complectentia textus originales, Hebraicos, cum Pentateucho Samaritano, Chaldaicos, Graecos, versionumque antiquarum Samaritanae, Graecae sept., Chaldaicae, Syriacae, Vulg. Lai., Arabicae, Aethiopicae, Persicae quicquid comparari poterat ... Ex mss. antiquiss. undique conquisitis optimisque ex impressis summa fide collatis, ed. Brian WALTON, 6 vols., London 1655-7. Walton was assisted, among others, by Pocock, Clarke, and Hyde. 786 Mr Huntington: i.e. Robert Huntington (1637-1701), orientalist, B.A. 1658, M.A. 1663 at Merton College; later bishop of Raphoe, DNB. 787 Mr Bernard: i.e. Edward Bernard (1638-96), student of oriental languages and mathematics, B.A. 1659, M.A. 1662 at St John's College, later appointed Savilian professor of astronomy; made proctor of the University of Oxford in 1667. DNB. 788 Mr Marsh: i.e. Narcissus Marsh (1638-1713), M.A. 1660, B.D. 1667, and D.D. 1671 at Exeter College, later archbishop of Armagh, DNB. 789 Mr Smith: i.e. Thomas Smith (1638-1710), B.A. 1661 and M.A. 1663 at the Queen's

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] Mr Nicholas Graves790 (or Edward) brother to John Graves who put out the Persian Grammar & some other bookes in Arabick; & some others whom I know not. The businesse of this, is particularly to desire that some notice be taken of the Spring-tydes which happen at this New-Moon; and the next Full moon; & (if it be not out of memory) what was the last Full-moon for at one of these three times are to bee (according to my hypothesis) the highest Tydes for this time of the year. (Higher than those in March next.) at London & Chatham, as well as on all the shore of Kent. I am Yours &c. J. Wallis. [2]

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To Count Ubaldini's last letter791, I doubt whether I shall give very good account from hence; but I shall not be unmindfull of it.

[1] These

For Mr Henry Oldenburgh in the Palmal near St James's London.

15

2 & some add.\ others 5 at add. 5 next (1) New (2) Full 7-13 for at one . . . unmindfull of it. written at 9(f to the left of first part of letter 1 to bee add. College, fellow of Magdalen College from 1666 onwards. He went to Constantinople as chaplain in 1668. DNB. 790 Mr Nicholas Graves: probably Thomas Greaves D.D. (1612-76), oriental scholar, fellow of Corpus Christi College, Oxford, rector of Dunsby in Lincolnshire, and prebend of Peterborough Cathedral in 1660, DNB. He had two elder brothers: Sir Edward Greaves M.D. (1608-80), physician to Charles II, and John Greaves (1602-52), Savilian Professor of Astronomy from 1643 to 1648. DNB. 791 Count Ubaldini's last letter: now missing. On Ubaldini see WALLis-OLDENBURG 12/[22].II.1666/7.

399

176. WALLIS to OLDENBURG, 8/[18] February 1667/8

175. HENRY OLDENBURG to WALLIS Oxford, 4/[14] February 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 8/[18].II. 1667/8. Reply to: WALLIS-OLDENBURG l/[ll].II.1667/8. Among other things, Oldenburg requested of Wallis that he forward him his translation of No. 16 of the Philosophical Transactions.

176. WALLIS to HENRY OLDENBURG Oxford, 8/[18] February 1667/8 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 38, 2 pp. (our source). Postmark on p. 2: 'FE/10'.—printed: OLDENBURG, Correspondence IV, 159-60. Reply to: OLDENBURG-WALLIS 4/[14].II.1667/8. Answered by: OLDENBURG-WALLIS 11/[21].II. 1667/8. Enclosure: WALLIS Solution to Dulaurens's Problem.

Oxford Febr. 8. 1667./8.

Sir,

5

I have sent you herewith, my solution792 of the French Problem in Rationails. Which doth not disagree from that of my Lord Brouncker793, though it's like his processe (which his paper doth not expresse) and mine, were not the same. I have allso sent you (according to yours794 of Febr. 4,) the Latin 5 processe (1) fe (2) (which 7 Latin translation of Numb. 16. originally deleted, but not replaced 792

solution: i.e. the enclosed Solution to Dulaurens's Problem. that of my Lord Brouncker: cf. BROUNCKER Solution to Dulaurens's Problem XII.1667-I.1667/8. 794 yours: i.e. OLDENBURG-WALLIS 4/[14]. 11.1667 793

400

176. WALLIS to OLDENBURG, 8/[18] February 1667/8 translation of Numb. 16. which went yesterday from hence (with some other things) to Dr Jenkins (Judge of the Admiralty) at Exeter house: whom I have desired to deliver it to you when you shall call for it. It is a smal parcell covered with brown paper & directed with an endorsement, for yourself. As to the Books of Steno, & Borellus795, which you mention: I like the designs well; & hope they may be well performed. As to the other796, I confesse I am suspicious whether it will answere the title (the rather, because the Probleme is so lamely expressed797, that it makes me doubt whether the Author have the dexterity to expresse his notions clearly.) I wish your Building798 good successe; but cannot promise that I shal here be able to get you subscriptions toward it; nor do I presume to deliver my opinion o(f) the design, (knowing so little of it,) but take for granted so many wise men see very good reasons to undertake it, & ways how to get it perfected. I am sorry I sent you not the advertisement of marking spring tides, a fortnight since; (because its possible that the last full moon might be the most considerable;) but as to that of this New moon, I hope it came time inough (because springtydes at London, usually happen 2 or 3 days after the change & full,) and for the next full-Moon, you have sufficient warning. No more at present but that I am Your most affectionate friend & humble servant, Jo: Wallis.

2 house: (1) it is (2) whom 3 it add. 10 successe; (1) though I (2) but 11 here add. 13 good add. 17 moon, (1) & the (2) I 795

Books of Steno, fe Borellus: i.e. STENSEN, Elementorum myologiae specimen, Florence 1667, and BORELLI, De vi percussionis liber, Bologna 1667. 796 the other: almost certainly DULAURENS, Specimina mathematica duobus libris comprehensa ..., Paris 1667. Oldenburg was still awaiting the two copies which Justel had dispatched no later than in December 1667. See OLDENBURG-WALLIS 10/[20].XII.1667. 797 the Probleme is so lamely expressed: cf. WALLIS-OLDENBURG 1/[11].II.1667/8. 798 Building: i.e. a college for the Royal Society, which the Society planned to build with the aid of subscriptions. The form for subscribing contributions was approved at the meeting of the council on 30 January 1667/8; see BIRCH, History of the Royal Society II, 243-4.

401

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177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February 1667/8 In my Lord Brounkers solution799, x is the same with my r — b — c; at d with him, is 2b in mine: and his BY, is my 2y. These For Mr Henry Oldenburgh, 5 at his house in the Palmal near St James's London.

[2]

177. WALLIS: Solution to Dulaurens's Problem Oxford, 8/[18] February 1667/8 Transmission:

W Paper sent: LONDON Royal Society Early Letters Wl, No. 39, 1 p. (our source).— printed: OLDENBURG, Correspondence IV, 160-2 (Latin original), 162-4 (English translation) . Enclosure to: WALLis-OLDENBURG 8/[18].II. 1667/8.

Oxon. Febr. 8. 1667./8.

10

Quod ad Problema ex Gallia propositum: Numero datis Circuli CBD, radio CG, et inscripta BD, extra circulum porrecta ad libitum; ita tamen ut ad ductam CF acta parallela BY, circuli circumferentiam secet in Y: Verum valorem exhibere lineae BY.

1-2 In my .. . my 2y. written at 90° on left margin 799

my Lord Brounkers solution: cf. Brouncker's Solution to Dulaurens's Problem. 402

177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February 1667/8

Solutio mea nupera800 haec fuit; Datis numeris r = CG, 2b = BD, et sumpto / = DF: Erit Quadratum semissis rectae BY.

Hoc est (extracta ubique radice quadratica) Petitur jam, ut BY sit rationalis. Dico 1°. Si id vellet Problema; dicendum fuisset, verum valorem numero rationali exhibere- saltern numero exhibere- non simpliciter, exhibere. 2°. Ut tamen hoc fiat; Expositis r, 6, (ex hypothesi) rationalibus; ita sumendus erit / rationalis (quo sit 262 — r2 + bf rationalis,) ut sit V/ : r2 + f2 + 2bf : rationalis. Hoc est, ut sit r2 + f2 + 2bf quadratus. Hujus autem quadrati radix (propter 6 < r) erit minor quam r + /. Esto r + / — c (sumpto c rational!:) Cujus Quadratus r 2 + / 2 + c2 + 2rf —

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2rc - 2cf = r2 + f2 + 2bf. Adeoque c2 + 2rf - 2rc - 2cf = 2bf. Hoc est,

2rf - 2bf - 2cf = 2rc - c2. Adeoque / =

15

Sumpto itaque c quovis rationali (saltern qui minor sit quam r —6, vel

major quam 2r;) positoque

Erit

CORR. ED.

3 800

Solutio mea nupera: see WALLIS-OLDENBURG 13/[23].XIL1667. 403

by,

177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February 1667/8

5

rationalis. Dico autem, qui minor sit quam r — b, vel major quam 2r: Quoniam, si major sit c quam r — b, erit 2r — 2b — 2c negativus; adeoque, nisi tarn magnus fuerit ut sit etiam Ire — c2 negativus, erit / negativus: Hoc est, Punctum F quod supponitur ultra D, continget citra D; ut in fig. 3 et 4.

Casus Figurae 2ae, (quern non animadvertisse videtur proponens,) tune continget quando BD exponitur minor quam subtensa Quadrantis, sumiturque BF non tanta ut sit arcus GB major quadrante. Atque sic habes plenariam Problematis solutionem. 10

Sin tibi videbitur expedire solutionem meam, (sine vel analyseos processu vel demonstratione,) nudam transmittere: Sic fiat. Datis numeris r = CG, 26 = BD, sumptoque ad libitum / = DF: Erit

BY Sin porro petatur, ut BY sit rationalis;

3 sit c add.\ quam 3 2r 2 - 26 - 2c corr. ed. 9 habes (1) Prob breaks off (2) plenariam 404

178. WALLIS to JENKINS, 11/[21] February 1667/8 Expositis r, b, rationalibus; sumatur c rationalis quilibet qui sit vel ajor qua minor quam r — 6, vel major quam 2r: fiatque / = Eritque rationalis. Jo. Wallis.

178. WALLIS TO LLEWELYN JENKINS Oxford, 11/[21] February 1667/8 Transmission:

W Letter sent: KEW The National Archives PRO SP 29/234, No. 145, 1 f. (originally folded; address on lower half at 180°; verso blank). Endorsement in Jenkins's hand: 'Dr Wallis about our suits with the Town. 10. Febr. 67.' Postmark: 'FE/12'. Answered by: JENKINS-WALLIS 13/[23].II.1667/8.

Oxford Febr. 11. 1667./8.

5

Sir,

I understand from Mr Vicechancellor801 that (at lest not before eleven of the clock) Fish Line's counsell had not moved for the Return of the Habeas Corpus. If they did not afterwards move, it will be yet our concernment (I conceive) that the return be put into the Court, & to move to have the Cause dismissed upon the Return. Which if it may be obtained, will be a good confirmation of the Precedent in Painters case802. I fear their artifice will be to put off the business by delay's; & if this return be not put into Court this Term, all this processe is lost. And if the cause should onely expire without a decision, wee shall have nothing remaining there on record to be a precedent for future cases. You will, I trust, excuse this trouble, from 7 understand (1) tha breaks off (2) from 8 clock) (1) the (2) Fish 12 good (1) fu breaks off (2) confirmation 801

Mr Vicechancellor: i.e. John Fell, q.v. Painters case: on the case of the bailiff Thomas Painter cf. Wallis's Notes relating to the allowance of the claim in the Exchequer enclosed in WALLIS-JENKINS 21/[31].I.1667/8. 802

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179. OLDENBURG to WALLIS, 11/[21] February 1667/8 Sir Yours to serve you Joh: Wallis.

5

These For Dr Lluellin Jenkins, Judge of the Court of Admiralty, at Exeter-house London.

179. HENRY OLDENBURG to WALLIS London, 11/[21] February 1667/8 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 31r-32v (f. 32r blank) (our source). Paper torn at bottom of f. 31, causing loss of (at least) one line of text (f. 31r) and of signature (f. 31V). Postmark on f. 32V: 'FE/11'.—printed: OLDENBURG, Correspondence IV, 172-3. Reply to: WALLis-OLDENBURG 8/[18].II.1667/8.

London Febr. 11. 1667/8. 10 Sir,

15

Mr Collins Joyns with MyLord Brounker and yourself in finding fault with the manner of expressing the French probleme; which perhaps the Author did to embarasse the facility of it. I hope, to see MyLord Brounker on Thursday next, and then He will see your solution and processe. Mean time, I had a letter803 from the Author, telling me, that he proposed it not, as if he Judged the solution of it very difficult, but that from it some fine proprieties of the Cercle might be discoverd, that no man yet had spoken off. And when he shall have seen our solutions, he saith, he will returne his owne and a second, found by a friend804 of his at 17 some (1) g breaks off (2) fine 803letter: without doubt DlILAURENS-OLDENBURG [2J/12.II.1667/8

(OLDENBURG,

Correspondence III, 335-6), which Dulaurens had mistakenly dated 1667, but was endorsed by Oldenburg as received on 10 February 1667/8. 804 friend: not identified. 406

179. OLDENBURG to WALLIS, 11/[21] February 1667/8 Paris, together with the consequences, he deduceth from thence. His new Book805, I find, is already on the way: When it corns, such persons as you will soon discover his abilities. There are now arrived two Exemplars of the New Petrus Blesensis806, to which you were pleased to contribute so much; one, I suppose, is for the publick library of Oxford, the other, for 5 yourself. They are not yet in my hands, though I have seen the packet at [31V] Mr Scots807, with whose {—}| then conveigh it to Oxford, God willing. Our friends here are careless enough in observing the Tydes. I went on Tuesday, Wednesday and Thursday last to Whitehall- and Westminsterstairs, to look the High-water mark of the Spring-tides of those dayes, and 10 understood by the Water-men, that on Tuesday-night and Wednesdaymorning the Spring-tide were between 2. and 3. foot higher than the ordinary Tides, though one of the Water-men pretended, the cause of it to be the then fierce blowing North-westerly wind, and the Landfluds: But on Wednesday night and Thursday-morning the Wind was westerly and 15 calme, and yet the Tydes continued as high as before. I shall endeavor to engage others to observe with me in other places the Tydes at the approaching full Moon, as you direct. I have not yet had leisure to call at Exeter house for your Latin Numb. 16.808 but I heartily thank you for it. 20 809 You will find in my letter to Mr Boyle a relation of a strange Earthquake in Persia, hapned not long since, which produced a Lake

10 High-water mark of (1) those (2) the 12 Spring-tide (1) was (2) were 13 the cause |of add. it to be the then (fierce add] blowing 21 find (.7) by (2) in 805 new Book: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667. 806 New Petrus Blesensis: i.e. PETRUS BLESENSIS, Opera omnia, ed. Pierre de Goussainville, Paris 1667, two copies of which Justel had sent from Paris; cf. JUSTELOLDENBURG [19J/29.II.1667/8 (OLDENBURG, Correspondence IV, 188-90, 188). For Wallis's contribution in collecting information on transcripts of works of Petrus Blesensis in English libraries see WALLIS-OLDENBURG 3/[13] JV.1666 and subsequent letters. 807 Mr Scots: i.e. the intermediary between Oldenburg and Justel. 808 Latin Numb. 16.: i.e. Wallis's Latin translation of Philosophical Transactions No. 16. Wallis had sent his translation to Jenkins, who delivered it to Oldenburg not later than on 13 February; see JENKINS-WALLIS 13/[23].II.1667/8. 809 letter to Mr Boyle: i.e. OLDENBURG-BOYLE ll/[21].II.1667/8 (OLDENBURG, Correspondence IV, 169-71).

407

180. JENKINS to WALLIS, 13/[23] February 1667/[1668] where was none, and dryed up another, that was before, and was full of fish. I am in hast

Sir Your very humble and faithf.

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[32V]

For his much honor'd friend Dr John Wallis Savilian Professor of Geometry in Oxford.

180. LLEWELYN JENKINS to WALLIS [London], 13/[23] February 1667/[1668] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 93r-94v (f. 93V and 94r blank). On f. 94V endorsement in unknown hand: '13 Feb. 1667', and postmark: 'FE/13'. Reply to: WALLIS-JENKINS ll/[21].II.1667/8.

13 Febr. 67.

Sir 15

Your Papers810 have been delivered to Mr Oldenbergh. Mr Serjeant Holloway811 is positive that we ought cannott goe on or doe any act this Terme: he having inform'd himself (as he sayes) by Mr Mandy the Protonotary that the first act or Motion regularly necessarily must be Line's812. What his opinion will be next Terme I know not: sure he must change or else we shall in a mischief otherwise inextricable. But he undertakes

810

Your Papers: i.e. Wallis's Latin translation of Philosophical Transactions No. 16; see OLDENBURG-WALLIS ll/[21].II.1667/8. 811 Holloway: i.e. Charles Holloway. 812 Line's: i.e. Fish Lyne. 408

181. WALLIS to COLLINS, early 1668? to satisfie Mr V. Chancellor813 & all Persons Concern'd when he comes down. I am

Sir Your most humble servant L. Jenkins

[94V] For the Reverend Dr Wallis D.D. at his House neer the Scholes in Oxford.

181.

WALLIS to JOHN COLLINS early 1668? Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-COLLINS 15/[25].II.1667/8. This letter was clearly written after WALLIS-COLLINS 5/[15].II. 1666/7 and before WALLIS-COLLINS 15/[25].II. 1667/8. The manner of reference suggests it had been sent recently.

182. WALLIS to JOHN COLLINS Oxford, 15/[25] February 1667/8 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, 196r-196av (f. 196ar blank) (our source). On right margin off. 196av in unknown hand: 'Dr Wallis de Angulo Contactus'. Postmark on f. 196av: 'FE/17'.—printed: RIGAUD, Correspondence of Scientific Men II, 485-8. Answered by: CoLLiNS-WALLis 25!I/[6!II]. 1667/8. 813

Mr V. Chancellor: i.e. John Fell, q.v.

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182. WALLIS to COLLINS, 15/[25] February 1667/8 Oxford. Febr. 15. 1667./8.

Sir, I have this evening received Leotaud's814 Cyclomathia815 which you sent mee; & have read over that part of his second Book816 which is particularly directed against mee817; and some of the former propositions which by 5 their titles seemed most to concern the business, as the 4 th , 5 th , 6 th , 7th (& some others) with their demonstrations. And I find (as my last818 did presage,) that, (beside the cavilling at some phrases & expressions, which hee would strein to such a sense as to make appear absurd,) there is 10 nothing material in it that seems to need any answere. The whole stress of what hee sayth depends on what he repeats out of Clavius819, that the Angle of Contact, & the Right-lined Angle, are Heterogeneal, & not capable of Proportion; which they would found on 5 Def. 5 Euclid820. Which they would have to be, not a definition of Homogeneal Quantities, (which is the whole drift of that Definition) but of such Homogeneal 15 Quantities as have proportion one to another: Supposing (contrary to the 4th definition,) that, of Homogeneous quantities, some have & some have not proportion to one another. To which I had spoken so fully in my Treatise de Angulo Contactus821, cap. 5, 6, 7, 8, as that the reading of those chapters is Answer inough to all of this kind. Of all which he takes very little or no notice. And if wee should grant that to be the meaning of the definition, as he would have it, (which indeed it is not;) yet this could serve him at the most, but onely as to the Angle of Contact; not as to that of a Semicircle, or other segment of a Circle. For it is manifest, that both a 4 particularly (1) written (2) directed 17 & some |have add. not 22 definition, (1) which hee would have, (2) as 814

Leotaud's: i.e. Vincent Leotaud, q.v. Cyclomathia: i.e. LEOTAUD, Cyclomathia sen multiplex circuli contemplatio tribus libris comprehensa, Lyons 1663. 816 second Book: i.e. 'Liber secundus. Explicatur Anguli contactus natura ex Euclidis, caeterorumque Geometrarum rnente: alienaque de eo aliquot Recentiorum sententia clare ac evidenter refellitur' (157-234). 8lr which . . . mee: cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668. 818 my last: i.e. WALLis-CoLLiNS early 1668 ?. 819 Clavius: i.e. Euclidis elementorum libri XV . . . , ed. CLAVIUS, 2 vols., Rome 1574, vol. I, 154r. 820 5 Def. 5 Euclid: i.e. EUCLID, Elements V, def. 5. 821 De Angulo Contactus: i.e. WALLIS, De angulo contactus, et semicirculi, disquisitio geometrica, Oxford 1656. 815

410

182. WALLIS to COLLINS, 15/[25] February 1667/8 Right Angle may be so multiplied as to exceed any Angle of a segment; & that the Angle of any segment may be so multiplied as to exceed a Right Angle: And therefore these at lest must be such Homogeneal quantities as have proportion one to another, by that definition. Which granted, doth directly overthrow his whole hypothesis; as himself is aware. And though I had urged this, clearly & strongly inough in my Treatise, especially cap. 7, yet of this (because it did pinch too close) hee takes no notice. For all those arguments drawn from the 1 prop. 10 Euclid822. & 2a prop. I1 Archimedis de Sphaera et Cylindro823, (which are the foundation of all those demonstrations whether of the Ancients or of the moderns,) which proceed (as they speak) by way of Exhaustion. (Which do all suppose; that, it is a sufficient proof of Equality, to prove that quantities differ less than by any assignable part.) Hee thinks it a sufficient evasion, to say, that a Right Angle may exceed that of a Semicircle, by such an excesse as is lesse than the Infinitesima pars of either. Which evasion, if it be allowed, will as well elude all those demonstrations by way of Exhaustion, [196V] or| by Inscription & Circumscription of Right-lined & Curve-lined figure, (so frequent amongst both the Ancients & the moderns.) For how easy is it to answer (if this be allowed) to that of Archimedes (for instance) de dimensione circuli; That, a Circle is bigger (or lesse, if you will,) than the Triangle which he proposeth824 as equal to it, but not by any assignable quantity, but by somewhat that is lesse than the Infinitesima pars thereof. And the like to all demonstrations of that kind. To my last Argument, from Opticks; he hath no other evasion, but to

1 any (1) seg breaks off (2) Angle 4 have |(even in his sense) del. proportion 6 especially add. 8 & (.7) 1 prop. 2 (2) 2 a prop. I1 10 all add. 10 ) add. ed.

16 allowed, (1) doth (2) will 16 those (1) Arg breaks off (2) demonstrations 20 (or ... will,) add. 21 not | bigger del] by 24 To (1) that (2) my 822

1 prop. 10 Euclid: i.e. EUCLID, Elements X, prop. 1. de Sphaera et Cylindro: i.e. ARCHIMEDES, De sphaera et cylindro I, prop. 2. 824 proposeth: i.e. ARCHIMEDES, De dimensione circuli, prop. 1. 823

411

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668]

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deny825, that, in speculis curvis, Angulus incidentiae est aequalis angulo reflectionis. Who, I presume, is the first that ever did deny it. To an Argument of Gregories, or Ainscombs, from that of Euclide826, si auferatur plusquam dimidium, atque ex reliquo plusquam dimidium, &c. (1 El. 10.) Hee tells827 (ad prop. 7.) that it is not to be left to the Demonstrator, to make such ablations, by what way he thinks fit; but that his Adversary is to direct, in what way such ablations shall bee made. Which discovers so great a weakness; as if either he did not know what it is to demonstrate; or else meant, of design, to prevaricate. But I suppose, hee might have been ingaged, unawares, in his former Examen828, against Ainscomb, or Gregory; which rather than to retract, hee resolved to defend as well as he might; & was obliged to take mee in by the way. But inough at present, from Sir

your friend & servant John Wallis

15

20

[196av]

For Mr John Collins, Accountant for his Majestic at the Excise Office in Bloomsbury London.

183. WALLIS to VINCENT LEOTAUD Oxford, 17/[27] February 1667/[1668] Transmission:

E1 First edition of (missing) letter sent: WALLIS, A Defense of the Treatise of the Angle of Contact, 79-88 (our source). E2 Second edition: WALLIS, Opera mathematica II, 638-45. 4 auferatur jplusquam add. \ dimidium, 825

deny: i.e. LEOTAUD, Cydomathia, 232-3. that of Euclide: i.e. EUCLID, Elements X, prop. 1. 82r tells: i.e. LEOTAUD, Cydomathia, 177. 828 former Examen: i.e. LEOTAUD, Examen circuli quadraturae hactenus editarum celeberrimae quam Apollonius alter, magno illo Pergaeo non minor Geometra R. P. Gregorius a Sancto Vincentio Societatis lesu, exposuit, 2 parts, Lyons 1653-5. 826

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] This letter, a reply to Leotaud's Cyclomathia, was first sent to Oldenburg in London who then transmitted it to his correspondent in Paris, Henri Justel. Cf. WALLIS, A Defense of the Treatise of the Angle of Contact, 79: 'It was some years after the Book was published, before I saw it: When I had seen it, I wrote a Letter to himself, in Answer to it. Which Mr. Henry Oldenburg (at my request) sent to his Correspondent at Paris, to transmit to Leotaud. This Letter, Mr. Oldenburg told me soon after, that his Correspondent had received at Paris, and would take care of it. But whether or no it came to Leotaud's hands, I am not certain; having since heard nothing of it: And, not long after, I heard that Leotaud was dead.' It would appear that this letter was conveyed by Henri Justel to Leotaud. See JUSTEL-OLDENBURG [25!V]/5.V.1668; OLDENBURG, Correspondence IV, 333-5. Cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668.

Clarissimo Viro D. Vincentio Leotaudo, Delphinati, Joannes Wallis, Oxoniensis, S. Oxoniae, Feb. 17. 1667. Stilo Angliae.

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Incidi, Vir Clarissime, nudius tertius, in Cyclomathiam tuam; ante quinque annos, ut videtur, impressam; sed nuperrime (quantum audio) hue allatam. Quam inspiciendam obtulit Amicus quidam meus830, harum rarum peritus; eo praesertim nomine, quod me ibidem animadverterit a te notatum. Quod fecit ut ad ejusdem Librum Secundum, qui me spectare dicebatur, me statim converterim; omissis Primo & Tertio, quibus de aliis rebus agitur. Quid feci, ut me ibidem in arenam vocaveris, nescio; neque solicitus inquire. Moleste forsan tuleris (utut vos inter vos dissentire non iniquum judicetis) quod contra Clavium vestrum (Jesuitam) ego (non Jesuita) nonnulla scripserim831. Adeoque vestra interesse putaveris, ut ex vestra Societate non-nemo causam ejus (sive justam sive injustam) utcunque defendendam susciperet. Quod tamen non erat necesse ut contra me faceres, qui non soleo Clavii vestri iniquus esse aestimator: Cujus etiam causam (ut & Gregorii San-Vincentiani vestri) contra Meibomium susceperim832; eumque alias, prout res tulerit, passim defendo. Quanquam enim in Re13 fecit E2 829

Cyclomathiam: i.e. LEOTAUD, Cyclomathia sen multiplex circuli contemplatio tribus libris comprehensa, Lyons 1663. 830 Amicus quidam meus: i.e. John Collins. Cf. WALLis-CoLLiNS 15/[25].II.1667/8. 831 scripserim: i.e. WALLIS, De angulo contactus, especially chapter 5-8. 832 susceperim: i.e. WALLIS, Adversus Marci Meibomii, De Proportionibus dialogum, tractatus elenchticus.

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ligionis negotio nos a vobis diversa sentiamus, non proinde necesse erit, ut dissentiamus in Mathematicis: Ubi non Authoritatibus res agenda est; sed, Demonstrationibus. Vel fieri potest, ut controversiis quae Tibi cum Aynscomio tuo intercesserant implicitus, non commode te expedire posse putaveris, nisi & me simul in partes vocaveris; cum ea, quae de Angulo Contactus & Semicirculi scripseram, Gregorii de Sancto Vincentio, Aynscomii, & Tacqueti, placitis quibusdam a te oppugnatis, favere videantur. Quaeque si prius vidisses, fieri fortasse posset, ut inde abstinuisses plane, & ne Aynscomium tuum ea de re solicitasses: (utut ubi temere forsan manum conseruisses, defensionem malueris utcunque moliri, quam videri palinodiam canere.) Quippe quam suscepisti contra Gregorii de S. Vincentio quadraturas controversiam833, non minus feliciter expedire potuisses, licet hanc intactam praeteriisses; quae cum ilia connexa non est: Praesertim, ubi sequiorem partem tuendam susceperis; cum, in ea de quadraturis, potiorem suscepisse videaris. Quanquam enim ego Gregorium San-Vicentianum, pro Mathematico minime imperito habeam; ut qui multa & acute & solide scripserit; (cujusque causam, ut dictum est, contra Meibomium, ne rogatus quidem, susceperim:) Quadraturas tamen absolvisse non existimo. Vel denique (quod potius speraverim) sine partium studio, mero veritatis intuitu, potueris hoc fecisse: Eadem libertate, qua & ego soleo ab aliis nonnunquam in paucis discedere, quos alias defendendos existimo: Praesertim cum te a probris, ut plurimum, abstinuisse videam. Quicquid sit quae contra me multis scripseris, paucis diluenda visum est: Neque enim prolixa Refutatione opus erit; sed potius brevibus stricturis. Quippe in angustum res redacta est: Num scilicet Angulus Contingentiae sive Contactus, ad Angulum Rectilineum (aut etiam ad alios Curvilineos & Mixtos) comparatus, pro Homogeneo habendus sit, an pro Heterogeneo (saltern quoad Rationem) & nullius Rationis capaci. Quippe in hoc unico, & Tu & Clavius Praesidium collocatis, & speratis Asylum; contra ea in contrarium prolata Argumenta, quae alias (ne vobis quidem diffitentibus) Demonstrativa habenda erunt. Si enim & Quantus fuerit (quod vos vultis) & Angulo Rectilineo Homogeneus; non erit quod Absurda ilia possitis declinare, quae non modo Pefetarras834 833

controversiam: i.e. LEOTAUD, Examen circuli quadraturae hactenus editarum celeberrimae quam Apollonius alter, magno illo Pergaeo non minor Geometra R. P. Gregorius a Sancto Vinventio Societatis lesu, exposuit, Lyons 1653. S3 *Peletarius: i.e. Jacques Peletier (1517-82). The arguments he employed in his con-

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] & Ego, (ne Savilium porro & Vietam memorem) sed & Vestrates SanVincentianus, Aynscomius, & Tacquetus, cumulate vobis objiciunt. Quam quidem rem, quanquam in meo De Angulo Contactus & Semicirculi tractatu, cap. 5, 6, 7, &c. me satis confecisse puto, ni praejudicium tibi oculos perstrinxisset: Quoniam tamen tu illud nondum assequi videri velis, te sequar, ut quae superest caligo, si fieri possit, etiam a tuis oculis discutiatur. Propositionem tuam primam835 ego hactenus concede: Nempe, Quo jure quis Quantitatem infinite extensam Imaginari velit, eodem & infinite diminutam imaginandam permittere debeat. Idque utroque sensu, quo Infiniti vox solet occurrere. Si enim, per Infinitum, intelligatur Indefinitum, seu quantumlibet magnum; quo sensu, apud Geometras, Recta Infinita hoc est, quantumlibet longa, vel quantum opus est longa, vel ducta supponitur, vel ducenda praescribitur: Quo jure Rectam quantumlibet Longam possibilem esse ponimus, eodem & quantumlibet Brevem possibilem esse, concedendum erit. Quippe prout supponitur Continuum posse in Infinitum continuari, ita & in Infinitum dividi; hoc est, nullos vel Continuationis vel Divisionis states esse terminos, ultra quos procedi sit impossibile. Si vero, per Infinitum, intelligatur id quod sit Absolute Infinitum Actu; (puta quod totam possibilitatem habeat in actum reductam:) Etiam hie concede, quo jure quis, hoc sensu, imaginari velit Infinite-Magnum, etiam Infinite-Parvum Imaginandum esse. Sed, Imaginandum potius dico, quam Datum iri. Ad Secundam Propositionem836 quod spectat; Concede, Infiniti ad Finitum, nullam esse [Finitam] Rationem: (Neque etiam Indefiniti ad Definitum rationem Definitam.) Dico tamen: Quo jure quis Quantum Infinitum imaginari velit, eodem & Infinitam Rationem imaginandam esse. Adeoque, Infiniti ad Finitum, aut etiam Finiti ad Infinite-exiguum, rationem esse dico Infinite-magnam: Finiti ad Infinitum, vel etiam Infiniteexigui ad Finitum, rationem Infinite- exiguam. Neque tibi in contrarium suppetias feret quinta Definitio quinti Euclidis837: Dicam utique, Infiniteexigui Infinite-Multiplum, expositum quodvis Finitum aequare posse, nedum superare. Atque ego pari jure admittendum postulabo Infinite-Mulflict with Clavius over the horn angle are quoted in CLAVIUS (ed.), Euclidis elementorum libri XV, 3rd ed., Cologne 1591, 133-45. 835 Propositionem tuam primam: i.e. LEOTAUD, Cyclomathia, 157-61. 836 Secundam Propositionem: i.e. LEOTAUD, Cyclomathia, 161-2. 83r quinta . . . Euclidis: i.e. EUCLID, Elements V, def. 5.

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tiplum, quo tu vel Infinite-Magnum vel Infinite-exiguum. Dum autem quaeris838, num Rationalis futura sit haec Ratio, an Irrationalis: Utrumvis dicas, perinde est: (quippe, quodcunque fuerit hoc Infinitum, quod ad Finitum unum habiturum est Rationem Rationalem; idem ad Finitum aliud, Rationem habebit Irrationalem; modo ilia Finita duo, Rationem habeant ad invicem Irrationalem:) Hujusmodi siquidem minutias omnes, absorbet ipsa Infinitude. Atque perinde est acsi peteres, si quis imaginari velit Numerum Infinitum, num futurus sit ille Par an Impar? Vel Tripartibilis, an secus? &c. At interim, quo jure tu vel Infinite-Magnum, vel Infinite-|Parvum imaginaberis; eodem ille: Vel [81] Infinite-Multum, vel etiam Infinite-Paucum (ipsa Unitate in Fractiones divisa) imaginabitur. Ad Prop. 3.839 Concede, Quantitates etiam Finitas esse, quarum nulla potest esse ad invicem Ratio. Tales utique sunt quantitates quae sunt ad invicem Heterogeneae, puta, Linea & Planum, Planum & Solidum, Linea & Solidum: item Angulus & Linea, Angulus & Superficies, Linea & Tempus, Tempus & Pondus, atque hujusmodi alia cum Heterogeneis comparata. Est utique Ratio (per def. 3. El. 5.840) Homogeneorum ea relatio quae est Kara Tr^AtKorr/ro;. Homogenea vero, seu (quod per def. 3. tantundem valet) Rationem invicem habentia, sunt (per def. 5.) ea quae Multiplicata possunt se mutuo superare. Quoniam itaque Hora Temporis, utcunque Multiplicata, nunquam aequabit vel superabit Libram Ponderis; Hora & Libra, seu Tempus & Pondus, Heterogenea censenda erunt, adeoque non Rationem ad invicem habentia. Atque de reliquis similiter. Estque haec unica Homogeneorum definitio, quae apud Euclidem uspiam extat. Hinc discas, Curvam quamvis & Rectam, utut Dissimiles, Homogeneas tamen esse; quoniam exposita Curva quaevis ita Multiplicari potest, ut expositam quamvis Rectam superet; & vice versa. Sic Curvilineum & Rectilineum; puta Circulum & Quadratum: Expositus utique Circulus si exposito Quadrato nondum major sit, erit saltern ipsius Duplum, Triplum, vel aliud aliquod Multiplum, quadrato illo majus; & vice versa. Lineam vero & Superficiem Heterogeneas esse; quoniam Linea, cum nihil habeat Latitudinis, quantumvis Multiplicata, necdum habebit; (quippe nihili Duplum, seu alias Multiplum, est adhuc nihil;) adeoque nee fiet Superficies. 838

quaeris: see LEOTAUD, Cyclomathia, 161. Prop. 3.: i.e. LEOTAUD, Cyclomathia, 162-5. 840 def. 3. El. 5.: i.e. EUCLID, Elements V, def. 3. 839

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Atque hinc speciatim discas, Angulos Pianos omnes, sive sint Rectilinei, sive Curvilinei, sive Misti, (qui ullius sint Magnitudinis,) invicem Homogeneos esse. Sunt utique vel Aequales, vel Majores, vel Minores exposito Rectilineo; & quidem si Minores, possunt saltern Multiplicati Majores fieri; & vice versa. (Quod ne tu quidem de quovis negaveris, excepto solo Angulo Contactus.) Cumque tu hoc in Angulo Contactus desideratum animadvertis; id non eo fit quod Heterogeneus sit, sed quod non sit Quantus. Miror autem ego te, hominem Mathematicum, existimare posse, turn totum Angulum Rectum, turn (quern hujus partem esse vis) Angulum Semicirculi, cuivis Recto Homogeneum (per 5 def. 5.841) reliquum vero quern facis Angulum Contactus Heterogeneum esse. Quasi quidem fieri possit, ut & Totum, & Ablatum, sed non & Reliquum, eidem alicui sit Homogeneum. Cum vero tu existimas, Quantitatum invicem Homogenearum, alias habere, alias non habere, rationem ad invicem; atque has ab illis Euclidem definitione 5. disterminasse: Hoc ipsum est quod & Clavio primum, atque post ilium Tibi, aliisque multis fraudi fuit. Euclides utique jam ante (def. 3.) turn Rationes omnes, Homogeneorum esse; turn & Homogeneorum omnium, esse ad invicem Rationem; non minus definiverat. Quoniam vero Homogenei vocem, non prius ab illo usurpatam, nedum in superioribus definitam, sed quae definitione omnino indigeret, hie usurpaverit; hanc alteram Homogeneorum, seu (quod ipsi est Iao5vva^tov} Rationem ad invicem habentium, definitionem subjungit. (Quae & in Graecis Codici[82] bus immediate subjungitur Tertiae; cur autem Clavius hanc| Quintam fecerit, interposita Quarta quae est in Graecis Octava, ego nescio.) Euclides itaque, neque ob eum finem quern tu insinuas, neque frustra tamen, sed justis de causis hanc quintam interposuit definitionem. Nempe, ut quid per Homogenea seu Rationem invicem habentia, significatum velit, definiret. Sed & per hanc ipsam Definitionem, & per 1 Prop. 10.842 determinat; Homogenei cujusvis nullam esse posse tarn exiguam partem, quae Multiplicata non possit totum superare. Quae itaque tu hie ex Aynscomio843 habes844, Suppono, inter duas 23 ipse corr. ed. 841

5 def. 5.: i.e. EUCLID, Elements V, def. 5. 1 Prop. 10.: i.e. EUCLID, Elements X, prop. 1. 843 ex Ayncomio: i.e. AYNSCOM, Expositio ac deductio geometrica, 5. 844 habes: see LEOTAUD, Cyclomathia, 163. 842

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Magnitudines esse aliquam Rationem, idem esse ac, duas Magnitudines secundum quantitatem (Kara rKr\\i^OTr\Ta) posse comparari: Rursum, illas Magnitudines sic posse comparari, de quibus did potest, haec major aut minor est ilia: Unde consequens est, Magnitudines illas, juxta de/. 5 5. lib. 5. Eucl.8^5 ejusmodi esse, ut una aliquoties sumpta possit alteram aequare vel superare: Adeo sunt & veritati & Euclidis menti consona, ut quo te vertas cunque nunquam sis evasurus. Propositionem quartam846, (quae est Euclidis Definitio Anguli Plani) ego admitto. Quod, Angulus Planus, est duarum linearum in piano se 10 mutuo tangentium, & non in directum jacentium, alterius ad alteram inclinatio. Adeoque haec saltern tria requiri judico, Quod in Piano lineae se mutuo tangant, Quod ad invicem inclinentur, Quodque non in directum jaceant: Et propterea, si vel in directum jaceant, (ut cum una sit alterius 15 continuatio,) vel nulla sit ad invicem inclinatio, (ut in Parallelismo, & cum altera alteri immergitur;) Angulum vel nullum fieri, vel nullius Magnitudinis: Sed &; horum alterum contingere, quoties ita concurrunt lineae, ut, licet continuentur, se mutuo non secabunt: Et propterea Contactus Angulum nullius esse Magnitudinis.

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Verbi gratia: Parallelae rectae A0, CD, Angulum non constituunt, turn quia nondum se mutuo tangunt, turn quia nulla est alterius ad alteram inclinatio, sed Parallelismus. Si vero AB, retento Parallelismo, deorsum ferri intelligatur donee ipsi CD occurrat: Tactus quidem net, sed non Angulus; propter nullam alterius ad alteram inclinationem: Nee, utcunque continuatae, se mutuo secabunt, sed altera alteri immergetur. 845

def. 5. lib. 5. End.: i.e. EUCLID, Elements V, def. 5. Propositionem quaxtam: i.e. LEOTAUD, Cyclomathia, 165-7.

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Sin, eodem retento Parallelismo, transferri intelligatur AB in situm DE- Tactus etiam sic net, non autem Angulus, cum altera sit alterius continuatio. Eadem vero AB, & (non Parallela) FG, invicem inclinantur; sed Angulum non constituunt, quia nondum se mutuo tangunt. Sin, eadem retenta inclinatione, sursum moveatur AB vel FG deorsum, donee invicem occurrant; Angulus net: (erit utique linearum occurrentium, nee in directum positarum, Inclinatio.) Sed &, propter illam ad invicem inclinationem, si continuentur se mutuo secabunt. In lineis Curvis, cum Curva Rectae non possit congruere, eadem [83] tamen Analogia accommodanda erit.j

Rectae HIK, subsit LMN Semicirculus; cujus supremum punctum, rectaeque HIK proximum, sit M: Manifestum est, turn varias Peripheriae partes varium respectu ipsius HIK rectae situm habere, turn quae propius sunt ad M propius ad Parallelismum accedere; ej usque propterea in ipso M situm, Parallelism! instar habendum; rectamque HIK, utpote huic situi Parallelam, si retento hoc Parallelismo deorsum ferri intelligamus, donee in M peripheriae occurrat; non tarn secabit ilia peripheriam, aut ad illam inclinabitur, quam super ipsum M punctum aKXivuq jacebit, Angulum vel nullum vel nullius Magnitudinis efficiens, (pariter atque AB recta ad rectam CD demissa,) propter nullam utrobique inclinationem. (Quern nullius Magnitudinis Angulum, Angulum Contactus dicunt,) Si vero ulterius adhuc demittatur eadem H IK recta; in binis semper punctis, 23 binis subinde punctis E 419

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(sed ubi alius est Peripheriae situs, ad illam rectam, quam in M fuerat) secabit; Angulos faciens Rectilineis vel aequales vel proportionales. Miraris autem tu, (pag. 209.) Tantae apud Me Authoritatis esse Peletarium, ut cum eo ausim affirmare, [eas solas Lineas indinari in puncto concursus, quae, si producantur, se mutuo secabunt,] quam cum Euclide sentire [duas quaslibet lineas quomodocunque concurrentes, mutuo indinari; sive, quod idem est, Angulum constituere.] Ego vero Peletarii Authoritate non moveor, (ut neque Clavii,) sed Argumentis, & rei necessitate. Miror autem ego te existimare posse847, Eudidem sentire, duas quaslibet lineas quomodocunque concurrentes mutuo indinari, seu (ut ais) quod idem est Angulum constituere. Ego (cum Euclide) duos casus excipio: Immersionem & Continuationem; (propter nullam utrobique Inclinationem, sed potius Parallelismum.) Si feratur AB in situm CD, non Angulum cum hac faciet (saltern nullius Magnitudinis,) sed Immersionem: Si ad situm DE; non Angulum, sed Continuationem: Si ad situm his intermedium; partim immergetur, partim continuabit ipsam CD: Angulum certe non constituet (saltern non ullius Magnitudinis) in sensu Euclidis, (qui non per Tangentium tactum, sed per Tangentium inclinationem, definit Angulum) cum nulla sit concurrentium Inclinatio. Tu si secus sentias, fruere tuo sensu. Sed & te male habet, (pag. 166.) quod dixerim ego, Recentiorum aliquot magnos viros, & ex veteribus fortasse nonnullos, de Angulo Contactus ita locutos esse, ac si haberet Anguli quantitatem; non dixerim Omnes. Atque exclamas, Notum esto ac definitum, Omnes Geometras turn Antiquos turn Recentiores, veritatis & observantiae gratia, Euclidis sententiae subscripsisse. Festina lente, Vir Clarissime; (quippe haec aXa^ovsiav potius sonant atque jactantiam, quam Mathematical!! Demonstrationem.) Tune Omnia Omnium, turn Antiquorum turn Recentiorum, scripta legisti? Omniaque ibidem lecta, turn animadvertisti probe, turn probe meministi? Ego certe, qui nee Omnia Legi, nee Lectorum omnia Memini, cautius loqui soleo. Sed nee observantiae gratia, sed Veritatis & Demonstrationum, soleo Geometris subscribere. Tu forte Clavio, observantiae gratia; ego Peletario, Veritatis tantum gratia subscribe. [84] 15 ad sit am DE corr. ed. 29 Tune corr. ed. 847

Miror . . . posse: see LEOTAUD, Cydomathia, 209. 420

183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Quaenam autem fuerit Euclidis sententia, nondum inter te & me convenit. Ego Euclidem saltern, & Apollonium, ex meis partibus stare existimo. Demonstrant utique Angulum Contactus (ille ad Circulos, hie ad Sectiones Conicas) saltern Minorem esse quam infinite exiguum, (neque uspiam dicunt, aliquid habere Magnitudinis.) Quod autem tale est, ego quidem (Euclidis authoritate fretus, Def. 5. El. 5. & Prop. 1. El. 10.) Non- quantum esse existimo. Verum quidem est, Euclidem non totidem verbis pro me pronunciasse: Nequid tamen in contrarium dicat (nedum demonstret) caute abstinet, turn ad Prop. 16. turn ad Prop. 31. Lib. 3.848 (quod Capite 2. ostenderam.) Atque Argumentis aliunde ex eo petitis, ad meas partes trahetur. Caeterique Graeci (quantum scio) omnes, uno excepto, vel de hoc negotio plane tacent, vel ita caute pronunciant, ut meis potius partibus favere videantur. Tu si uno illo plures ex Graecis noveris, (quos ego vel non legi, vel non animadvert!, vel non memini) qui Angulum Contactus positive-quantum esse, aperte dixerint: Opitulare, quaeso, nescientiae meae; mihique benignus indica. Fieri quidem potest ut plures sint, (ideoque dixi, fortasse nonnullos:) Ego, praeter unicum, neminem novi. Ex Recentioribus Latinis, plures agnosco tecum sentire: Non Omnes tamen. Quippe praeter Peletarium (quern mihi, credo, concedes) Tres saltern ex tuis, San- Vincentianus84^9, Aynscomius850, atque Tacquetus851, (quos ut magnos viros praedicas) secus atque tu sentiunt: Atque ex aliis, SAVILIUS852 saltern & VIETA853 (Viri certe tuis non minores) quibus & Flussatam854: addas (nobilem Euclidis Interpretem.) Qui quamvis non eodem modo se omnes expediant, a Te tamen Omnes diversa sentiunt (atque a Clavio tuo.) Atque in hoc saltern omnes consentiunt mecum; Quod impossibile est, ut & Angulus contactus positivam habeat Anguli 848

Prop. 16. . . . Lib. 3.: i.e. EUCLID, Elements III, prop. 16 and 31. San-Vincentianus: i.e. ST-VINCENT, Opus geometricum, lib. 8, 870-1. 850 Aynscomius: i.e. AYNSCOM, Expositio ac deductio geometrica, cap. 2, 5-13. 851 Tacquetus: i.e. TACQUET, Elementa geometriae planae ac solidae. Quibus accedunt selecta ex Archimede theoremata, Antwerp 1665, 86-93. 852 SAVILIUS: i.e. SAVILE, Praelectiones ire.sde.cim in principium Elementorum Eudidis, Oxford 1621, 54-5. 853 VIETA: i.e. VIETE, Variorum de rebus mathematicis responsorum, liber VIII, Tours 1593, cap. 13; Opera, mathematica, 386-7. 864: Flussatam: i.e. FRANCiscus FLUSSATES CANDALLA (Francois de Foix, Comte de Candale), Elementa geometrica: Libris XV; ad germanam geometriae intelligentium e diversis lapsibus temporis iniuria contractis restituta . . ., Paris 1566. 849

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Magnitudinem, quae tamen utcunque Multiplicata nunquam vel minimum aequet rectilineum, nedum excedat; & simul constent Def. 5. Lib. 5. & Prop. 1. Lib. 10. cum Def. 3. Lib. 5. Euclidis. 5

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Sed & Catoptricos, ad unum omnes, existimo ex meis partibus esse. Quippe qui & uno ore consentire videntur, Angulum Incidentiae & Angulum Reflectionis aequales esse, turn qui ad Speculum fiunt, turn qui ad Planum Tangens. Quod fieri non potest, nisi vel Anguli Contactus nullius sint Magnitudinis, vel semper aequales: Quorum utrumvis, in speculis Parabolicis, Ellipticis, Hyperbolicis, & (praeter Sphaerica) curvis omnibus, tibi pariter adversantur, atque pro me concludunt. Dum vero tu (Pag. 232.) negas, in hujusmodi Curvis speculis Angulum Reflexionis Angulo Incidentiae aequalem esse: Tu certe primus es qui hoc dixeris, nee eris audiendus. Ubi vero tu Euclidem existimas (Pag. 164.) ne syllabam quidem perperam tradidisse: Sed nee Libraries quicquam vel addidisse vel immutasse, (quod Tacquetum subdubitasse dicis, Pag. 200.) sed erga hunc Geometriae parentem observantiores semper fuisse, quam ut ejus opus tarn absolutum quoquo modo temerare non vererentur, (Pag. 204.) Nae tu homo credulus es atque EvrrsiOr/s, qui haec sentis! (quod itaque in Theologicis minus mirabor.) Videris tu certe, Euclidis Codices Manuscriptos nunquam vidisse; (quorum vix duos reperias, qui non ab invicem multum different) sed neque Graecum editum; Cujus editor saepius innuit, turn Codices suos variasse (ut ne de ordine vel numero propositionum semper consentiant) turn se nonnunquam, praeter omnium quos habuit Codicum fidem, nonnulla immutasse. Ego certe Euclidem, siquis alius, maxime veneror, (nee apud eum quicquam scio cui non assentior; tantum abest ut me neglecti Euclidis insimulasse debeas.)| Agnosco utique Celebrem Geometram; sed & Homi- [85] nem, nee •deb'Kvevaov. Quod vero ne syllabam posuerit ipse, quae possit in melius mutari; neque librariorum vel incuria vel audacia mutatum quicquam: Rhetorice forsan dici poterit; certe non Geometrice. Quippe ego non pauca, (in libris quos habemus) & omissa, & addita, & loco mota, nullus dubito. Et quanquam mihi non necesse sit, ad rem praesentem, ut haec dicam; cum nihil apud Euclidem occurrat quod mihi adversatur, (nisi tu Clavii paraphrasin & additamenta pro Euclide habeas:) Tua tamen vel maxime interest haec dicere. Nisi enim & 5. Def. 5. & 1. Prop. 10. obliterentur, tua constare non poterunt. 422

183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Ad Prop. 5.855 Concedo tibi, Angulo competere Quantitatum Affectiones: Sed & ego tibi permitto, ut Quantitatem rotunde appelles. Quo verbo nunc vulgo dicimus, quod Euclides p,£j£'&os dixerit. Cum enim Anguli sint ad invicem Rationum capaces; etiam nejefir] dicendi erunt, per 3. Def. 5. El. Ad Prop. 6.856 Ego tibi non concedo, eum Angulum majorem statim esse, cujus crura, post aliquam a puncto concursus distantiam, magis divaricantur: (manifestum utique est, Acutum rectilineum, minorem sic futurum Angulo Contactus:) Nisi, retenta ea quae in concursu fuerat directione utriusque, idem fiet. Rationes ego in meis Cap. 3. & 4. attuli: Nee opus est ut hie repetam; cum tu nihil hie affers quod eorum vires imminuat. Adeoque & Prop. 7.857 Ut falsam rejicio: Tuisque Gregorioh Aynscomiohactenus saltern assentior, ut impossibile dicam Angulo Contactus positivam Magnitudinem concessum iri, quin Geometrica Principia destruantur. (ut autem aliis Angulis vera Magnitude concedatur, nihil impedit.) Eorumque Argumenti aciem (quo probant, Semicirculorum omnium Angulos aequales esse, adeoque & Contactus Angulos vel saltern aequales esse vel potius nullius Magnitudinis) tu nullis viribus obtundes. Quod utique tu opponis (Pag. 177.) In exhaustionibus, (ubi plus quam dimidium aufertur, atque ex residue plus quam dimidium, atque sic deinceps) subtractiones illas, non pro suo Demonstratoris arbitrio, sed arbitrio Adversarii iniri debere: Ridiculum est sophisma. Quotusquisque (quaeso) est, ex Demonstratoribus per exhaustiones, qui Adversarium consulit, quo pacto velit ille ablationes fieri? Num Archimedes, in Dimensione Circuit vel in Quadratura Parabolael vel de Sphaera & Cylindrol vel uspiam alibi, hoc facit? Num Euclidesl Num quispiam alius, seu veterum seu recentiorum? Apage has ineptias! Consule tu primam decimi Euclidis, & discas inde exhaustiones ineundi methodum. Prop. 8.858 Duae Magnitudines inaequales, quarum discrimen tale est, ut quantumlibet Multiplicatum neutram possit superare vel adaequare; nullam inter se rationem habere possunt: si pro nullam inter se rationem habere possunt; dixisses, sunt impossibiles; vera fuisset propositio; quam demonstrasses ex Prop. 1. El. 10. Sed prout tu illam enuncias, absurda 10 fiat E2 855

Prop. Prop. 857 Prop. 858 Prop. 856

5.: 6.: 7.: 8.:

i.e. LEOTAUD, i.e. LEOTAUD, i.e. LEOTAUD, i.e. LEOTAUD,

Cyclomathia, Cyclomathia, Cyclomathia, Cyclomathia,

167-8. 168-73. 174-7. 178-84.

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est, & sui destructiva. Quippe quae Magnitudines inaequales sunt, Rationem habent. Ipsa enim inaequalitas est ratio. Ipsaque illarum Differentia (quod tu Discrimen vocas859) qua altera alteram superat; Homogeneas esse indicat. Heterogenea quippe inter se non comparantur: Vel die tu, si possis, quo excessu, Hora Temporis, superat Libram Ponderis?| [86] Neque aliud supponit Euclides, (Prop. 1. El. 10.) quam ut Magnitudines sint inaequales, quo affirmet, turn ipsas, turn earum per continuam subductionem ortas Differentias, ita Multiplicari posse ut utramvis superent. Quippe si inaequales sint, Rationem habent; adeoque per 3. Def. 5. sunt Homogeneae, (turn ipsae quidem, turn partes suae) adeoque poterit utriusvis quaelibet particula sic Multiplicari, ut reliquam superet, per 5. Def. 5. Dico, per 5. Def. 5. Non per postulatum lib. 10. Quippe quod tu memoras libri decimi postulatum, non Euclidis est, sed Clavii, postulatum: Et quidem plane superfluum. Continetur utique in 5. Def. 5. Et non nisi ob hanc definitionem perperam intellectam, a Clavio insertum. Dum vero tu Propositionem, ut a te propositam, in lineis demonstrare satagis, operam ludis. Impossibile utique est, ut sit lineae pars aliqua (nisi tu Punctum vis esse Partem lineae) quae vel ad totam vel ad reliquam (Homogenea ad Homogeneam) non habeat Rationem, vel etiam tantilla sit, ut non possit multiplicata totam superare: per Prop. 1. El. 10. vel 5. Def. 5. Quodque tu San- Vincentianum& Aynscomiumsibi persuasum habere dicis, Duas quaslibet Magnitudines, quibus competit inter se comparari secundum majus & minus, eo ipso rationem aliquam inter se habere, adeoque debere, per Def. 5. se mutuo superare si saepius repetantur: Omnino verum est. Quodque tu in contrarium profers, nullius est momenti. Illud speciatim quod habes, de Homogeneis quoad quantitatem, sed non quoad rationem; haberi forsan possit inter Sophistarum A070'paxi&S acuta distinctio, (ubi verbis tantum agitur) sed non in Geometrarum schola; ubi non nuda vocabula, sed rerum pondera & demonstrationes spectantur. Nam eo ipso quod sint, quoad quantitatem Homogenea, ra9 turn |ut del. ed.\ earum 10 utrumvis corr. ed. 23 El add. ed. 859

vocas: see LEOTAUD, Cydomathia, 179.

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] tionem habent, per 3. Def. 5. Item inaequalia esse, nee tamen Rationem habere, est contradictio in terminis. (Nisi quo sensu nihil & aliquid sunt inaequalia.) Item Datis Magnitudinibus, datur eorum Ratio; Dataque Ratione Totius ad Partem suam, datur ejusdem & ad Reliquam Ratio; per 1. & 5. Datorum Euclidis Item Angulus Rectus ad Angulum Semicirculi, etiam te judice, rationem habere debet, (per 5. Def. 5.) quoniam utervis ita multiplicari potest ut reliquum superet: Sed & per tuam hanc, Prop. 8. Rationem non haberet, utpote quorum differentia (quam tu facis Angulum Contactus) non potest ita multiplicari ut utrumvis superet. Habebit igitur, & non habebit: Quod est Absurdum. Prop. 9.860 Vera est, Si A ad B rationem habeat, atque B ad (7, etiam A ad (7, rationem habebit, sed & ad B + C. (sunt utique omnes Homogeneae) Sed mihi non officit. Prop. 10.861 Anguli Segmentorum similium nullam inter se rationem habere possunt: Falsa est. Sunt utique Aequales. Quod quidem, in Circulis Aequalibus; ipse fateberis. Ego etiam in Circulis inaequalibus affirmo: Nee tu potis eris refutare. (Propositiones utique praecedentes aliquot unde hoc infers, nihili sunt.) Sed & possunt multiplicati se mutuo superare: Ergo [87] rationem habent; per 5. Def. 5. Prop. II.862 Anguli duorum segmentorum inaequalium ejusdem Circuli; & segmentorum dissimilium in Circulis diversis; rationem inter se habere non possunt: Falsa est. Possunt utique multiplicati se mutuo superare: Ergo Rationem habent; per 5. Def. 5. Demonstratio tua nihili est; quia futilibus superstruitur. Prop. 12. & 13.863 Verae sunt: Sed mihi non officiunt. Prop. 14.864 Quae Definitio est: & Prop. 15.865 Quae illi accommodatur: satis inter se conveniunt: Sed non, cum aliorum loquendi formulis. Sed mihi non officiunt. Prop. 16.866 Nullus Angulus diversae speciei lineis comprehensus, ad

860

Prop. Prop. 862 Prop. 863 Prop. 864 Prop. 865 Prop. 866 Prop. 861

9.: i.e. LEOTAUD, Cyclomathia, 184-5. 10.: i.e. LEOTAUD, Cyclomathia, 185-6. 11.: i.e. LEOTAUD, Cyclomathia, 187-8. 12. fe 13.: i.e. LEOTAUD, Cyclomathia, 188-9 and 189-91. 14.: i.e. LEOTAUD, Cyclomathia, 191-3. 15.: i.e. LEOTAUD, Cyclomathia, 193-7. 16.: i.e. LEOTAUD, Cyclomathia, 197-234.

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668]

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alium quemvis Angulum, Rationem ullam habere potest: Omnino falsa est: Turn quia futilibus superstruitur; turn propter 5. Def. 5. Certum utique est, ita multiplicari posse utrumvis comparatorum, ut reliquum superet. Vides itaque, quam ampla seges propositionum falsarum, (etiam contra 5. Def. 5. tuo sensu intellectam) ex infelici tuo lolio pullulaverit. Tuo, inquam: Quamquam enim Clavius tibi in aliquibus praeiverit; non tamen sustinuit ille tot monstra proferre. Die igitur in posterum, quod omnino dicendum est; Anguli Contactus Magnitudinem nullam esse: atque videbis haec omnia monstra protinus disparere, omniaque in Geometria belle convenire. Vel si tu id malis, die esse Minorem quam infinite-exiguum; utpote minimo possibili Rectilineo minorem; (quod ab Euclide demonstratum esse, ne tu negaveris) quod mini perinde satisfaciet. Quippe, quod demonstratum est, minus esse quam infinite-exiguum, haberi solet pro nonquanto, unde tota Exhaustionum doctrina pendet. Vel etiam, (quo tibi maxime faveam) si circulum haberi vis pro Polygono Rectilineo laterum numero-infinitorum; & Tangentem, pro recta per Polygoni Angulum transeunte, rectae ab ejus Centro Perpendiculari: Die Angulum Contactus esse, Infinitesimam partem duorum Rectorum: (seu -^R-} Quippe tantus erit uterque Angulus externus contactu illo factus; per Calculum a me, Cap. 12. institutum. (Quo tamen minor esse debet, certe non major, Angulus Contactus Circuli.) Verum si tu hoc dixeris; dicendum etiam erit, Peripheriam Circuli non habendam pro una linea in directum continuata, (prout tu, Pag. 221.) sed, totidem Angulorum esse quot est Laterum: Hoc est, in quovis Peripheriae puncto Angulum constitui, aequalem duobus rectis dempta infinitesima parte quatuor rectorum, (vel 2R — ^R.) Quippe tantus erit quilibet Angulus istius Polygoni. Sin tu velis (ut Pag. 221.) ut haec Tra/j.TroXvjui'ia, in Peripheria, evanescere censeatur in Non-Angulum, sed continuam ejusdem lineae directionem, (pariter atque cum duo crura Anguli Rectilinei explicata, cessante Angulo, fiant continua recta:) pariter censendus erit externus ille Contactus Angulus, quasi complicatis cruribus, etiam in non-angulum, seu Angulum nullius magnitudinis, transire. Dumque Peripheria pro una continuata linea censeatur; censendus erit Angulus Contactus pro non1 ullum corr. ed. 2 superstuitur corr. ed. 3 utramvis corr. ed. 32 fiunt E'2

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] angulo. Argumenta mea non repeto; (ex TVactatu meo de Angulo Contactus petenda) ut quae adhuc inconcussa manent, nee opus habeant ut denuo statuminentur. Nam (praeter quasdam verborum captiunculas, quae nullius momenti sunt, neque responsionem merentur) totum illud, quod tu contra eorum aliqua (omissis reliquis) movere satagis, hoc unico nititur fundamento, Quod (ex 5. Def. 5. perperam intellecta) existimes, Magnitudinum invicem Homogenearum (etiam finitarum) alias habere, alias [88] non habere, rationem ad\ invicem. Quod quidem fundamentum, cum in praecedentibus subversum sit, plurimisque absurdis gravatum, quae tu ut justas inde consequentias deducis; quae huic superstruuntur, simul ruunt. Sed nee Argumenta nova superaddo (quae tamen in promptu sunt) utpote supervacanea; cum res ipsa jam abunde sit confecta. Hanc unam tamen, de novo, adjungam demonstrationem.

Curvam quamvis AE, recta contingens AT, Angulum Contactus faciet EAT; qui immotus maneat. Atque huic congruus, motu continue ferri

13 supervacanda corr. ed. 14 adjunjam corr. ed. 15-16 faciat E2

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184. OLDENBURG to WALLIS, 25 February/[6 March] 1667/8

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intelligatur, a situ CAR, per DAS, ad EAT, porroque ad FAV. Manifestum est, (propter Angulos Curvilineos CAD, DAE, EAF, aequales rectilineis RAS, SAT, TAV,) quantum hoc motu rectae AR, demitur angulo RAT, tantum motu curvae AC continue demi angulo CAT: totisque tandem demptis, transitum iri ad angulos ex contraria parte positos, ut TAV, TAP. Fiet autem hie transitus (ab angulo a sinistra ad angulum a Dextra) totis demptis, vel eodem utrobique momento; adeoque (propter aequales utrobique ablationes) aequales ab initio fuerint CAT, RAT, (utpote aequalibus ablationibus absumpti) & propterea CAR nullius magnitudinis; (quod nos dicimus) vel non eodem momento. Quo autem momento AR ad AT pervenit, (Adeoque AC ad AE) exhauritur angulus RAT: Si autem non eodem momento exhauritur totus CAT; esto hoc paulo serius, (quippe citius fieri, ne tu dixeris) recta AR existente in AX, (quippe rectam AT transiisse, necesse erit, cum serius sit quam dum AR fuerit in AT) & AC in AC. Erit igitur angulus CAR seu EAT, (quo CAT superat rectilineum RAT) aequalis ipsi EAG, seu TAX: Angulus Contactus, rectilineo: Quod est absurdum. Eodem igitur momento fit utrobique transitus: Adeoque angulus Contactus est nullius magnitudinis. Quod erat demonstrandum. Tu interim vir Clarissime, aequo animo feras velim, quod non iniquo raptim exaravi. Vale.

184. HENRY OLDENBURG to WALLIS 25 February/[6 March] 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in the postscript of OLDENBURG-BOYLE 25!I/[6!II]. 1667/8; OLDENBURG, Correspondence IV, 205-8, 207. Answered by: WALLis-OLDENBURG 29.II/[10.III]?. 1667/8. This letter accompanied two copies of PETRUS BLESENSIS, Opera omnia, ed. Pierre de Goussainville, Paris 1667, which Justel had sent from Paris for Wallis and for the Bodleian Library. In return, as Oldenburg pointed out, Justel desired a copy of Wallis's 4 tantundem E2 12 memento corr. ed. 17 memento corr. ed.

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185. COLLINS to WALLIS, 25 February/[6 March] 1667/8 Grammatica linguae Anglicanae. See OLDENBURG-BOYLE 25.II/[6.III].1667/8; OLDENBURG, Correspondence IV, 205-8, 207.

185. JOHN COLLINS to WALLIS 25 February/[6 March] 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-COLLINS 27.II/[8.III]. 1667/8. This letter apparently contained an account of DULAURENS, Specimina mathematica, duobus libris comprehensa, Paris 1667.

186. WALLIS to JOHN COLLINS Oxford, 27 February/[8 March] 1667/8 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 197r-197v (our source). On left margin of f. 197V at 90° to address in Collins's hand: 'About Dulaurens and Leotaud', and at top at 180°, again in Collins's hand: 'Vera Circuli et Hyperbolae quadratura in propria sua Proportionis Specie inventa et demonstrata a Jacobo Gregorio Ab S Patavii'.—printed: RlGAUD Correspondence of Scientific Men II, 488-90. Reply to: COLLiNS-WALLis 25.II/[6.III].1667/8.

Oxford Febr. 27. 1667./8. Sir,

Yours867 Febr. 25. I received this morning. The account which you give mee of Du Laurens his Algebra868, answers my expectation. For by the 3 account (1) of (2) which 867

Yours: i.e. COLLINS-WALLIS 25.II/[6.III].1667/8. Algebra: i.e. DULAURENS, Specimina Mathematica, duobus libris comprehensa, Paris 1667. 868

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glorious Title which was represented; & the great weight layd upon a slight Probleme869, & that so lamely proposed that it was hard to pick out what he meant: made mee think there was no great matter to be expected from him. The problem you mention about the length of a line cutting an Ellipse given in specie, & cutting one of the Axes with an angle given: You well observe, to be none of mine870; but one proposed (it seems) by such an other Algebrist, as Du-Laurens. And it may be solved by the same method in an Hyperbola, as I have done in the Ellipse, with little more then changing the signes + and —. And in a Parabola, yet more easyly. The streightening a Curve871 was done by Mr Neil, (& after him, by Dr Wren & my Lord Brounker,) a good while before Heurat872: (& I suppose both proceeded upon the grounds I mentioned in my Arithm: Infinit:) And it was commonly known to divers of our English mathematicians before Heurat's came abroad. With my answer873 to Leotauds Cavills, I perceive you are satisfyed. I confesse I thought Leotaud (though I had not read much of him) had been a better Geometer, till I saw that piece. But since I find that hee assents to opinions in Geometry venerationis gratia', I shall take him to be as Orthodox in Geometry, as in Divinity, confining his opinions in the one to father Clavius, as in the other to the Popes authority. Of Slusius874 I have a much better opinion. And did you not tell mee that hee is reprinting875 this summer, I should have desired you to have bought his works for mee, (because I have them not,) & sent down. But I think, I shall see you at London about the middle of Easter term. Fabry's Opticks876 I have not yet 17 hee (1) takes up (2) assents to 21 not add. 22 mee, (1) fe (2) (because 23 (because |I add. ed.\ have 869

Probleme: on Dulaurens's problem and Wallis's solution cf. OLDENBURG-WALLIS 10/[20].XIL1667 and subsequent letters. 8ro none of mine: cf. WALLIS-OLDENBURG 30.III/[9.IV].1668. 871 streightening a Curve: see WALLIS, Tractatus duo, 91-6; Opera mathematica I, 5514. This controversy originated in 1659 and was rekindled in 1673 by the publication of Huygens's Horologium oscillatorium. 872 Heurat: i.e. Heuraet's Epistola de transmutations curvarum linearum in rectas of 13 January 1659 (new style), printed in DESCARTES, Geometria, ed. F. v. Schooten, vol. I, Leiden 1659, 517-20. 873 answer: i.e. WALLIS-LEOTAUD 17/[27].II.1667/8. 874 Slusius: i.e. Rene Francois de Sluse (1622-85). 875 reprinting: i.e. the second edition of Sluse's Mesolabum, printed in Liege in 1668. 876 Opticks: i.e. FABRI, Synopsis optica, Lyons 1667. 430

186. WALLIS to COLLINS, 27 February/[8 March] 1667/8 had leisure to examine. I doubt allso hee admits much in Mathematicks venerationis gratia, as well as Leotaud; which makes him insist on the Earths stability; But it is in him more pardonable; because here the Popes authority is interposed, (having condemned the Copernican Hypothesis of Heresy;) But whether an Angle of Contact and of a circular segment, be homogeneous or heterogeneous to right-lined Angles, the Pope hath not yet determined. Mr Gregory's peece877 about the Quadrature of the Circle, Ellipse, & Hyperbola; I have looked over: And, for ought I discern upon a slight view of it, (not having strictly examined every proposition,) it seemes to be truly inough performed according to the method proposed. And his Prop. 32. shews his way of calculating an Hyperbolick Space. I am of your opinion, that it would advance the sale of the book, to have a clear & easy methode layd down for the operation, for any space proposed. But I think it best to bee done by himself, who is most Master of his own Notion therein; & hath allready (I presume) considered of abbreviating Methods; For, to calculate a Table after the method of this 32d proposition, would be a long work. What you mention, of solving equations of the 3d 4th or superior degrees, Trigonometrically; I do well inough approve of. For the process of Angular Sections, resolving themselves into such Equations; The one will be but the inverse of the other. This at present from, yours &c. John Wallis. [197V] These For Mr John Collins, his Majesties Accountant, at the Excise Office in Bloomsbury, London.

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1 allso add. 3 stability; (1) & (2) But 6 to (1) that of (2) right-lined 9 ought add. 12 an (1) Elliptick (2) Hyperbolick 13 sale |of add. ed.\ the 14 methode (1) pro breaks off (2) layd 877

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peece: i.e. GREGORY, Ve.ro, circuli et hyperbolae quadratura, Padua 1667.

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189. WALLIS to MORAY?, ? February 1667/8

187.

WALLIS to HENRY OLDENBURG 29 February/[10 March] ? 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in OLDENBURG-BOYLE 3/[13].III. 1667/8; OLDENBURG, Correspondence IV, 223-5, 225. Reply to: OLDENBURG-WALLIS 25!I/[6!II]. 1667/8.

188. ROBERT MORAY ? to WALLIS ? February 1667/8 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-BROUNCKER 4/[14].XL1668. Answered by: WALLIS-MORAY? 7.II.1667/8. In this letter, almost certainly from Sir Robert Moray, Wallis was asked to supply his opinion on Gregory's Vera circuli et hyperbolae quadrature, which accompanied it.

189. WALLIS to ROBERT MORAY ? [Oxford], ? February 1667/8 Transmission:

W Part quotation (from memory) of missing letter sent in WALLIS-BROUNCKER 4/[14]. XI.1668: LONDON Royal Society Early Letters Wl, No. 65, p. 1 of 6 pp. Reply to: MORAY 7-WALLis 7.II.1667/8. The member of the Royal Society who sent Wallis a copy of Gregory's Vera circuli et hyperbolae quadratura in February 1667/8 is almost certainly Sir Robert Moray. Like Gregory, with whom he shared episcopalian principles, Moray had studied at St. Andrews and it was probably through his favour that the young mathematician was appointed to the newly-created professorship there in 1668. Evidence suggests that they had been in contact during Gregory's stay in Italy and possibly also beforehand. See

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189. WALLIS to MORAY?, ? February 1667/8 COLLINS-GREGORY ?.II.1667/8 (RIGAUD, Correspondence of Scientific Men II, 174-9) and COLLINS-GREGORY 30.XII.1668/[10!.1669] (TURNBULL, James Gregory, 54-8). Since Moray had for a long time been a friend of Huygens, and was also acquainted with Wallis, it is conceivable that he suggested to Gregory that he submit copies of his work on the quadrature of the circle and the hyperbola to these two mathematicians for their opinion. It appears that Gregory sent two copies of the book to England and that these arrived in early February 1668. Probably they were addressed to Moray, although Collins suggests that they were sent to Gregory's friend, the bookseller Samuel Thompson. See COLLINS-GREGORY ?.II.1667/8; RIGAUD, Correspondence of Scientific Men II, 174-9, 174. One of these, intended as a copy for reprinting, was given to Thompson, who lent it to Collins. Collins in turn showed it to Brouncker, who then gave a short account of the work at the meeting of the Royal Society on 27 February 1668. See BIRCH, History of the Royal Society II, 253. The other copy would clearly have been sent to Wallis at the same time. Assuming Moray was the sender, he would as patron understandably have asked the Savilian professor to give his opinion privately, thus explaining the reference to a 'familiar letter' in WALLIS-BROUNCKER 4/[14].XL1668. Wallis also gave a short account to Collins at the same time. See WALLIS-COLLINS 27.II/[8.III].1668. Huygens published his opinion under the title 'Examen de Vera Circuli & Hyperboles Quadratura' in the Journal des Scavans of 2 July 1668 (new style), 52-6. See HUYGENS-WALLIS [3]/13.XL1668.

/... ] a Book lately published at Padua by Mr James Gregory now a Member of this Society, entituled, Vera Circuli et Hyperbolae Quadratura in propria sua proportionis specie. This Book soon after it came over into England, was by another Member of this Society sent to mee, desiring (in general terms) my opinion of it. And after a slight perusal of the whole, to see what matter it conteined, and examining the Demonstration of the more leading propositions, the account I gave him in a familiar letter (of which I kept no Coppy) was, as I remember, to this purpose; that it seemed to mee to contein divers things (so far as I could judge upon a slight perusall) ingeniously demonstrated though obscurely; amongst which was a new methode of approximation for the squaring of the Circle, which was allso equally applicable to the Ellipsis and Hyperbola. And more than to this purpose I do not remember that I did write. /".../

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190. WALLIS to OLDENBURG, 7/[17] March 1667/8 (i)

190.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] March 1667/8 (i) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 40, 2 pp. (our source). Endorsed by Oldenburg on page 2 beneath address: 'Rec. March 9. 67.' Postmark on p. 2: 'MR/9'.—printed: OLDENBURG, Correspondence IV, 229-30, 230-1 (English translation). Enclosure: WALLIS-OLDENBURG 7/[17].III.1667/8 (ii).

Oxford. March. 7. 1667./8. Sir,

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I was desired878 by Mr Collins (a member of our Society) to look over Monsr Fabry's Synopsis Optica879, & to give you an account of the Contents thereof. Which is as follows. Honorati Fabri Synopsis Optica, Eorum plurima quae in Opticis, Catoptricis, et Dioptricis tractari solent, breviter attingit. viz. De modo visionis, Angulo visorio, atque Magnitudinum et Distantiarum aestimatione; a prop. 1, ad 10. De Perspectivae et Scenographiae fundamento, 11 — 15. De Objecto in motu, vel per foramen conspecto, aliisve accidentibus affecto 16. — 19. De Speciebus et Lumine per foramen trajectis; umbrarum projectione, et circulorum caelestium; Saturnique phaenomenis per principia optica explicandis. 20 — 24.

6 plurima add. 7 breviter (1) exhibet, viz. (2) attingit. 8 visionis, (1) et (2) Angulo 11 motu, (1) vel (2) vel 13 foramen (1) c breaks off (2) trajectis (a) 20 — 21. (6) ; (oa) et (66) umbrarum projectione, et circulorum caelestium; (oao) 20 — 22. (bbb) Saturnique 878

desired: possibly in CoLLiNS-WALLis 25!I/[6!II]. 1667/8. See WALLIS-COLLINS 27.II/[8.III].1667/8. 879 Synopsis Optica: i.e. FABRI, Synopsis optica, in qua ilia omnia quae ad opticam, dioptricam, catoptricam pertinent, . . . demonstrantur, Lyons 1667. 434

191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) De Luminis Reflexione, speculisque planis, concavis et convexis, Sphaerico, Cylindrico, Conico, Parabolico, Elliptico, Hyperbolico; figuraeque in illis conspectae loco, situ et magnitudine. 25 — 40. De Luminis diffusione et Refractione, in aqua, in vitris, planis, sphaericis, Ellipticis, Hyperbolicis; ubi de Telescopic, Perspicillis ocularibus, Microscopic, Polyscopio, et de coloribus. 41 — 53. De Refractionibus, et Parallaxibus, aliisque phaenomenis caelestibus, et speciatim cometis. 54 — 58. In appendice, agitur, de Hypothesi Motuum caelestium quam ipse commentus est, ut Systemati Ptolemaicae consentaneam, sed demptis orbibus solidis. Accedunt tres Epistolae; in quibus agitur, de nuperis nonnullis Saturni, Jovis, et Martis observationibus; de Maculis in Jove et Marte observatis, unde colligitur eorum circa suos axes conversio; umbrisque Satellitum in Jovis disco observatis.

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[2] These

For Mr Henry Oldenburg, at the Pellmell, near St James's London.

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191.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] March 1667/8 (ii) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 43, 2pp. (our source). Endorsements at top of p. 1: 'read April. 2. 1668. enterd L.B. 2. 197.' at top right, and 'a translation of the foregoing.' at top left. On right margin of p. 2 Oldenburg has noted beneath signature at 90°: '(./) Dr Wallis's letter concerning tydes, read before the Society April 2. 1668. and orderd to be registred (2) An Extract (additional del] 1 convexis, (1) sive Sphaericis (2) Sphaerico 5 Hyperbolicis; (1) lentibusque in (2) in (3) ubi de 6 de (1) d breaks off (2) coloribus. 7 Refractionibus, (1) et (2) et Parallaxibus (a) caelestibus, (b) , aliisque 10 ut (1) tamquam { — } (2) Systemati 14 Satellitum (1) Jovialium (2) in

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191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) of a letter of Dr Wallis to M. Oldenburg Concerning Tydes.'—printed: OLDENBURG, Correspondence IV, 231-3. wl Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 197-200. w2 Copy of wl: LONDON Royal Society Letter Book Copy 2, pp. 236-9. E First edition of letter sent (with alterations according to the instructions given in WALLIS-OLDENBURG 17/[27].III.1667/8 (i)): Philosophical Transactions No. 34 (13 April 1668), 652-3 ('A Letter. Written by Dr. John Wallis to the Publisher, concerning the Variety of the Annual High-Tydes, as to several places; with respect to his own Hypothesis, deliver'd No. 16, touching the Flux and Reflux of the Sea.'). Answered by: OLDENBURG-WALLis 10/[20].III.1667/8. Enclosure to: WALLIS-OLDENBURG 7/[17].III.1667/8 (i). As was requested by Oldenburg in his letter of 10/[20].III.1667/8, Wallis, who had not kept a copy of the present letter (see WALLIS-OLDENBURG 17/[27].III.1667/8 (i)), prepared a Latin version on 17/[27].III. 1667/8. Correspondingly, that version differs from this in a number of respects. See WALLIS-OLDENBURG 17/[27].III. 1667/8 (ii). Contrary to the endorsement, it was the Latin and not the English version, which was read before the Royal Society on 2 April 1668; see BIRCH, History of the Royal Society II, 262.

March. 7. 1667./8. Oxford. Sir

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In my Hypothesis for Tydes880, you may remember, that I cast the Annual High-Tydes not on the Two Aequinoxes, about the 11th of March & September; nor yet on the Apogaeum & Perigaeum of the Sun, about the middle of December & June; but (as proceeding from a complication of those two causes) on a Midle time between the Perigaeum and the two Aequinoxes; (like as is the greatest inaequality of the Natural days, preceding from a complication of the same causes:) And, particularly, for the coast of Kent (& consequently the rivers of Thames & Medway) about the beginning of November, & February: Which agrees with obser-

1-3 Sir, In my Hypothesis E 4 High-Tydes |for the Coast of Kent, (fe consequently in the Rivers of Thames, & Medway,) del. \ not 6 middle of (1) Jul breaks off (2) December 7 between |Dec. 11, fe Mart. 11; & between del. the Perigaeum and the two Aequinoxes; (1) and particularly (2) (like 9 from (1) the lik breaks off (2) a complication of (a) both (b) the same 880

Hypothesis for Tydes: i.e. WALLIS-BOYLE 25.IV/[5.V].1666, printed in Philoso ical Transactions No. 16 (6 August 1666), 263-81. The relevant sections follow the paragraph beginning with 'But it is now . . . ' . 436

191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) vations on those coasts: and particularly with that of yours881 of Febr. 5. this year. The last year, when I was present in the Royall Society, I remember, an account882 was brought us, of the Annual High-Tydes on the Severn, & at Chepstow-bridge, to bee about the beginning of March & end of September. Which though they do not agree with the particular times on the coast of Kent; yet in the general they agree thus far; that the one is about as much before the one Aequinox, as the other is after the other Aequinox. You now acquaint mee883 with High-tydes about Febr. 22. about the coast of Plimouth (if I mistake not:) which is later than that of the Coast of Kent, but sooner than that on the Severn. And I doubt not but in other parts of the world, will be found other varietyes. The reasons of these varieties are (as I have formerly signified,) to bee attributed, to the particular position of those parts, rather then to the Generall Hypothesis. Of which this, in brief, may serve for some account at present. The General Hypothesis of the Earths diurnal motion from West to East, would cast that of the Waters (not following so fast) from East to West; which causes the constant current within the Tropicks (where the Circles are greatest) westward from the Coast of Africk to the Coast of America, (which is allso the cause of the constant Eastern Brize blowing in those

1-2 and particularly . . . this year. add. 2 The last (1) I (2) year, 6 they (1) th breaks off (2) have this (3) agree 8 with (1) a very (2) High-tydes about Febr. 22. about the (a) coasts (6) coast 9 (if I mistake not:) missing in E 15 diurnar corr. ed. 881

yours: see OLDENBURG-WALLIS 11/[11[.II1667/8 account: i.e. the account given by Henry Powle at the meeting of the Royal Society on 12 December 1666, upon which a paper was read at the meeting on 19 December 1666; BIRCH, History of the Royal Society II, 133-4. Wallis, who commented on these observations in WALLIS-OLDENBURG 19/[29].1.1666/7, had probably been present at one of these meetings. 883 You now acquaint mee: i.e. by OLDENBURG-BOYLE 3/[13].III. 1667/8 (OLDENBURG, Correspondence IV, 223-5, 225). Oldenburg explicitly told Boyle to pass on to Wallis the information on Colepresse's latest observation on high-tides at Plymouth, contained in COLEPRESSE-OLDENBURG 22.II/[3.III]. 1667/8 (OLDENBURG, Correspondence IV, 197-9, 199). Already in an earlier letter (COLEPRESSE-OLDENBURG 7/[17].1.1667/8; OLDENBURG, Correspondence IV, 105-7) Colepresse had sent the Royal Society observations on the tides, which were subsequently printed in Philosophical Transactions No. 33 (16 March 1667/8), 632-3. 882

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191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii)

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parts:) But the sea thus beating on the Coast of America is cast back, as with an Eddy, on either hand; & consequently returnes from the American shoar Eastward towards the coast of Europ; where, the Parallel circles to the Equator being lesse, & consequently the diurnal motion slower, doth not cast the Waters so strongly westward, as between the Tropicks, & so not strong inough to overcome the Eddy, which it meets with from the other motion. Which gives the Sea a North-Easterly motion, (on these coasts,) as to its usuall course: (which allso I take to be the cause, why North-Easterly winds are more frequent then any others on our coasts.) The Current therefore of our Seas being North-Easterly; we are next to consider at what times it runs more to the North, & at what more to the East. When it runs most Northerly, it runs up the Irish sea, & so up the Severn: when most Easterly, it runs straight up the Chanel, & so to the coast of Kent: when between these; it beats against Devonshire & Cornwall, & those parts. Wee are therefore to consider (as to the annuall periods) that the Annual motion of the Earth in the Zodiack, & the Diurnal in the Aequator, are not precisely in the same direction, but make an Angle of 23^ grad. at the Aequinoxes, but run as it were parallel at the Solstices; and as they be near or farther from these points, so is the inclination varied. Which several directions of motion, do cause the compound motion of both, to vary from the East & West, more or lesse according as the suns position is further or nearer the Solstices: And therefore nearer to the Aequinoxes, this inclination doth cast the| constant current of our Seas more to the North & South; & further from it, more to the East & West. Which is the reason why the current up the Irish sea, is near to the Aequinoxes, (at the beginning of March & end of September;) and up the Chanel or narrow seas, farther from it,

1 as add. 3 shoar add. 5 as (1) under the (2) between the Tropicks, & (a) therefore (6) so 7 a (1) Norther breaks off (2) North-Easterly 8-9 (which allso . . . on our coasts.) del. Oldenburg (on Wallis's instructions) The Current therefore E 12 When it (1) lies (2) runs 13 straight add. 17 direction, add. 25 current (1) up t breaks off (2) up 438

course:

[2]

191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) (at the beginning of Febr. & of November;) and against the Coasts of Devonshire & thereabouts, at some intermediate time. And thus much I thought convenient to signify upon this occasion. Desiring allso that the Spring tides now next coming may be observed; whether they be not less than those at work about the 5th of February; & that you will please to keep notes of such observations as you meet with of this kind. I remember that the great wind884 which happened about Febr. 18. about 5 or 6 years agoe; was a North Easterly wind; which came off from Sea about Devonshire & Cornwall, & so crosse the land by Oxford & so to London. Whether it were at the time of the menstrual Spring-tydes, I do not remember. And I do not much question, but that the same things which cause extraordinary Tydes; may likewise cause extraordinary winds: though not allways at the same times at the same place: the one being caused by an Eddy of the water beating against Land; the other by a like reverberation of the Air against Mountains or high hills. But in the water it is more observable then in the Ayr. This in hast from Yours &c Joh: Wallis.

I and (1) on the (2) towards (3) against the Coasts of Devonshire (a) , at s breaks off (&)& 2-19 I thought fit to signifie upon this occasion. Dat. Oxford the 7. of March An. 1667/8. E 3 convenient (1) so (2) to 8 about (1) 5 (2) 5 or 6 II much add. 15 hills. (1) An breaks off (2) But in the water it is more observable then in the (a) water. (6) Ayr. 884

great wind: i.e. the storm of 17/18 February 1661/2; see The Diary of John Evelyn, ed. DE BEER, III, 316.

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193. WALLIS to OLDENBURG, IT/[27] March 1667/8 (i)

192.

HENRY OLDENBURG to WALLIS 10/[20] March 1667/8 Transmission:

Manuscript missing. Existence and date: Referred to in OLDENBURG-BOYLE 17/[27].III. 1667/8 (OLDENBURG, Correspondence IV, 248-50, 250) and WALLIS-OLDENBURG 17/[27].III. 1667/8 (i). Answered by: WALLIS-OLDENBURG 17/[27].III.1667/8 (i). In this letter, Oldenburg apparently requested that Wallis send him a Latin version of his latest letter concerning his hypothesis of the tides (WALLIS-OLDENBURG 7/[17].m.l667/8(ii)).

193.

WALLIS to HENRY OLDENBURG Oxford, 17/[27] March 1667/8 (i) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 42, 2pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Rec. March. 18. 1667.' Postmark on p. 2: 'MR/1{8}'.—printed: OLDENBURG, Correspondence IV, 252. Reply to: OLDENBURG-WALLIS 10/[20].III. 1667/8. Enclosure: WALLIS-OLDENBURG 17/[27].iII.1667/8 (ii).

Oxford March. 17. 1667./8.

Sir,

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I thank you for yours885 of Mar. 10. with the information in it. I have according to what you did therein intimate, sent you886 the summe of mine887 of March 7. (for the words I have not) in Latine. In my English, which before I sent you, you may please to blot out that Parenthesis [which may bee the occasion why the North-East Winds blow so frequently on our coast] and likewise that concerning the High winds in February 4 or 885

yours: i.e. OLDENBURG-WALLis 10/[20].III.1667/8. sent you: i.e. the enclosed WALLIS-OLDENBURG 17/[27].III.1667/8 (ii). 88r mine: i.e. WALLIS-OLDENBURG 7/[l7].III. 1667/8 (i). 886

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) 5 years ago: For it was (I think) in Febr. 18. 1662./3. and, if so, it was near the time of Neap-tides, not of the Spring-tydes; & so nothing to the purpose. The Latine though I now send it; yet I am not willing to venture it to the publike, till my Lord Brounker have first well weighed it, & do approve it. I expect to hear, how the Spring-Tydes happened at London, at the Aequinox last past. Which I expect should be higher then ordinary, but not so high as at the beginning of February. I think it would now be time to think of Printing Mr Horrocks's papers888. By Eddy, I mean, Aqua (repercussa) per circuitum retrocurrens. At Billingsgate, (because of the Graves-end Barges,) they do constantly attend the High-water, both by night & by day. Were it not to get some porters, or water-men, or some appertaining to Innes near it, who use there to attend, to keep a Diary of the Time & Hight of each Tyde, for a Year or more together? This in hast from Yours John Wallis. [2] These For Mr Henry Oldenburgh, in the Old Palmal, near St James's London.

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194.

WALLIS to HENRY OLDENBURG Oxford, 17/[27] March 1667/8 (ii) Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 41, 2pp. (our source). At top left of p. 1: 'read April 2: 68. enter'd LB. 2. 163.'. At top middle of p. 1 Oldenburg has noted: 'Prom Dr Wallis to M. Oldenburg (1) Concerning (2) About the hapning of I Febr. |18. add] (1) 166{3./4.) (2) 1662./3. 4-6 & (1) ap breaks off (2) do approve it. I expect to hear, how the (a) Tydes (6) Spring-Tydes happened at London, (oa) upon (66) at II or some appertaining to Innes near it, add. 888

papers: i.e. astronomical papers of Jeremiah Horrox. See Wallis's letters to Oldenburg of 6/[16].IV.1664, 30.IV/[10.V].1664, and 21.IX/[1.X].1664.

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) the (a) Tydes (6) highest Annual Tydes in the Interniediat times between the Perigee of the Sun and the Aequinoxes.' wl Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 163-5. w2 Copy of wl: LONDON Royal Society Letter Book Copy 2, pp. 196-9. E First edition (with several major alterations): WALLIS, Opera mathematica II, 755-6 ('De aestu maris, epistola tertia. Ad D. Oldenburgium scripta, 7° Martii, 1667. Stilo Angliae.'). Enclosure to: WALLis-OLDENBURG 17/[27].III. 1667/8 (i). Already on 7/[17] March, Wallis had sent Oldenburg a letter in English in which he gave an account of the latest observations on the high tides reported to him and showed how they were accommodated by his hypothesis. Oldenburg subsequently requested this Latin version, which Wallis produced without a copy of the former and which therefore differs from that. It was probably written on or shortly before 17/[27] March; nevertheless Wallis gave it the date of the original English letter. The present letter was read at the meeting of the Royal Society on 2 April 1668. See BIRCH, History of the Royal Society II, 262.

Oxoniae Martii 7. 1667./8.

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In mea (Vir Clarissime) de Marinis Aestibus Hypothesi889, conjicio ego (uti meministi) maximos Tumores Annuos, non in bina Aequinoctiorum tempora (circa Martii et Septembris diem undecimum, stil. vet. contingentia;) neque in Apogaei et Perigaei Solis tempora (circa medium Junii et Decembris:) Sed, in tempora Perigaeo et Aequinoctiis intermedia; (Oriuntur utique ex earundem causarum complicatione; turn hi Tumores Annui, turn Dierum Naturalium Inaequalitates maximae, sub eadem tempora contingentes;) Et speciatim, ad Kantiana littora, (et propterea in Fluviis Thamesino et Medwayano,) sub initia Februarii et Novembris. Quod quidem Observatis satis respondet; eisque nominatim quae tu modo ad dies Febr. 3, 4, 5, te Londini habuisse refers890. Dum autem, praeterita Aestate, in Regia Societate nostra aderam;

4 (circa |undecimum del] Martii W (circa 11 Martii, & 13 Septembris, St. vet. E 5 Apogaei aut Perigaei E 7 Annuui W corr. ed. 9 propterea, credo, in Fluviis etiam Thamesino E 10 Medwayano, sub W corr. ed. 889

Hypothesi: i.e. WALLIS-BOYLE 25.IV/[5.V].1666, printed in Philosophical Transactions No. 16 (6 August 1666), 263-81. 890 refers: see OLDENBURG-WALLIS ll/[21].II.1667/8.

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) ibidem nunciatum891 memini, Annuos hos Tuniores ad Sabrinae fluvium, et adjacentem Ponteni Chepstowensem, sub Martii initium, et Septembris finem, contigisse solitos, (quos St Davids stream et Michaelmas stream appellant:) Non quidem eisdem plane temporibus quo ad littora Kantiana contingunt (quos vocant Candlemas stream et Allholland stream) sub initia Februarii et Novembris: sed saltern (quod utrobique convenit;) alterum quasi tantundem ante Aequinoctium Vernum, quantum alterum post Aequinoctium Autumnale. Tandem mini nuncias892, in proximis literis, circa diem 22 Februarii jam elapsi, praetumidos Aestus in littore Devoniensi (prope Plimutham) contigisse: Hoc est, serius quam pro littore Kantiano, citius autem quam pro Sabrinae Ostiis. Sed et alias alibi terrarum varietates contingere solitas, nullus dubito. Hujusmodi autem varietates, ubique terrarum contingentes, cum non tarn ex generali Hypothesi, quam particular! locorum et littorum situ, aliisque circumstantiis mille modis variatis dependeant; earum omnium causas reddere, ob rationes jam aliquoties dictas, non in me suscipio.

1 ad Sabrinae (1) fl breaks off (2) fluvium, W 3 contingere solitos, E 3-4 et (1) appe breaks off (2) Michaelmas stream appellant:) (a) Qui quamquam (b) Non quidem eisdem jplane add.\ temporibus W 4-9 appellant:) non eo tempore quo ad littus Kantianum, (quos vocant Candlemas Stream & Allholland Stream,) sub initia Februarii & Novembris contingunt. || Hactenus tamen convenit; tantundem quasi Ante Aequinoctium Vernum alteros, atque alteros Post Aequinoctium Autumnale, utrobique contingere. 11 Tandem milii narras, in E 6 sed |saltern add] (quod (1) utrob breaks off (2) in utrisque (3) utrobique W 13-444, 1 dubito. || Cur ego hujusmodi varietatum, ubique terrarum contingentium, causas reddere, nolim in me suscipere; jam anteliac dictum est aliquoties: Utpote quae non tarn ex generali Hypothesi, quam ex particular! locorum & littorum situ, aliisque circumstantiis, mille modis variatis, dependent. || Quod autem E 15-16 locorum (1) et littorum (2) et littorum situ, (a) dependeant (6) aliisque circumstantiis mille modis variatis dependeant; earum (ao) omnes (66) omnium W 891

ibidem nunciatum: a reference either to the account of observations concerning tides given by Henry Powle at the meeting of the Royal Society on 12 December 1666 or to Powle's paper read at the meeting on 19 December. See the comment on the corresponding passage in WALLis-OLDENBURG 7/[17].III. 1667/8 (ii). 892 nuncias: i.e. in OLDENBURG-BOYLE 3/[13].III.1667/8 (OLDENBURG, Correspondence IV, 223-5, 225), which contained information on Colepresse's observations on high tides at Plymouth, made on 22 February 1667/8. See the comment on the corresponding passage in WALLIS-OLDENBURG 7/[17].III.1667/8 (ii). 443

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii)

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Quod autem ad modo memoratas attinet: haec forte non incommode dici poterunt. Cum Tellus diurno motu ab Occidente Orientem versus volvatur; Aquaque propterea, (quae tardius sequitur,) quasi retro relicta, ab Oriente versus Occidentem rejici videatur; maxime intra Tropicos, ubi, propter Parallelos majores, diurnus Telluris motus est concitatior; (quod in causa esse creditur, cur molliores aurae ab Oriente quasi quotidie flantes, ibi terrarum sentiri soleant:) Ordinarius aquarum cursus esse debet ab Africano littore Americanum versus. Sed adverse littore Americano repercussae, (utrinque versus Polos declinantes,) circuitu facto, inde ad Europaeum littus, contrario quasi motu retrocurrunt: utpote ubi, propter parallelos minores, adeoque tardiorem Telluris motum diurnum, non tarn fortiter Occidentem versus rejiciuntur aquae, ut ab Americano littore repercussis obsistere valeant. Unde fit, ut Maris cursus, tamquam inde ad nos delati, quasi Euro-Boream spectare debeat: (ita tamen ut, ubi hue prope advenerit, circuire possit juxta nostra littora, Aequatorem versus, ut hoc circuitu restituatur id quod ab Africano littore continue cursu Occidentem versus decedit: Hoc utique postulare videtur elementi fluidi Libramentum.)! [2] Id autem porro inquirendum est, Quibus anni temporibus cursus hie, Orientem et Septentrionem interjectus, magis ad Septentrionem declinat;

3 versus (pro Copernicana hypothesi) censetur converti: Aquaque E 5 ferri videatur E 9 versus, sed aquae adverse E 10 facto (Eddy nostri vocant,) inde E 11 quasi |motu add.\ (1) feruntur: (2) retrocurrunt: utpote W quasi motu feruntur. Utpote E 13 repercussis resistere E 15-20 debeat. Indeque factum judico quod retrocurrens hie motus, postquam Norwegiae Promontorium praeteriverit (in Septentrionem rejectus) in Littore Suecico (& quod inde ad Orientem porrigitur) vix ullum efficit notabilem Aestum. Prout etiam in mari Caspio nullus observatur Aestus Marinus (ut quod cum Oceano non conjungitur:) nee quidem notabilis in Mari Baltico (quo per Fretum Sundicum haud facile penetrat retrocurrens Aqua quae Aestum faceret:) nee (ob similem causam) in Mediterranei Maris partes interiores, & Euxino Ponto. || Id autem E 20 hie (ab Americano littore repercussus) Orientem E 444

194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) et, quibus magis ad Orientem. Quippe cum magis ad Septentrionem declinat; in Mare Hibernicum directe fertur, adeoque ad Sabrinae Ostia: Ubi autem magis ad Orientem vergit; directe fertur in Fretum Britannicum, adeoque ad littus Kantianum: Interea vero, dum cursu his intermedio fertur; Devoniae littus, tractumque adjacentem, fortius impetit. Animadvertendum insuper est, Telluris motum Annuum (per Zodiacum,) et Diurnum (secundum Aequatorem) non eodem vel parallelis planis fieri; sed ad invicem inclinari, angulum facientes grad. 23^ circiter. Sed et, Tellure circa puncta Solstitialia versante, motus hi sunt quasi parallel!; alibi vero se mutuo plus minusve decussant, prout propius aut remotius ab Aequinoctiis Tellus abest. Et consequenter, qui ex ambobus componitur motus, (sive accelerando, sive retardando, quorum illud noctu fit, hoc interdiu,) cum utroque participans; prope Solstitia, in Orientem directe tendit; alibi, ad Septentrionem vel Meridiem vergit, (alterum de die, alterum de nocte,) et maxime circa Aequinoctia. Atque hinc fieri videtur, quod Tumores Annui ad Sabrinae Ostia, (propter Mare Hibernicum, inter Euro-boream et Austro-zephyrum porrectum,) propius ad Aequinoctia contingunt, sub initium Martii et finem Septembris: sed in Kantiano littore, (propter Fretum Britannicum, inter Orientem et Occidentem porrectum,) longius utrinque ab Aequinoctiis (adeoque ad Solstitium Brumale prope,) sub Februarii et Novembris initia:

1 Quippe cum magis add. ad W Quippe (nos quod spectat) quum magis ad E 2 Hibernicum (1) fertur (2) directe W 4 Kantianum, fe (quod vocatur) Rumney Marsh: Interea E 5-6 impetit: Indeque factum, ut citius in Hibernicum mare & Sabrinae ostia, serius in Mare Britannicum fe littus Kantianum, (post Aequinoctium Autumnale) pertingant Annui aestus maximi, fe intermedio tempore ad littus Dcvonicnse. \ \ Animadvertendum E 6 motum (1) Diurnum (per Zodiacum,) et Annuum (2) Annuum (per Zodiacum,) et Diurnum W 9 Sed, Tellure E 10 mutuo (1) decussant (2) plus W decussantes E 11-16 qui ab ambobus componitur motus, (cum utroque participans,) prope Solstitia, in Orientem directe tendit: alibi, ad Septentrionem vel Meridiem vergit, fe maxime circa Aequinoctia. || Atque hinc fit, quod E 12 motus add. W 14 (alterum de die, alterum de nocte,) add. W 20 Aequinoctiis, sub Februarii E

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 Atque ad Littus Devoniae cum tractu adjacente, temporibus intermediis.

195.

HENRY OLDENBURG to WALLIS 26 March/[5 April] 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS-OLDENBURG 2/[12].VII. 1668 (ii). In this letter, Oldenburg requested that Wallis provide an account of Dulaurens' Specimina mathematica. To this end he sent a copy of that work with the letter. See WALLIS-OLDENBURG 30.III/[9.IV].1668 and WALLIS-OLDENBURG 2/[12].VII. 1668 (ii).

196.

WALLIS to HENRY OLDENBURG Oxford, 30 March/[9 April] 1668 Transmission:

Wl Letter sent: LONDON Royal Society Early Letters Wl, No. 44, 4 pp. (our source).— printed: OLDENBURG, Correspondence IV, 285-91. W2 Latin translation of W1: LONDON Royal Society Early Letters Wl, No. 45, 4 pp. E First edition of Latin translation (with some passages omitted): Philosophical Transactions No. 34 (13 April 1668), 654-5 ('Another Letter Written by the same Hand, concerning some Mistakes, to be found in a Book lately publish'd under the Title of Specimina Mathematica Francisci Du Laurens, especially touching a certain Probleme, afnrm'd to have been proposed by Dr. Wallis, to the Mathematicians of all Europe, to solve it.') (first part), and Philosophical Transactions No. 38 (17 August 1668), 748-50 (second part). In the present letter, Wallis provides an account of Dulaurens's Specimina mathematica as requested by Oldenburg. Oldenburg chose at first to publish only part of the Latin translation (up to the paragraph ending '... in the face of all the World, a thing so false.' in the English original), in which Wallis rejects the idea that he was the author of the probleme on the ellipse. Only after Dulaurens printed his reply entitled Responsio . . . ad epistolam D. Wallisii ad darissimum virum Oldenburgium scriptam (Paris [1668]), rejecting Wallis's account of the history of the problem, did Oldenburg publish 'what 1 intermediis. Quippe in eas partes, illis temporibus, ab Americano littore ad Europaeum Recurrens maris motus, composite Telluris motu, detorquetur. || Atque haec est, quae impraesentiarum occurrit, hujus Phaenomeni non improbabilis ratio. Vale. E 446

196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 (out of respect of the same [i.e. Dulaurens]) was supprest ever since that Vindication was printed, with which it then came joyned' (Philosophical Transactions No. 38 (17 August 1668), 748). See also DULAURENS-OLDENBURG [13J/23.V.1668 (OLDENBURG, Correspondence IV, 398-400) and JUSTEL-OLDENBURG [13J/23.V.1668 (ibid., 402-4, 402) for preliminary comments by Dulaurens on the first part of Wallis's letter. Dulaurens's printed reply was sent to Wallis in a packet accompanied by OLDENBURG-WALLIS 30.VI/[10.VII].1668. See also JUSTEL-OLDENBURG [27.VJ/6.VI.1668 (OLDENBURG, Correspondence IV, 427-9, 428).

Oxford March. 30. 1668.

Sir, I received from you893, a few days since, Dulaurens his Specimina Mathemafe'ca894; which I did presently peruse, that I might (as you desire) give you some account of it. The Title, mythinks, promises much more than the book doth perform. A great part of his first booke, seems to be taken out of Mr Oughtred895 & my self896, (though he doth not there so much as name either of us,) & that so evidently, that he doth many times not onely retain the peculiar phrases & forms of speaking, but the very same Notes &; Symbols. And much of the second book, out of Vieta, Schooten, & the Tracts by him published897, (but these he thinks fit to mention898.) There are, in it, divers things unsound; &; many more, unaccurate inough. But what are those genuine principles, & new Elements 9 not onely add. 893

received from you: i.e. with OLDENBURG-WALLis 26.III/[5.IV].1668. Dulaurens his Specimina Mathematical i.e. DULAURENS, Specimina mathematica duobus libris comprehensa. Quorum primus syntheticus agit de genuinis matheseos principiis in genere, in specie autem de veris geometriae elementis hucusque nondum traditis. Secundus vero de methodo compositionis, atque resolutionis fuse disserit, & multa nova complectitur, quae subtilissimam analyseos artem mirum in modum promovent, Paris 1667. 895 Oughtred: presumably OuGHTRED, Clavis mathematicae, London 1631 (and subsequent editions). 896 my self: presumably WALLIS, Arithmetica infinitorum, Oxford 1656. 897 Vieta, Schooten, fe the Tracts by him published: i.e. VlETE, In artem analyticam isagoge, Tours 1592, reprinted in VIETE, Opera mathematica, ed. Fr. v. Schooten, Leiden 1646; SCHOOTEN, Principia matheseos universalis, ed. E. Bartholin, Leiden 1651, reprinted in DESCARTES, Geometria, ed. Schooten, II, Leiden 1661, 1-47; SCHOOTEN, Appendix de cubicarum aequationum resolutione, in DESCARTES, Geometria, ed. Schooten, I, Leiden 1659, 345-68; HUDDE, Epistola de reductione aequationum, in DESCARTES, Geometria, ed. Schooten, I, Leiden 1659, 401-506. 898 mention: see e.g. DULAURENS, Specimina mathematica, 165, 172, 174, 191, 211, 213-17. 894

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of Geometry, hitherto undiscovered, (which his Title promises) I do not find. Much less can I be of his opinion, when he tells us, pag. 141, that he thinks there is no man who would not prefer his Few, before All Euclides, Elements. In the close, he doth mee manifest injury; in affirming of mee what is grossely false. Hee hath, there, a particular Appendix899, with this specious Title, Solutio Problematis a D. Wallisio totius Europae Mathematicis propositi, sed prius ad generale revocati, Anno MDCLVIII, eodem tempore quo propositum est. After this Title, it thus follows. Problema D. Wallisii. Datis Ellypseos maximis Diametris, turn puncto in transversa ejus Diametro assignato, reperire in numeris segmenta lineae intra Ellypsim terminatae, et per datum punctum transeuntis, atque datum angulum cum dicta Diametro facientis. Verum quia praepositae quaestionis solutio aeque facilis est in numeris, ac in lineis, (ut postea apparebit,) melius facturum me judicavi, si prius demonstrationem Analyticam hie afferrem, ex qua tarn Numerica, quam Geometrica sequeretur ad Problematis solutionem pertinens effectio. Atque ut haec solutio cum faenore detur, speciale D. Wallisii Problema, ad generale sic revoco.

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(After which Praeface, he proceeds to give as a Probleme in his own words, with a solution of it, for 7 pages together.) To this I say, 1. The proposing a chalenge of this nature, to all the Mathematicians in Europe; is a Vanity, that I was never yet guilty of: & 'tis like inough, that I never may be. 2. If I had thought fit (for Ostentation) to make such a chalenge; certainly I should have made choise of something, which should have been either of more difficulty, or of more concernment, than I take this to be: which a very indifferent Algebrist, at first sight, may solve in half an hour. 3. I never yet did, nor perhaps ever shall, propose either that Problem, or any to that purpose, (to the best of my remembrance,) to any one person: much less to all the Mathematicians in Europe. Nor is there any thing of truth in what he doth here affirm of mee. 4. There was indeed once a question somewhat like it, proposed to mee, (& I gave a present solution of it,) but never by mee proposed to any. 899

Appendix: i.e. DULAURENS, Specimina mathematica, 249-55.

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 And how far I am concerned in it, you have in a Letter900 of mine to the Lord Vicount Brounker, dated May 11, 1658, (the same year that Dulaurens mentions) and the same year Printed in my Commercium Epi901 [2] stolicum , pag. 171. in these words,| Sub initium Februarii jam proxime elapsi, amicorum non nemo cui forte occurrebam sero vesperi, quaestionem sequentem mihi porrexit in scriptis: quam jam nuperrime, ab eodeni intelligo, typis vulgatam esse, cum hac Epigraphe, Spectatissimos viros Matheseos Professores, et alios praeclaros in Anglia Mathematicos, ut Problema solvere dignentur, Jean de Montfert maxime desiderat. Extremis Ellipseos Diametris, distantia centri ab aliquo puncto in axi transverso, ubi linea eundem secet sub angulo dato, in numeris datis: segmenta ejusdem lineae (si opus est) productae, et intra transversum axem et ellipsin terminata, in numeris exhibere. Hanc ego quaestionem, suam ratus (neque enim vel innuebat ille, vel ego turn sciscitabar, cujus erat,) paulo adhuc universalius expositam, sub hac fere quae subest forma (neque enim ipsissima verba memini) postero mane solvebam; neque eram de ilia ultra solicitus, (ut quae res nee magnae difficultatis videbatur, nee momenti:) quam et, quod jam audio, varii variis modis solvebant, utut eorum solutiones nondum viderim. (After which follows, a short & clear solution of mine, with the Demonstration of it, of that Probleme more universally proposed: allowing onely for the Errors of the Presse, which the Reader who understands it, will easyly rectify.) And this is all that I ever had to do in it. But how Mons. Dulaurens can pretend this to be, a proposing of that Probleme, by mee, to all the Mathematicians in Europe; I leave to the judgement of any who can understand Latine, though he should understand nothing of mathematicks. The person who shewed it mee, was Dr Rawlinson902, (first, in writ900

Letter: i.e. WALLis-BaouNCKER 11/[21].V.1658. Printed in my Commercium Epistolicum: i.e. WALLIS, Commercium epistolicum, Oxford 1658, 168-74. The passage quoted by Walk's is to be found on 171-2. 902 Dr Rawlinson: i.e. Richard Rawlinson (d. 1668), fellow of the Queen's College, Oxford and associate of Christopher Wren and Samuel Hartlib. 901

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ing; afterwards, in Print.) What that Jean de Montfert903 is, who did propose it, I do not know; But his printed Paper, at that time, was in London common inough, and proposed to divers other Mathematicians beside my self, (as a Probleme sent out of France;) and was solved by divers of ours in London. And one of them (Dr Christopher Wren, then Astronomy Professor at Gresham-College; now, at Oxford,) printed & published his Solution; and, with it, (in the same Paper,) by way of Return, a Probleme by him proposed, to the Mathematicians in France, (as this had been from thence to us in England;) to which there is none of them have yet (that I hear of) returned any Solution. His own Solution of it, is since published904 in my Treatise De Cycloide, pag. 72, 73. Now when all this is publike & notorious; I cannot but wonder, with what confidence (or gross negligence,) he should publish, in the face of all the World, a thing so false. But the truth is, Mons. Dulaurens, is much more negligent of what hee writes, than doth become a Mathematician. Of which we need go no further for instance then this Probleme of Mons. De Monfert, put into Dulaurens's Words. Where why he should (constantly) write Ellypsis with y, I will not examine, (permitting him to spell as he please;) Nor, why he gives us one turn without another to answere it, (because this concerns the Grammar of it, not the Geometry.) But why he should, instead of Extremis Ellipseos Diametris, (that is, the Greatest & the Lest,) substitute Diametris Maximis, (as if an Ellipse had more Greatest diameters then One;) he ought to give us some good account, or else it will be thought a Negligent expression. With the like negligence he puts in transversa ejus Diametro, instead of in Axe transverso; as if either an Ellipse had no other Transverse Diameters but the Axes; or else, that a point assigned in any other Transverse Diameter would indifferently serve; where no other Diameters are given but the two Axes. He might as well have sayd, ubivis intra Ellipsin assignato; for there is no Point within the Ellipse, which is not in some Transverse Diameter. Again, the Segments of the Line sought being required in Numbers; the things given were to be designed

5 them (1) (Dr Wren (2) (Dr 7 by way of Return, add. 18-21 Where why . . . not the Geometry.) omitted in E 903 904

Jean de Monfert: possibly a pseudonym of Pascal. See PASCAL, (Euvres VIII, 136. published: i.e. WALLIS, Tractatus duo, Oxford 1659, 72-3.

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 [3]

in Numbers allso:| And so, instead of datis Ellypseos diametris Maximis, he should have sayd, Ellipseos Diametris Extremis (not Maximis] per Numeros designatis; or, in Numeris datis. And instead of turn assignato puncto in transversa ejus diametro; (where puncto in Numeris assignato would not be proper:) he should have sayd, Punctoque in utrovis Axe transverso (not transversa diametro) per suam vel a Centra, vel a Vertice, distantiam, Numero designatam assignato, (or words to that purpose.) And instead of Segmenta lineae intra Ellipsim terminatae, (which doth not design either end, either of the Line, or of its Segments;) it should have been Segmenta rectae Ellipsi (not intra Ellipsim) terminatae, in puncto illo sectae; or Segmenta rectae per punctum illud transeuntis, huic Axi (sen Puncto) et Ellipsi interjecta; or, Rectae Segmenta Ellipsi et puncto illo terminata; or something to that purpose, which might design as well the End of the streight Line, as the Point in which it is divided: neither of which are by his words determined, but left to conjecture. And so many negligent expressions, in one short proposition, are so much the more unpardonable, because De Montferts probleme, which Dulaurens doth thus imperfectly represent, was much better expressed. Which therefore he should not have varied, unless it had been for the better. Nor do I see what excuse he can pretend: For where ever he mett with this Probleme, (whether in De Montfert's printed Probleme, or in Dr Wren's printed Solution, or in my printed Letter,) he could not but see the Problem better expressed; and, withall, that it was Jean de Monfert, (not Johannes Wallisius,) that did propose it. But there is again as great a negligence, in that which next follows. For when he tells us presently, that it is equally easy to solve the question in Numbers and in Lines, (& therefore would advance his skill, by shewing how it may be done in Lines allso;) It is much otherwise. For to do it in lines, is of no more difficulty, then, to draw a streight line which shall cut another at angles given; (for if a streight line be so drawn, which in the point assigned shal so cut the Axis; the Ellipse, without further construction, doth terminate its Segments:) Which De Montfert, it seemes, was not so weake a Geometer, as to think it a Problem worth proposing;

4 ejus add. 20 For (1) whethe breaks off (2) where 32-452,1 Which De Monfert . . . to solve it. omitted in E 32 De Monfert, (1) was (2) it

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though Dulaurens think it a credit to be able to solve it. I might adde also; that, where as it is proposed, to find the Segments; his whole process doth onely give us the Greater Segment; not the Lesser, nor the Whole Line. 'Tis true, the Lesser Segment may be easyly found, (& so might the Greater allso;) but he should have told us How, since he undertook so to do. The applying this Problem to other Conick Sections; is no new difficulty. For the same process which I shew in the Ellipse, will (mutatis mutandis) be equally applicable to any curve line, whose ordinates are known: (and that whether the Diameter be cut within, or without, the Curve:) and a very indifferent Algebrist will know how to accommodate it to the particular case. I do not trouble you with so diligent a Collection of the like Negligences throughout the Book; Because I am not now writing a Confutation, but Negligentiae Specimina; & these (for so short a Probleme) may suffice. I might shew the like Negligence in that Probleme of his own; which you were pleased to send mee, a while since, to solve. Which was so ambigiously, & imperfectly expressed, (and so unlike a Mathematician,) that it was more difficult to determine the sense of it, than to solve the Probleme. But of this I have given you some account heretofore. | [4] I shal onely adde a few Specimina of the same Negligence which appeares frequently throughout his book. As pag. 67. where he tells us, that as Two streight Lines, do not comprehend a Space, so neither Two Plains. Hee should have sayd, Nor Three Plains. For, as Fewer Streight Lines than Three cannot conclude a Superficial; So neither can Fewer Plains than Power, conclude a Solide Figure. Nor is it very accurate, when he tells us (in the next words) that Two Plains cannot meet in Two Places: For in All Places of the same Line which is their Common Section, they do meet; but no where else. But his Description of an Angle, (in the same page,) is yet more negligent: which he defines to be Duarum pluriumve, ejusdem Speciei, magnitudinum, (ad unum punctum collectarum, &c,) brevissima distantia. For, (to say nothing of Time, Weight, Strength, &c, which he doth elsewhere905 allow to be Magnitudes; &; are, in Euclide's phrase, p,ejedri;)

14 Book; (1) (which would be endless (2) Because 16-20 I might shew .. . some account heretofore, omitted in E 905

elsewhere: see DULAURENS, Specimina mathematica, I.

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 taking Magnitude in the strictest sense, for Extensive Corporeal Magnitude; Hee doth himself acknowledge, that it is not to be understood of Bolides, but onely of Lines & Surfaces] & therefore gives us a Retractation, pag. 89. Again, instead of ejusdem Speciei, he should rather have sayd ejusdem Generis, (for so he means,) that is Homogeneous quantities: For a streight line & a Curve, (being Homogeneous, or ejusdem Generis,) though not ejusdem Speciei, do make an Angle: & the like of superficies specie differentes: And even a Line to a Plain, (or other Surface,) may have its Angle of Inclination; (as well as a Plain to a Plain;) though not so much as Homogeneous. Again, having thus defined an Angle, he tells us in the next page, (pag. 68.) that, if those magnitudes be two Lines, the Angle is a Plain Angle; as if he had never heard of Spherical Triangles; or as if no Superficial Angle could be made but in a Plain. Again, Duarum pluriumve, is not safely sayd of Lines: For such meeting of Three Lines, are not an Angle, but Two Angles at lest; more then Two, not concurring to make One Angle. Again, having defined his Angle in generall, (pag. 67,) by Duarum pluriumve; he tells us particularly of a Solide Angle, pag. 68, that it may be comprehended Una vel pluribus, (giving instance for that of One, in the Vertex of a Cone;) as if One were duae vel plures. Again, what doth he mean by brevissima distantial In the Point of Concurse, there is no Distantia; out of it, there is no Minima: (For there can be none assigned, than which there may not be a Lesse.) But indeed his whole Notion of an Angle, is unsound: which is not to be defined by Remotio, or Distantia, but by Inclinatio; as he might have learned from Euclide's Definition. So, pag. 171. He tells us, that in these two forms of Quadratick Equations, aa — ca + dd — 0, and aa + ca + dd — 0, the Roots are both Affirmative: which is far otherwise. In the Former of them the Rootes are indeed both Affirmative; but in the Latter, they are both Negative. If these few, for a tast, do not suffice; it will be easy to furnish a more ample service. But whether they proceed from Incuria or Inscitia, (as he

1 strictest (1) strict breaks off (2) sense, 5 (for so he means,) add. 5 quantities: (1) For streight fe crooked (2) For 6 Generis,) (1) do make an Angle, though, not ejusdem Speciei: (2) though 7 of (1) Surfaces (2) superficies 18 be (1) compl breaks off (2) comprehended 21 there I is add. ed.\ no Distantia

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197. BROUNCKERto OLDENBURG, March? 1667/8 distinguisheth of Errors, pag. ult.90&) I will not determine: (I doubt, there is somewhat of Both:) nor give you, at present, any further trouble. Yours &c. John Wallis.

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For my honoured friend, Henry Oldenburg, Esquire.

197. WILLIAM BROUNCKER to HENRY OLDENBURG [London], March? 1667/8 Transmission:

E1 First edition of (missing) letter sent: Philosophical Transactions No. 34 (13 April 1668), 646-9 ('The Squaring of the Hyperbola by an infinite series of Rational Numbers, together with its demonstration') (our source). E2 Latin translation of E1: WALLIS, Opera mathematica III, 656-9. The occasion for publishing this quadrature was without doubt the publication of Gregory's Vera circuli et hyperbolae quadratura, on which Brouncker himself spoke at the meeting of the Royal Society on 27 February 1668/9; BIRCH, History of the Royal Society II, 253. The introduction to E1 suggests that Brouncker had long been in possession of the result: 'What the Acute Dr. John Wallis had intimated, some years since, in the Dedication of his Answer to M. Meibomius de proportionibus, vid. That the World one day would learn from the Noble Lord Brounker, the Quadrature of the Hyperbole; the Ingenious Reader may see performed in the subjoyned operation, which its Excellent Author was pleased to communicate, as followeth in his own words.' Cf. OLDENBURGBOYLE 17/[27].III.1667/8; OLDENBURG, Correspondence IV, 248-51, 248.

My Method for Squaring the Hyperbola is this:

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Let AB be one Asymptote of the Hyperbola EdC; and let AE and BC be parallel to th'other: Let also AE be to BC as 2 to 1; and let the Parallelogram ABDE equal 1. See Fig. 1. And note, that the Letter x every where stands for Multiplication.

1-2 (I doubt, . . . further trouble, omitted in E 906

pag. ult.: i.e. DULAURENS, Specimina mathematica, 256 (not paginated).

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Supposing the Reader knows, that EA. a£. KH. CB. &c. are in an Harmonic series, or a series reciproca, primanorum sen arithmetice proportionalium (otherwise he is referr'd for satisfaction to the 87, 88, 89, 90, 91, 92, 93, 94, 95, prop. Arithm. Infinitor. Wallisii907: I say

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in infinitum For (in Fig. 2, & 3) the Parallelog.

And (in Fig. 4.) the Triangl.

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907 Arithm. Infinitor. Wallisii: i.e. WALLIS, Arithmetic/1 infinitorum, Oxford 1656, prop. 87-95.

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| And that therefore in the first series half the first term is greater than the sum of the two next, and half this sum of the second and third greater than the sum of the four next, and half the sum of those four greater than the sum of the next eight, &c. in infinitum. For \dD = br + bn; but bn > fG, therefore \dD > br + fG, &c. And in the second series half the first term is less then the sum of the two next, and half this sum less then the sum of the four next, &c. in infinitum. That the first series are the even terms, viz. the 2 d , 4 th , 6 th , 8 th , 10th, &c. and the second, the odd, viz. the 1st, 3d, 5 th , 7th, 9 th , &c. of the following series, viz. ^ ^3, ^, ^ ^, g^, &c. in infinitum = 1. Whereof a being put for the number of terms taken at pleasure, -^TT^ is the last, ^n- is the sum of all those terms from the beginning, and ^kthe sum of the rest to the end. 456

[647]

197. BROUNCKERto OLDENBURG, March? 1667/8 That | of thefirstterme in the third series is less than the sum of the two next, and a quarter of this sum, less than the sum of the four next, and one fourth of this last sum less than the next eight, I thus demonstrate. Let a = the 3d or last number of any term of the first Column, viz. of Divisors,

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And 48o4 — 192a3+240a2—96a = Excess of the Numerator above Denomin. But The affirm. That is, 48o4 + 240a2 Because a4 + 5a2 a3 + 5a

> > > >

the Negat. 192a3 + 96a 4a3 + 2a 4a2 + 2

if a > 2. 15

Therefore Therefore | of any number of A: or Terms, is less than their so many [648] respective B, that is, than twice so many of the next Terms. Quod, &c.

By any one of which three Series, it is not hard to calculate, as near as you please, these and the like Hyperbolic spaces, whatever be the Rational Proportion of AE to BC. As for Example, when AE is to BC, as 5 to 4. (whereof the Calculation follows after that where the Proportion is, as 2 to 1. and both by the third Series.) First then when (in Fig. 1.) AE.BC :: 2.1.

10

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197. BROUNCKER to OLDENBURG, March? 1667/8

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2 x 3 x 4) 1. (0.04166666664 x 5 x 6) 1. (0.00833333336 x 7 x 8) 1. (0.00297619048 x 9 x 10) 1. (0.001388888810 x 11 x 12) 1. (0.000757575712 x 13 x 14) 1. (0.000457875414 x 15 x 16) 1. (0.000297619016 x 17 x 18) 1. (0.000204248418 x 19 x 20) 1. (0.000146198820 x 21 x 22) 1. (0.000108225122 x 23 x 24) 1. (0.000082345224 x 25 x 26) 1. (0.000064102626 x 27 x 28) 1. (0.000050875128 x 29 x 30) 1. (0.000041050930 x 31 x 32) 1. (0.000033602132 x 33 x 34) 1. (0.000027852034 x 35 x 36) 1. (0.000023342636 x 37 x 38) 1. (0.000019756638 x 39 x 40) 1. (0.000016869140 x 41 x 42) 1. (0.000014518042 x 43 x 44) 1. (0.000012584344 x 45 x 46) 1. (0.000010979346 x 47 x 48) 1. (0.000009636148 x 49 x 50) 1. (0.000008503450 x 51 x 52) 1. (0.000007541552 x 53 x 54) 1. (0.000006719354 x 55 x 56) 1. (0.000006012556 x 57 x 58) 1. (0.000005401458 x 59 x 60) 1. (0.000004870460 x 61 x 62) 1. (0.000004406862 x 63 x 64) 1. (0.0000040002-

0.0416666666 0.0113095237

0.0029019589

0.0007306482

0.0001829939

21 42 x 43 x 44) 1. (0.0000025843- corr. ed.

458

197. BROUNCKERto OLDENBURG, March? 1667/8 0.0416666666 0.0113095237 0.0029019589 0.0007306482 3) 0.0001829939 (0.0000609980 0.05679179 + 0.00006100 0.05685279 < EdCy But 0.0007306482 0.0001829939 0.0000458315

10

Therefore 0.05679179 + 0.00004583 + 0.00001528 0.05685290 > EdCy.

15

For, it has been demonstrated that | of any terme in the last Column is less than the terme next after it; and therefore that, of the last terme, at [6491 which you| stop, is less than the remaining terms, and that the total of these is less than | of a third proportional to the two last. And therefore ABCyE being =0.75 0.75 and EdCy > 0.05685279- and < 0.05685290 And ABCdE is < 0.69314720- and > 0.69314709 But when AE.BC :: 5.4. or as EA. to KH. then will the space ABCE. or now, the space AHKE (AH = \AB.) be found as follows. 8 x 9 x 10) 1 (0.0013888888 16 x 17 x 18) 1 (0.0002042484 18 x 19 x 20) 1 (0.0001461988 32 x 33 x 34) 1 (0.0000278520 34 x 35 x 36) 1 (0.0000233426 36 x 37 x 38) 1 (0.0000197566 38 x 39 x 40 1 0.0000168691

5

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25

0.0003504472

0.0000878204

459

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197. BROUNCKERto OLDENBURG, March? 1667/8

5

0.0013888888 0.0003504472 3) 0.0000878204 (0.0000292735 0.0018271564 + 0.0000292735 0.0018564299 < Eab But 0.0003504472 0.0000878204 0.00002200737

10

15

20

Therefore 0.0018271564 + 0.0000220074 + 0.0000073358 0.0018564996 > #06 Therefore EMb. (Fig. 4.) being =0.025 Eab > 0.0018564299EMba (Fig.4.) or EKM (Fig.l.) > 0.02685643AHKM< 0.22314356-

0.025 & < 0.0018564996 < 0.02685650 > 0.22314349

Therefore 3 ABCdE= 2.07944154 Therefore the Logar. of 10 and AHKE = 0.2231435is to the Log. of 2 ABCdE (when AE.BC:: 10.1.) = 2.3025850as 2.302585 to 0.693147

1 0.003888888 corr. ed. 21 ABCdE (when AE.BC :: 10.1.) = 2.025850 corr. ed. 460

198. [WALLIS?] to OLDENBURG, March/April ? 1668

198. [WALLIS ?] to HENRY OLDENBURG [Oxford], March/April? 1668 Transmission:

w Copy (in unknown hand) of (missing) note sent: LONDON Royal Society Early Letters Wl, No. 37, 1 p. At top of page, in the same hand: 'I have communicated an Extract of yours of the 10th of march last, to one of our Analysts, whose opinion concerning it is this.' On reverse in Oldenburg's hand: 'Answer to Du Laurens about the Analyt. way of Demonstrating.' This short paper constitutes a reply to a question raised by Dulaurens in DULAURENSOLDENBURG [29.IIj/10.III.1667/8, OLDENBURG, Correspondence IV, 214-15. Oldenburg translated the mathematical section of Dulaurens's letter into English {Royal Society Early Letters L5, No. 15), possibly to disguise the author. The style employed in the answer suggests that it might have originated from Wallis.

The Solution of a geometricall Probl. requires Construction & demonstration, & that synthetically by arguing with knowne quantities onely, not analytically, which argueth by knowne & unknowne mixt together. The proper office of the Analysis in solving a Probl. is to find out a Canon to direct how the Construction may be made, but the Canon doth not alwaies discover all the Determinations requisite to make the Probl. & its Construction possible, for experience will shew that oftentimes in forming the Construction, even of plane Problemes, such objections will start & rise up against the possibility of the Construction, which cannot be solved by any thing apparent either in the Canon or the Analysis. So if a Triangle be to be made of three right lines, although one of them be rightly found out by Construction according as the Canon directs & the other two were first given in the Probl. yet before the Triangle can be described it must be proved synthetically that every two of those 3 lines taken together as one right line are longer than the third, which many times is a worke of greater difficulty than the finding out of the Canon. The Analyticall Resolution indeed if geometrically argued may be evidence enough to convince a Geometrician whether he be an Algebrician or not that the Canon or Theor. deduced from it is true. But a Geometrician will hardly allow a

14 as one right line add. 19 allow a (1) pro breaks off (2) geometricall

461

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199. PHILIPS to WALLIS, 6/[16] April 1668

5

geometricall Probl. to be sufficiently solved unlesse it be performed by Composition, towards the finding out whereof Analysis is an admirable guide, but not sufficient of it selfe to make a compleat solution of a geometricall Probl. And as to a Syntheticall regresse by the steps of the Analysis of a geometricall Probl. it is not alwaies so easy to goe backwards as forwards, nor is it possible, when any terme in any Equation or Analogy of the Analysis exceeds geometricall dimensions.

199.

HENRY PHILIPS to WALLIS [London], 6/[16] April 1668 Transmission:

C First part of letter sent (figure and remaining part containing tables and last two paragraphs missing): LONDON Royal Society Early Letters PI, No. 52, 2 pp. At top of p. 1 in Oldenburg's hand: 'A letter written to Dr John Wallis by Mr Henry Philips, containing his Observations about the True Time of the Tydes.', and: 'Read before the Society Apr. 9. 1668.' Endorsement on left margin: 'Entered L.B. 3.367.' c1 Copy of letter sent (first part in scribal hand, tables and last two paragraphs in Oldenburg's hand): LONDON Royal Society Letter Book Original 3, pp. 367-71. c2 Copy of c1: LONDON Royal Society Letter Book Copy 3, pp. 454-9. E First edition of letter sent: Philosophical Transactions No. 34 (13 April 1668), 656-9 ('A Letter written to Dr. John Wallis by Mr. Henry Philips, containing his Observations about the True Time of the Tides') (our source). The present letter, in which Philips gives an account of his observations concerning the true time of the tides, was read at the meeting of the Royal Society on 9 April 1668; BIRCH, History of the Royal Society II, 264.

Worthy Sir, Being desired by Mr. O.908 to give in, what informations I could, concerning the Tides, I have made bold to present this Paper to 10 your Consideration; which though it have little or no relation to your more curious Philosophical Experiments, yet, I hope, will be of very good

2 an (1) eas breaks off (2) admirable 6 it add. 908

Mr. O.: i.e. Henry Oldenburg.

462

199. PHILIPS to WALLIS, 6/[16] April 1668 use for the finding out the True time of the Tides at all times of the Moone, which is (I conceive) of as great concernment, as any thing in the Motion of the Tides. For, this time of the Tides, though it be a very necessary thing to be known, yet is very rudely and slightly reckoned up by most Seamen and Astronomers; most of them reckoning, as if the Moone being upon such a set point of the Compasse (as the Seaman calls it) or so many houres past the Meridian (as the Almanack-Makers reckon) it were High-Tide in such and such a Port at all times of the Moone. And thus they reckon the Tides every day to differ constantly 48 m. As for instance; A South- West Moone makes a full Tide at London, that must be understood, that it is High-Tide at London when the Moon is three hours past the Meridian. Now this is true indeed at the New and Full Moon, but not at other times of the Moone, which few take any notice of: only Mr. Booker909 had wont to give this Caveat, that about the first and last quarters of the Moone, the Neap-tides did not flow so long as the Spring-tides by one point of the Compasse; but he gives no rule to proportion the difference. But observing this more narrowly, I find, that at London the Tides fall out at the least two points, that is, one hour and an half sooner, in the Quarters then in the New and Full Moone. Now this being a very considerable difference of time, which might very well make many Seamen and Passengers to lose their Tides, I set my self to watch this difference of the time of the Tides, and to find out some Rule, how to proportion the time of the Tides between the Spring-tides and the Neap-tides, and I [657] found by many trialls, that the true time of the Tides might be found| out to be somewhat shorter and shorter, from the New and Full Moone unto the Quarters; yet not in an equall manner, neither gradually decreasing from the New and Full Moone untill the Quarters; but rather, that there was some little difference of alteration both at the New and Full Moones, and also at the Quarters; and that the greatest difference fell out in the midst between them, agreeing very well to a Circular proportion, after this manner: (See Fig. 5.) First, Divide a Circle into 12 Equal parts, or hours, according to the Moones motion or distance from the Sun, from the New Moone to the Full. Secondly, Let the Diameter of the Circle be divided into 90 parts or 909 Booker: i.e. John Booker, an almanack-maker. See TAYLOR, Mathematical Practitioners, 362.

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199. PHILIPS to WALLIS, 6/[16] April 1668

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min. that is, according to the time of the difference of Tides between the New or Full Moone, and the Quarters; which is one hour and an halfe. Thirdly, Make perpendicular lines cross the Diameter of the Circle, from hour to hour. Fourthly, Reckon the time of the Moones coming to the South in the circumference of the Circle, and observe the Perpendicular-Line, that falls from that point upon the Diameter; and the proportionall Minutes, cut thereby, will shew, how many Houres, or Minutes are to be substracted from the time of Hightides at the New and Full Moone, that so you may have the true time of the Tides that present day. For Example; At London, on the day of New and Full Moone, it is High-Tide at London at 3 of the Clock, that is, when the Moone is three hours past the Meridian: and so by the Common Rule, the Moone being about four dayes old, it will be South about three of the Clock, and it will be High-tide three houres afterwards, that is, at 6 of the Clock. But now by this Rule, if you count this time of the Moones coming to the South in 464

199. PHILIPS to WALLIS, 6/[16] April 1668 the Circumference, the perpendicular-line, which comes from 3 to 9, cuts the Diameter in the halfe, at 45 min. which shews, that so much is to be abated from the time of High-tide in the New and Full Moones; So that it is High-tide 45 min. before 6 of the Clocke, that is, at 5. hours 15 min. and not at 6 of the clock, according to the common-Rule. The like you may do for any other Port or place, knowing the| time [658] of High-water at the New and Full Moon in that place: And you may do it the more readily, if you set down the time of Highwater at the New and Full Moon under the Diameter, as I have done for London, where it is high-tide at III. of the clock. So that when the Moon is South at III. of the clock, the perpendicular cuts the diameter at II. hours 15. m. which added to the time of the Southing makes it V. hours 15. m. and so when the Moon is South at IX. of the clock, by adding 2 h. 15. m. you have the time of highwater, which is XL of the clock and 15m. And thus you may easily make a Table, which by the Southing of the Moon, shall readily tell you the time of High-tide at any time of the Moone, as I have done here for London: To which all other places may be reduced to correspond. Moon South

Tide

Moon

Tide

Moon

Tide

London

London

H. 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5

South H. M. VI 0 10 20 30 40 50 VII 0 10 20 30 40 50 VIII 0 10 20 30 40 50

London

H. M. XII 0 10 20 30 40 50 I 0 10 20 30 40 50 II 0 10 20 30 40 50

South H. M. Ill 0 10 20 30 40 50 IV 0 10 20 30 40 50 V 0 10 20 30 40 50

M. 0 9 18 27 36 45 54 2 9 16 23 30 37 44 50 57 3 9

H. 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7

M. 15 21 27 33 40 46 52 59 6 13 20 28 36 44 53 2 11 20

465

H. 7 7 7 8 8 8 8 8 9 9 9 9 9 10

M. 30 41 52 4 14 25 36 48 0 13 26 39 53 6

10 10 10 11

20 33 47 1

Moon South

H. M. IX 0 10 20 30 40 50 X 0 10 20 30 40 50 XI 0 10 20 30 40 50

5

10

15

Tide London

20

H. M. 11 15 11 11 11 12 12 12 12 1 1 1 1 1 2 2 2 2 2

29 43 57 10 24 37 50 3 16 29 42 54 3 16 27 38 49

25

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199. PHILIPS to WALLIS, 6/[16] April 1668 [659]

20 23 30 31 33 34 35

In In In In In In In

column column column column column column column

'Octo.': 3 34 c1 E corr. ed. 'Sept.': 8 38 c1 'Jan.': 6 5 corr. ed. 'Sept.': 4 44 c1 'Mar.': 12 12 c1 'Jan.': 11 42 c1 In column 'July': 4 06 c1 'Aug.': 5 2 c1 In column 'Dec.': 8 36 c1

466

200. CONSTANTIJN HUYGENS to WALLIS, [21 June]/I July 1668

These things I have found to fall out right at London for many years, and so I suppose they may in other places. If the difference be not so much between the Neap-tides and Spring-tides in other places, the Diameter must be divided into fewer parts. As for the highest Tides to happen two or three dayes after the full Moon, I have not made much observation of it, and see little reason for it, but the time thereof agrees here with. And high Spring-tides are not alwayes alike; this year I have not observed any. I should be glad to hear, how these rules hold in other places, that so this true time of the Tides may be more punctually known.

5

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April 6. 1668.

200. CONSTANTIJN HUYGENS to WALLIS

The Hague, [21 June]/l July 1668 Transmission:

C Draft of (missing) letter sent: THE HAGUE Koninklijke Bibliotheek KA XLIV, pp. 618-19, No. 507 (our source). At top left of p. 618 in Huygens's hand: 'Doctor! Joanni Wallisio. Oxon. 1. Jul. 1668.'—printed: CONSTANTIJN HUYGENS, Bnefwisseling III, 233. c Copy of C: THE HAGUE Koninklijke Bibliotheek KA XLV, f. 177v-178r. The present letter is a letter of recommendation written by Constantijn Huygens for his nephew, Maurits le Leu de Wilhem, at the time on a journey through Europe. Cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668.

Amplissime doctissimeque vir, de tot humanitatis vestrae officiis, quibus me indignum hospitem excepistis, cum ante triennium ad vos excurrerem910, praecipuum illud existimavi, quod rogare me, et quidem serio, ut videbatur, dignati estis ut

5 higest corr. ed. 13 indignum (1) (—) (2) hospitem 15 ut videbatur, add. 910 excurrerem: i.e. Constantijn Huygens's visit to Oxford in 1664. Cf. WALLISOLDENBURG 3/[13].XI.1668; WALLIS-HUYGENS 31.VIII/[10.IX].1668.

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200. CONSTANTIJN HUYGENS to WALLIS, [21 June]/I July 1668

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opusculorum, quae in lucem subinde aliud agens dedi, exemplaria illustrissimae vestrae Bibliothecae sinerem inferri. Memini equidem me respondere, magnam eorum partem mihi vernaculam esse, vobis barbaram, non destitistis tamen ab incepto: ut qui complurium lucubrationum exocticarum copiam in illo omnis disciplinae et eruditionis horreo dicebatis asservari. Parui itaque, et recepi facturum me quod hodie praesto. Non expecto ut quaeratis, quamobrem toto triennio malum nomen fuerim. Dicant hoc dissidia nostra, et acerrima et pudendissima fratrum odia, quibus invicem nos laceratum ivimus. Nimirum, viri amicissimi ne in amicorum numero essetis mihi, neque Regnorum vestrorum, neque hujus Reip. ratio passa est. Quod felix, faustum ac perpetuum sit, deferbuit ille aestus, et, ut voveo speroque, amantium ira amoris redintegratio erit. Redeo itaque ad vos eodem vultu atque animo, quo ante non solos tres, sed et paene quinquaginta annos Oxonios colles salutavi911. Accipite, rogo, constantis amicitiae leves tesseras, quando hoc voluistis; par est quid intrivistis etiam vos exedere. Mihi non exiguum operae pretium erit in turba tot insignium virorum vestra autoritate censeri, nisi quidem vos oblati loculi in illo Theatro tandem coeperit poenitere. Quod si factum est, quaeso et obtestor per candorem vestrum ne id latere sinatis Juvenem912 hunc, meum e sorore nepotem, qui his munusculis passus est se onerari. Non gravabitur autori reportare si quid vobis agnoverit minus gratum esse. Ipsum interea vestrae omnium comitati ex animo commendo:| Tuae im- [619]

2 equidem add. 6 recepi (1) hoc (2) facturum |me add. quod (a) ecce (6) hodie 8 et ( 1 ) acus (2) acerrima 9 amicissimi (1) in. hostium (2) ne in amicorum 10 neque (1) hui Reip. conditio (2) hujus Reip. ratio 13 atque animo add. 15 est (1) exedere vos quid intrivistis. (2) quid intrivistis etiam vos exedere. 16 pretium (1) est (2) erit 17 virorum (1) (—} (2) |vestra autoritate add.\ censeri, nisi jquidem add.\ vos 18 in (1) Theatro vestro dum poeniteret (2) illo Theatro tandem coeperit poenitere 18 factum |est, add.\ (1) rogo (2) quaeso et obtestor 20 qui (1) (his) (2) his 20 onerari. (1) Ipsum (2) Non gravabitur (a) domum referre (6) autori reportare 911

salutavi: i.e. Constantijn Huygens's visits to Oxford in 1618 and 1622. Juvenem: i.e. Maurits le Leu de Wilhem (b. 1641), nephew of Constantijn Huygens the Elder. Cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668. 912

468

201. OLDENBURG to WALLIS, 30 June/[10 July] 1668 primis, mi Wallisi, et excellent! Collegae Bibliothecario vestro. Gestit in magnorum virorum amicitias et consortia admitti et, si qua fides Avunculo, sua virtute, doctrina, morumque suavitate ac modestia invenietur illis non indignus. Vale vir maxime et me ama, quasi nunc ex postliminio. Scrib. Hagae Com. Calendis nostris quintilibus MDCLXIIX.

201. HENRY OLDENBURG to WALLIS 30 June/[10 July] 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 2/[12].VII. 1668 (i) and WALLIS-OLDENBURG 6/[16].VIL1668. This letter contained a copy of DULAURENS, Responsio ... ad epistolam D. Wallisii, [1668]. It was received by Wallis on 2/[12] July.

202. WALLIS to JOHN COLLINS early July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS-COLLINS 21/[31].VII. 1668 and COLLINSPELL 18/[28].VIL1668. Evidently written before COLLINS-BRERETON 11/[21].VII.1668. Answered by: CoLLiNS-WALLis 14/[24].VII. 1668. As emerges from COLLINS-PELL 18/[28].VII. 1668, this letter contained as an enclosure Wallis's catalogue of 145 errata which he had found in proofs of Brancker's Table of Incomposit Numbers, less than 100,000. Collins made a copy and sent this to Pell as an enclosure to COLLINS-PELL 18/[28].VII. 1668. Pell in turn sent a list of those

1 in add. 2 consortia (1) intromitti (2) admitti 2 Avunculo (1) est, (2) , sua 3 invenietur (1) ( } (2) illis non indignus 4 ama, (1) vel (2) quasi nunc

469

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203. WALLIS to OLDENBURG, 2/[12] July 1668 (i) ten errata which had not been recognized previously to Brancker (PELL-BRANCKER 21/[31].VII. 1668). However, they arrived too late to be incorporated into the table as printed (with separate pagination) at the end of An Introduction to Algebra, London 1668. The errata (with one exception) were first published by Wallis in A Discourse of Combinations, Alternations, and Aliquot Parts, London 1685 (136), together with twenty others, corresponding to the list Collins added to the foot of his copy.

203. WALLIS to HENRY OLDENBURG Oxford, 2/[12] July 1668 (i) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 46, 2 pp. (our source). On p. 2 Oldenburg has noted to the right of address at 90°: 'For Transactions 1. The writ of Stubs913. 2. Gregory of sight914. 3. Dr. Wallis'.—printed: OLDENBURG, Correspondence IV, 488-9. Reply to: OLDENBURG-WALLis 30.VI/[10.VII].1668. Enclosure: WALLis-OLDENBURG 2/[12].VIL1668 (ii).

Oxford July. 2. 1668.

Sir,

5

I received yours915 this morning (for which I thank you) & have since finished & transcribed the inclosed916, as here you have it; in answere to the paper917 inclosed in yours. Which I purpose to send you to morrow morning by the Coach, together with the Printed paper & the former

5 the (1) inclosed (2) paper 913

writ of Stubs: probably STUBBE, 'The Remainder of the Observations made in the formerly mention'd Voyage to Jamaica, publisht Numb. 36', printed in Philosophical Transactions No. 37 (13 July 1668), 717-22. 914 Gregory of sight: possibly 'An Extract of a Letter concerning an Optical Experiment, conducive to a decayed Sight, communicated by a Worthy person, who found the benefit of it himself, printed in Philosophical Transactions No. 37 (13 July 1668), 727-9, and 'An Extract of another Letter from the same hand, confirming the contents of the former; and adding some other Observations about Sight', printed ibid., 729-31. 915 yours: i.e. OLDENBURG-WALLIS 30.VI/[10.VII].1668. 916 inclosed: i.e. WALLIS-OLDENBURG 2/[12].VII.1668 (ii). 9lr paper: i.e. DULAURENS, Responsio . . . ad epistolam D. Wallisii ad darissimum virum Oldenburgium scriptam, [1668]. 470

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) letter918; rather then by Post, both because this comes sooner, & because the whole would be too big a packet for the Post. This I now send, with the former letter, is as much as will be convenient for you to insert into one occurrence. If he919 have not inough of this; he may have more hereafter if there be occasion. If he have too much of this, he may thank himself. I

5

am Sir yours &c John Wallis. [2] These For my worthy friend, Mr Henry Oldenburgh, at his house about the middle of the Old Palmal near

10

St. James's

15

London.

204.

WALLIS to HENRY OLDENBURG Oxford, 2/[12] July 1668 (ii) Transmission:

W Paper sent: LONDON Royal Society Early Letters Wl, No. 47, 3 pp. (our source). — printed: OLDENBURG, Correspondence IV, 489-92 (Latin original), 492-5 (English translation). E Printed version (with corrections and amendments incorporated): Philosophical Transactions No. 38 (17 August 1668), 744-7 ('Some Animadversions, written in a Letter by Dr. John Wallis, on a printed Paper, entitul'd Responsio Francisci du Laurens ad Epistolam D. Wallisii ad Cl. V. Oldenburgium scriptam.'). Enclosure to: WALLIS-OLDENBURG 2/[12].VII.1668 (i). The present letter, although addressed to Oldenburg, is in fact a reply to Dulaurens's Responsio ad epistolam D. Wallisii, which in turn was a response to the pub1 Post, (1) as well (2) both 1 & (1) the (2) because 918 former letter: i.e. WALLIS-OLDENBURG 30.III/[9.IV].1668, the remaining parts of which were now to be printed. 919 he: i.e. Dulaurens.

471

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) lication of the first part of WALLIS-()LDENBURG 30.III/[9.IV].1668, containing Wallis's review of Dulaurens's Specimina mathematical, in Philosophical Transactions No. 34 (13 April 1668). The printed version of the present letter, including the corrections and amendments given in WALLIS-()LDENBURG 4/[14].VII. 1668 and in WALLISOLDENBURG 6/[16].VII. 1668, appeared in Philosophical Transactions No. 38 (17 August 1668), where it is followed by the second part of WALLIS-OLDENBURG 30.III/[9.IV].1668.

Oxonii Julii 2°. 1668. Doctissimo Amicissimoque D. Hen. Oldenburgio, S. 5

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Epistolam tuam920 (Vir Clarissime,) quae Dulaurensii impressam chartarn921 (quae me spectabat) habuit inclusam; accepi hodie. Cui et statim respondendum censui: neque enim longa deliberatione opus est. Expetebas Tu a me non ita pridem literis tuis922, (quod meministi probe,) ut, quid ego de Dulaurensii libro923 turn nuper edito (quern cum illis literis in eum finem mittebas) sentirem, tibi paucis exponerem. Quod cum ego privatis ad te literis924 post quatriduum missis fecerim, (quippe hoc amico expetenti negandum non putavi,) tu harum partem aliquam typis vulgandam paulo post curasti925, (earn nempe quae illatam mihi injuriam expostulabat,) reliqua reticendo. Haec homini bilem movent. Quibus ego haec summatim repono. Turn mihi fuisse liberum, amico expetenti, libere quid sentirem exponere: turn et Te arbitrio tuo usum esse. Speciatim vero, quod ad injuriam spectat quam mihi factam quere6 longa missing in E 7 Expectabas E 7 (quod meministi probe,) missing in E 10 post quatriduum missis missing in E 12 typis vulgandam paulo post curasti, add. W 13 reliqua, in sui, credo, gratiam, reticendo. E 920

Epistolam tuam: i.e. OLDENBURG-WALLIS 30.VI/[10.VII].1668. impressam chartam: i.e. DULAURENS, Responsio . . . ad epistolam D. Wallisii, [1668]. 922 literis tuis: i.e. OLDENBURG-WALLIS 26.III/[5.IV].1668. 923 libro: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667. 924 literis: i.e. WALLIS-OLDENBURG 30.III/[9.IV].1668. 925 harum partem aliquam vulgandam paulo post curasti: i.e. the publication of the first part of WALLIS-OLDENBURG 30.III/[9.lV].1668 in Philosophical Transactions No. 34 (13 April 1668). 921

472

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) bar; dum me tanquam Thrasonem926 aliquem inducit, Problema leviculum (et quidem, prout ipsius verbis exponitur, ridiculum,) totius Europae Mathematicis proponentem; quo de me triumphum ageret, monstrando, quam ille potis sit solvere: Non diffitetur, errasse se927. (Et quidem res ipsa clamat: quippe non ego, sed Montfertius nescio quis Gallus, illud quod innuit Problema, Anglis proposuerat: quod varii variis modis solutum dederunt; inter quos et ego.) Hoc tantum causatur, quod Amiens928 nescio quis tale quid (non dicit, id ipsum,) ipsi retulerat. Sed quorsum est, ut Amicum advocet; cum, quid ego ea in re fecerim, jam palam prostet in scriptis meis editis. Et quidem suspicor suum tale quid quod ab Amico acceperat, non aliud fuisse, quam, hujusmodi Problema in scriptis meis a me Solutum extare; pro quo ille (pari atque in reliquis negligentia) a me Propositum substituit; additque de suo, Totius Europae Mathematicis, quo et jactantior Thraso, et Triumphus suus sit illustrior. Utut sit: hoc eum male habet, quod non simpliciter negaverim me istius Problematis authorem esse; sed, (quod garrulitatem vocat,) mea verba cum suis juxta ponendo, ostenderim, quam mihi manifesto fuerit injurius. De caeteris autem; Non placet ei, quod de suis Ego Censuram instituerem: Hoc est, Nollet ut ego tibi petenti dicerem, quid de libro edito [2] sentirem. Sed quidni liceat? Nam et idem alios sentire, tu etiam nosti.| Male habet etiam929, quod eum censuerim plus fronte polliceri, quam opere absolvent; (nempe hoc mollius sonare putaram, quam si dixissem Parturiunt monies, &c.930) Sed et tu alios juxta mecum sentire, Fastuoso Titulo931 Librum haud satis respondere, non ignoras; nedum in exauctorati Euclidis vices successurum. Neque prius illis fidem faciet, rem se-

7 dederint E 1 Amicus quidam tale E 14 suus (1) sit ill breaks off (2) sit W 17 ponendo, (1) ostenderem (2) ostenderim W 21 polliceri, (1) quam (2) quam W 22 putaram, add. W 926

Thrasonem: i.e. the boastful soldier in Terence's Eunuchus. Non diffitetur, errasse se: see DULAURENS, Responsio ad epistolam D. Wallisii, 6-7. 928 Amicus: i.e. Prenicle; see DULAURENS-OLDENBURG [13]/23.V.1668 (OLDENBURG, Correspondence IV, 398-400, 399). 929 Male habet etiam: see DULAURENS, Responsio ad epistolam D. Wallisii, 1. 930 Parturiunt montes, &c.: HORACE, Ars poetica 139. 931 Fastuoso Titulo: cf. the complete title given in WALLIS-OLDENBURG 30.Ill/ [9.IV].1668. 92r

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204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii)

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cus esse; quam viderint, Genuina Matheseos Principia, et Elementa Vera, (quae hucusque nondum tradita insinuat,) ab eo felicius tradi, quam ea tradiderint superiores. Dixeram, partem magnam ex Oughtredi meisque scriptis (utut neutrius meminerit) desumptam VIDERI; (nempe propter multa quae nobiscum habet communia, et peculiares loquendi formulas ipsaque eadem Symbola passim retenta;) item ex Vieta, Schotenio, aliisque ab eo editis, quorum et subinde meminit. Sed quo animo quove consilio haec dixerim, dicit932 se plane non capere. Dicam: (imo res ipsa dicit, quippe hoc inde directe sequitur:) Nempe ut tibi ostenderem, sua non esse Nova omnia, et hucusque nondum tradita. Rem ipsam quoad caeteros agnoscit, (quibus et Hariotum accenset;) Mea tantum scripta, dicit933 se non legisse; quod excusatum petit. (Esto. Et habeo excusatum. At interim non eo magis inter hucusque nondum tradita censenda erunt, quod ipse non legerit.) Et quidem, Oughtredum quod spectat, enumerat aliquammulta quae jam fatetur ex ipso quasi verbatim variis in locis transcripta; atque excusatum it quod Authorem non nominaverit. (Unde me conjecturam non temere fecisse, satis constat.) Sed negat ea partem magnam (respectu totius) dicenda esse. (Patior itaque ut, pro parte magna, modo id dictum malit, partem potissimam rescribat.) Verum ego, non Numero Verborum, sed rerum pondere, partem magnam aestimo; nee ea tantum ex Oughtredodesumpta Videri existimo, quae totidem Verbis apud eum extant; sed totam earn doctrinam, utut aliis verbis expositam, quae ab ipso jam ante multos annos tradita fuerit, quamque ex eo hausisse videri possit hie Author: licet hie pluribus forte paginis, quam ille lineis, rem eandem explicaverit. Id itaque dictum velim; Magnam partem earum Rerum quae hie traduntur, apud Oughtredum(ne et mea scripta interponam) vel totidem verbis extare, vel verbis tantundem significantibus, vel inde posse levi negotio deduci; ut non pro Rebus hucusque nondum traditis censeri debeant: Sed et hinc desumptam (rem ipsam quod spectat, licet variata nonunquam verborum formula,) videri' propter easdem non raro peculiares loquendi

2 ab ipso E 7 ab (1) eod breaks off (2) eo W 10 et (1) hactenus (2) hucusque W 24 licet |hie add.\ pluribus (1) fortasse (2) forte W 932 933

dicit: see DULAURENS, Responsio ad epistolam D. Wallisii, 5. dicit: see DULAURENS, Responsio ad epistolam D. Wallisii, 4. 474

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) [3]

formulas retentas, ipsaque eadem non raro symbola. Quae quidem tam| aperta sunt vestigia unde haec desumpta sint; ut jam non possit ipse non fateri, utut nomen prius retinerit. Atque eadem, servata proportione, de caeteris quos dixeram intellige. Non enim ego utrobivis intenderam crimen Plagii, (quod ipse amoliri vellet,) sed ut tibi dicerem, quod res est, Principia sua, quatenus sana sunt, turn et aliis fuisse pridem cognita, turn et ab aliis dudum tradita, rem ipsam quod spectat, utut sub aliis verborum formulis; neque jam primitus detecta, atque hucusque nondum tradita. Sed et tibi digitum intendi, apud quos authores haec eadem negotia reperias ipse; et quidem, prout ego sentio, non minus feliciter exposita. Cui concinit, quern ad me de illo characterem scripto misit Vir quidam Mathematicus934, tibi non ignotus, (Dulaurentio non inferior,) priusquam ego librum videram, nee interogatus quidem: Algebram (inquit) Dulaurentii, ad D. Oldenburg transmissam, vidi: qui autem Tua Chartesiique et hujus Interpretum scripta viderunt, Authorem credo hunc non sunt admiraturi: Quasi quidem ego, non tarn Censere dicendus sim, quam Consentire. Sed conqueritur porro935, quod dixerim inibi reperiri aliqua parum sana, et minime accurata multo plura. Quorum alterum jam fatetur ipse, (ut non tibi fuerim hac ex parte iniquus index;) alterum non-dum. Neque tarn conqueritur quod haec censuerim, quam quod hujus censurae causas in publicum non protulerim; quippe si hoc fecissem, turn de publico, turn de seipso, gratiam (inquit) meruissem. Hoc autem crimine tuum est me levare. Quippe ego neutrum horum in publicum protuli; sed at Te utrumque. Rogatus utique a te sententiam; ego privatis ad te literis936, et quid censerem paucis indicavi, et cur ita. Addebam scilicet (non quidem justam totius libri confutationem; neque enim id agebam: sed) pauca specimina eorum quae cursim legenti occurrebant vel parum sana, vel minus 1 tarn missing in E 3 reticuerit. E 10 Cui sententiae concinit, E 12 (Dulaurensio, credo, non inferior) E 18 alterum add. W 19 iniquus Judex E 21 non (1) protulerit (2) protulerim W 23 levare. Rogatus enim a Te sententiam, Ego datis ad Te literis E 934

Vir quidam Mathematicus: almost certainly John Collins, who had sent an account of Dulaurens's Specimma mathematica in COLLINS-WALLIS 25.II/[6.III].1667/8. 935 conqueritur porro: see DULAURENS, Responsio ad epistolam D. Wallisii, 5. 936 privatis ad te literis: i.e. WALLIS-OLDENBURG 30!II/[9!V].1668.

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accurate dicta. Quo autem consilio tu, cum partem horum in publicum emiseris (quo forte illatam mihi injuriam utcunque elueres,) reliquum reticueris, (quod in illius gratiam factum putaverim,) cum et totum vel emittere vel reticere potuisses; tu melius noveris. Quoniam vero et ille hoc expetit; per me licebit ut tota Epistola, prout scripta fuerat, quae et tuae potestatis (utpote ad te scripta) facta est, simul prodeat. Ut judicet orbis literatus, numnon justas habuerim ita censendi causas, utut stricturis brevibus insinuatas. Neque enim novas ego jam adjungo; turn quod liber ipse mihi nunc prae manibus non sit; turn, si esset, non nova hanc ob causam recensione censerem indigere. Opprobria, reliquamque quam habet maledicendi copiam, non attingo: quoniam haec non aliud demonstrant magis, quam impotentem scribendis animuin: et me minus quam ilium feriunt. Tu interim (Vir Clarissime) Vive, et Vale. Tuus &c Johannes Wallis.

3 cum et . . . potuisses; missing in E 3 vel emittere vel add. W 5 et (1) ju breaks off (2) tuae W 8 insinuatas; atque resciscat ipse, postquam iram decoxerit, quam inibi & libere & candide egerim; libere tecum, & cum illo, satis candide. Novas ego jam non adjungo, turn E 10 recensione add. W 10-11 indigere. Neque enim mihi tune erat in animo ad vivum omnia resecare, necdum est. Id olim forte fiet, si necesse videbitur; quod non fore autumo, quippe non tanti res est. Quod ad Problema spectat, quod a me Freniclio, ut difficile, propositum innuit937, atque ab ipso solutum; rem secus atque est narrat. Patet utique, Scriptis editis, neque Freniclio a me propositum fuisse, neque ut difficile, Problema quod insinuat, aut etiam ut magni momenti; sed apud alium (cum Ego de Freniclio nih.il inaudiveram) obiter insinuatum, tanquam Fermatiano simile; (Vid. Commercium Epstolicum pag. 35. lin. 4. & seqq.) Quod autem Ego Problema meum depreciatum iveram, arripuit Freniclius, sponte sua, ut satis elegans, & solutione sua dignum. Quae quam aliena sint ab iis, quae hie narrat Du Laurens, cum ipse videas, non possum non rogare, ut imposterum velit ille in Historicis enarrandis fidelius agere, atque in tradendis Mathematicis accuratius. || Opprobria E 12 magis, add. W 937

Problema . . . , quod a me Freniclio ... propositum innuit: see DULAURENS, Responsio ad epistolam D. Wallisii, 7. Cf. WALLIS-DIGBY 21.XI/[1.XII].1657, especially the passage, to which Wallis explicitely refers in the following. 476

205. WALLIS to OLDENBURG, 4/[14] July 1668

205. WALLIS to HENRY OLDENBURG Oxford, 4/[14] July 1668 Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 48, 2 pp. (our source). Postmark on p. 2: 'JY/6'.—printed: OLDENBURG, Correspondence IV, 495-6.

July. 4. 1668. Oxford. Sir,

I sent you, on Friday morning by Moor's Coach, a packet conteining, a Letter938 in answere to Dulaurens's his printed paper, with the other papers returned as you desired. Which I hope was safely received. You may in that Answere, make these few alterations, if you think fit. In the end of the second paragraph; for reliqua reticendo, you may say reliqua (in sui, credo, gratiam) reticendo. In the fourth paragraph, about the middle, after the word Amicus, instead of nescio quis, put quidam. Toward the end of the 2d page; for Dulaurensio non inferior, say rather Dulaurensio, credo, non inferior. And presently after, for hujus interpretum, say ejus interpretum. Toward the end of the next paragraph, after these words, stricturis brevibus insinuatas, adde (in the same period,) atque resciscat ipse (postquam iram decoxerit) quam inibi et Libere, et Candide, egerim. (Libere, tecum, et, cum illo, satis Candide.) And begin the next period thus, Novas ego jam non adjungo, (instead of Neque enim novas jam adjungo,) and in the end of that paragraph, after indigere. adde Neque enim mihi tune erat in animo ad vivum omnia resecare; necdum est. To which you may subjoin this Paragraph. 6 may add. 8 gratiam (1) reti breaks off (2) reticendo 9 middle, (1) for (2) after 19 of (1) it, after in breaks off (2) that paragraph, 938

Letter: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii). Wallis dispatched this letter on 3 July 1668 in a packet together with WALLIS-OLDENBURG 2/[12].VII.1668 (i), Dulaurens's Responsio ad epistolam D. Wallisii, and WALLIS-OLDENBURG 30.III/[9.IV].1668. 477

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Quod ad Problema spectat, quod a me Freniclio, ut difficile, propositum innuit939, atque ab ipso solutum: rem secus atque est narrat. Patet utique (scriptis editis) neque Freniclio a me propositum fuisse, neque ut Difficile, Problema quod insinuat; aut etiam, ut magni momenti: sed apud alium (cum ego de Freniclio nihil inaudiveram) obiter insinuation, tanquam Fermatiano simile, (vide Commercium Epistolicum, pag. 35. lin. 4 et seqq.) Quod autem ego Problema meum depreciatum iveram; arripuit Freniclius, sponte sua, ut satis elegans, et solutione sua dignum. Quae, quam aliena sunt ab eis quae hie narrat Dulaurens, cum ipse videas: non possum non rogare, ut in posterum velit ille, in Historicis enarrandis fidelius agere, atque in tradendis Mathematicis accuratius. And what other alterations you, or My Lord Brouncker (to whom my service) shall think necessary: I permit to your discretions. Resting, Your very humble servant, John Wallis.

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[2]

These For Mr Henry Oldenburgh, in the Pelmell, near St. James's London.

3 utique (1) in scriptis (2) (in scriptis editis) (3) (scriptis editis) (a) nee (6) neque 4-7 apud (1) alium (2) alium |(cum ego de Freniclio nihil inaudiveram) add] obiter insinuatum, (a) ut Perm breaks off (b) tanquam Fermatiano simile. |(vide Commercium Epistolicum, pag. 35. lin. 4. et seqq.) add] Quod 7 depreciatum (1) (—) (2) iveram; arripuit Freniclius, |sponte sua, add. ut 10 velit (1) illius (2) ille, 11 in (1) tra breaks off (2) tradendis 13 think add. ed. 939

Problema . . . , quod a me Freniclio ... propositum innuit: see DULAURENS, Responsio ad epistolam D. Wallisii, 7. Cf. WALLIS-DIGBY 21.XI/[1.XII].1657, especially the passage, to which Wallis explicitely refers in the following. 478

206. OLDENBURG to WALLIS, 4/[14] ? July 1668

206.

HENRY OLDENBURG to WALLIS 4/[14] ? July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 6/[16].VII. 1668. This letter apparently contained a report on new notions about mean proportionals from Dulaurens, of which Oldenburg had been informed in JuSTEL-OLDENBURG [27.VI]/7.VII.1668 (OLDENBURG, Correspondence IV, 477-9, 478).

207. WALLIS to HENRY OLDENBURG Oxford, 6/[16] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 49, 2 pp. (our source). At top of p. 1 beneath date: 'Extr: enter'd LB. 2. 234'. At top right of p. 2 above address in Oldenburg's hand: 'An Extract of Dr Wallis's letter concerning Dr Wilkins's universal Character.', and beneath address, again in Oldenburg's hand: 'Received July 8. 68.' Postmark on p. 2: 'JY/8'.—printed: OLDENBURG, Correspondence IV, 508-9. w1 Extract of letter sent: LONDON Royal Society Letter Book Original 2, p. 234. w 2 Copy of w1: LONDON Royal Society Letter Book Copy 2, p. 274. Reply to: OLDENBURG-WALLIS 4/[14]?.VII.1668.

Oxford July. 6. 1668.

Sir, To your packet940 of Tuesday last (for which I thank you) which came to mee on Thursday before noon (when our Post was ready to go away) I dispatched an answere941 the same day before supper, & sent it you the next morning by Moor's coach; which should have been with you

6 should (1) be (2) have been 940 941

your packet: i.e. OLDENBURG-WALLIS 30.VI/[10.VII].1668. answere: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (i) and (ii).

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on Saturday: But I perceive by yours942 of (I suppose) Saturday night (though without date) you had not then received it. I gave you since notice of it (that it may be called for, if not timely delivered,) by mine943 of the last post: which also intimated some smal alterations which you may make. Nor do I think of adding more. Yours, which I received to day, doth not seeme to mee to alter the case at all. The Means Proportionall which the Letter944 intimates, do (certainly) not concern this book945. For I do not remember any thing of that nature (if the meaning be, of 2 mean proportionals between two extremes given, to be found Geometrically:) and having since sent for the book (which was out of my hands) I do not find any such thing in it. But it is (I suppose) some other thing that passeth in writing amongst them946. However; if he undertake that task, I am ready to venture one the Mathematicians side; qui gageroint qu'il se tromperoit9^7. I was thinking after those words, in the former Additions, — ad vivum omnia resecare; Necdum est; of adding Id olim forte fiet, si necesse videbitur. But I think they may be as well left out, unless you adde this allso, Quod non fore autumo; quippe non tanti res est. For I think time may be better imployed, than in large answering of such Books; though there bee matter inough to furnish it. If I must needs be put to that work (or to more replies) it would be convenient that wee know what those new notions are about mean proportionals, which he pretends to: which wil, I suppose, furnish us with new matter. Hevelius's Cometography948 I shal be glad to see.

5 more (1) , unlesse (2) . Yours 12 if lie undertake that task, add. 14-15 words, (1) tow breaks off (2) in the former Additions, — \ad vivum omnia resecare; add. Necdum est; of adding (a) Id olim (b) Id 17 think (1) time (2) time 22 which wil, I suppose, furnish us with new matter add. 942

yours: i.e. OLDENBURG-WALLis 4/[14]?.VIL1668. mine: i.e. WALLis-OLDENBURG 4/[14].VII. 1668. 944 Letter: i.e. JuSTEL-OLDENBURG [27.VIJ/7.VII.1668 (OLDENBURG, Correspondence IV, 477-9, 478). 945 book: presumably Dulaurens's Specimina mathematica. 946 them: i.e. Dulaurens and Huygens. See JUSTEL-OLDENBURG [27.VIJ/7.VII.1668 (OLDENBURG, Correspondence IV, 478). 947 qui gageroint qu'il se tromperoit: see JUSTEL-OLDENBURG [27.VIJ/7.VII.1668 (OLDENBURG, Correspondence IV, 478). 948 Hevelius's Cometography: i.e. HEVELIUS, Cometographia, Danzig 1668. 943

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208. WALLIS to OLDENBURG, 8/[18] July 1668 I suppose Dr Millington949 may be willing to have one of his Selenography950 ; of which therefore I shall speake to him. Dr Wilkins951 his Book952 I have perused, & judge well of it. The thing doubtless is fesible, that is, that a Philosophicall Language may be contrived, explicable either by sounds, by letters, or by other marks: which may have many advantages of any language yet extant. But that any one such Language (for there may be infinites of such possible) shal so obtain, as to become Universal!, I must say (as he doth) that I have but very slender expectations. However; what he hath done may be of very good use, though That should never come to pass. I was (before I came out of town) to seek you at your lodgings, to have come out of your debt; but found you not at home. I shall be ready to do it on the first occasion. I am Sir Your affectionate friend & humble servant Joh: Wallis. [2] These For Mr Henry Oldenburgh, at his house in the Palmal, near St James's London.

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208. WALLIS to HENRY OLDENBURG Oxford, 8/[18] July 1668 Transmission:

W Letter sent (with corrections, amendments, and instructions for the printer entered later by Oldenburg): LONDON Royal Society Early Letters Wl, No. 50, 4pp. (our source). At top right of p. 1 in Oldenburg's hand: 'Dr Wallis's Account of Mr Mercators Logarithmotechnia produced and read at the Society July 16. 1668.' Beneath date: 'Entered R.B. 4. 01.' On p. 4 in Oldenburg's hand: 'Produced and read in the Society July 16. 1668. and ordered to be entered.' 949

Dr Millington: i.e. Thomas Millington. Selenography: i.e. HEVELIUS, Selenographia, Danzig 1647. 951 Dr Wilkins: i.e. John Wilkins, q.v. 952 Book: i.e. WILKINS, Essay towards a Real Character and a Philosophical Language, London 1668. 950

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208. WALLIS to OLDENBURG, 8/[18] July 1668 w1 Copy of letter sent: LONDON Royal Society Register Book Original 4, pp. 1-4. w2 Copy of w1: LONDON Royal Society Register Book Copy 3, pp. 82-6. E Printed version, based on WALLIS-BROUNCKER VII/VIII.1668 (missing) containing the amended and slightly expanded text of W and bearing its date: Philosophical Transactions No. 38 (17 August 1668), 753-6 ('Logarithmotechnia Nicholai Mercatoris: discoursed of in a letter written by Dr. J. Wallis to the Lord Vis-count Brouncker') (our source). Three days after Wallis had sent the present letter, in which he gives an account of Mercator's Logarithmotechnia, he wrote to Oldenburg again, requesting that a number of alterations be made in the case of its being accepted for publication in the Philosophical Transactions. He also suggested that the amended version of the letter be communicated to the Society and registered. Oldenburg duly carried out the instructions and the paper was read at the meeting of the Society on 16/[26] July and entered in the Register Book. See BIRCH, History of the Royal Society II, 306. Wallis later sent a copy of the letter, incorporating alterations and additions, to Brouncker (WALLIS-BROUNCKER VII/VIII.1668). He reports on this in WALLISOLDENBURG 3/[13].VIII. 1668, and therefore it is probable that the letter was sent shortly before this time. There, Wallis indicates to Oldenburg that he would like this amended version to be published in the Philosophical Transactions. It appeared in the August issue, but with the date of the original letter to Oldenburg. Also published in that issue was Wallis's letter to Brouncker of 5/[15] August 1668, containing the proof to which he had referred at the end of the earlier letter.

(W)

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Clarissimo Doctissimoque Oxoniae Julii 8°. 1668. D. Henrico Oldenburgh. S. Incidebam heri Vir Clarissime in D. Mercatoris Logarithmotechniam953 nuper editam954. Quae ita mihi perplacuit ut non prius dimiserim quam perlegissem totam. Phraseologiam quam sub initio habet, seu loquendi formulam, unam aut alteram, mutatam mallem; quippe terminorum ali4 heri (Illustrissime Domine) E 5 placuit E 6-483, 4 totam. Et quamquam pauca quaedam, Phraseologiam quod spectat seu loquendi formulas nonnullas, mutata mallem; sunt tamen ipsa sensu suo sana: Eaque E 953

Logarithmotechniam: i.e. MERCATOR, Logarithmotechnia, London 1668. nuper editam: Mercator's Logarithmotechnia was first communicated to the public on 5/[15] August 1667. The printed edition of 1668 included additional propositions on the quadrature of the hyperbola and the discovery of sums of logarithms (prop. 17-19). It was bound with Ricci's Exercitatio geometrica de maximis et minimis [Rome 1666] at the request of Collins. See COLLINS-PELL 5/[15].IX.1668, British Library MS. Add. 4278, f. 342 r ~ v . 954

482

208. WALLIS to OLDENBURG, 8/[18] July 1668 quot levis immutatio, quantum ego judico, turn rem ipsam, turn sensa sua, aptius explicata daret. Verum illud necesse non est; nam et ea quam habet loquendi formula, rite intellecta, (atque ita ut ipsam intellectam velit,) sensum sanum habet: Eaque quae superstruitur doctrina Logarithmos expedite atque subtiliter construendi, perspicue satis atque ingeniose traditur: ut non videam quid illic additum optarem. Et (quae huic subjungitur955) Quadratura Hyperbolae, elegans admodum est atque ingeniosa. Nempe in hunc sensum. Postquam in Hyperbola MBF, (cujus Asymptotae AN, AH, ad angulum rectum coeant) ostenderat (prop. 14.) Rectangula BIA, FHA, spA etc. (ductis BI, FH, sp, etc. parallelis asymptotae AN,) invicem esse aequalia; adeoque latera habere reciproce proportionalia; (quae nota est Hyperbolae proprietas:) Positis AI = BI = 1, et HI = a; ostendit (prop. 15.) FH — ^5 (Nempe propter Analogiam AH. AI :: BI .FH: hoc est, 1 + a .1 :: I . ^q^-) Sed et (quod dividendo 1, per 1 + a, ostenditur,) j^ = 1 — a + a2 — a3 + a4 &c. (continuatis deinceps in infinitum, ipsius a potestatibus, alternatim negatis et affirmatis.)

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Cumque hoc perinde obtineat ubicunque ultra punctum / ponatur

6 ut non videam quid illic additum optarem. missing in E 7 Quae huic subjungitur Quadraturae Hyperbolae E 8 Nempe ad hunc sensum. V. Fig. 1. E 9 AN, AE corr. Oldenburg (on Wallis's instructions) 10 Rectangula BIA, FHA, spA, (reliquaque hujusmodi,) invicem alt. Oldenburg (on Wallis 's instructions) 16 +a4 add. 16 in infinitum missing in E 955

quae . . . subjungitur: i.e. Prop. 17, Quadraxe Hyperbolam. 483

208. WALLIS to OLDENBURG, 8/[18] July 1668

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H: Positis, ut prius, AI — 1; hujusque continuatione qualibet, ut IT — A; quae intelligatur in aequales partes innumeras dividi, quarum quaelibet, ut Ip, pq, &c, dicatur a; adeoque Ip, Jg, &c, sint a, 2a, 3a, &c, usque ad A: Quae his respondent rectae (seu parallelogramma totidem, latitudinem a habentia,) ps, qt, &c, usque ad ru, (spatium BIru complentia,) erunt 1 -a +a2 -a3 +a4 &c. 1 -2o +4o2 -8o3 +16o4 &c. 1 -3a +9a2 -27a3 +81a4 &c. et sic deinceps usque ad 1 -A +A2 -A3 +A4 &c.

Cum itaque sint 1 + 1 + 1 &c (usque ad ultimum) a + 2o + 3o &c (usque ad A) a2 + 4a2 + 9a2 &c (usque ad A 2 ) a3 + 8a3 + 27a3 &c (usque ad A 3 ) a4 + 36o4 + 81a4 &c (usque ad A4) et sic deinceps:

= = = = =

(quod ostendit ille, prop. 16. estque a me alibi demonstratum:) Recte colligit, (prop. 17) expositum spatium Hyperbolicum BIru = A— gA 2 + jjA3 — |A4 + |A5 &c. Adeoque si (assignato ipsi A — Jr, valore suo in numeris, ut res postulaverit,) distribuantur in duas classes, A, |A3, ^A 5 , &c (potestates Amrmatae,) et ^A 2 , |A4, &c. (potestates negatae;) harumque Aggregatum ex Aggregate illarum| subducatur: Residuum erit [2] ipsum BIru, spatium Hyperbolicum. Nequis autem operam lusum iri existimet, propter addendorum seriem in utraque classe infinitam; adeoque non absolvendam: Huic incommodo medelam (tacitus) adhibet: ponendo A = 0,1, vel = 0,21, aliive fraction! Decimali aequalem, quae minor est quam 1: (hoc est, sumpta Ir minore quam AI — 1:) Quo fit, ut posteriores ipsius A potestates, tot gradibus infra integrorum sedem descendant, ut merito negligi possint. Exempli gratia; Positis AI — 1, et Ir — 0, 21: erit

4 rectae add. 4-5 rectae ps, qt, &c. usque ad ru, (spatium BIru complentes) sunt, E 15 a 4 + . . . = |A5 missing in E 25 Hinc E 27 aequalem, adeoque minorem quam E 484

208. WALLIS to OLDENBURG, 8/[18] July 1668

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Quae est brevis Synopsis, Quadraturae suae satis elegantis. Dissimulandum interim non est; siquis totius BIHF spatii, (cujus latus IH longius intelligatur quam AI,) quadraturam postulet; rein non ita feliciter successuram; propter medelam, quam modo diximus, malo minus sufficientem. Cum enim jam ponenda sit A > 1; manifestum est, posteriores ipsius potestates, altius in Integrorum sedes penetraturas, adeoque minime negligendas. Huic autem incommode, levi constructionis immutatione, facile subvenitur. Caeteris utique ut prius constructis; Quadrandum exponatur HFur spatium; (cujuscunque fuerit longitudinis AH assignata, puta major minorve quam AI; sumptoque ubivis inter A et H puncto r; puta ultra citrave punctum I.) Ponantur autem (non, ut prius, AI = 1, et Ir = A; sed) AH = 1, et Hr = A, quae intelligatur in aequales partes innumeras dividi, quarum quaelibet sit a. Erunt itaque, post AH = 1, reliquae deinceps decrescentes, 1 — a, 1 — 2a, 1 — 3a, &c usque ad Ar — I — A. Item, propter aequalia rectangula FHA, urA, BIA, &c, puta = 62; erit 1,2 i,2 k2 k2 k2 HF = y, reliquaeque deinceps j^, jz^;, fir^ &c, usque ad ru = ^ZTA? spatium HFur coniplentes. Factaque divisione,

13 Synopsis Quadraturae suae satis elegans. E 20 subvenitur. Vid. Fig. 1. E 23 assignata missing in E 24 AI; vel huic aequalis: sumptoque E 25 punctum I, vel in ipso I puncto:) Ponantur E 26 partes add. 29 aequalia add. 31 coniplentes. (Quae oninia ostensa sunt, in niea Arithmetica Infinitorum, prop. 88, 94, 95.) E

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208. WALLIS to OLDENBURG, 8/[18] July 1668

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reperietur ^ = b2 + b2a + b2a2 + b2a3 + 62o4 &c, Hoc est b2 in 1 + 10 a + a2 + a3 + a4 &c, (sumptis ipsius a potestatibus continue sequentibus, affirmatis omnibus.) Cumque de reliquis idem sit judicium: Erunt rectae omnes, ipsis HF & ru interjectae,

15

b2 in et sic deinceps, usque ad [3]

Omniumque summa, 20

Exempli gratia: Erunt A= 0, 21 Positis AH = I . Hr = A = 0,2l AI = b = 0,l adeoque D BIA = b2 = 0,01

25

18 Omniumque Aggregatum, A + f A2 + f A3 + \A* + \A6 fee, in 62 = FHru. (per Arithm. In fin. prop. 64.) E 486

208. WALLIS to OLDENBURG, 8/[18] July 1668

Horumque summa ducta in Est

5

Qualium

Rhombo AHGN

Atque haec sunt quae Tibi hac de re ideo scribenda duxi, quoniam ipsi D. Mercatori minus familiariter notus sum. Quae tamen si ipsi impertire velis; non displicebit, credo, haec suae Quadraturae facta accessio. Quae autem de Hyperbola cujus Asymptotae angulos rectos faciunt dicta sunt, alii cuivis tarn facili negotio accommodantur, ut non sit quis ea de re quicquam ultra monere. Vale. Tuus ex animo, Johannes Wallis.

6 Quadr: AHGN corr. Oldenburg (on Wallis's instructions) 9 accessio. (1) Vale. (2) Quae autem 487

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208. WALLIS to OLDENBURG, 8/[18] July 1668 (E, ending)

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Exempli gratia: Positis AH = I . Hr = A = 0,21 AI = b = 0,1 Adeoque b2 = 0,01

Erunt

[755]

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Horum summa ducta in b2 Exhibet

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[756]

Rectus. Quadrato, si angulus A sit ^ Rhombo, J ^ Obhquus. Quae quidem tarn absoluta est tamque expedita Hyperbolae quadratura, ut nesciam an melior sperari debeat. Atque haec sunt quae hac de re scribenda duxi. Quae si D. Mercatori impertiveris; non displicebit, credo, haec suae Quadraturae facta accessio. Posse haec ad Logarithmorum inventionem accommodari, non est quod moneam: Sed & ad Summam Logarithmorum inveniendam956: quam inquirit ille prop. 19.) Nempe, Positis AH = 1, AI = IB = 6, (ut prius) planoque BIHF = pi. Erit pi - b2 + b3 = BIps + BIqt + BIru, &c. usque ad BIHF. Si autem non ab ipsa BI incipiatur; sed ultra citrave, puta a ps: Posita pH = a & psFH = pi. erit (universaliter) pstq + psur &c (usque ad psFH) = pi — ab2: qualium 1, aequetur cubo ipsius AH.) Quod

Uuahum 1. = AHGJM

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5 |A4 = 0,00048623- corr. ed. 11 ±A10 = 0,0000000017- corr. ed. 16 Qualium 1. = ANGN corr. ed. 956

Summam ... inveniendam: i.e. Prop. 19, Invenire Summam Logarithmorum.

488

209. WALLIS to OLDENBURG, 11/[21] July 1668 alias, si opus erit, demonstrabitur. Tu interim, Illustrissime Domine, Vale. Oxon. d. 8. Julii, 1668.

209.

WALLIS to HENRY OLDENBURG Oxford, 11/[21] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 51, 2 pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Ace. jul. 13. 68.' At bottom of p. 2 Oldenburg has noted at 180° to address: 'not to be entered'. Postmark on p. 2: 'JY 13'.—printed: OLDENBURG, Correspondence IV, 525-6.

Oxford July. 11. 1668. Sir,

I have given you the trouble of so many letters of late, that I am allmost ashamed to trouble you with this. Which is but to desire, that in case you think fit to print that957 which I sent you by the last Post: You would please to adde (if it be not too late) just before Vale, (or instead of it.) Nempe pro Substituendum erit,

Qualium 1. = AHGN, Quadrato; Qualium 1. = AHGN, Rhombo.

5 you (1) so (2) the 957

that: i.e. WALLIS-OLDENBURG 8/[18].VII. 1668.

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209. WALLIS to OLDENBURG, 11/[21] July 1668

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Which few words, make the quadrature Universal to All Hyperbole's; which in Mercator's is particularly fitted to that onely where A is a, Right Angle. And now I hope the Quadrature of the Hyperbola is so compleat; as that wee can hardly expect a better. And I think it will be no dishonour to our Society, to have this work perfected by members of our own. In conformity to this Addition; it will be convenient, in the line just before the last paragraph, to insert the letters AHGN- & to read it thus, Qualium 1.00000000000 = AHGN, Quadrato rectae AH. And, if the scheme be not drawn, let it be onely such as this inclosed;958 (for the line DE, (& the rest of Mercator's scheme,) I make no use of at all.) But then; (in the beginning of the 3d Paragraph, Postquam &c: in stead of Asymptotae AN, AE, you must say Asymptotae AN, AH, (putting the letter H for E.) And, in the next line; instead of (reliquaque hujusmodi) put, &c. (ductis BI, FH, sp. &c. parallelis Asymptotae AN.) This being done; you have both Mercators Quadrature, & my Additional; full, clear, & short: & the Figure allso cleared of what could well be spared. If these amendments should come too late (which I am apt to think they will not, unless the other did so;) it is not much material; for all was sound before, this onely makes it a little the more neat. Yours &c J. Wallis. I think it will not be amisse to communicate my letter (amended as is here directed) to the Society at their next meeting; whereby it may be Registred, & thereby become the more Theirs.

25

These For Mr Henry Oldenburg, at his house in the Palmal, near St James's London.

[2]

6 convenient, |in the line add] just before the last (paragraph, add] to 12 must add. 13 line; (1) instead (2) instead 15 both add. 958 this inclosed: not found. Cf. the figure contained in the printed version of WALLISOLDENBURG 8/[18].VII. 1668.

490

210. COLLINS to BRERETON, 11/[21] July 1668

210. JOHN COLLINS to WILLIAM BRERETON London, 11/[21] July 1668 Transmission:

C Copy of (missing) letter sent in Pell's hand: LONDON British Library MS. Add. 4398, f. 148r.

From Brooke house f? July XL 1668 Dr Wallis hath examined the table of Incomposits & found many faults therein more than are mentioned in the Errata959. Whereof I intend to send Dr Pell the particulars960. Dr Wallis is sorry that seeing Dr Pell hath done so well, he hath done no more. I have received a letter961 from Dr Pell directed to my house, and hope by the next Poste to return an answere. In the interim may I beseech your Lordship to represent my humble service to the Dr. I should gladly know the Drs opinion962 of Mr Mercators Logarithmotechnia. Hugenius hath wrote something963 in the French transactions against Mr Gregories assertion that the Circle & Hyperbole are quantities non-Analytick that is, whose Dimensions cannot be expressed by any Aequation. To which Mr Gregorie was not long a drawing up a Reply964, 1 f? add. 2 table of (1) L breaks off (2) Incomposits 9 of (i) Dr (2) Mr 959

hath examined . .. Errata: see WALLIS-COLLINS early VII. 1668. I . . . particulars: He did this in COLLINS-PELL 18/[28].VIL1668. 961 letter: i.e. PELL-COLLINS 6/[16].VII. 1668 (LONDON British Library MS. Add. 4278, f. 126v-127r). 962 Drs opinion: In fact, Pell did not approve of Mercator's book. See PELL-COLLINS 29.VIII/[8!X].1668 and PELL-COLLINS 6/[16].IX.1668. Cf. BIRCH, History of the Royal Society II, 306. 963 wrote something: i.e. Huygens's critical review of J. Gregory's Vera circuli et hyperbolae quadratura, entitled 'Examen de Vera Circuli & Hyperboles Quadratura, in propria sua proportionis specie inventa & demonstrata a Jacobo Gregorio Scoto, in 4°. Patavii', Journal des Scavans (2 July 1668), 52-6; HUYGENS, (Euvres completes VI, 228-30. This piece was written as a letter to Gallois, dated [22.VIJ/2.VII.1668. See ScRIBA, 'Gregory's Converging Double Sequence', Historia mathematica 10 (1983), 274-85 and HUYGENS-WALLIS [3]/13.XI.1668; HUYGENS, (Euvres completes VI, 27881. 964 Reply: i.e. GREGORY, 'Mr. Gregories Answer to the Animadversions of Monsieur 960

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212. WILKINS to WALLIS, 13/[23] July 1668

5

which may come into the English Transactions. He hath also squared the Conchoide, Cissoeid and hath found an Hyperbole equall to a figure made of all Tangents standing as perpendiculars on a line of equall parts. Hence, as the adding of Secants made a Logarithme tangents: So the adding of tangents makes a Logarithme Secant. Dr Wallis hath duply'd965 to Du Lauren's Vindication.

211.

HENRY OLDENBURG to WALLIS 13/[23] July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 16/[26].VII. 1668. This letter apparently concerned the question of when best to publish Wallis's animadversions on Dulaurens's Responsio.

212. JOHN WILKINS to WALLIS London, 13/[23] July 1668 Transmission:

C Letter sent (in real character, except address): OXFORD Bodleian Library Savile A.4 (iv)r-(iv)v (our source). Postmark on f. (iv) v : '(JY}/14'. W Wallis's transliteration of (7: OXFORD Bodleian Library Savile A.4 (iv)r (our source). Answered by: WALLIS-WILKINS 16/[26].VII. 1668.

2 made |of add] all |such del] Tangents Hugenius upon his Book, De vera Circuli & Hyperbolae Quadratura; as they were publish'd in the Journal des S§avans of July 2. 1668', Philosophical Transactions No. 37 (13 July 1668), 732-5. This piece was written as a letter to Oldenburg, dated 13/[23].VIL1668. 965 duply'd: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii). This letter was published in Philosophical Transactions No. 38 (17 August 1668), 744-7.

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212. WILKINS to WALLIS, 13/[23] July 1668 Beneath Wilkins's letter, written in his 'real character' is Wallis's transliteration in Latin letters. Underneath this is the draft of his reply to Wilkins in 'philosophical language', acompanied by the transliteration thereof (WALLIS-WILKINS 16/[26]. VII. 1668).

(C)

t(iv) v ] For the Reverend Doctor John Wallis at Oxford

(W)

5

Friend (beloved person) I-am-saluting-you in-the-Character real,-that-by-this-means I-maybe-provoking an-answer-from-you of-thesame-kind which-you-can-be-performing without-difficulty, if-you-not-have-been -wanting leisure for-reading my-book966, in-hope of-an-Epistle 7 may- (1) provoke-you (2) be-provoking 10 of- (1) a-letter (2) an-Epistle 966

book: i.e. WlLKINS, An Essay towards a Real Character, and a Philosophical Language, London 1668. This was licensed by the council of the Royal Society on 13/[23].IV.1668. See BIRCH, History of the Royal Society II, 265.

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213. WALLIS to MORAY, 14/[24] July 1668 from-you, I-am-under-writing-my-self Lyndyn 1668 month 7th day 13th

Your-friend most-loving Dzhon Wilcinz

213.

WALLIS to ROBERT MORAY Oxford, 14/[24] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 52, 2pp. (p. 2 blank) (our source). At top of p. 1: 'Read July 16: 68. Entered LB 2. 238.' At top of p. 2 Oldenburg has noted: 'A Letter of Dr Wallis to Sir R. Moray, Concerning some Experiments of Motion.' w1 Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 238-9. w2 Copy of w1: LONDON Royal Society Letter Book Copy 2, pp. 277-8. The present letter, concerning Wallis's thoughts on Borelli's De vi percussionis, was read by Sir Robert Moray himself at the meeting of the Royal Society on 16/[26] July 1668. See BIRCH, History of the Royal Society II, 306-7.

Oxford July. 14. 1668.

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Sir,

10

I have tired Mr Oldenburg with so many letters of late, that I am willing to give him a little respite & take this opportunity of congratulating your return967 from Scotland; which, I doubt not, will add some vigour to the proceedings of the Royall Society, who could not but be sensible of the absence of so active a Member. And (that my letter may not look like a bare complement) I take the liberty hereby to suggest the repeating of some experiments, which Borellus (in his late book968 de vi percussionis pag. 269, 270, 271, 272,) tells us that he hath made. The result 12 the (1) tryall (2) repeating 96r

return: Since 1661, Moray had been Lord of the Exchequer for Scotland, but often spent time in London. He presented to the Royal Society 'some curiosities which he had brought with him out of Scotland' on 2/[12] July and therefore had probably arrived in London shortly before then. See BIRCH, History of the Royal Society II, 302-3. 968 book: i.e. BORELLI, De vi percussionis, Bologna 1667.

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214. COLLINS to WALLIS, 14/[24] July 1668 of them, is to this purpose: That, which way so ever a heavy body be violently cast, (upward, downward, horizontally, or at any angle of inclination thereunto,) the naturall motion of Descent by reason of its gravity, (which with the motion of projection makes up its compound motion,) is still the same. It is a thing which I have oft had thoughts of, as very well worthy a severe Examination by triall; & what I meet with in him, doth renew those thoughts. The particular ways of tryall I will not take upon mee to prescribe, (whether that of Borellus himself, or any other;) But if you think it as proper, as I do, that the thing be tryed by a committee of the Society; you will (I doubt not) find proper ways of doing it. To these I may adde another of his experiments, pag. 114. 115. about a Pendulum; whose string being stopped about the middle in its motion, the bullet swinging onely by the lower half of the string shall (he sayth) spend the same time in finishing its excursion, as if the string at its whole length had continued to move freely. I doe not inlarge in describing the experiments, but refer to the places where they are more fully described, & subscribe myself Your very humble servant John Wallis.

214.

JOHN COLLINS to WALLIS 14/[24] July 1668 Transmission: Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-COLLINS 21/[31].VII. 1668. Reply to: WALLIS-COLLINS early VII.1668. Collins evidently sent Wallis news on James Gregory, who was spending the summer in

1 way add. 1 heavy (1) be (2) body 6 in add. 9 tryed (1) I doubt not but (2) by a committee of the Society; 15-19 con|tinued . .. Wallis. at 90° in left margin 16 experiments, (1) because they are sufficient (2) but

16 are add.

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215. WALLIS to WILKINS, 16/[26] July 1668 London, in particular concerning the emerging controversy between him and Huygens following the publication of Vera circuli et hyperbolae quadratura (Padua 1667). He also commented on Wallis's catalogue of corrections to Brancker's table of incomposite numbers.

215.

WALLIS to JOHN WILKINS Oxford, 16/[26] July 1668 Transmission:

Wl First draft (in philosophical language) of (missing) letter sent: OXFORD Bodleian Library Savile A.4 (iii)r. W2 Second draft (in philosophical language) of (missing) letter sent: OXFORD Bodleian Library Savile A.4 (vi) r . W3 Final draft (in philosophical language with transliteration) of (missing) letter sent (written on WILKINS-WALLIS 13/[23].VII. 1668): OXFORD Bodleian Library Savile A.4 (iv)r (our source). Reply to: WiLKiNS-WALLis 13/[23].VIL1668.

Sir,

I have-been receiving your letter, written in the character real, and I-am sending to you an answer, written in the language Philosophical, that you may-be-able-to know I-am understanding this and that, and I am 5

Oxford, year 1668. month 7. day 16.

6 year add.

496

Your friend and servant John Wallis

216. WALLIS to OLDENBURG, 16/[26] July 1668

216.

WALLIS to HENRY OLDENBURG Oxford, 16/[26] July 1668 Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 53, 2 pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Ace. 17. julii 1668.', and at bottom at 180°, again in Oldenburg's hand: 'not to be entered'. Postmark on p. 2: 'JY/17'.—printed: OLDENBURG, Correspondence IV, 553-4. Reply to: OLDENBURG-WALLis 13/[23].VII.1668. The letter enclosed a now missing specimen of a reply to Dulaurens's Responsio ad epistolam D. Wallisii, which was more extensive than WALLIS-OLDENBURG 2/[12].VII. 1668 (ii).

July. 16. 1668. Oxford.

Sir, I received yours969 of the 13th instant. Notwithstanding which I am still of opinion that it were more proper to put that brief answere to Dulaurens into this months transactions. Because I am not of opinion to make a solemn business of it; (such things being better answered by neglecting.) and should I do so, it would be too long to insert into a transaction. I send you here a specimen970 of somewhat on the first three or four leaves, which I have done since your letter. And should I on this occasion go to make a new collection, it would be expected it should be full; & in the nature of a just confutation, which I do not think the Book to deserve. You may adde, if you please, somewhere to this purpose; Quae autem tanquam Nova, a se inventa enumerat; patior ut ii qui haec prius non noverunt, atque tanti aestimant, ab illo discant. Qui autem trita vident et trivialia vel etiam falsa, pro novis et magnis venditata; ita censebunt ut eos sentire par est; etiam me non monente. But I am no way of opinion, 5 transactions. (1) Both that it (2) Because 6 (such things . . . neglecting.) add. 9 which I ... letter add. 13 tanquam \Nova, add] a se inventa (1) enumer breaks off (2) enumerat; 15 vel etiam falsa add. 969 970

yours: i.e. OLDENBURG-WALLIS 13/[23].VIL1668. specimen: i.e. the now missing enclosure.

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217. WALLIS to OLDENBURG, 18/[28] July 1668

5

either to defer it; or to make a solemn business of it: But shall rather let it by as it doth. The Paper I send, I have no copy of; & therefore must have it again if there be occasion. I have onely time to adde (lest the post be gone) that I am Yours &c John Wallis.

These For Mr. Henry Oldenburgh 10 in the Pallmall near St. James's London.

[2]

217. WALLIS to HENRY OLDENBURG Oxford, 18/[28] July 1668 Transmission:

W1 First draft (broken off): LONDON Royal Society Early Letters Wl, No. 55, pp. 1-4 of 8 pp. (our source). At top of p. 1 in Oldenburg's hand: ' ( 1 ) Other Animadversions Written (2) A second letter (a) by (b) of Dr John Wallis on (ao) a (66) the same printed Paper (aoa) entitul'd Responsio Francisci Du Laurens ad Epistolam D. Wallisii ad Cl. Oldenburgium scriptam (bb) of Franciscus Du Laurens mention'd in the next foregoing Transactions.' W2 Paper sent: LONDON Royal Society Early Letters Wl, No. 56, 4 pp. and No. 55, pp. 5-8 of 8 pp. (our source). Foot of both leaves of No. 56 cut off (now W3). W3 Two fragments of the first two leaves of W2: LONDON Royal Society Early Letters Wl, No. 57, middle and lower of three fragments (our source). C Oldenburg's introduction to E2 and part transcript of middle fragment of W3 (verso): LONDON Royal Society Early Letters Wl, No. 57, upper fragment (our source). El First edition of first part of paper, based on W1: Philosophical Transactions No. 39 (21 September 1668), 775-9 ('A second Letter of Dr. John Wallis on the same printed Paper of Franciscus Du Laurens'). E2 First edition of second part of paper, based on W2 and W3, middle (verso only) and lower (recto and verso): Philosophical Transactions No. 41 (16 November 1668), 825-32 ('A Continuation of Dr. Wallis his second Letter, publish't in Numb. 39, to the Printed Paper of Mr. Du Laurens'). Enclosure to: WALLis-OLDENBURG 20/[30].VII. 1668. The background to the existence of the two main manuscripts W1 and W2 is explained

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217. WALLIS to OLDENBURG, 18/[28] July 1668 in WALLIS-OLDENBURG 7/[17].XI.1668. On recognizing that Wl was too long, Wallis produced a shorter version W2 and then inadvertently sent both versions to Oldenburg as enclosures to WALLIS-OLDENBURG 20/[30].VII. 1668. Wallis did not become aware of the mistake until Oldenburg returned the sheets concerned in OLDENBURG-WALLIS early XI.1668. In that letter, Oldenburg asked how to proceed with the publication of Wallis's paper, the first part of which had in the meantime been printed in Philosophical Transactions No. 39 (21 September 1668), based on the incomplete draft W1. Wallis sent both versions back to Oldenburg on 10/[20] November 1668 (see WALLISOLDENBURG 7/[17].XI.1668) with the instruction to 'omitt what is in the first leaf of the two sheets [i.e. of W2], till you come to Porro (ut minutiora quaedam praeteream) &x. And print on, the next (in this months transactions) as being a part of the former letter of July. 18. but omitted in the last either by a mistake, or for want of room. [... ] For my meaning was that the whole letter of July 18. should have come together. Only (as it now happens) the first part was intended to be a little more contracted then as it is now printed'; WALLIS-OLDENBURG 7/[17].XI.1668. A corresponding introductory comment ((7) was inserted by Oldenburg before the continuation of Wallis's letter, now based on W2, in Philosophical Transactions No. 41 (16 November 1668), 825-6.

(W1) Dixeram971, Vir Clarissime, in fine cujusdam ad Te Epistolae, nonnisi ex multis pauca, ea esse specimina, eorum quae apud Dulaurensium occurrunt vel parum sana vel minime accurate tradita. Si vis, ut paucis illis annumerem plura, Obsequor, modo ne petas, ut Omnia. Ab initio itaque ut ordiar; Quantum (inquit §1.) vocamus id omne, quod Extensionem vel Distinctionem in se recipit; eadem videlicet Quantitatis denominatione duabus his rerum affectionibus significandis accommodata. Quantitatis itaque vocem jam definivit, qua nempe significatione

3 eorum add. 4 tradita. (1) Petis jam (2) Si vis, ut paucis illis (a) adjungam (&) annumerem plura, (oa) quo (bb) Obsequor (aaa) itaque, (sed tantum (bbb) , |(sed tantum non invitus,) add. and del.\ modo 5 Omnia. jQuanquam etiam illi petition! tantum non invitus obsequor turn quod ego non aliorum sphalmatis recensendis delector; (1) turn (2) (nee, nisi provocatus, cum aliis severus esse soleo;) turn quod (a) Erratis (b) in Mathesi minus est (oa) periculi necess breaks off (bb) ab Erratis periculi; cum nemo, paulo attentior, ab illis ipse non carere possit; saltern si monitus ut cum delectu agat. del] 6 §1. add. 971

Dixeram: i.e. WALLIS-OLDENBURG 4/[14].VII. 1668. Already in the Latin version of WALLIS-OLDENBURG 30.III/]9.IV].1668, published in Philosophical Transactions No. 34, Wallis remarks on Dulaurens's Specimina mathematical 'Occurrunt inibi aliqua parum sana, fe, minime accurata multo plura'.

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intellectam vult. Eandem (§2) dividit (ut alii) in Continuum, et Discretam seu Separatum, quam Multitudinem vocat. Mox autem (§.3.) Quantitas, inquit, Separata, etiam in rebus omni quantitate destitutis locum habet. Die, quaeso, Quomodo possunt res illae Quantitatem habere, quae sunt omni quantitate destitutael vel (quod statim occurrit) Subjectum Quantitatis Discretae, Quantitatis expers essel Sed exempla subjicit, Duo Puncta, Tria Momenta, Decem Spiritus, &c. At vero annon Duo Puncta, (etiam juxta ilium,) Distinctionem habent? adeoque Quantitatem discretaml Quomodo igitur Omni quantitate destituta dicit? Vel, si non habent, Quomodo dici poterunt discretae Quantitatis Subjectum? Dices forsan, haec eum velle, Puncta Singula non esse Quanta, esse autem Bina. Esto. At peto, an in Punctis singulis locum habeat Discreta Quantitas? sintve singula discretae quantitatis Subjectum? Si sic; turn non sunt omni quantitate destituta: Si non; quomodo hinc ostenditur, quod Subjectum quantitatis discretae, sit Quantitatis expersl Sed et, ut de Singulis prospiciamus. Dicit ille §3, Quantitatis ideo esse expertia, eo quod in unoquoque subjectorum nee ulla distinctio sit, nee extensio. Taceo hie unumquodque dici quantitatis subjectum (quod tamen quantitatis expers esse vult:) Quoniam id expressius dicitur §4 Numerorum, inquit, natura, non simplicem magnitudinem repraesentat, sed ex pluribus Quantitatibus aggregatam. Si itaque Duo Puncta sint ex pluribus Quantitatibus aggregata; horum singula sunt Quantitates, non omni quantitate destituta. Vides tu, quam haec inter se belle conveniunt. Porro, §2, Distinctio, inquit, arguit plurium rerum aggregatum, Dis-

1 (§2) |mox del. dividit 2-3 vocat. (1) Hanc autem Quantitatem Separatum, etiam in rebus omni quantitate destitutis locum habere dicit. (2) Mox autem (§.3.) Quantitas, inquit, Separata, etiam . . . locum habet. 9 disc.re.tam1 (1) Cur (2) Quomodo 13 an (1) Puncta singula (2) in Punctis singulis 14 Quantitas'! (1) sintve Discretae si sic (2) sintve 15 ostenditur, (1) subjectum quantitatis discretae esse (2) quod subjectum quantitatis discretae, sit 17 Quantitatis (1) esse expertia, quoniam (2) ideo esse expertia, eo quod 19 dici (1) subjectum quantitatis (2) quantitatis subjectum, 24 Vides tu, quam haec inter se belle conveniunt. add. 24 bene El

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217. WALLIS to OLDENBURG, 18/[28] July 1668 junctis inter se partibus compositum; ut est Populus, Acervus, et unumquodque eorum, quorum Paries propriis extremitatibus terminantur et ab alterius fine discretae sunt: Atque hujusmodi rerum congeries peculiari nomine Multitudo dicitur. Habes hie turn Distinctions, turn Multitudinis definitionem: Et utrobique requiruntur Paries disjunctae, et quae suis extremitatibus terminatae ab aliis sint Discretae, non Continuae: Et §5, Quantitati Discretae, nimirum Multitudini, praecipue convenit Divisio: Dividi autem nihil aliud est quam Separari, sen Distingui: Multitudinis autem naturam in Distinctions positam esse vidimus. At statim §6, Dis[2] tinctio, inquit, a Divisione differt,\ &c. unde si Distinctio pro quodam genere accipiatur, erit Divisio pro ejus specie assumenda. At vero, si Dividi nihil aliud sit quam Distingui; etiam Divisio nihil aliud erit quam Distinction Quomodo igitur Distinctio a Divisione differtl Aut, altera Genus, altera Species ejusdem censenda erit? Eo, inquit, differunt, quod Distinctio quamcunque Pluralitatem exprimit; Divisio vero earn solam, quae a rerum separabilitate exoritur. Imo vero non quamcunque Pluralitatem exprimit Distinctio; sensu suo, sed rerum tantum, quae sunt inter se Disjunctae, extremitatibus propriis terminatae, atque ab aliis discretae; Quippe hoc arguere modo dixerat Distinctionis vocem. Quae vero ita sunt, non modo Separabilia, sed et acta Divisa sunt et separata: Nee erit Distinctio Divisionis genus quoddam, sed Divisionem in sua significatione includet. Subdit §7. Neque Actualis tantummodo Divisio Multitudini concedenda, verum etiam Potentialis tribuenda videtur; non enim unquam ita divisa est, ut pluribus aliis modis secari non possit: Veluti numerus Duo-

1 compositum; (1) et max, cujus, inquit, partes pr breaks off (2) ut 4 hie turn (1) Disju breaks off (2) Disting breaks off (3) Distinctionis 5 disjunctae, (1) quaeque (2) et quae 9 statim add. 12-13 etiam (1) Distinctio nihil erit quam Divisio (2) Divisio nihil jaliud add] erit quam Distinction Quomodo igitur (a) Distinctione (b) Distinctio a Divisione differt? (aa) vel (bb) Aut 17 Pluralitatem (1) exprimit Distin breaks off (2) exprimit Distinctio; sensu suo, 19 atque add. 19 discretae; (1) D breaks off (2) exprimit D breaks off (3) Quippe 20 sed et acta (1) separata (2) separata sunt (a) et (b) seu divisa (3) Divisa sunt et separata: 23 Divisio (1) multitud breaks off (2) Multitudini

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217. WALLIS to OLDENBURG, 18/[28] July 1668 denarius non ita divisus est in paries suas Duodecimas, ut in tertias, quartas, sextas, et adhuc alias quasdam sine nomine dividi nequeat. Verum mihi dicat velim, (ut ab exemplo suo non recedam,) an Duo Puncta (quam esse Multitudinem satis diserte dixerat,) sint hujusmodi Divisionis capacia? Nempe, ut nunquam ita divisa esse possint, quin et pluribus adhuc 5 modis secari possint! Vel etiam, an Duodecim puncta (quoniam et hunc numerum jam insinuat) postquam in multitudinis partes duodecimas, (nedum in tertias, quartas, sextas, aliasque Assis partes, quae suis nominibus censeri solent, quales sunt Quincunx, Septunx, Bes, Dodrans, Decunx, 10 et Deunx,) distributa fuerint; in alias adhuc partes sine nomine dividi queant? Quippe si non possint, retractandum videtur quod hie dicitur: Si possint, retractandum erit quod modo dixerat §3. in horum unoquoque subjectorum nullam esse vel distinctionem vel extensionem; secandum utique erit, punctorum quodvis. Postquam autem haec dixerat, §7, quodque hanc actualem, virtua15 lemque divisionem Multitudo sibi principaliter assumit, (cum Magnitudo potentialem tantum derivatamque participet;) ne sibi non statim contradiceret; subjungit §8. Ex hoc discrimine sequitur, Cunctam vim Multitudinis, (quae certo, inquit, determinatoque partium numero conflatur,) modum in Divisione recipere, sive Divisionis terminos habere, ultra quos 20 sectio amplius procedere nequit; Magnitudinem vero Divisionem in infinitum admittere. Die, quaeso, (modo Oedipus sis,) qui haec simul constare possint, Nunquam ita dividi posse multitudinem, quin et pluribus aliis modis secari possit; Habere tamen certos divisionis terminos, ultra quos sectio nequit amplius procederel Sed et eadem pluribus prosequitur, Nam, 25 inquit, quando Multitudo dividitur, quia in ilia partium numerus determinatur, necesse est divisionis modos in eodem determinari; et consequenter potentiae divisivae vim, quae tune ad certos illos modos contrahitur alligaturque, aliquando exhauriri et tandem omnino sisti, quando videlicet ad

5 esse (1) pos breaks off (2) possint 5 possint, quia El 8 sextas, (1) aliasque (2) aliasque Assis (a) qu breaks off (&) partes 10 Deunx,) (1) in alias (2) distributa 12 horum add. 16 assumit, (1) Magnitudo (2) (cum Magnitudo \vero del.\ potentialem 18 §8. (1) Cunctam vim Mu breaks off (2) Ex 22 quaeso, |tu, del] (modo

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217. WALLIS to OLDENBURG, 18/[28] July 1668 [3] ultimam usque divisionem perventum erit. Atque ad hunc sensum plura.j

Putan' haec sana esse posse omnia, quae ita contradictionibus scatent? Turn §9.10. Multitudinis ortum et incrementum, a Continuae Magnitudinis Divisione, arcessere satagit. Satis frigide. Quippe ex suis omnino principiis oritur Multitude, a continui natura plane diversis. Nisi et decem spirituum (quos modo memorabat) Multitudinem, ex Continui Sectione ortam censeri velit. Rectius Euclides972' Multitudini, non ex divisione Continui, sed ex Unitatum compositione, ortum asscribit; 2 def. 7. Et quidem, si quando accidit, continuum aliquod in membra dividi; non eo magis Multa dicenda erunt, quam si continua non fuissent: est utique ad hoc, omnino accidentarium, quod aliquando fuerint unum. Imo vero: Nunierus Quadrantum in eodem Continue, Quaternarius non minus jam est, quam post sectionem erit. Nee rectius §11.12. a Divisions rerumque Distinctions, Aequalitatem, Inaequalitatem, aliaque similia, primum orta, dicit. Quippe eorum, quae ab invicem satis distincta sunt, fieri potest ut nulla sit vel Aequalitas, vel Inaequalitas, (puta Temporis, et Lineae;) Aequalitas enim atque Inaequa-

1 perventum erit. (1) Itaque (2) At in magnitudinis divisione, idem accidere nequaquam potest &c. Itaque manifestum est, Multitudinis divisibilitatem longe aliam esse ab ea divisibilitate quae in Magnitudine reperitur: haec enim infinitam cum sit virtutis nullam unquam secandis corporibus finem imponit: alteram vero limitatem habens potentiam &c (3) Atque ad hunc sensum plura. 2 (1) Putas tu (2) Putan' tu corr. ed. 4 incrementum, (1) arcessere satagit a continui (2) a Magnitudinis Div breaks off (3) a Continuae Magnitudinis 5 Quippe (1) omnino (2) ex suis omnino principiis oritur (a) a cont breaks off (b) Multitude 6 Nisi (1) valet ut (2) et 8-10 ortam (1) vel breaks off (2) censeri velit. jRectius Euclides; ... asscribit; 2 def. 7. add] Et quidem, (a) quo (6) si 10 in (1) partes (2) membra 11 ad hoc, add. 12 Imo vero: (1) equidem Quadrantum numerus, (2) Numerus Quadrantum in eodem continue, (a) non minus est Quaternarius, quam postquam (&) Quaternarius non minus 17 sunt, (1) fieri potest ( ( 2 ) fieri 972

Euchdes: i.e. Elements VII, def. 2.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 litas, non est nisi inter Homogenea; cum tamen et Heterogenea sint inter se satis Distincta. Sed et ejusdem Continui Duo semisses sunt invicem Aequales, et Triens Quadrante Major, quamquam (suo sensu) neque Distincta sint neque Divisa; Quippe Dividi nihil aliud est, quam separari sive distingui, §5; hoc est, §2. inter se disjungi, propriisque extremitatibus 5 terminari, atque ab invicem discreta esse, non continua. Et quidem Aequale esse, vel Inaequale, omnino abstrahit ab eo quod res continuae sint vel disjunctae. Duae Horae Continuae, non minus sunt inter se aequales, quam si essent totis annis Disjunctae. Sed et Idem est sibi-ipsi Aequale, 10 (quippe tantundem est quantum ipsum est; quae vera est Aequalitatis notio:) sed a seipso Distinctum esse vel Disjunctum, non erit dicendum. Item, si Aequalibus A, B, addatur commune C, tota fiunt aequalia A + C et B + C (propter communem notionem, Si aequalia aequalibus addantur, tota erunt aequalia;) cum tamen C, quod utrobique additur, non sit a se distinctum. Item, aggregata ilia A + C, B + C, aequalia esse posse nemo 15 negabit, quamquam non sint ita propriis extremitatibus terminata, ut ab alterius fine sint discreta, (quod ad Distinctionem requiritur §2.) sed sint in eodem C communicantia. Sed et Totum parts, sua majus esse, nemo non dixerit; utut ab ea Distinctum atque Disjunctum non sit. Perperam itaque concludit, Aequalitatis atque Inaequalitatis principium non aliunde 20 quam ab ipsa Divisions petendum. Quippe Aequale esse vel Inaequale, non minus est immediata Entis affectio, quam Continuum esse vel Discretum. Neque illud ab hoc ortuni, magis quam hoc ab illo. Quod autem ait, §13. Aequalitatem atque Inaequalitatem certas esse Quantitatum inter se habitudines; omnino verum est. Sed novum non est; 25 quippe hoc ipsum norunt omnes; ipseque jam olim definiverat Euclides973, nenipe turn has, turn rationes alias, Homogeneorum invicem habitudines esse, 3 def. 5. Quod vero mox sequitur, Sola quantitatum inter se compara-

l Homogenea; (1) sed et (2) cum 3 quamquam (1) Distincta non sint: suo sensu (2) (suo sensu) neque (3) (suo 4 nihil (1) esse ait (2) aliud est 5 §5; (1) sive (2) inter (3) sive (4) hoc est, §2. 14 tamen jipsum del] C 19 utut (1) Totum a parte sua (2) ab 22 Discretum (1) ; ut et vel Unum esse vel Multa Neque alterum ab (2) . Neque 26 Euclides, |3 def. 5. del] nempe 27 habitudines \certas del] esse 973

Euclides: i.e. Elements V, def. 3. 504

217. WALLIS to OLDENBURG, 18/[28] July 1668 tio, non sufficit ut aequales inaequalesve dicantur; sed inter se secundum corporis sui molem conferri debent: Quapropter Aequalitatem Inaequali[4] tatemque certas esse Quantitatum\ inter se juxta corporis molem amplitudinemve comparatarum habitudines, colligere licet: Novum quidem est; sed Verum non est: nee omnino admittenda est haec sua Aequalitatis et Inaequalitatis definitio. Certum utique est, Aequalia esse posse atque Inaequalia, quae corporis molem nullam habent, adeoque non possunt secundum hanc comparari: Sic duo tempora, aequalia possunt esse vel inaequalia; Sed et duo puncta duobus punctis, tria momenta tribus momentis, decem spiritus decem spiritibus; numero aequales esse possunt; utut non habeant, secundum quam comparentur, corporis molem. Requiritur utique ad Aequalitatem atque Inaequalitatem, non quidem sola quantitatum comparatio; sed et ut quantitates comparatae sint Homogeneae; (quod ex Euclide modo diximus;) ut juxta corporis molem amplitudinemve comparentur, non requiritur: ob causas modo dictas. Neque tarn ego ipsi hac in re contradico, quam ipse sibi. Quippe (tanquam dictorum §13 oblitus,) subjungit §14, inter duos numeros, duasve lineas, (quae molem corpoream non habent,) aequalitatem vel inaequalitatem reperiri; non vero inter numerum atque lineam. At vero, qui fieri potest, ut numerus numero (qui corporis molem non habent) aequalis sit vel inaequalis, si ad hoc requiratur, ut juxta corporis molem comparentur? Imo vero, quae molem habent corpoream, non est necesse ut quoad hanc comparentur, quoties Aequalia dicantur vel Inaequalia. Possunt utique duo corpora, longitudine aequalia, vel aeque alta, vel aeque lata, vel aequaliter inclinata, vel etiam aeque gravia dici; quae quoad corporis molem sunt inaequalia. Sic duo montes, duobus muribus, sunt aequales numero seu multitudine; utut non magnitudine seu corporis mole.

8 compaxaxi add. 8 tempora, (1) duae lineae, duo numeri, &c (2) aequalia possunt esse vel inaequalia; (a) item duae (&) Sed 11 molem. (1) \\ Neque tarn ego sibi (2) Requiritur (a) quidem, (&) utique 13 comparatio; verum etiam, ut alt. Oldenburg comparatio; verum etiam ut E1 14 diximus;) (1) apud (2) ut juxta

18 lineas, ( (1) utut (2) quae 19 fieri (1) potest (2) possit (3) potest, ut numerus numero (a) aequ breaks off (b) (qui corporis 22 habent add.

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Verum quidem est, inter lineam et numerum (quatenus tales) aequalitatem aut inaequalitatem non reperiri; (utpote quantitates heterogeneas:) neque inter alias, quam quae sunt ejusdem generis quantitates. Verum hoc non ille primus docuit, sed Euclides olim; (ut modo dictum est;) atque post ilium alii, nedum qui illo fuerunt superiores. (Quamquam hoc ipsum a Dulaurensio vix tuto dicitur; ut qui Aequalitatem atque Inaequalitatem, non in ipsa Quantitatum Homogenearum natura fundatam esse vult, sed in rerum Distinction^ adeoque Heterogeneis, utpote non minus inter se distinctis, pariter convenire debent, atque Homogeneis:) Id autem requiritur, ut quoad saltern illud mensurae genus, quo comparantur, sint homogeneae; ut autem illud sit moles corporea, non requiritur. Sic Triangulum Pyramidi aeque altum dici posse, (sin altitudine aequale,) nemo non dixerit; utut haec corporis molem habeat, illud non habeat: Quoniam quoad altitudinem (secundum quam comparantur) Homogenea sunt, utut alias heterogenea. Dum vero ille, ad Aequalitatem aut Inaequalitatem requiri docet, ut juxta corporis molem comparentur; tu mecum juxta judicabis, credo, hoc minus Sanum esse. (Sufficit utique ut juxta Longitudinem, Latitudinem, Altitudinem, angulum Inclinationis, Curvedinem, Durationem, Numerum, Vim, Pondus, Celeritatem, aut aliud quodcunque demum fuit quantitatis genus comparentur, quod utrique comparatorum commune sit; non minus quam juxta molem corporis.) Vides itaque quo tendunt ipsius nova Principia, hucusque nondum tradita, quorum hoc unum est.

1-3 aequalitatem (1) et (2) aut inaequalitatem non (a) intercedere; (b) reperiri; (utpote quantitates (ao) heterog breaks off (bb) heterogeneas:) (aoa) cum (666) neque inter alias, quam quae sunt (aaaa) unius (bbbb) ejusdem, 5-9 (Quamquam hoc . . . atque Homogeneis:) add. 9 Id (1) saltern (2) autem requiritur, ut quoad jsaltem add.\ illud 12 Pyramidi (1) Altitudine aequale (2) aeque altum (a) (sin (b) dici 13 corporis add. 17 minus (1) esse (2) Sanum 18 Altitudinem, (1) Inclinationis (2) angulum 19 Numerum, |Vim, add.\ Pondus, jCeleritatem, add.\ aut aliud (1) atque (2) quodcunque 20 demum (1) fuerit (2) fuit 506

217. WALLIS to OLDENBURG, 18/[28] July 1668 <^ 2 > [56, p. 1] Doctissimo Amicissimoque Viro D. Henrico Oldenburg S.

Oxoniae Julii 18. 1668.

Dixeram974 (Vir Clarissime) in fine cujusdam ad Te Epistolae; nonnisi ex multis pauca ibidem esse specimina, eorum quae apud Dulaurensium occurrunt, vel parum sana, vel minime accurate tradita. Petis, ut paucis illis plura annumerem: quo illud constet. Obsequor itaque; modo ne petas ut Omnia, sed tantum non invitus, quia bonas horas melius collocatum iri puta quam in his expandis. Ut itaque ab initio exordiar; Quantum definit (§1) id omne quod Extensionem vel Distinctionem in se recipit: extensione innuens Quantitatem Continuum] Distinctione, Discretam, (quam et Quantitatem Separatam, et Multitudinem appellat,) in quae duo genera (sicut alii) Quantitatem distribuit. Mox tamen subjicit §3. Discretam quantitatem, etiam in rebus omni quantitate destitutis locum habere; et Discretae quantitatis Subjectum, quantitatis expers esse. Quae contradictiones in terminis videntur implicare. Et, ne se expediat dicendo, ea sigillatim sumpta non esse quanta, quorum Aggregatum quantum sit: non modo se hinc excludit dicendo, in Uniquoque subjectorum, nee ullam distinctionem esse nee extensionem; ubi unumquodque (sigillatim sumptum) hujus Subjectum esse

5 pauca (1) ea (2) ibidem 6 accurate (1) dicta (2) tradita 7 quo illud constet. add. 9 quam (1) altercando (I?) in 12 et | Quantitatem add] Separatam 18 quanta, |(ut duo puncta, trio, momenta, decem spiritus,) del] (1) non modo sibi hie excludit eadem (2) quorum 19 subjectorum, (1) nee extensionem (2) nee 20-508, 2 sumptum) jliujus add. Subjectum esse innuitur: sed et (1) idem expresse dicitur §4, (a) Numerum (b) Numerorum naturam, (aa) divisam esse atque compositam, et (66) divisam esse atque compositam, (aoa) et (bbb) et ex pluribus Quantitatibus aggregatam, repraesentare: (2) expresse §4, Numerorum, inquit, natura ex pluribus Quantitatibus aggregatam, repraesentat: (aaaa) Si itaque Numeri (bbbb) ubi 974

Dixeram: i.e. WALLIS-OLDENBURG 4/[14].VII. 1668. Already in the Latin version of WALLIS-OLDENBURG 30!II/[9!V].1668, published in Philosophical Transactions No. 34, Wallis remarks on Dulaurens's Specimina mathematical 'Occurrunt inibi aliqua parum sana, fe, minime accurata multo plura'. 507

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innuitur: sed et expresse §4, Numerorum, inquit, natura ex pluribus Quantitatibus aggregatum, repraesentat: ubi eadem, etiam sigillatim sumpta, Quantitates dicuntur, quorum Aggregatum est multitude. Unumquodque igitur et Quantitatis Subjectum est et Quantitas; et tamen omni quantitate destitutum, ipsum Quantitatis subjectum est quantitatis expers. Similiter, cum Distinctionem, (qua Multitudinis, seu discretae quantitatis naturam designat,) sic definiverat §2, Distinctio arguit rerum aggregatum, Disjunctis inter se partibus compositum; ut, inquit, est populus, acervus, et quicquid eorum, quorum paries propriis extremitatibus terminantur, et ab alterius fine discretae sunt: dixeratque §5, Dividi, nihil aliud est quam separari, sive distingui. Quo sibi statum contradicat; subdit §6. Distinctio a Divisione differt: et, si Distinctio pro quodam Genere accipiatur; erit Divisio pro ejus specie. Item, §7, Multitudo, inquit, nunquam ita divisa est, ut pluribus aliis modis secari non possit. Veluti Numerus Duodenarius &c. Cui statim contradicit §8. Ait enim, cunctam vim multitudinis certo determinatoque partium numero conflari; et modum in divisione recipere, sive divisionis terminos habere, ultra quos sectio procedere nequit; Item, Quando multitude dividitur, quia in ilia partium numerus determinatur, necesse est etiam divisionis modos determinari; et consequenter potentiae divisivae vim, (quae ad certos illos modos contrahitur, alligaturque,) aliqando exhauriri, et tandem omnino sisti, quando videlicet ad ultimam divisionem perventum erit: Et mox, Multitudinis, inquit, divisibilitas; limitatam habens potentiam, circa praefinitas quasdam divisionis rationes solummodo versatur, et determinatis certarum partium spatiis inclusa\ coercetur. Quae omnia, [57 (m)1 supra dictis, directe repugnantur.

3 dicuntur, add. 3-5 Unumquodque igitur . . . quantitate destitutum, (a) et quantitatis expers (b) ipsum Quantitatis subjectum est quantitatis expers. add. 6 (1) Item (2) Similiter 7 aggregatum, (1) distinctis (2) Disjunctis 12 Divisione add. 15 possit. (1) veluti Numerus duodenarius, non ita divisus est in (2) cui statim contradicitur §8. (3) Veluti 21 exhauriri, et tandem add. 26 repugnantur. (1) \\ Multitudinis ortum et incrementum, a continua Magnitudinis Divisione arcessere videtur (quasi lianc origi breaks off (2) \\ Et §6,

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217. WALLIS to OLDENBURG, 18/[28] July 1668 Et §6, Distinctio^ inquit, quamcunque pluralitatem exprimit, et non earn solam quae a rerum separabilitate exoritur. Quod supra traditis repugnat. Si enim Distinctio arguit (ut earn ipse definiverat §2) paries inter se disjunctas- et quidem ita propriis extremitatibus terminatam quamlibet, ut ab alterius fine discreta sit: Quam, quaeso, aliam pluralitatem [56, p. 2] potest exprimere quam quae a rerum separabilitate oritur?| Porro, §10, Multitudinis originem et incrementum, a Divisione continui, petitum Numerique infinitudinem, hinc esse vult, quia continui divisibilitas est interminabilis. Satis frigide. Quippe Multitude ex suis principiis oritur, non minus quam Magnitude ex suis. Neque ex sectione continui, sed ex compositions Unitatum, numerum ortum esse, jam olim definiverat Euclides975, 2 def. 7. Numerique infinitude, non tarn ex infinita continui divisibilitate oritur, quam quia tanta non potest esse Unitatum multitude quin adhuc Unitas alia componi possit, nedum plures. Certumque est, ea Multa esse posse, quae Continua nunquam fuerunt, (ut duo Puncta, tria momenta, decem spiritus, &c, ne ab exemplis suis §3 discedam.) Et quidem siquando continget continuum aliquod in membra dividi; hoc, quantum ad Multitudinis naturam, est plane accidentarium; quippe haec non minus fuissent Multa, etiamsi non fuissent aliquando continua. Et quidem in eodem continue, nondum secto, Quadrantum numerus non minus est Quaternarius, quam olim erit cum fuerit divisum. Abstrahit atque simplicior Numerorum natura, ab eo quod, quae Numerantur, vel

I exprimit, (1) et non earn (2) non autem earn. (3) et 3 enim (1) Distinguit (2) Distinctio 5 pluralitatem (1) exprimit, (2) potest exprimere 7 a (1) Dist breaks off (2) Divisione 10 non minus add. 10 suis. (1) Neque ex (2) Neque II olim (1) docuit (2) definiverat 12 2 def. 7. (1) Possuntque ea Multa esse, quae Continua nunquam fuerunt: (ut duo puncta, tria momenta, decem spiritus, fee, ne ab illius exemplis §3. discedam) (2) Numerique 13 quia (1) tanta Unitatum multitude nulla sit cui non possit adhuc alia U breaks off (2) tanta 14 Unitas add. 14 possit (1) . (2) , nedum plures. 22 quod, (1) Numerata (2) quae Numerantur 975

Euclides: i.e. Elements VII, def. 2. 509

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continua sint, aut aliquando fuerint, vel divisa. Nee rectius §11, a Divisione rerumque distinctione, aequalitatem et inaequalitatem primum oria, dicit; et §12, aequalitatis et inaequalitatis principium non aliunde quam ab ipsamet Divisione petendum. Certum utique est, Aequalia multa esse, vel inaequalia, quae neque Divisa sunt neque Distincta; saltern prout ipse Distinctionem definit; hoc est, ita ab invicem disjuncta, et propriis extremitatibus terminata, ut alterum ab alterius fine Discretion sit: Quippe, ejusdem continui, duo semisses, sunt invicem aequales, utut non discreti, sed continui. Et duae Horae continuae non minus inter se aequales sunt, quam si totis annis interpositis Discretae fuissent. Sic Totum quodvis est majus sua parte, quamvis nee disterminentur: Et aequale omnibus simul sumptis; hoc est, idem sibi ipsi aequale (quamvis hie §12, neget hoc esse posse et distinctionem suppositorum requiri asserit.) Item A + B, et B + C, aequalia possunt esse, vel inaequalia, utut in eodem C communicent. Sed et ipsius B est aequale sibi, (hoc est, tantumdem est atque ipsum est; quae vero est aequalitatis notio;) utut a se distinctum atque divisum non fuerit. Quotusquisque enim est qui non hujusmodi demonstrationem apud Euclidem et alios passim viderit; positis A et C invicem se aequalibus, addatur utrique idem B; eruntque A + B et B + C invicem aequalia, propter communem notionem Si aequalia aequalibus adduntur &c. Sed et hie ipse §16 Aequalitatem Identitatis quandam speciem esse dicit; siquid itaque eum ipse B, non modo sit ejusdem magnitudinis (quod ad aequalitatem sufficit) sed et idem ipsum Ens; erit certe, a fortiori, ipse B aequale. Sed et e contra; Numerus et

2 (1) Similiter (2) Nee rectius §11, |12 del] a 5 sunt add. 7 disjuncta, et (1) terminis (2) propriis 7 ut (1) ab (2) alterum 9-14 Et duae ... requiri asserit). add. in margin 11 fuissent. (1) Sic Totum quodvis (2) Sic Totum quodvis 15 B est (1) sibi (2) aequale 17 qui |non add] hujusmodi 22 itaque (1) hie (2) eum 24 contra; (1) quae |sunt add.\ ab invicem satis Distincta, non tamen Aequalia statim sunt vel Inaequalia. (2) Numerus

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217. WALLIS to OLDENBURG, 18/[28] July 1668 Linea, sunt quidem inter se satis Distincta, non tamen sunt vel Aequalia vel Inaequalia; utpote quantitates invicem Heterogeneae, quarum ad invicem rationem non esse, definiverat olim Euclides976, 3 def. 5. sed et hie fatetur §14. Perperam itaque asserit, non aliunde quam ab ipsamet divisions petendum esse, Aequalitatis et Inaequalitatis principium. Et quidem non minus est immediata Entis affectio, Aequale esse vel Inaequale, (utpote quod ex Homogeneae quantitatis essentia fluit,) quam vel Continuum v [57 (m) ] esse vel Divisum; non illud ex hocce derivatum. | Porro (ut minutiora quaedam praeteream, ne nimius sim, quae tamen ipsa reprehensionem merentur) inter alia, quibus Aequalitatem ob Moderationis virtutem laudat, Inaequalitati interim vitio vertens, quod Excessu et Defectu laboret; haec occurrunt §18. Tanta est aequalitatis moderatio; ut eas non solum quas afficit quantitates augeat, minuat, multiplied, atque dividat, nulla facta in ipsis quoad aequalitatem mutatione; sed etiam, ut quantitates ab inaequalitate affectas per similes operationes tractando, in[56, p. 3] tactam in illis inaequalitatis notam relinquat. Id credo vult, (nisi velit rhetoricando fucum facere,) eandem, quae 9 [In left margin Oldenburg's instruction for the printer:] Here you are to begin to print on.

1 non tamen (1) vel Aequalia sunt vel Inaequalia (2) sunt vel Aequalia vel Inaequalia 4 quam ab (1) (—} (2) (—} divisione (3) ipsamet divisions petendum esse, (a) pr breaks off (b) Aequalitatis 5 quidem (1) minus immediata est Entis affectio (2) non 7 essentia (1) fluere (2) fluit 16 relinquat. |Et paulo post, Sic ergo Aequalitas seipsam primo, deinde inaequalitatem per quaelibet augmenta vel decrementa modo aequalia deducere valet, nullo aut aequalitatis aut inaequalitatis detrimento. del] 17 (1) Id credo vult, (nisi velit rhetoricando fucum facere,) (a) si expositae (6) si 976

Euclides: i.e. Elements V, def. 3.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 prius erat vel aequalitatem vel inaequalitatem, manere immutatam; quaecunque facta fuerit utrinque vel aequalium additio aut subductio, vel per aequalia multiplicatio aut divisio. Missis autem Aequalibus, de inaequalibus dispiciamus. Inaequalitatis nota quam vult, (ni fallor,) est ipsa inaequalium Differentia; et, hanc intactam relinqui, est, eandem manere quae 5 prius fuerat. (Quippe hoc turn ipsa verba spectare videntur, turn argumentum ejus.) Quod quidem in Additione et Subductione, verum est; puta, si expositis 10 et 6, addantur utrinque 2, ut fiant 12 et 8; vel subducantur 2, ut fiant 8 et 4; eadem intacta manet inaequalitatis nota, seu Differentia 10 (4.) Non autem in Multiplication et Divisione; Quippe si per 2 vel multiplicentur, ut fiant 20 et 12; vel dividantur, ut fiant 5 et 3; Differentia fit illic 8, hie 2; neutrobique (quae prius erat) 4. Argumentum ejus est merum sophisma, (quod plus habet in conclusione quam in Praemissis.) Hoc (inquit) facile colligitur ex inaequalitatis ad aequalitatem revocandi ratione: ut enim inaequales quantitates ad aequalitatem perveniant, necesse est 15 addi minori, vel a majori detrahi, ipsarum quantitatum Differentiam: sed,

expositae quantitates, sive aequales fuerint, sive inaequales, aequalium admittent vel additionem vel subductionem, aut per aequalia vel multiplicationem vel divisionem; eadem quae prius fuerat vel aequalitas vel inaequalitas immutata manet. (oa) Ego vero, (ut de aequalibus taceam;) in inaequalibus rem secus esse pronuntior (bb) Missis autem Aequalibus, (aaa) de Inaequalibus paulo videamur quo (bbb) de Inaequalibus despiciamus; num. eadem quae prius fuerat semper maneat Inaequalitas. Inaequalitatis nota, quam vult, (ni fallor,) est ipsa Inaequalium Differentia: Hanc autem intactam manere, est eandem quae prius esse. (Quippe hue spectat, quod affert argumentum.) Hoc autem (aaaa) Aequalium Additionem et Subductionem, (bbbb) ubi Aequalium facta est Additio et Subductio, vero est: non autem ubi per aequalia facta est vel Multiplicatio vel Divisio. (aaaaa) vel (bbbbb) Puta si expositis 10, et 6, fiat utrobique binarii vel additio, ut fiant 12, et 8; vel subductio, ut fiant 8 et 4; eadem quae prius manet diferentia 4. Si vero (aaaaaa) fiat utrinque vel m breaks off (bbbbbb) utraque per 2, vel multiplicentur, ut fiant 20 et 12; vel dividantur, ut fiant 5 et 3: differentia (aaaaaaa) saltern sit (bbbbbbb) fit illic 8, hie 2, non (aaaaaaaa) (ut prius) (bbbbbbbb) (quae prius erat) 4. (2) Id credo 1-3 immutatam; |quaecunque facta . . . aut divisio. add. (1) Inaequalitatis (2) Missis 5 intactam (1) manere, (2) relinqui, 10 Divisione; (1) puta (2) Quippe 12 neutrobique (1) (pri breaks off (2) (quae prius 14 ex (1) aequalitatis (2) inaequalitatis 15 enim (1) inaequalitates (2) inaequales

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217. WALLIS to OLDENBURG, 18/[28] July 1668 per communem aequalium Additionem vel Subductionem, (vides de Multiplicatione et Divisione nihil dici,) neque minor quantitas, majoris differentia augetur, neque major eadem differentia contrahitur; cum idem utrique inaequalitatis parti adjiciatur vel dematur; (quod in Multiplicatione et Divisione non fit:) Haec sunt praemissa; videamus conclusionem: [57 an Ergo (inquit) sive aequalium additione, aut Multiplicatione, sive aequalium detractione vel Divisione, inaequales quantitates augeantur minuanturve, (vides quomodo in conclusione se insinuant Multiplicatio et Divisio, quae in praemissis non erant;) nunquam hac rations in aequalitatem incident, hoc est, manebit semper in ipsis inaequalitas; (vides quomodo jam languet illud, manere intactam, in nudum manere; sed mox resumet vires: Verum hoc non est quod erat probandum, inaequalitatem manere aliquam, sed manere intactam; et praemissa, quatenus quicquam probant, hoc probant, propter idem utrique parti adjectum vel demptum: Sed pergit rhetoricando;) Sic ergo, inquit, Aequalitas seipsam primo, deinde inaequalitatem, per quaelibet augmenta vel decrementa, modo aequalia, (quod interim per aequalia multiplicando, vel dividendo non obtinetur, ut ipse putaverat,) deducere valet, nullo vel aequalitatis vel inaequalitatis detrimento: (vides, resumptis viribus, languidum illud manere, in cum nullo detrimento manere, jam erigi: quod, per rhetoricam variationem phraseos, idem significet, quod prius, intactam relinqui, et nulla facta mutatione.} [56, p. 4] Haec autem fusius aliquanto deduxi, ut| videas, quam, in Demonstrando vacillet hie Mathematicus Rhetoricaster. Mox autem §19, quoniam Inaequalium quantitatum una Major, sive Excedens; altera minor, sive Deficiens dicitur; haec autem Excessus atque

I communem (1) Aequalium (2) aequalium 3 cum idem (1) (—} (2) utrobique (3) utrique 6 aequalium additione, (1) vel (2) aut 7 inaequales quantitates augeantur minuanturve, add. in margin II illud, |manere add. intactam (1) per (2) in 14 parti \vel add] adjectum 14 Sed (1) (—} (2) pergit 16 quod (1) aequaliter (2) interim 17 obtinetur, (1) quo (2) ut 19 (vides, |jam, del.\ resumptis 20 rhetoricam (1) phraseos immutationem, (2) variationem phraseos, 24 autem §19, add. 25 dicitur; (1) sint (2) haec

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Defectus nomina, aberrationes a media significant, (quod vitiorum est;) hoc est, ab Aequali, (cujus itaque modo laudata Moderatio, virtus erit:) quo tandem in vituperatae Inaequalitatis favorem se insinuet; missis his (quae imperfectam Inaequalitatis naturam respiciunt) nominibus; alia (inquit) hi termini nomina sortiuntur; nam qui Major est, Totum dicitur; qui Minor, Pars: (quasi quidem Partis nomen, non pariter imperfectam naturam insinuet, atque Minoris:) Adeoque (novis definitionibus) Totum definit esse quantitatem majorem ad minorem et homogeneam collatam; Partem vero, minorem esse quantitatem ad majorem et homogeneam comparatam. Sed omnino fallitur hie novus Definitor; qui Totum et Partem, tantundem significare autumat, atque Majus et Minus: Verum quidem est Totum sua Parts, majus esse, (et Partem Toto minorem:) Sed non vice versa, Omne Minus cujusque Majoris Partem esse, quod hie insinuat. Lunam ego Tellure Minorem existimo; Sed non existimo Telluris Partem esse. Hoc ilium forte decepit, quod videret apud Euclidem977, 1 def. 5, Partis nomen, peculiar! signification, prout Multiplo opponitur, pro eodem atque Submultiplo, seu aliquota parte, (uti nunc loquimur,) usurpari. Sed aliud significat Pars prout, peculiar! sensu, est correlatum Multipli: (I def. 5.) aliud, prout, vulgata signification, opponitur Toti, (9 ax. I.)978 (nempe, illud quod, cum reliquo, componit Totum.) Atque ex his, inquit, manifestum fit, Totum majus esse sua parte; (quod est Euclidis Axioma Nonum:) Omnino quidem; nempe si Totum et Pars, idem significant atque Majus et Minus. Sed et inde pariter manifestum est, Euclidem fuisse Asinum; Nempe si in illo Axiomate, hoc solum dictum velit, Majus, majus

7 definitionibus), (1) definit (2) Totum 8 ad minorem add. 13 versa, (1) omne Maj breaks off (2) Omne 14 non add. 16 pro (1) submulti breaks off (2) eodem 18-19 prout (1) Multiplo opponitur (2) , peculiari sensu, est (a) Multipli (b) correlatum Multipli: (1 def. 5.) aliud, prout (ao) opponitur (66) , vulgata 20 reliquo, jactu del. componit 21 Axioma (1) 9 um (2) Nonum:) 24 si (1) Axiomate illo (2) in illo Axiomate 24 velit, (1) Majus est majus (2) Majus, majus est 977

Euclidem: i.e. Elements V, def. 1. (9 ax. 1.): i.e. Elements I, ax. 9.

978

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217. WALLIS to OLDENBURG, 18/[28] July 1668 est minore. Quod, si forte, pro Definitione, ferri posset; saltern Axioma esset plane ridiculum. Deinde §.21, Commensurabilitatis et Incommensurabilitatis fontem aperire satagit, (eadem felicitate, qua Multitudinis, et Aequalitatis sive Inaequalitatis originem quaesivit:) Nempe, Quando pars aliquoties sumpta totum suum praecise constituit, Aliquota dicitur: Atque haec pars est toti suo commensurabilis: Belle quidem. Annon vero est hie egregius Defini[57 an tor, qui Partem commensurabilem,\ eandem esse Definit, atque Partem aliquotam? Verum, quid si Pars aliqua non possit aliquoties sumpta totum suum praecise constituere? (puta si sit ut 4 ad 6;) An propterea non erit commensurabilis? Quid item, si duae sumantur quantitates quarum altera alterius non sit pars? Num propterea non possunt esse commensurabiles? Vel etiam duae quantitates invicem aequales; (quarum itaque altera alterius pars esse non possit, cum non sit minor;) Annon erunt commensurabiles? Die tu potius; Duas pluresve quaslibet quantitates (sive altera alterius pars aliquota sit, sive non aliquota, sive ne pars quidem,) commensurabiles esse, si ulla quantitas assumi possit (utut ab eis omnibus diversa) quae singulas possit aliquoties repetita adaequare. Noli autem commensurabilitatem coercere ad earn solam, quae est inter Partem aliquam aliquotam, Totumque illud cujus ea pars sit. Quippe hoc non est [55, p. 5] Commensurabilitatis fontem aperire, sed obturare.| Mox autem §24, Partem Aliquantam (quae ab Aliquota distinguitur)

1 Quod (1) fortasse (2) , si forte 7 commensurabilis: \Illa enim, velut mesura, seipsam primo, deinde aliquoties repetita totum suum mensurat; Quae cum ipsummet hac ratione adaequet, necesse est mensuram et commensurabilitatem in Aequalitate fundari. del. \ Belle quidem. 7 est hie (1) Definitor (2) egregius Definitor, qui Partem commensurabilem, (a) tantundem (6) eandem 9 aliquotam? (1) Verum si paxs aliqua aliquoties sumpta (2) Verum, 9 Verum quod E2 12 pars? (1) Annon possunt liae com breaks off (2) Num 12 commensurabiles? (1) Si sumantur (2) Vel 13 quarum jitaque add. \ altera alterius (1) non sit pars: ne suo quidem sensu; hoc est, non sit minor;) (2) pars esse non possit, cum non sit minor;) 15 Duas (1) quaslibet pluresve (2) pluresve quaslibet 17 esse, (1) quas (2) si 17 ab (1) utraque diversa (2) eis omnibus diversa) quae (a) utramque (b) singulas 19 commensurabilitatem (1) ad (2) coercere

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sic Definit. Quando vero Pars, aut quantumlibet exigua hujus partis portio aliquoties sumpta, toti suo aequalis fieri nequit, sed vel ipsum semper excedit, vel ab eo semper deficit, tune Aliquanta vocatur. Atque haec pars, inquit, est Toti suo Incommensurabilis. Si ego cum singulis, quae passim occurrunt, verbis imperite positis, litem movere vellem; infinitus essem. Haec autem Definitio ita multis scatet mendis, ut ea prius amovenda sint, quam id dicat, quod ille dictum vellet. 1°. Perperam dicitur, sed vel ipsum semper excedit, vel ab ipso semper deficit] et satis absurde. Impossible enim est ut pars ea, ejusve portio, sic sumpta, vel semper excedat, vel semper deficiat. Verbi gratia; 1 ad \/5, talis pars est, qualeni ille vellet. Sed non vel semper excedit aliquoties sumpta, (nam 1, semel, vel bis sumpta, minor est quam -\/5;) vel semper deficit, (nam ter vel pluries sumpta, major erit; est enim \/5, major quam 2, et minor quam 3:) sed aliquando excedit, aliquando deficit, semper autem vel excedit vel deficit, nunquam aequalis est; atque hoc ipsum est quod ille dictum vellet. Pro his itaque verbis, vel semper excedit vel semper deficit; reponendum erit, semper vel excedit vel deficit. 2° Perperam etiam, disjunctive, dicitur Quando pars, aut hujus partis portio, nequit, &c. Quippe hoc semper contingit, ut vel ipsa pars, vel saltern hujus aliqua Portio, nequeat aliquoties sumpta toti aequalis fieri: Adeoque per hanc definitionem, pars omnis dicenda esset turn Aliquanta, turn Incommensurabilis cum toto suo. Verbi gratia, si Pars sit ad Totum suum, ut 4 ad 6; non potest ea toties sumi ut toti sit aequalis; nam semel sumpta, minor erit; bis sumpta, major: Si sit ut 4 ad 8; pars quidem ea bis sumpta, toti aequabitur, sed ejus portio, 3, nequit ita sumi ut aequalis fiat; nam bis sumpta, minor erit; ter sumpta, major

5 essem add. 8 semper (1) excedat (2) excedit 10 talis est, quam E2 11 aliquoties sumpta, add. 12 \/5;) (1) neque (2) vel 132, (1) sed (2) et 17 deficit. (1) 2° Quod disjunctive (2) 2° Perperam 19 hujus (1) Portio (2) aliqua 20 per hanc definitionem, add. 22 6; (1) nequit haec aliquoties su breaks off (2) non 23 Si (1) pars (#) sit 24 quidem add.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 quam 8: et quidem semper, vel pars ipsa, vel ejus aliqua portio, (saltern in quantitate continua;) ita se habebit. Itaque pro eo quod disjunctive dicitur, Pars, aut hujus partis portio, nequit; dicendum erat copulative, neque pars ipsa, neque hujus partis portio, potest. 3° Neque hoc sufficit; fieri enim potest, ut turn ipsa pars, turn ipsius aliqua portio, (nedum aliquammultae portiones,) ita se habeant, nee tamen ea pars sit incommensurabilis. Verbi gratia, Si pars sit ad totum, ut 4 ad 5; non potest ipsa pars sic sumi (nam semel sumpta, minor est; bis sumpta, major illo Toto:) sed neque ipsius portio 3 vel 2; (nam portio 3 semel sumpta, minor est quam 5; bis sumpta, major: et 2; bis sumpta minor; ter sumpta, major:) potest tamen ejus alia portio, nempe 1, sic sumi; (nam portio 1 quinquies sumpta, toti 5 aequatur.) Neque hie opem feret, inferta clausula quantumlibet exigua; certum enim est, in parte, quae vel maxime commensurabilis sit, sumi posse portiones quantumlibet exiguas, quae non modo totum non metiantur, sed ne commensurabiles sint. Dicendum igitur, neque pars ipsa, neque ulla hujus partis portio, &c. (Quod ita limitandum erit ut mox dicetur.) 4° Superest adhuc aliud mendum, quod majoris est momenti, et imperitiam arguit. Quippe si haec constet definitio, omnino nulla pars erit cum toto suo incommensurabilis. Nam in ea quae vel maxime sit incommensurabilis, sumi poterit portio aliqua (nedum innumerae) quae Totum mensurant. Verbi gratia, Latus Quadrati ad Diagonium suum, est incommensurabile; vel (ut hie loquitur) est pars ejus incommensurabilis: Sumi tamen potest Lateris aliqua portio, quae Diagonii Dimidio, vel Quadranti aequetur: quae itaque bis aut quater sumpta, Toti aequabitur. Quod videtur hie Definitor non animadvertisse; cui vel maxime prospiciendum erat.

1 (saltern. (1) si Totum sit (2) in quantitate 2 eo quod disjunctive dicitur, add. 3 nequit; (1) repondendum erit, (2) dicendum erat copulative, 7 potest (1) vel (2) ipsa pars 8 sumpta, |neque del. minor est; bis sumpta, |neque del. major 9 (nam | portio add] 3 10 major:) (1) sufficit si ulla ejus (2) potest 11 nam portio add. 13 enim est, add. 16 (Quod . . . dicetur.) add. 17 mendum, (1) majoris adhuc momenti, quod (2) quod majoris est momenti, et 18 haec jconstet add] definitio, (1) nulla (2) omnino 23 potest (1) in Latere (2) Lateris

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Non enim sufficit ad commensurabilitatem, ut partis aliqua Portio mensuret Totum, (quod semper net,) sed ut partis aliquae Pars aliquota totum mensuret: Pro portio itaque reponendum erit pars aliquota. Suntque haec quatuor menda, tanti momenti singula, ut eorum nullum non evertat totam definitionem: et quartum omnium maxime; quod ego non Incuriae, sed Inscitiae (prout ipse distinguit) imputandum existimo. [55, p. 6] Sed esto Definitio, vel maxime ad mentem suam, sic reformata; Quando Pars ad Totum suum ita se habeat, ut neque pars ipsa, neque ulla hujus partis pars aliquota, quantumlibet exigua, possit, aliquoties sumpta; Toti suo aequalis fieri, sed semper vel ipsum excedit, vel ab eo deficit, tune Aliquanta vocatur. Atque haec pars est toti suo Incommensurabilis. Haec, inquam, Definitio sic reformata (quae apud ipsum erat misere deformis) admitti potest pro Partis Incommensurabilis definitione. Si vero sit etiam definitio Partis Aliquantae: Die tu mini, quaeso, (modo Oedipus sis,) Qualem ego partem dicam, numerum 4, numeri 6? Pars Aliquota non est, per §21, quia non aliquoties sumpta totum praecise constituit, (nam semel sumpta, minor est; bis sumpta, major:) Neque est Aliquanta Pars, per jam defmita; quamquam enim non possit ipsa, potest tamen ipsius aliquota pars, ut 2 vel 1, aliquoties sumpta, toti aequalis fieri; (nam 2 ter sumpta, vel 1 sexies, aequantur toti 6.) Cum itaque neque Pars Aliquota sit, nee Aliquanta, (partem autem omnem vel Aliquotam vel Aliquantam dicendam, hactenus censuerint homines,) Die mihi, Quam dicam? Sed neque Pars Commensurabilis est, per §21, (quippe commensurabilem non aliam definit ille, quam Aliquotam;) Nee Incommensurabilis, per jam definita. Ecqua igitur? At interim hie Definitor; qui Partem Commensurabilem, idem esse facit cum Aliquota; et partem Aliquantam, idem cum Incommensurabili; male se habitum conqueritur, quod apud eum nonnulla reperiri parum sana dixerim.

4 ut (1) n breaks off (2) eorum nullum non (a) totam (&) evertat 5 omnium (1) maximum (2) maxime 11 vocatur. (1) Estque (2) Atque 13 vero sit (1) et (2) etiam 18 enim (1) p breaks off (2) non 19 pars, (1) puta (2) ut 23 Pars add. 23 (quippe (1) non aliam (2) commensurabilem 518

217. WALLIS to OLDENBURG, 18/[28] July 1668 Statim vero, §26, (ne sibi non, ut solet, contradiceret,) Numeros omnes, invicem esse commensurabiles, affirmat; quoniam omnes mensural Unitas. Quae quidem vera sunt; sed prius traditis contraria. Quippe ille non alias definiverat Commensurabiles quantitates, quam quarum altera sit altering aliquota pars: multi autem numeri ita se non habent; puta 4 et 6. Neque illas commensurabiles dixerat, quas aliqua tertia commensurat, (quod definivisse oportuit,) sed quarum altera mensural reliquam, sitque ejus aliquota pars. Adeoque utut 1 sit ad 4 et ad 6, commensurabilis, (quoniam utrumque metitur,) non tamen erit (per illius tradita) numerus 4 ad 6 commensurabilis, quorum neuter metitur reliquum, sitve ipsius aliquota pars. Eandem enim ille, et Partis Aliquotae, et partis Commensurabilis, defmitionem fecerat, §21; sicut et (illi contradistinctam) partem Aliquantam, eandem esse definit atque Incommensurabilem, §26. Quae quidem ego inter ipsius Nova Principia, hucusque nondum tradita (necdum recipienda,) annumeranda censeo. Sed et §25, Commensurabilitatis et Incommensurabilitatis fontes, porro investigatum it. Omnis, inquit, numerus juxta possibiles quae sunt in eo sectiones divisus, tandem relinquit Unitatem, sen particulam sui minimam. Docuimus enim, inquit, omnem numerum divisibilitatis suae terminos habere, ultra quos sectio non procedit. Fateor haec dixisse, (docuisse, non dico: Ecquis enim ante nescivit.) Sed et contraria docuit, (nempe, si quis Discere velit,) Ait enim, §7, Multitudo nunquam ita divisa est, ut pluribus aliis modis secari non possit. Veluti numerus Duodenarius non ita divisus est in paries duodecimas, ut in tertias, quartas, sextas, et adhuc alias quasdam sine nomine dividi nequeat. Sed esto; ea jam dicit. Quid postea? Ergo (infert) ex naturali numerorum structura commensurabilitas exurgit. Commensurabilitas, inquam, Numerorum, ex sua numerorum natura exsurgit, (non minus quam ex sua Linearum natura, Commensurabilitas Linearum;) Hoc est, ex numerorum natura fit, quod illis (quae et aliis quantitatibus convenit) conveniat Commensurabilitas;

2 esse add. 5 pars: \(1) cum (2) multi autem numeri ita se non habent; puta 4 et 6. add.\ Neque 9 numerus add. 16-17 Commensurabilitatis ... investigatum it. add. 27 exurgit. (1) Commensurabilitatis Num. breaks off (2) Commensurabilitas, inquam, Numerorum, (a) ex eorum natura exsurgit (b) ex 29 Linearum; (1) sicut ex omnium omnino rerum natura exsurgit, quod eas, quas habent, habeant affectiones.) (2) ) Hoc est,

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(sicut et ex omnium omnino rerum natura oritur, quod eas, quas habent, habeant affectiones;) et quidem omnibus; (quoniam omnes mensurat unitas.) Sed Commensurabilitatis simpliciter (quae et aliis quantitatibus cum numero communis est) non minus ex sua cujusque quantitatis natura, vel ipsa Quantitatis, quae est omnibus communis, petenda est| ratio. Sed, et [55, p. 7] ait, ex naturali Magnitudinis constitutions, Incommensurabilitas exoritur. Recte quidem. Sed et Commensurabilitas. Sed et pariter ea quae in Sonis est, et quae in Ponderibus, vel Durationibus, turn Commensurabilitas turn Incommensurabilitas, ex ipsa Sonorum, Ponderum, Durationum, &c, constitutione exoritur. Quippe omnium horum naturae ita sunt comparatae, ut Soni, Pondera, Tempora, &c, sint Incommensurabilitatis capacia: sed et Commensurabilitatis non minus. Quod vero ille persuasum iret; Incommensurabilitatis quae in Magnitudinibus est, rationem ex magnitudinum natura petendam; illius autem quae in eisdem est Commensurabilitatis non ex ipsa magnitudinis, sed ex Numerorum natura oriri: omnino est ridiculum. Non minus enim est ex Magnitudinis natura, ut possit in partes Commensurabiles dividi, quam ut possit in Incommensurabiles. Quod et eo magis absurdum est, quod ea quae jam est numerorum constitutio, ex humano institute oritur. Sed et, si ipsi adhibenda fides, ipsa Numerorum natura (adeoque et horum Commensurabilitas) ex continui divisione oriri putanda erat. §10. Sed, caeteris missis, videamus quam hie Demonstrator probat, (non quidem Incommensurabilitatem ex magnitudinis natura ortam, sed) omnino ullas esse posse magnitudines Incommensurabiles. (Quamquam enim ego illud non negem, sed aliunde probari posse sciam: Nego tamen eum, etsi hoc probandum suscipiat, omnino probasse.) Omnis, inquit, Magnitudo in infinitum divisa non relinquit particulam, quae propterea quod

4 non minus add. 5 quae (1) omnibus communis est, (2) est omnibus communis, 5 et add. Sed ait E2 7 et (1) Incommensurabilitas (2) Commensurabilitas 9 Durationum, (1) et (2) fee, 14 petendam; (1) Illius a breaks off (2) illius 14 est add. 18 numerorum (1) instit breaks off (2) constitutio 19 Sed et, si ... erat. §10. add. 20 ex (1) Magnit breaks off (2) continui divisione oriri (a) di breaks off (b) putanda 22 probat, (1) magnitudines omnino (2) (non

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217. WALLIS to OLDENBURG, 18/[28] July 1668 parva sit secari non possit, quin ilia in infinitum secta infinitas efficit particulas, quarum singulae in infinitas minores sectiles sunt, ut res finem habitura non sit, si quis minutias omnes consectari velit. (Quippe hoc est, quod aliter dici solet, continuum esse divisibile in semper divisibilia.) Nunquam igitur, inquit, ex infinita magnitudinis divisions, ad aliquam particulam devenietur, quae minima dici debeat: quae pro communi omnium mensura sumi queat. Esto. Hallucinatur autem omnino si hinc oriri sentiat Incommensurabilitatem: Non enim ex sections interminabili, sed ex modo sectionis, probasse oportuit Incommensurabilitatem. Certum enim est sectionem in infinitum continuari posse, sine ulla incommensurabilitate (Crassamque arguit naturae Incommensurabilitatis ignorationem, hoc nescire.) Verbi gratia. Si exposita recta (aliave magnitude) intelligatur continua bisectione dividi quousque libet: certum est, commensurabilem illam esse dimidiis suis, et dimidiorum dimidiis, et sic deinceps in infinitum, utut ad minimum nunquam pervenitur, (quod Tyro quilibet in mathematicis, facile demonstrabit; tantusque Magister non debuit ignorare.) Nam aliquotae partis aliquota pars (quantumlibet continuetur sectio) erit et Totius aliquota pars; et omnes invicem commensurabiles. (Quod-

3 velit. |Esto. del] (Quippe 4 divisibilia.) (1) Sed quid porro. Nunquam ex infinita igitur, inquit, Sed omnino (2) Hallucinatur autem omnino, (3) Nunquam 6 quae pro . . . queat. add. in margin 12-13 Si exposita (1) magni breaks off (2) recta ( (a) aliaque (&) aliave magnitude) intelligatur continua bisectione (aa) in infinitum (bb) dividi 14 dimidiis, (1) & si breaks off (2) et sic 15 infinitum, (1) utcun breaks off (2) utut 17 ignorare.) jSimiliter in Trisectione, Quadrisectione, aliisque in infinitum innumeris sectionum formis; (1) utut ad mini breaks off (2) sectionibus rationalibus quibuscunque. Sed pergit Nee propterea, inquit, certa determinataque dabitur in magnitudine particula quae pro communi omnium magnitudinum mensura sumi queat, nam si talis daritur, earn minimam (a) fore (b) esse necesse foret, quandoquid in minima seetitur atque determinatur. (Estque haec tota ipsius probatio) sed nee est necesse ut ad (aa) minimum (bb) minimum possibile pervenitur. Sufficit ut demonstretur omnia bisegmenta, et bisegmentarum bisegmenta, et quae sequuntur in infinitum, fore semper partes (aaa) rationales, quod satis (constat) (bbb) toti commensurabiles nee unquam bisecando ad partem (aaaa) irrationalem (bbbb) inncommensurabilem perveniri posse (aaaaa) , sed (bbbbb) . del]

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que de Bisectione dicitur; de aliis sectionibus in partes commensurabiles, pariter ostendi potest, etiam in infinitum continuatis.) Nunquam igitur, hac ratione, ad Incommensurabilitatem pervenietur. Adeoque argumentuni ejus, ab interminabili divisibilitate continui, ad partium Incommensurabilitatem; non modo non probat quod susceperat probandum; sed probat eum Commensurabilitatis et Incommensurabilitatis naturam non satis intelligere. Quod ex proxime dicendis confirmabitur. | [55, p. 8] Nam §29. Utrum fortuito oblata Problemata sive Theoremata, in quibus Commensurabilitas vel Incommensurabilitas ex ipsis terminis non statim apparet, Geometrica solum an vero Numerica simul sint, id est utrum solis magnitudinibus, an et numeris etiam accommodari possint, hac (inquit) ratione dignosces. Si ad illorum constructionem arbitraria tantum requiratur quantitatum Divisio, vel Multiplicatio, indubitabile signum est, ipsa de utraque quantitatum specie simul exponi: Si vero per appositam in quaestione conditionem determinatae vel multiplicationes vel divisiones necessariae sint ad quaesitum efficiendum, tune generales Commensurabilitatis vel Incommensurabilitatis regulae docebunt, utrum numerorum essentiae talibus multiplicationibus ferendis idoneae sint. Admodum imperite. Quippe nullae vel Additiones, vel Subductiones, vel

1 de (1) quavis alia sectione pariter obtinet, quae sit in partes divisa (2) aliis sectionibus in partes commensurabiles, pariter (a) obtinet (b) ostendi potest, 2 Nunquam . . . pervenietur. add. 4 continui, (1) non modo non probat (2) ad partium 7 satis add. 15 quaestione add. 16 generales (1) Commensurabilitates vel Incommensurabilitates (2) Commensurabilitatis vel Incommensurabilitatis 18 Admodum imperite. missing in E2 19 imperite. jEstque hoc indubitale signum (ut cum eo loquar) Commensurabilitatis et Incommensurabilitatis naturam, liuic minime perspectam esse. del. \ Quippe 19 Quippe (1) nulla unquam vel Additio, vel Subductio, vel Multiplicatio, vel Divisio (2) nullae vel Additiones, vel Subdictiones, vel etiam Multiplicationes, vel Divisiones, (inter

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217. WALLIS to OLDENBURG, 18/[28] July 1668 etiam Multiplication.es, vel Divisiones, (inter terminos invicem commensurabiles peractae,) ullam unquam Incommensurabilitatem inducent. Oritur utique haec ex Radicum extractionibus' (quoties nempe faciendae requirantur, nee absolvi possunt.) Adeoque si nulla requiritur Radicum extractio, (seu quod huic tantundem est;) sed Additionibus, Subductionibus, Multiplicationibus, et Divisionibus, (inter terminos commensurabiles peragendis,) quaecumque demum illae, vel qualescunque fuerint, peragenda sint omnia: nullus erit Incommensurabilitatis metus. Admodum igitur imperite, et absurde satis, de Multiplicationibus et Divisionibus, hac in re, praecepta tradit. Quod et indubitabile signum est, (ut cum ipso loquar) Commensurabilitatis atque Incommensurabilitatis naturam, huic minime perspectam esse. Porro §31. Rationem definit esse, determinatam quandam aequalitatis, inaequalitatisve, speciem. Cujus contrarium verum est. Sunt enim Aequalitas et Inaequalitas, species rationis. Mox §32. Cum Rationem in Arithmeticam et Geometricam divideret; De Ratione indiscriminatim pronunciat Rationis terminos in infinitum augeri posse, manente semper eadem ratione: quasi idem in Arithmetica ratione (quae Differentiis aestimatur, non Quotientibus,) pariter verum esset atque in Geometrica. Sed taedet plura commemorare. Haec interim eorum aliqua sunt (nee tamen omnia) quae in ipius Libri primi Capite primo, notanda censui. Ex quibus possis de reliquis conjecturam facere. Totum vero librum ita recensere atque ad examen vocare, mini neque vacat, neque animus est: Sed neque operae pretium fore autumo. Haec autem sunt de quibus gloriatur;

2 inducent. (1) ; (ut |adeoque add] absurde admodum ad de Multiplicationibus et Divisionibus hac in re praecepta tradit.) (2) . Oritur 3 ex (1) nuper at breaks off (2) Radicum 3 nempe add. 10 ut add. 13 Rationem (1) seu Propor breaks off (2) definit esse (a) defin breaks off (6) determinatam 18 idem (1) Arithmetica ratione seu proportione (2) in Arithmetica ratione 24-524,1 animus est: (1) Tu vero vides (a) quaenam ea sunt de (6) qualia sunt ea de quibus ipse gloriatur, quae adhuc observavit nemo, quae hucusque nondum tradita. Tu (2) Sed neque .. . gloriatur; quae (oa) adhuc observavit nemo (inquit) (66) adhuc observavit nemo; quae hucusque nondum tradita.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 quae adhuc, observavit nemo; quae hucusque nondum tradita. Tu vero boni consulas; et Vale. Tims, Johannes Wallis.

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(C] A Continuation of Dr Wallis his second Letter, publish't in Numb. 39, to [57 (u) r ] the Printed Paper of M. Du Laurens.

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This other part of Dr Wallis's second Letter to M. Du Laurens, though written and sent to the Publisher at the same time, when the first part was, yet came not then abroad, upon a consideration intimated in Numb. 38, p. 750; and the same could not find room in these Tracts, till this month, when 'tis publish't, rather from a desire, further to comply with the said Du Laurens, demanding the reasons of the Animadvertor's Censure, than from any propension to disputes. The Publisher can bona fide assure the Author of the Paper, here further animadverted upon, of the reality of what is here affirm'd and profess'd by him, and in particular, that the originall of this, what follows, came to his hands but a day or two after its Date, which was July 18. 1668. the same, which was mentioned Numb. 38. p. 750. | [57 (u) r ] The Letter itself is thus continued, Porro (ut minutiora quaedam praeteream, ne nimius sim, quae tamen ipsa reprehensionem merentur) inter alia, quibus Aequalitatem ob Moderationis virtutem laudat, Inaequalitati interim vitio ver-

6 publish't add. 7 of (1) Monsr (2) M. 8 This (1) second (2) other 11 till (1) now, (2) this month, 12 publish't (1) much (2) rather 12 desire, (1) to {—} (2) further to comply with (a) the demand of the said Du Laurens, insisting to (b) the said Du Laurens, demanding 14 any (1) Inclination to (2) propension 17 the originall of add. 17 follows, (1) hath been in his hands ever since the first (2) came 20 itself (1) follows thus, (2) continues thus, (3) is thus continued,

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218. OLDENBURG to WALLIS, 18/[28] July 1668 tens, quod Excessu et Defectu laboret, haec occurrunt §18. Tanta est aequalitatis moderatio; ut eas non solum, quas afficit quantitates, augeat, minuat, multiplicet atque dividat, nulla facta in ipsis quoad aequalitatem mutatione, sed etiam, ut quantitates ab inaequalitate affectas per similes operationes tractando, intactam in illis inaequalitatis notam relinquat. Id, credo, vult

218.

HENRY OLDENBURG to WALLIS 18/[28] July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 20/[30].VII. 1668. In this letter, Oldenburg apparently informed Wallis that his short reply to Dulaurens (i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii)) could not be included in the Philosophical Transactions for July. At the same time, he appears to have offered to announce that Wallis's piece would appear in the subsequent issue. In addition, the letter contained news from France on Huygens's disclosure of mistakes in Dulaurens's Specimina mathematica. Of this, Oldenburg had been informed by JUSTEL-OLDENBURG [4]/14.VII.1668 (OLDENBURG, Correspondence IV, 499-502, 499-500).

219. JOHN COLLINS to JOHN PELL London, 18/[28] July 1668 Transmission:

C Letter sent: LONDON British Library MS. Add. 4278, f. 340r-340v. On f. 340r beneath date in Pell's hand: 'Received Jul. 20 th .' Postmark on f. 340V: 'JY/18'. Enclosure: Wallis's Catalogue of Errata in Brancker's Table of Incomposite Numbers. Answered by: PELL-COLLINS 29.VIII/[8.IX].1668.

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219. COLLINS to PELL, 18/[28] July 1668 Reverend Sir

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I yesterday received a Letter979 from Mr Branker in answer to Mr Merry's980 Scruples981. Mr Branker supposeth I kept a Coppy of his obiections, but I assure you I did not, nor had leisure to read them at all. Mr Branker likewise sent up 19 faults982 more (saith he) to be added to those in page 198, and am now so confident that my table is thoroughly corrected that out of it I dare Collect the enumeration of Incomposits alone mentioned page 193 and thinkes of sending it by some other opportunity. The last 5 lines in the former Page are the Errata as sent by Mr Branker. All the rest above, are the Errata as examined by Dr Wallis who you see hath enlarged them to a greater Number. This I promised sooner to send, but had not leisure. You write to know what you are in my Debt983. I answer my Lord Brereton hath promised to pay it all, and entred into Bond so to doe. Whereto there is but one Witnesse who is his Lordships Servant. And that being paid there will be nothing due. I should be glad to receive your censure of Mercators Logarithmotechnia and of Riccio984. And when his Lordship intends to returne, be pleased to Mind him to bring up all his bookes of Trade. As Baker985 of the Smyrna trade &c, and also the

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Letter: i.e. BRANCKER-COLLINS c.!6/[26].VII. 1668 (not found). Merry's: i.e. Thomas Merry (d. 1683), English mathematician. Following his death, Wallis received from Collins a large manuscript tract, in which Merry expounds Hudde's rules. Wallis's plan to publish the manuscript was not carried out, though he did print excerpts. See WALLIS, Algebra, 142; Additions and Emendations, 157-62; Opera mathematica II, 150-5. See also COLLINS-PELL 9/[19].IV.1667 (RiGAUD, Correspondence I, 127) and COLLINS-WALLIS 21/[31].III.1670/1. A list of Merry's works is given in BIRCH, History of the Royal Society IV, 326-7. 981 Scruples: i.e. Merry's objections to RAHN, An Introduction to Algebra, translated out of the High Dutch into English by T. Brancker ... Much altered and augmented by D[r]. Pfell]., London 1668. 982 19 faults: Collins has copied these at the foot of Wallis's Catalogue of Errata. 983 Debt: Collins appears to have regularly sent money to Pell. 984 Mercators Logarithmotechnia, and of Riccio: i.e. MERCATOR, Logarithmotechnia and RlCCI, Exercitatio geometrica de maximis et minimis, which were printed together at London in 1668; see PELL-COLLINS 29.VIII/[8.IX].1668. 985 Baker: not identified. 980

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220. COLLINS to PELL, 18/[28] July 1668, enclosure Seiur de Taneur986 on the 10 of Euclid gallice which I have a desire to see. I cease enlargement and subscribe my selfe Brooke house July 18th 1668

Your most affectionate servant JC

[340V] To the Reverend Doctor John Pell at the house of the right honourable the Lord Brereton at Brereton in Stone bagg Cheshire

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JOHN COLLINS to JOHN PELL 18/[28] July 1668, enclosure: Wallis's Catalogue of Errata in Brancker's Table of Incomposite Numbers Transmission:

C Copy of (missing) paper sent in Collins's hand: LONDON British Library MS. Add. 4398, f. 175r. Enclosure to: COLLINS-PELL 18/[28].VII. 1668. Some time after receiving Wallis's catalogue of errata, which were enclosed in WALLISCOLLINS early VII.1668, Collins produced a copy which he enclosed in his letter to Pell. Beneath the table he has entered five lines of errata received from Brancker himself, all of which had also been recognized by Wallis. Pell has marked off those ten errata in Wallis's catalogue which are not contained in Brancker's earlier lists. Cf. PELLBRANCKER 21/[31].VIL1668 and PELL-COLLINS 29.VIII/[8.IX].1668, where Pell points out that Brancker had mistakenly written 40597 for 40599.

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Seiur de Taneur: i.e. Jacques-Alexandre Le Tenneur (fl. 1639-48). His Traite des quantites incommensurables, ou sont decidees plusieurs belles questions des nombres rationaus et irrationaus. Les erreurs de Stevin refutees. Et le Dizieme livre d'Eudide, illustre de nouvelles demonstrations plus faciles et plus succinctes que les ordinaires, et reduit a 62 propositions. Avec un Discours de la maniere d'expliquer les sciences en francais was published in Paris in 1640. See PELL-COLLINS 28.X/[7.XI.]1668 and note to CAVENDISH-MERSENNE 22.VIII/[1.IX].1640 (MERSENNE, Correspondance X, 99).

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220. COLLINS to PELL, 18/[28] July 1668, enclosure

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Page Number for set 3 5579 7 P 9211 5 19 61 9287 19 37 9799 40 41 6 10199 7 P 11 10813 13 7 —47 37 47 13201 23 43 8 14873 73 107 9 17563 3 7 41 17981 P 10 18903 7 3 18907 3 7 18909 7 3 11 20983 3 P 12 23203 3 P 23381 193 103 13 24011 3 13 25093 13 23 25873 23 P 14 27233 31 113 27517 7 P 15 28201 3 P 28203 3 P 29599 blank P 16 30167 71 97 31001 -29 29 17 32297 71 P 33259 97 79 33409 47 P 33591 7 3 33593 3 7 18 34089 7 3 34209 23 3 35089 3 P 37583 19 13 7 3 5 8 11 16

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set 7 11 29 P P 3 P 3 P 43 17 7 3 466 3 P P 7 3 7 7 7 3 P 17 P 3 7 79 P 151 13 73 173 P 3 139 P 7 29 3 P

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for 72383 3 72557 73 72601 97 73023 P 73051 7 73481 179 73493 P 73913 P 75653 151 78199 P 80333 67 80561 17 80663 P 83123 103 84311 57 85909 137 86699 281 86993 79 91189 P 91707 P 91793 23 92701 3 92703 7 92773 163 93101 151 52 93161 93719 7 94769 41 95371 281 95797 P 96109 3 97487 3 97903 3 98099 26 98551 39 99443 17

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137 79 7 41

set P 37 79 3 11 197 7 7 P 11 11 13 11 101 59 P 181 P 7 3 13 7 3 113 157 59 P 97 283 13 13 13 13 263 139 277

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221. WALLIS to OLDENBURG, 20/[30] July 1668

221.

WALLIS to HENRY OLDENBURG Oxford, 20/[30] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 54, 2 pp. (our source). At top of p. 2 at 90° to address in Oldenburg's hand: 'not to be entered', and beneath address: 'Ace. d. 21. julii 68.'—printed: OLDENBURG, Correspondence IV, 560. Reply to: OLDENBURG-WALLis 18/[28].VII.1668. Enclosure: WALLIS-OLDENBURG 18/[28].VII.1668.

Oxford July. 20. 1668. Sir,

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I had finished the inclosed987 on Saturday, but because the pacquet would have been too big to send by the Post, I send it not till this morning by the carrier. If you have thought fit to insert what I sent before988, into the Transactions of this Month, I think that will be enough for once. And if that be not inough for the whole, you have this in store against another month if there bee occasion.

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Yours &c. J. Wallis.

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Since I wrote that above I received yours989 of July. 18.1 am not much displeased that what I wrote, is left out this time; though I had rather it had been in. Chiefly because I would not have too much of one thing at a time in it. The expedient you offer, will serve the turn; especially if it were mentioned in the transactions. That of Mr Gregory990 might possibly 4 too big (1) for (2) to send by 8 occasion. (1) whi breaks off (2) Yours 15 in the I in the del. ed.\ transactions 98r

inclosed: i.e. WALLIS-OLDENBURG 18/[28].VII.1668. sent before: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii). 989 yours: i.e. OLDENBURG-WALLIS 18/[28].VIL1668. "°That of Mr Gregory: i.e. Gregory's written reply to Huygens's review of his De vera circuli et hyperbolae quadratura, printed in Philosophical Transactions No. 37 (13 July 1668), 732-5 ('Mr. Gregories Answer to the Animadversions of Monsieur Hugenius upon his Book, De vera Circuli & Hyperbolae Quadratura . . . ' ) . Cf. COLLINS-BRERETON 11/[21].VIL1668. 988

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222. WALLIS to COLLINS, 21/[31] July 1668 require more hast then mine. I find by your note from France991; that I was not mistaken in what I conjectured992 of his mean proportionals. Certainly he is no competent undertaker of such designs: & but a very indifferent superficial mathematician. And I think it would be more honour then his book deserves: For mee to confute it. Of the inclosed, I keep no copy. [2] These For my worthy friend, Mr Henry Oldenburgh, at his house, about the middle of the Pelmel, near St James's London.

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222. WALLIS to JOHN COLLINS Oxford, 21/[31] July 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 198r-198av (f. 198V and 198ar blank) (our source). At top right off. 198av in Collins's hand: 'About finding the Area of an Hyperbola by Logarithmes'. Postmark on f. 198av: 'JY/22'.—printed: RIGAUD, Correspondence II, 490-2. Reply to: CoLLiNS-WALLis 14/[24].VIL1668.

Oxford July 21. 1668.

Sir, I received yours993 of July 14. & thank you. What I wrote994 of Mr Mercator, was not intended to his disadvantage at all; but the contrary. For, though I know him not, I have a good esteem for him. What concerns the 1 by yours (1) of (2) note corr. ed. 4 it (1) is (2) would 991

your note from France: i.e. JusTEL-OLDENBURG [4]/14.VII. 1668, of which Oldenburg had informed Wallis in his letter of 18/[28].VII. 1668. 992 conjectured: see WALLIS-OLDENBURG 6/[16].VII. 1668. 993 yours: i.e. COLLINS-WALLIS 14/[24].VIL1668. 994 wrote: presumably in WALLIS-COLLINS early VII. 1668.

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222. WALLIS to COLLINS, 21/[31] July 1668

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Logarithms, I did so far onely consider (as I read it over once, in some hast,) whether what was done were well; & it seemed to mee so to be, & very ingenious; & so much as would be sufficient for a good expedite way of calculating Logarithms; but I did not so far study it as to consider whether or how it might bee further improved. For the whole was but an afternoons work. Onely having found the Quadrature of the Hyperbola, a little lame (that is, not so full as wished it were,) I did that night consider how it might be improved, & the next morning wrote that to Mr Oldenburg995. Because I was unwilling, to leave to foreigners the perfecting of that which was by ours carried on so far. That this expedient in squaring the Hyperbola, must need afford a proportional help in the Logarithms; is evident: because they depend one upon the other. And his last proposition, of finding a summe of Logarithms; may be much expedited from some other principles that I have not room here to mention. In sum if (as I put it) we make AH = 1. AI = IB = b. HI = A. and the plain BIHF = pi. then is pi - b2 + 63 = BIps + BIqt + BIrn &c to BIHF.

What I wrote996 of the Errata in Mr Brankers Table; was not to find fault with the correcting; but to make a supply of his table of Errata, which in such a businesse is material to the Reader: & cost mee allmost as much pains as calculating the whole Table anew (saving the time of writing it over) for I used the same method to examine as I would have done to calculate a new one. I can say nothing to the comparing997 of Mr Gregorie's with that M. Huygens, because I do not remember that I have seen that of Huygens. 16 BIHF (1) ; which is summe of the Logarithms (2) . \\ What 20 pains as add. calculating 995

wrote . . . Oldenburg: i.e. WALLIS-OLDENBURG 11/[21].VII.1668. wrote: i.e. in WALLIS-COLLINS early VII.1668. 997 the comparing: cf. COLLINS-BRERETON 11/[21].VII.1668. 996

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223. PELL to BRANCKER, 21/[31] July 1668 But wonder that Huygens should write against him unprovoked, being himself an ingenious modest man, as I have hitherto apprehended him: & certainly Gregorie's piece is not contemptible. The other particulars you mention of Mr Gregories, are good; &, I suppose, likely inough to be true. In printing my things998: I had rather you make use of Mr Oughtreds note of Multiplication x than that of *; the other being the more simple. And if it be thought apt to be mistaken for X, it may helped by making the upper & lower angles more obtuse x . The Post is going; so that I can adde no more but that I am yours &c. John Wallis. [198av] These For Mr John Collins at his house next the three Crowns in Blomesbury-market. London.

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223. JOHN PELL to THOMAS BRANCKER Brereton, 21/[31] July 1668 Transmission:

C Letter sent: LONDON British Library MS. Add. 4278, f. 81r-82v (f. 81V and f. 82V blank). Answered by: BRANCKER-PELL 25.VII/[4.VIII].1668. These additional errata, taken from Wallis's catalogue, were too late to be incorporated into RAHN, An Introduction to Algebra, translated out of the High Dutch into English by T. Brancker . . . Much altered and augmented by D[r]. P[ell]., London 1668. Apaxt from the first, and with the addition of twenty others, they were published by Wallis in A Discourse of Combinations, alternations, and Aliquot Parts, London 1685, 136.

5 of add. 998

my things: probably the manuscript of Wallis's Mechanica sive de motu tractatus geometricus, which Collins was seeing through the press. See COLLINS-PELL 21!V/[1.V].1668, British Library MS. Add. 4278, f. 334r and WALLIS-COLLINS 8/[18].IX.1668.

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224. BRANCKER to PELL, 25 July/[4 August] 1668 July 21 Mr Collins in his letter999 of July 18th sent Dr Wallis his Catalog1000 of 145 Errata in your Table. That is, TEN more than your first & second Collection containes. They are these Page

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7 16 28 31 33 34 41 49

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For

Set

—47 37 47 31001 -29 29 54449 71 P 55609 3 P 60701 01 101 64499 13 P 1 67993 P 80561 17 13 96109 3 13 97487 3 13

He addes that he hath your answere to Mr Merry's scruples. I assure you, saith he, I did not keep a coppy of his objections nor had leisure to read them at all. Brereton, 7° 21 20

[82r]

to T.B. at Newchurch to be left with Mr Peter Yates mercer in Middle Wiche

224.

THOMAS BRANCKER to JOHN PELL 25 July/[4 August] 1668 Transmission:

C Letter sent: LONDON British Library MS. Add. 4278, f. 87r-87v. At bottom off. 87r Pell lias written: 'I received this on Munday August 3.'

19 Brereton in cipher 999

letter: i.e. COLLINS-PELL 18/[28].VII.1668. Catalog: see enclosure to COLLINS-PELL 18/[28].VII. 1668.

1000

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224. BRANCKER to PELL, 25 July/[4 August] 1668 Reply to: PELL-BRANCKER 21/[31].VII.1668. Honoured Sir Yours1001 of the 21 I had the 22d with Drs Ws. ten Errata not observed by me. I am very glad a Person of such eminent Fame and Industry hath been at the pains to review the Table. And yet perhaps it is not correct. But for these ten, the first of them I saw but sleighted: the 2d and 5* are, in my copyes, not errata: the 3 last are all that I yet know to have escaped my red pen unduely: the remaining 4 were left un-redded, but I saw them not when I went over all to find all that were not marked red. I note these things so particularly that you might the better conjecture at the reason of these oversights, and direct me to any other way more sure, and more likely to maintain attention in the proof. Which if you do I shall once more prove the table: for I yet doubt its exactnesse. I purpose to be at Br: as soon as I well can: But, partly bee: of Mr F. Ch: Mathem:1002 and partly bee: our Lad is gone from us, I think it will not be till the middle of the 2d week to come. I pray be pleased to tell Mrs Linds:1003 that I yesterday found St Johns wort in great plenty, and in flour. If she desire it I can send her as much as in reason she can use this year. But I must find somebody here to employ about it. Possibly it were better for her to send some one early in the morning hither and he need not be above \ an hour gathering of a convenient burden. I am briefe bee: I know not through what hands this is to passe. If you please to send at any time to Mr Peter Yates Mercer in Middle Wiche. He can soon convey it to me. (or, it comes safely by Joh: Faringdon when you have that opportunity.) We are all well and both of us prefer our respective salutations &c. d

There hath been a 2 attempt but lesse violent July 25. 68.

Your ever obliged T.B. V.S.S.

5 for (1) this (2) these 12 prove (1) it (2) the table 12 yet (1) fear (2) doubt 21 burden. (1) But if she will (2) I 1001

Yours: i.e. PELL-BRANCKER 21/[31].VIL1668. 1002 p QJ^ Mathem: j e Frallcis Champante. 1003 Mrs Linds: presumably an apothecary at or near Brereton.

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225. WALLIS to BRAMSTON, 27 July/[6 August] 1668 [87V]

For Dr Pell at Brereton Hall

225.

WALLIS to JOHN BRAMSTON Oxford, 27 July/[6 August] 1668 Transmission:

W Draft letter: OXFORD Bodleian Library MS. Savile a. 11, f. 89V.

Oxford July. 27. 1668. Sir 5

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Being so wholly a stranger to those of the Gentry in Essex; I was incouraged by the Warden1004 of Wadham College to make my applications to yourself (though a Stranger) as a civil & generous person; in a business wherein my self & Dr Christopher Wren (the two Savilian Professors of Mathernaticks in this University) are concerned. Trusting that (the business being just) you will excuse the confidence that a Stranger takes in giving you this trouble. There lyes in Rettenden, a farm called Littlehays, now in the occupation of one Henry Spitty1005; which is part of the Lands given by Sir Henry Savile for the maintainance of his two Professors. The whole Rent whereof being part of our Stipend; wee have taken it to be exempted from the Assessements for the Monthly Contribution, by a Proviso (confirmed, for about twenty years last past, in all Acts of that nature; as well since his Majesties Restauration as before;) whereby it is Provided, That nothing contained in the sayd Acts shall be extended to charge any - Reader - in either of the Universities, for or in regard of 10 the (1) trouble (2) confidence 14 whole add. 18 is (1) Prop breaks off (2) Provided 18 nothing (1) therein contained (2) contained in the sayd Acts 1004

Warden: i.e. Gilbert Ironside, the younger (1632-1701), warden of Wadham College 1667-92; vice-chancellor of the University of Oxford 1687-9. Thereafter bishop of Bristol (1689-91) and of Hereford (1691-1701). DNB. 1005 Spitty: see WALLIS and CHRISTOPHER WREN-ESSEX COMMISSIONERS 27.VII/[6. VIII].1668. Cf. WALLIS-? 3/[14].IX.1700.

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225. WALLIS to BRAMSTON, 27 July/[6 August] 1668 any Stipend, Wages, or Profit whatsoever arising or growing due to them in respect of their places & imployments in the sayd universities. Which hath been allowed a good Plea, & the Professors for the time being have, from the first, constantly injoyed this exemption, now for about twenty years; not only in Essex, but in all other counties wherein wee are con- 5 cerned; viz. in the Counties of Northampton, Kent, & Glocester. But wee are now informed from our Tenant, that now very lately wee have been assessed for our rents issuing out of those Lands, & the mony levied by distresse. (Occasioned, I suppose, by the endeavours of some, who, by charging of us, would ease themselves.) Wee have, upon this occasion, 10 addressed a Letter1006 to the Commissioners for that purpose. Which our Tenant hath direction, if he have opportunity, to show you; & to receive your instructions. I do not know, whether, about six years since, My Lord chancellor1007 did not write you a letter in our behalf, (the same thing being then in question;) Hee did, at lest, promise us so to do if there 15 should bee need. But this I know, that it was then adjudged for us; &, since that time till now, wee have not been troubled. And if to the justice of our cause, you shall please to adde the favour of your assistance; I trust we shal find the same successe again; and you will very much oblige

Sir, For the Right Worshipful Sir John Brampston in Essex

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Your very humble servant John Wallis.

1 them (1) by virtue of (2) in respect of their \sayd del] places (a) & (b) or im breaks °ff (c) & imployments 2 universities. (1) And wee have, upon this plea, [constantly add.\ injoyed that exceniption, (2) Which 3 have (1) injoyed this exemption, now (2) , from 4 for (1) about (2) about 7 that (1) wee have been very lately assessed (2) now very lately wee have been assessed 11 for (1) redresse (2) that 12 opportunity, (1) to let you see (2) to 13 your (1) d breaks off (2) instructions, (a) Not (b) I 13 years (since, add] (1) (the same thing (2) My Lord chancellor did not jthen del] write 15 question (1) & adjudged for us;) (2) ;) Hee 1006 1007

Letter: i.e. WALLIS and WREN-ESSEX COMMISSIONERS 27.VII/[6.VIII].1668. Lord chancellor: i.e. Edward Hyde, first earl of Clarendon (1609-74).

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226. WALLIS et al. to the ESSEX COMMISSIONERS, 27 July/[6 Aug.] 1668

226.

WALLIS and CHRISTOPHER WREN to the COMMISSIONERS OF THE MONTHLY ASSESSMENTS IN THE COUNTY OF ESSEX Oxford, 27 July/[6 August] 1668 Transmission:

W Draft letter: OXFORD Bodleian Library MS. Savile a. 11, f.89r.

Oxford July. 27. 1668. Gentlemen,

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We understand from our Tenant Henry Spitty1008 of Rettendon in the County of Essex, that a distress hath been lately taken for monyes assessed toward the Monthly contribution, for the farm which he there hath, called Little-hays. And though, wee beleeve, it was not by any direct order of yours to that purpose; but rather by the mistake of the Collectors or Assessors: Yet you are the persons to whom wee are to apply for redresse. You may please therefore to understand, that the lands aforesayd were given by Sir Henry Savile to the University of Oxford (not to any particular college) in trust1009, for the maintenance of the two Mathematick Professors, or Publike Readers of Geometry & Astronomy in the sayd University; & that the whole rent thereof, be it more or lesse,

3 in the County of Essex add. 4 assessed (1) to the (2) toward 5 faxm (1) of Little-hays which he hath. (2) which he there hath, called Little-hays. 6 wee |do del.\ beleeve 7 rather add. 10-11 to the University . . . in trust, add. 12 Publike add. 12 of (1) Astron breaks off (2) Geometry 13 sayd add. 1008

Spitty: see WALLIS-BRAMSTON 27.VII/[6.VIII].1668. Cf. WALLIS-? 3/[14].IX.1700. given by ... in trust: see Statutes of the University of Oxford, ed. Griffiths, Oxford 1888, 249-50. 1009

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226. WALLIS et al. to the ESSEX COMMISSIONERS, 27 July/[6 Aug.] 1668 is the stipend or wages of the sayd publike Readers; & that whatsoever is charged on the sayd rents, is charged on the sayd Stipend or wages. You may please allso to consider that both in the Act of Parliament for the Royall Ayd (to which that allso for the Additional supply doth referr,) and all other Acts for monthly contributions not only since his Majesties happy restitution, but even in the late times of Usurpation, it is & hath been airways expressely Provided That nothing contained in the sayd Acts shall be extended to charge any — Reader in either of the Universities, for or in regard of any Stipend, Wages, or profit whatsoever, arising or growing due to them in respect of their places & imployments in the sayd Universities. By virtue of which Proviso (continued from time to time for about twenty years last past) the Rents of the sayd lands (as being the Stipend of the sayd Publike Readers) have airways been exempted from such assessements (& so reputed) as well in the late times of usurpation, as ever since his Majesties happy Restitution, even to this time. And when at any time, by mistake, (as about eighteen years ago,) the sayd lands have been assessed; upon application to the Commissioners for that purpose it hath airways been redressed, & the mony (when any hath been levyed) repayd again. And allthough the Proviso hath not been thought to extend to all College lands, (because the rents thereof are not onely for

1 whatsoever is (1) assessed (a) thereon, is (6) on the sayd rents, (2) charged on the sayd rents, is 2 on the (1) Stipend or wages, of the sayd Professors (2) sayd Stipend or wages. 3 that (1) as well (2) both 4 allso (1) of (2) for 4 referr,) (1) as (2) and 6 it (1) is (2) is fe hath been allways 8 shall (1) ext breaks off (2) be extended 10 sayd (1) University. (2) Universities, (a) By virtue of which Proviso (oa) , the sayd lands (66) (continued from time to time) the Rents of the sayd Lands have been (oao) ther breaks off (bbb) hitherto (aaaa) as breaks off (bbbb) exempted, (& so reputed,) from (b) By 12 the (1) sa breaks off (2) Rents 13 allways |been add] exempted (1) hitherto (& so reputed) from such assessements, (2) from such assessements (& so reputed) 15 as (1) ever since (2) since (3) ever since 15 even add. 16 (as about eighteen years ago,) add.

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226. WALLIS et al. to the ESSEX COMMISSIONERS, 27 July/[6 Aug.] 1668

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the Stipends of the Heads & Fellows of the sayd Colleges, but due to the Colleges at large, as well for those as for divers others uses;) yet to these (which are not College lands, but of another nature) it hath allways been allowed to extend; because here the whole rent, bee it more or lesse, is the Stipend of the sayd Publike Readers, & not applicable to other uses; & therefore clearly within the words of the Act. The same justice & favour hath been in like manner allowed as all along in all other Counties where wee have been concerned, viz. in the Counties of Northampton, Kent, & Glocester. And when, upon the change of the Commissioners, presently after his Majesties return, the Rents of these lands in Essex were once assessed: The commissioners who then were (the same wee suppose, for the most part, with those that now are,) upon a like application with this, did us the favour (without any further attendance) to discharge that Assessement, & give such order as that wee have never since been troubled till this time. Since therefore it is no new case, but what hath been so often, & in so many places adjudged in favour of us, & which is in itself so clear: wee trust, that those worthy persons who are now commissioners, will not be found either less favourers of learning, or less friends to the Universitie, than those who have been Commissioners, in worse times, or in other places, or lesse than yourselves have formerly been. We have therefore directed our Tenant to wait upon you; & desire you will please to give him & us such redresse in this case, that wee may

2 to these . . . another nature) add. 4 extend; (1) to these (which are not (a) College) (6) in the nature of Coll: lands) (2) because |here add.\ the 6 Act. (1) Nor hath it been onely so allowed (2) The same (a) favour fe (b) justice 7 as all along add. 7 Counties (1) wherein (2) where 9 when (1) as presently upon the change of the Commissioners in the County of Essex, (2) upon the change of the Commissioners, presently 10 the Rents of add. 13 this, (1) {—} (2) did 13 attendance) (1) to give us such redresse (2) to 14 order as add. that (1) till now (2) wee 21 therefore (1) giv breaks off (2) directed

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227. WALLIS to BROUNCKER, July/August 1668 not our selves be put to the trouble of so long a journy to look after it. And you shall oblige, Gentlemen, Your very humble servants, John Wallis D.D. Professor of Geometry 1 Christopher Wren D.L. Professor of Astronomy > in the University of Oxford. For the Right Worshipful the Commissioners of the Monthly Assessements in the County of Essex.

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227. WALLIS to WILLIAM BROUNCKER Oxford, July/August 1668 Transmission:

E First edition of missing letter sent: Philosophical Transactions No. 38 (17 August 1668), 753-6 ('Logarithniotechnia Nicholai Mercatoris: discoursed of in a letter written by Dr. J. Wallis to the Lord Vis-count Brouncker') (see WALLIS-OLDENBURG 8/[18].VII. 1668, where alterations and modified ending of E are reproduced). In this letter, Wallis apparently sent Brouncker this amended and expanded version of his letter to Oldenburg of 8/[18].VII.1668 at the end of July or the beginning of August, as he reports in WALLIS-OLDENBURG 3/[13].VIIL1668. There he also expresses the wish that this version be printed in Philosophical Transactions. It was duly published as a letter to Brouncker, though with the date of the earlier letter, together with the demonstration he had promised at the end; see WALLIS-OLDENBURG 8/[18].VII. 1668. See also WALLIS-BROUNCKER 5/[15].VIII. 1668.

1 our selves add.

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229. WALLIS to OLDENBURG, 3/[13] August 1668

228.

HENRY OLDENBURG to WALLIS July/August 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 3/[13].VIII. 1668. In this letter, Oldenburg apparently reported on the response to Wallis's letter to Moray of 14/[24] July, which had been read at the meeting of the Society on 16/[26] July. He also requested Wallis's opinion on Huygens's review of Gregory's Vera circuli et hyperbolae quadratura and of Gregory's reply, recently published in Philosophical Transactions. Probably both pieces were enclosed to this purpose. See WALLIS-BROUNCKER 4/[14].XL1668.

229.

WALLIS to HENRY OLDENBURG Oxford, 3/[13] August 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 58, 2 pp. (our source).— printed: OLDENBURG, Correspondence V, 3-4. Reply to: OLDENBURG-WALLS VII/VIII.1668.

Oxford Aug. 3. 1668.

Sir,

5

I received yours1010 by your friend1011; & shall be ready to serve him for your sake, & his own; for hee seems to be an ingenious person. The particulars of your letter, most of them, I have answered in one1012 to my Lo. Brouncker. The experiments1013 of Borellus I am well contented should 6 well add. 1010

yours: i.e. OLDENBURG-WALLIS VII/VIII.1668. friend: not identified. 1012 one: presumably WALLIS-BROUNCKER 3/[13].VIIL1668. 1013 experiments: i.e. pendulum experiments described in BORELLI, De m percussionis, Bologna 1667 (see WALLIS-MORAY 14/[24].VII. 1668), and discussed at the meeetings of the Royal Society on 16 and 23 July 1668 (BIRCH, History of the Royal Society II, 1011

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229. WALLIS to OLDENBURG, 3/[13] August 1668 be true, & I am ready inough to beleeve them, (as others who write de motu projectorum are wont to do:) yet (if it have not been done allready) I thought they might have been repeated for confirmation, that wee may (in point of experiment) trust our own eyes. But, it seems something of that kind hath been done allready. Borellus his book I have read; &, like him well. And, though I have not so far examined him, or am so satisfyed, as to avouch it all: yet there is, at lest, very much of it good: &, I hope so of the rest, though I cannot say that I am satisfyed in all. Mr Gregories reply1014 to M. Hugens, I am well inough satisfied with: having compared it with what was before written, & what M. Hugens objected. I do a little wonder, that M. Hugens should write against him unprovoked, unlesse there had been more cause then what I yet see: For (except a little slip in one example, which was not materiall,) the rest is Hugens mistake, not his. And, that there was a conformity between the Hyperbolick spaces, & the Logarithmes; tis true, is not new in Mr Gregory; but neither was in M. Hugens; But was, before either of them, in Gregorius de Sancto Vincenfo'o1015; from whence I suppose both of them, and others allso, have taken the notion first; though managed it variously. You will see it in my Commercium Epistolicum Epist. 39. 401016. What you have of mine1017 concerning Du Laurens. I did beleeve

4 of that (1) hath bee breaks off (2) kind 5 &, (1) find that (2) like him well. And, though I have |no corr. ed. so fax examined him, or (a) so (b) am 7 much |of it add] good; (1) (into) (2) (in) (3) &, 16 Gregorius (1) the (2) de 18 first; (1) they (2) though managed it (a) differently (6) variously. 20 did (1) fear (2) beleeve 306-9). Oldenburg presumably had sent Wallis much the same information on these experiments as he did in his letter to Boyle of 27. VII/[6. VIII]. 1668 (OLDENBURG, Correspondence IV, 569-71, 571). 1014 Mr Gregories reply: i.e. 'Mr. Gregories Answer to the Animadversions of Monsieur Hugenius upon his Book, De vera Circuli fe Hyperbolae Quadratura . . . ' , Philosophical Transactions No. 37 (13 July 1668), 732-5, in reply to Huygens's review of Gregory's De vera circuli et hyperbolae quadratura, printed in Journal des Scavans (2 July 1668 new style), 52-6. 1015 in Gregorius de Sancto Vincentio: see SAINT-VINCENT, Opus geometricum, 'liber sextus de hyperbola', esp. 586 (prop. 109) and 594-7 (prop. 125-30). 1016 Epist. 39. 40: i.e. WALLIS-DIGBY 5/[15].V.1658 and WALLIS-BROUNCKER 11/[21]. V.1658. 1017 mine: i.e. WALLIS-OLDENBURG 18/[28].VII. 1668.

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229. WALLIS to OLDENBURG, 3/[13] August 1668

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would be more then would bee fit for one time to insert: and therefore could have been contented you had put part of it in this month, & referred the rest to the next. Mr Gregories 10th Proposition1018, as to what it undertakes (if I mistake not his meaning) seems well inough demonstrated; viz. that those converging series, cannot bee so terminated by Analytical operation as he proposeth. But (what Mr Hugens exceptions do oppose) that there can be no other Analytick means of squaring the Circle or Hyperbole; is not (at lest there) affirmed, & therefore was not to bee proved. What you cite out of the Italian1019, concerning the business of Transfusions (unless there be more than what is transcribed) seems onely to speak of the thing as a ridiculous fansy; not as if any such thing were then practised, or attempted; nor proposed in order to any experiment to be made. My Letter1020 about Mr Mercator, I have a little altered in the Copy1021 sent to my Lord; according to which you may please to insert it: And, whether, as sent to my Lord or to yourself, you may do as you please. No more but that I am Yours &c. J. Wallis.

20

These For Mr Henry Oldenburg, in the Palmal near St. James's London.

[2]

I insert: (1) which made mee wish rather (2) and therefore could have been contented 3 the rest add. 8 other add. II is (1) d breaks off (2) transcribed) 14 Letter (1) to (2) about 1018

Mr Gregories 10th Proposition: i.e. GREGORY, De vera circuli et hyperbolae quadratura, prop. 10. 1019 cite out of the Italian: i.e. in Philosophical Transactions No. 37 (13 July 1668), 731-2, where Oldenburg had quoted from a tract printed at Rome, in which it was stated that blood transfusion had been known to Andreas Libavius (15407-1616). 1020 Letter: i.e. WALLis-OLDENBURG 8/[18].VII.1668. 1021 Copy: i.e. WALLis-BROUNCKER VII/VIII.1668.

544

230. WALLIS to BROUNCKER, 3/[13] August 1668

230. WALLIS to WILLIAM BROUNCKER 3/[13] August 1668 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-BROUNCKER 6/[16].VIII. 1668 and WALLISOLDENBURG 3/[13].VIII. 1668. Enclosure: Investigations on Fermat's Negative Theorem. Wallis used the present letter to send Brouncker his investigations on Fermat's negative theorem for n — 3, which holds that a rational cube cannot be divided into two further rational cubes, and on the indeterminate quadratic equation otherwise known as Pell's equation. The form of presentation clearly indicates that they were sent as an enclosure to a now missing letter written at the same time, but only posted the following day. Cf. WALLIS-BROUNCKER 6/[16].VIIL1668.

231. WALLIS to WILLIAM BROUNCKER 3/[13] August 1668, enclosure: Investigations on Fermat's Negative Theorem Transmission:

Wl Final draft of paper sent (Part I): OXFORD Bodleian Library MS. Don. d. 45, 116r-117r (paginated 219-221) (our source). At top right of f. 116r in Wallis's hand: 'anno 1666 or 1667'. W2 Early draft of paper sent (Part II): OXFORD Bodleian Library MS. Don. d. 45, f. 117V-119V (paginated 222-226). W3 Early draft of paper sent (Part II): OXFORD Bodleian Library MS. Don. d. 45, f. 135r-135v (paginated 257-258). At top right off. 135r in Wallis's hand: 'vid. pag. 222.' W* Final draft of paper sent (Part II): OXFORD Bodleian Library MS. Don. d. 45, f. 136r-137v (paginated 259-262) (our source). At top off. 136r in Wallis's hand: 'Aug. 1. 1668. Sent to my Lo. Brouncker (with that of pag. 219) Aug. 3. 1668.' W5 Draft of paper sent (Part II, English): OXFORD Bodleian Library MS. Don. c. 49, f. 14r-14v. W6 Draft of paper sent (Part II, English): OXFORD Bodleian Library MS. Don. c. 49, f. 15r-15v. El Reworked version of paper sent (Part I, English): WALLIS, Algebra, 372-4. E2 Latin translation of E1: WALLIS, Opera mathematica II, 427-9. Enclosure to: WALLIS-BROUNCKER 3/[13].VIII. 1668.

545

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure As indicated by Wallis's note on W1, the earliest drafts of this paper probably date from 1666 or 1667 and were possibly motivated by plans in France to have all or part of Fermat's works printed; see JuSTEL-OLDENBURG [22.IVJ/2.V.1668 (OLDENBURG, Correspondence IV, 330-2). The content of the paper of which the first part deals with Pell's equation and the second part with Fermat's negative theorem for n = 3, largely corresponds to parts I and II of WALLIS-BROUNCKER VIII?. 1668. The latter was probably intended for publication in the Philosophical Transactions.

(Wl) Theorema Fermatianum, cum Problemate annexe.

5

Exposito n numero quovis integro non-quadrato: Dantur quadrat! in integris innumeris, ut a 2 , qui in expositum non-quadratum ducti, assumpta unitate, efficient na2 + I numerum quadratum. Proponitur Theorema demonstrandum; et construendum Problema; Exposito n, numeros a invenire. Investigatio Theorematis.

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2

Si sit na + 1 numerus quadratus integer; Erit ^/ : na2 + 1, (ejusdem Latus) Numerus integer, qui major sit quam o\/n, et minor quam a\Jn + (Quippe quadratus ab illo, esset na2; quadratus ab hoc, no2 + 1 + Cumque horum differentia sit unitate minor (propter a, n, nu2 meros integros;) oportet ^/ : na +1, ilium esse integrum qui surdum a^/n proxime superat; et quidem excessu minore quam Sit m, numerus integer, surdo ^Jn proxime major: hujusque ad ilium complementum p — m — ^/n, < 1. Adeoque surdi a^/n, complementum ad am, erit ap = am — a^/n.

4 (1) Sit n expositus numerus integer quilibet non-quadratus (2) Exposito n numero quovis integro non-quadrato 4 quadrat! (1) numero infiniti, qu breaks off (2) in integris innumeris, ut 6 Proponitur (1) demonstrandum Theorema; (2) Theorema demonstrandum; 10 quadratus (1) ; Erit hujus latus, ^/ : na2 +1 : (2) integer; Erit \/ : na2 + 1, (ejusdem Latus) Numerus (a) quadr breaks off (b) integer

15 ilium (1) in breaks off (2) esse 16 quam (1)

(2)

546

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure

Sit /, numerus integer, surdo a^n proxime major. Cujus quidem surdi, ad am, complementum ap, cum possit esse majus unitate, (utut enim sit p < 1, possit tamen ap unitatem superare;) Sit z, integer proxime minor quam ap; qui itaque hinc demptus, relinquit surdi a^/n ad I (integrum proxime majorem) complementum ap — z = I — a^/n < < 1. 2 Sit r = 2^; erit itaque ^ > ap — z; et a p — az < r. Et propterea a < adeoque p < Erat autem et z < ap, adeoque | Est ergo ^ < p < Qu°d omnino possibile esse constat; quoniam ita sumi possunt a, z, integri, ut sit |, turn minor quam p, turn hinc tamen deficiat differentia quae sit data minor; adeoque tali ut sit Dato autem ejusmodi uno numero o, habentur alii infiniti, per ea quae ostendimus in Commercio Epistolico1022, Epist. 14. 16. 17. 19.1023 et alibi. Ejusque (ut problemati satisfiat) quomodo inveniantur, ibidem V [116 ] ostendimus. | Demonstratio. Numero surdo ^Jn (utpote radice non-quadrati) sit m numerus integer proxime major: Cujus itaque excessus, unitate minor erit; nempe 1 quidem (1) jsurdi add.\ complementum (2) surdi, 5 complementum add. 6 r = 277^, (1) adeoque (2) erit itaque 15 problemati (1) satisfaciat (2) satisfiat 1022

Commercio Epistolico: i.e. WALLIS, Commercium epistolicum, Oxford 1658. Epist. 14. 16. 17. 19.: i.e. BROUNCKER-WALLIS 22.X/[1.XI].1657 (letter 14), WALLIS-DIGBY 21.XI/[1.XII].1657 (letter 16), WALLIS-BROUNCKER 17/[27].XIL1657 (letter 17), and WALLIS-BROUNCKER 20/[30]. 1.1657/8 (letter 19). 1023

547

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231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure m — \Jn < 1. Ponatur p — m — ^/n; et r — ^THSumantur porro integri duo z, a, ita comparati, ut sit | < p < 5

' Z' k°c es^' u^ §it P maJor quam |, sed minor quam Quod fieri omnino posse, constat. Certum utique est, sumi posse, in numeris rationalibus, fractum ^, qui tarn prope deficiat ab exposito irrationali p, ut hunc superet y=p;) ut sit Hoc set (posito

Hoc est (demptis aequalibus) 10

majori excessu quam sit y,(qui Poterit supert Hoc est; ut esse quantumvis exiguus, seu dato minor:) Hoc est (sumptis quadratis) superet -jj^ majori excessu quam ^ + y 2 ; hoc est, ut sit ^ > ^+y2; seupr > zay+a2y2: putapr — s = zay+a2y2; Hoc est (resolvendo

15

aequationem) radix aequationis ay = ^/ : ^z2 + pr — s : —\z. Hoc itaque diviso per y ita utcunque sumpto ut a sit numerus rationalis, atque tarn magnus ut | minor sit p- habebitur a qui sumpto z respondeat. Q a sic inventus sit fractio; reducantur z, a, ad eandem denominationem, et communis denominator abjiciatur.

5-549,1 constat. (1) \\ Cum enim ita sumi possint z, a, ut ^-, turn data quantitate p minor sit, turn ab ea deficiat differentia quantumvis exigua, seu quae sit data minor: Esto ilia quantumvis exigua differentia, y: Adeoque ^ + y — p. Id porro requiritur, ut sit f + y < V--z2+*pr-.+z. Hoc egt; 2z + 2ay < ^/ : z2 + 4pr : +z; seu z + lay < ^J : z2 + 4pr : Hoc est (sumptis quadratis) z2 + 4zay + 4a 2 y 2 < z2 + 4pr; Hoc est 4,zay + 4a 2 j/ 2 < 4pr; seu zay + a2y2 < pr. Quod fieri posse, ex Aequationum doctrina constat. Esto enim zay + a2y2(< pr) = pr — s. Manifestum est, (propter pr — s quantitatem positivam, adeoque s < pr,) turn aequationem illam zay + a2y2 — pr — s, non esse impossibilem, (quippe in ilia forma, nulla est impossibilis;) turn et radicem ay, verum habere valorem affirmativum, sumptis z, a, conformem: Nempe (a) a^. possun^ igitur ita sumi z, a, numeri, ut dictum est. \\ Sumptis itaque (2) jCertum utique . . . denominator abjiciatur. in margin\ \\ Sumptis itaque 5 est, |ita del. sumi 7 est, |ut ^ tarn prope deficiat ab |p, ut ^pr > eundem superet del.\ (posito 9 poterit |poterit del. esse 11 sit add. 12 seu pr > zay+a2y2: ( 1 ) Hoc est (per aequationum doctrinam) sumpto z ad placitum, radix ay < *J : |z + pr : —z (2) puta 13 aequationis (1) ay + a2y2 (2) ay = ^/ : |z2 + pr — s : \ — ^z add.\. Hoc itaque (a) per y (b) diviso

548

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure

[nr]

Sumptis itaque, ut dictum est, z, a, numeris: Propter | < p; ad z < an: erit an — z, quantitas positiva, seu maior quam 0. Item; propter p < erit lap < ^/ : z2 + 4pr : +z. et 2 2ap — z < -y/ : z + 4pr. Hoc est, (sumptis utrinque quadratis) 4o2p2 ^apz + z2 < z2 + 4pr: Adeoque 4a2p2 — 4apz < 4pr: Hoc est, a2p — az seu am ~ aVn ~ z < 9^7n- -^rS° am-z a^/n. Cum itaque sit am — z (quern I dicemus) numerus integer (propter m, a, z, integros;) major quidem quam o-^/n, sed minor quam a^/n + 2g*~/n; Quadratum exhibebit numerum integrum, majorem quam na2(= Q : a-^/n;) minorem vero quam na2 + 1 + -^^(— Q : a^/n + 2a7n^ Cumque inter hos duos numeros na 2 , et no2 + 1 + 4^?, nullus intercedat integer praeter na2 + 1; (quern, propter o, n, 1, integros, integrum esse constat, et quidem proxime majorem quam na 2 , et -^j minorem esse unitate:) Sequitur quadratum numeri I = am — z, esse ipsissimumna2 + l, quadratum. Constat igitur esse ejusmodi numerum a2 quaesitum. Adeoque et infinites. Uno enim habito, quotlibet alios suppeditari posse, alibi demonstravimus; in Commercio Epistolico, Epist. 14. 16. 17. 19. et alibi. Atque ibidem Problematis Constructionem fuse docuimus; quomodo istiusmodi quadrati a2 inveniantur, exposito quovis numero non-quadrato, n.

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(W4) Theorema Fermatianum Negativum: quod Demonstrandum proponitur.

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Non potest Numerus Cubus Rationalis in duos Cubos Rationales dividi. Demonstratio. Esto, si fieri potest, in Rationalibus. Potest igitur et, in Integris, idem fieri. Quippe, sublato communi Denominatore, erit, in Integris, a /3 + orb — d . 10 (quern I dicemus) add. 20 suppeditaxi |posse add.\ , alibi demonstravimus; (1) in Epi breaks off (2) in

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231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure

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10

Sunt autem hi tres Cubi, (a3/?3, a363, rf3,) vel singuli singulis inter se primi; vel saltern, facta ad communem mensuram applicatione, ad tales reduci possunt. Non potest enim eorum duos aliquos eadem mensura metiri, quin et tertium metietur. Quae enim Totum et Ablatum metitur, metitur Reliquum; Quaeque Ablatum et Reliquum metitur, metitur Totum. Sunto igitur, si fieri potest, ejusmodi tres Cubi, in Integris, singuli singulis inter se primi: Puta a3 + 63 — c3. Adeoque a, 6, c, similiter singuli singulis inter se primi erunt. (Nam siquis horum duos mensuret numerus, idem et eorum Cubos mensurabit.) Est autem c, major utrovis duorum o, 6, (propter c3 = o3 + fe3:) sed simul Utroque a + b, minor. (Nam si c aequalis sit vel major quam a + b; erit c3 aequalis vel major quam a3 + fe3 + 3o26 + Safe2: ponitur autem C3

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= 03 + fe3_}

Esto igitur c — a+b—e. Eritque e numerus integer (utpote integrorum differentia) minor utrovis o, 6; (secus enim, non esset c utrolibet a, 6, major.) Ponamus b—e = /, et a—e = g. Adeoque c(= a+b—e) = a+f = b+g. Turn, propter a + f = c, erit a3 + / 3 + 3a2/ + 3a/2 = c3 = a3 + ft3 Adeoque fe3 = / 3 + 3o2/ + 3a/2. Cujus aequationis pars posterior, est divisibilis per /: Ergo et prior. Adeoque y = / 2 + 3a2 + 3a/, numerus integer. Cum igitur / dividat fe3; idem etiam dividet 6: (vel illius saltern aliquota pars, major quam 1: de quo casu, post dicetur.) Adeoque (propter a, 6, inter se primos) idem / non dividet a; (nee illius ulla pars aliquota major quam 1.) nisi sit / — 1. Esto igitur b = f h ; adeoque 63 = /3/t3. Ergo bj - / 2 = /i3/2 - / 2 = 3a2 + 3a/; et h3 - 1 - ^- + ^ numerus integer. Est autem /, vel divisibilis per 3, vel non est. Si sic; esto / = 3fc. eritque h3 - 1(= y^ + y) = ^ + f - Ergo 3 dividit a. Quodfierino potest: quoniam f = 3k diviserat 6; sunt autem o, 6, inter se primi. [136V] Si autem / non sit divisibilis per 3: turn, propter h3 — I = erit yp^ — j j + j numerus integer. Ergo / dividit a. Sed et diviserat b. Est igitur (propter a, 6, inter se primos) / = 1. 4 enim add. 22 idem etiam dividet b: add. 24 (nee illius ulla pars aliquota major quam 1.) add. 26-27 — 3a2 + 3a/; (1) numerus in breaks off (2) et

550

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure Similiter; propter b + g = c, adeoque 63 + g3 + 3b2g + 3bg2 = c3 = o3 + fe3: Erit a3 = #3 + 362# + 36#2. Cujus aequationis cum pars posterior sit divisibilis per g; sic erit et prior. Adeoque ^- — g2 + 362 + 36g, numerus integer. Cum igitur g dividat a3; idem etiam dividet a: (vel illius saltern aliquota pars, quae major sit quam 1; de quo casu; mox dicetur.) Adeoque (propter o, b, inter se primos) idem g non dividet b: nisi sit g = 1. Esto a = gl- adeoque o3 = # 3 ^ 3 , et a ~9 — fig2 — g2 — 362 + 36g, et p_ numerus integer. Est autem (/, vel divisibilis per 3, vel non est. Si sic; esto g = 3m. Eritque / 3 - l ( = fr + f ) = 3 ^ 2 + ^ - Ergo 3 dividit 6: quod fieri non potest; quoniam g = 3m diviserat o; sunt autem a, 6, inter se primi. Sin g non sit divisibilis per 3: turn (propter I3 — 1 — ^V + —;) erit

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l

—^ = ^2 + -„ numerus integer. Ergo g dividit b. Sed et diviserat a. Ergo (propter o, 6, inter se primos) g = I. Erat autem et / — 1. Ergo b — e — (f — g —)a — e. Adeoque a — 6, et 3 o = fe3, et o3 + 63 = 2o3 = c3. Quod impossibile est. Non potest utique, in Rationalibus, dari, (nedum in Integris,) Cubus Cubi duplus. Sin dicatur, fieri posse (quod modo insinuatum est) ut / dividat ft3, nee tamen dividat 6; (et similiter g dividat a3, nee tamen dividat o:) Quoniam, licet nullus sit numerus primus (vel etiam ex solis primis semel positis compositus) Cubum dividens, qui non et Latus dividat: fieri tamen potest, ut numerus ex primi quadrato vel cubo compositus, Cubum dividat, nee tamen dividat Latus. Puta, si ponatur / — rs 2 , et b — rsi, [1371 adeoque 63 — r3s3t3:| Qui quidem Cubus, sed non item ejusdem Latus, ab / = rs2 dividitur. Non tamen minus procedit Demonstratio. Quippe, si non ipse /; ipsius saltern pars aliquota, quae major sit quam 1, eundem 6 dividet. Nam, propter —y^- = 3a2 + 3a/: Positis b = rst, et / = rs2; erit = 3a2 + 3ars2: adeoque r3s3t3 - r3s6 = 3a2rs2 + 3ar2s4.

1 propter add. 7 dividet add. 27 minus (1) valet (2) procedit Demonstratio. jQuamquam enim id verum sit in thesi; tamen, in hypothesi, similiter ostendetur casus hie impossibilis esse. del.\ Quippe 28 pars add.

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231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure Cujus aequationis pars prior cum divisibilis sit, non quidem, ut prius, per /3 = r3s6, sed saltern per hujus aliquotam partem quae major sit quam 1; nempe per r3s3: erit et posterior pars ita divisibilis: Adeoque t3 5

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_ S3 =

numems Integer.

Unde ostendetur (ut prius) quod, vel 3 dividet a, qui prius diviserat 6; (nempe si 3 dividat r:) vel, (si secus,) idem r2s (vel hujus aliquota par major quam 1) qui prius diviserat 6, dividet a. (Quod fieri non potest, propter a, 6, inter se primes.) Unico saltern casu excepto; nempe si r = 1, et s = 3. Quippe turn t3 - s3 = t3 - 27 = a2 + 9a; seu t3 = a2 + 9a + 27. Eritque hoc casu / = rs2 = 9. Atque eadem eodem modo ostendenda sunt, si dicatur g dividere a3, sed non a. Nempe, vel casus erit impossibilis, propter a, 6, inter se primes; vel saltern erit g — 9. Jam vero si duorum /, (/, uterque dicatur = 9; perinde est atque si sit uterque = 1. Quippe turn, ut prius, b — e = (f = g =)a — e; adeoque a — 6, et a3 + b3 — 2a3 — c3; quod est in rationalibus impossibile. Id unicum ostendendum restat; nempe, si duorum /, g, dicatur alter = 9, alter = 1, rem adhuc esse impossibilem. Puta si / = 9, et g = 1; vel vice versa: utrumvis autem dicatur, eodem res recidit. f=b-e = 9 g =a — e = I

f-g=b-a=S 25

Id si ponamus: erit b — a = 8, (ut patet;) adeoque b = a + 8, et fe = o3 + 24a2 + 192a + 512. Sed et, propter r = 1, s = 3, et r3s3t3 = fe3, erit 4&3 = t3 = 2 a + 9a + 27: Adeoque 27a2 + 243a + 729 = 63 = a3 + 24a2 + 192a + 512. Et propterea o3 - 3o2 - 51a - 217 = 0; seu a3 - 3o2 - 51a = 21 3

2 quae (1) imitate major sit; (2) major sit quam 1; 4 numerus Integer add. 10 t3 - s3 = add. 12 sunt, (1) d breaks off (2) si 17 in rationalibus add. 18 restat; (a) num., (b) nempe, 19 = 1, (1) num res sit adhuc impossibilis. (2) rem adhuc esse inipossibileni. 552

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure a3+ 24a2 +192a +512 = b3. 2 3 27a +243a+729 = b . a3- 3o2 -51o -217 = 0. 3 2 a - 3a -51a = 217. Hoc autem fieri non potest, ita ut a sit numerus rationalis integer, prout supponitur. Posito enim a3 — 3a2 — 51a = 217; erit a major quam 10, sed minor quam 11. Quippe, si ponatur a = 10: erit o3 — 3o2 — 51o = 190. Si a = 11: erit a3 - 3a2 - 51a = 407. (Eadem, paucis mutatis, eadem methodo ostenderentur, si, pro / = 2 rs , poneretur / — rs3: ut non sit opus eadem repetere.) Non est igitur, in Integris, o3 + b3 = c3. Adeoque (per ante demonstrata) nee in Rationalibus. Quod erat demonstrandum.

9-10 (Eadem . . . repetere.) add. in margin 12 demonstrandum. | || Aliter. || Si potest unus aliquis Cubus rationalis in duos rationales dividi; potest quilibet sic dividi. Nam, si c3 = a3 + b3, erit et c3e3 = a3e3 + b3e3: quicunque autem sit cubus c3, potest hie per cubum aliquem (1) , q breaks off (2) (integrum vel fractum,) quern e3 dicamus, multiplicatus, quemlibet constituere cubum. || Potest igitur (verbi gratia) 8 sic dividi. (a) Idque vel in duos integros, puta (b) Puta, 8 — a3 + b3, ut sint a3, b3, cubi in integris (quod fieri non posse certum est; cum, praeter 1, nullus sit cubus eo minor:) vel saltern 8 = ^ + 4j, ut (ao) per (bb) dividatur in duos Cubos fractos. |(qui intelligantur in minimis terminis descripti; adeoque a, a, inter se prinii; et siniiliter &, /?.) add. in margin] \\ Sed neque hoc potest esse. Quippe turn erit 8a3/33 = a3/33 + a363. Qui quidem cubi tres cum non sint add] singuli singulis inter se primi, possunt omnes (facta ad communem divisorem applicatione) sic reduci ut ita fiant, (nam siquis numerus Totum et Ablatum metitur, metitur et Reliquum; et siquis metitur Ablatum et Reliquum, metitur Totum.) Ergo, propter 8a3/33 et a3/?3 divisibiles per /33, etiam a363 sic erit divisibilis, adeoque Si^- numerus integer; |(aoa) (et quidem (666) (adeoque et j% numerus integer, propter b f3 inter se primes:) adeoque j3 dividit a: Pariter ostendetur quod a dividit (3 (propter 8(33 — ^f—h b3 in integris.) Ergo a = (3. Ergo 8a3 = a3 +6 3 in integris. add. in margin\ et propterea 8a3 = a3 + ^-p- in integris. Item, propter 8a3 et ^-p- divisibiles per a3, erit et a3 sic divisibilis, adeoque ^5- numerus integer; et propterea 8 — ^ + -p- in integris. Quod fieri non posse ante demonstratum est. || Non potest igitur 8 sic dividi, ergo (per ante demonstrata) nee cubus ullus alius. Quod erat demonstrandum, del; note in margin: Demonstratio haec non valet; adeoque deletur.|

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233. WALLIS to BROUNCKER, 5/[15] August 1668

232.

WILLIAM BROUNCKER to WALLIS 3/[13] August 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-BROUNCKER 5/[15].VIII. 1668. Apparently Brouncker requested that Wallis demonstrate his method for finding the sums of logarithms, which he had referred to in his letter to Oldenburg of 8/[18] July 1668.

233.

WALLIS to WILLIAM BROUNCKER Oxford, 5/[15] August 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 59, 2 pp. (our source). The address, place, and date have been crossed out by Oldenburg. He has also added an introduction for the Philosophical Transactions: 'The demonstration promised at the end of the foregoing Letter, follows in another from the same Author to the same Noble Lord, thus;' as well as an instruction for the printer: 'This is to follow (1) after (2) immediately after the letter of the same Author of July 8.' w1 Copy of letter sent: LONDON Royal Society Register Book Original 4, pp. 5-8. w2 Copy of w 1 : LONDON Royal Society Register Book Copy 3, pp. 86-90. E First edition of letter sent: Philosophical Transactions No. 38 (17 August 1668), 756-9. Reply to: BROUNCKER-WALLIS 3/[13].VIII. 1668. Enclosure to: WALLIS-BROUNCKER 6/[16].VIII. 1668.

Excellentissimo Domino, D. Gulielmo Vicecomiti Brouncker; Regiae Societatis Londini, Praesidi. 5

Oxoniae. Aug. 5. 1668.

Petis (Illustrissime Domine) per literas tuas1024 Aug. 3. datas, (quas hesterna nocte accepi,) ut demonstrare velim methodum meam, Logarith1024

tuas: i.e. BROUNCKER-WALLIS 3/[13].VIIL1668.

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233. WALLIS to BROUNCKER, 5/[15] August 1668 morum Summam inveniendi, quam literis meis1025 Julii 8 datis, brevissime insinuaveram. Quae quidem, cum sit cum Ungularum doctrina (quam alibi trado) connexa; opus erit ut utramque simul exponam; sed et rem totam (quam D. Mercatoris figurae et methodo quantum res ferebat accommodaveram) ad principia mea revocatam ab origine repetam. Ostensumest1026, in mea Arithmetica Infinitorum, prop. 95. Spatium Hyperbolicum AD[3[36 (in infinitum continuatum a parte (3d, sed a parte D/3 ubivis terminatum,) Figuram esse quam ex Primanorum Reciprocis conflatam appello1027, Prop. 88. definitam: Cujus nempe Ordinatimapplicatae d/3, d[3, sint Primanis (seu Arithmetice proportionalibus) db, db, (Triangulum complentibus,) adeoque ipsis dA, dA, (suis a vertice distantiis) Reciproce Proportionales. Hoc est, (posito AD = d; et rectangulo AD/3 — b2; particulisque infinite exiguis a, a, &c;) Si a vertice Ad incipias, ^-, ^-, 1^, |^, &c, usque ad ^- — Df3: vel, si a base Df3 incipias, ¥' ePa> 5^2a> eP3a' ^C' usclue ad j^ = AS infinitae, (nempe, si ad Verticem usque processum continuaveris;) vel, usque ad -£^ — C(3, (posito DC = A,) si continuaveris usque ad C/3, ubivis intra AS et D(3 sumptam. (Adeoque omnium Aggregatum, ^ + -^ + ^^ + ^^ &c, est ipsum DC(3(3 planum.) Manifestum itaque est, (et ibidem prop. 94, ostensum1028,) si intelligantur singulae df3, in suas a vertice distantias Ad, ductae; hoc est, -^in a, ^ in 2a, (et sic de reliquis;) erunt omnia rectangula Ad(3, hoc est, rectarum d(3 momenta respectu AS, (intellige, facta ex magnitudine in distantiam ducta;) seu plana semiquadrantalem Ungulam (cujus acies AS] complentia, (eisdem d/3 rectis perpendiculariter insistentia;) invicem aequalia. Quippe singula — b2. (Quorum cum unum sit AIV8 quadratum, erit AI = b.) Adeoque, Totius AD/3/36 (plani infiniti,) seu omnium d/3 illud complentium, momentum respectu rectae AS (ut axis aequilibrii;) seu Ungula semiquadrantalis eidem ADfifiS insistens (aciem habens AS;) sunt totidem 6 repetam. |V. Fig. 2. add. Oldenburg] 8 sed (1) ex. (2) a parte 1025 literis meis: i.e. WALLIS-OLDENBURG 8/[18].VIL1668. Cf. WALLIS-BROUNCKER VII/VIII.1668. 1026 Ostensum est: i.e. WALLIS, Arithmetica infinitorum, 71 (prop. 95). 1027 appello: i.e. WALLIS, Arithmetica infinitorum, 68-9 (prop. 88). 1028 ostensum: i.e. WALLIS, Arithmetica infinitorum, 71 (prop. 94).

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233. WALLIS to BROUNCKER, 5/[15] August 1668

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b2' hoc est, db2. (Ungula magnitudinis finitae piano infinitae magnitudinis insistens.) Ejusque pars piano AC(3(3'8 insistens (propter AC = d — A.) similiter ostendetur aequalis ipsi d — A in b2 ductae; hoc est, db2 — Ab2. Adeoque pars reliqua, ipsi DC(3'[3 insistens, aequalis ipsi Ab2. Quod itaque est ejusdem DC/3/3 momentum respectu Ad. Atque hoc momentum per plani DC/3/3 magnitudinem, puta per pi, divisum; exhibet plani distantiam Centri gravitatis ab A8, ^-: adeoque distantiam ejusdem a D/3, d — Haec itaque a Df3 distantia, in pi (plani magnitudinem) ducta; exhibet dpi — Ab2 ejusdem DC/3/3 momentum respectu D/3; seu Ungulam eidem DC'/?/? insistentem cujus acies sit D(3. Haec denique Ungula (cujus altitude, in D/?, nulla sit, sed, in C/3, ipsi DC aequalis;) si ex planis ipsi DC[3[3 parallelis conflari intelligatur; erunt ea, CD(3(3 , Cd[3[3, et sic deinceps; hoc est, aggregatum omnium Cd/3/3, Cd/3/3, usque ad CD/3/3. Sunt autem ea plura (ut ex Gregorii de Sancto Vincentio, aliorumque post ilium, doctrina1029 constat,) tanquam Logarithm! Arithmetice proportionalium Cd, Cd, &c. usque ad CD] (seu a, 2a, 3o, &c usque ad A.) Ergo ungula ipsa, est eorundem Aggregatum. Hoc est (posito D = 1,) [2] dpi — Ab2 = pi — Ab2. Quod ostendendum erat.l Porro; cum sit &c. (Quod dividendo 2 b per d — a, patebit:) vel, posito d = 1, (quo ipsius d potestates omnes deleantur, seu 1 &c in b2. et similiter in 1 + 2o + 4a + 8a &c; et similiter in reliquis: Erunt omnes d(3, (spatium DC[3[3 complentes,) Adeoque (per Arithm. Infin. prop. 641030) omnium Aggregatum, seu ipsum DC[3[3 spatium, erit

infr 2 et sic deinceps usque ad

4 aequalis add. 16 ut ex (i) doctrina (2) Gregorii 24 4a262 + 8aV con: ed. 1029

ex . . . doctrina: see SAINT-VINCENT, Opus geometricum, 'liber sextus de hyperbola', esp. 586 (prop. 109) and 594-7 (prop. 125-30). 1030 per . . . prop. 64: i.e. WALLIS, Arithmetica infinitorum, 52-3 (prop. 64).

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233. WALLIS to BROUNCKER, 5/[15] August 1668 Ideoque, Plani DC(3(3 momentum respectu D/3; seu semiquadrantalis Ungula eidem insistens cujus acies sit D(3~ seu planorum aggregatum ipsam constituentium; seu Loe;arithmorum summa quos ea repraesentant, Qualium Cubus (seu Rhombus solidus) ACEo sit 1. Si vero non ponatur d = 1, sed cujuscunque magnitudinis: erit saltern

5

in ft2. = pi. Qualium 2

d = ADE8 Quadrato vel Rhombo. Ungulaque (ut prius) dpi — Ab2. Qualium e?3 — AD Ed Cubo, vel (si angulus A sit obliquus) Rhombo solido. Cumque A (posito d = 1) vel e (quicunque ponatur valor ipsius d) sit minor quam 1, (propter A < d:) illius potestates posteriores ita continue decrescunt, ut tandem negligi possint; planique valor pi exhibeatur quantumlibet vero propinquus. Atque haec est (Illustrissime Domine) Methodi, quam innuebam, ex meis principiis deductio, et demonstratio brevis. Vale. Tui observantissimus Johannes Wallis.

1 respectu add. 8 Qualium d2 . . . Rhombo. add. 10-11 Qualium d3 . . . Rhombo solido. add. 12-15 Cumque A ... propinquus. add. in margin 17 Vale. |Oxon. d. 5. Aug. 1668. add. Oldenburg]

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234. WALLIS to BROUNCKER, 6/[16] August 1668

234.

WALLIS to WILLIAM BROUNCKER Oxford, 6/[16] August 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 60, 2pp. Enclosure: WALLIS-BROUNCKER 5/[15].VIII.1668. Clearly, Wallis originally intended to enclose a letter for Oldenburg in the present letter. There is no other evidence to suggest that such a letter was sent, as an enclosure or otherwise.

Oxford Aug. 6. 1668 My Lord,

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This1031 should have come by Thursday's Post1032 (according to the first date) but company coming in, deteined mee so long that before I had transcribed it, the Post was gone. I hope your Lordship will find the Demonstration without mistake, & such as will satisfy your Lordship (& then, I know it ought to satisfy others:) If it prove otherwise; Your Lordship will oblige mee by letting mee know so much. That which your Lordship suspected, (when it came without the Demonstration,) I think was not an error: for in pi — ab2, since that pi doth as well increase as a, the whole quantity is not diminished by the increase of a, as it would have been if pi had been a standing quantity. I sent your Honour a former Pacquet1033 by Tuesday's post, which I hope your Lordship received: Your Honour will excuse mee that I trouble you with another so soon. I am My Lord,

15

Your Lordships very humble & obliged servant, John Wallis. 12 quantity. |I have left Mr Oldenburg's open, that I need not transcribe what is sayd to him. del.\ I sent 14 excuse (1) you (2) mee 1031

This: i.e. WALLIS-BROUNCKER 5/[15].VIII. 1668. should . . . Post: i.e. should have arrived on 6/[16] August, which was a Thursday. 1033 former Pacquet: i.e. WALLIS-BROUNCKER 3/[13].VIII.1668, enclosing Wallis's investigations on Fermat's negative theorem. 1032

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235. WALLIS to BROUNCKER, 8/[18] August 1668 [2] These For the Right Honourable William Lord Vicount Brouncker, President of the Royall Society London.

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235. WALLIS to WILLIAM BROUNCKER 8/[18] August 1668 Transmission:

Manuscript missing. Existence and date: Covering letter to Wallis's demonstration of Fermat's other negative theorem, which holds that there is no right-angled triangle in numbers whose area is a square.

236. WALLIS to WILLIAM BROUNCKER 8/[18] August 1668, enclosure: Demonstration of Fermat's Other Negative Theorem Transmission:

W Draft of paper sent: OXFORD Bodleian Library MS. Don. d. 45, f. 138r. At top right Wallis has written: 'Sent to my Lo: Brounker Aug. 8. 1668.' Enclosure to: WALLIS-BROUNCKER 8/[18].VIII. 1668. The content of the paper largely corresponds to part IV of WALLIS-BROUNCKER VIII?.1668, which was probably intended for publication in the Philosophical Transactions.

Aliud Theorema Fermatianum Negativum, Quod ibidem demonstrandum proponebatur. Nullum est in numeris Triangulum Rectangulum cujus Area

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236. WALLIS to BROUNCKER, 8/[18] August 1668, enclosure sit numerus Quadratus.1034 Demonstratio. Demonstrationem hujus, in Commercio Epistolico Epist. 44.1035 dederam his verbis.

In exposito schemate (cujus constructio patet) Trianguli Rectanguli BCD latera non possunt esse numeris effabilia, nisi AD DE sint inter se ut numeri plani similes, (secus enim, qui ab ipsis fit, non erit numerus quadratus, ejusque radix BD efFabilis,) hoc est, ut numeri quadrati inter se. Esto ut 2a 2 , 2e2. Erunt igitur CB, CD, BD, ut a2 + e 2 , a2 - e 2 , 2ae. Adeoque CD, ^BD, ut a2 — e 2 , ae. Et proinde (cum duorum quadratorum differentia, atque eorundem medius proportionalis, non possint esse plani similes,) qui ab ipsis fit, (hoc est, area trianguli,) non potest esse numerus quadratus. Quod erat demonstrandum.

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Haec demonstratio quo minus pro legitima habeatur, hoc tantum objicitur; quod non demonstraveram assumptum Lemma, Duorum quadratorum differentiam, eorumque medium proportionalem, (nempe a2 — e 2 et ae,) non posse esse numeros pianos similes.

3 Epist. 44. (1) hunc (2) dederam 7se (1) aut (2) ut

8 ejusque (1) latus (2) radix 18 differentiam, 1034

(1) et eo breaks off (2) eorumque

Aliud . . . Quadratus: see FERMAT-DIGBY [28.III]/7.IV.1658. Epist. 44.: i.e. WALLIS-DIGBY 20/[30].VI.1658.

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237. COLLINS to WALLIS, 15/[25] August 1668 Putaram ego, Lemma illud, apud tantos viros quibuscum res ag batur admissum iri, neque expressa demonstratione indigere. Id autem, quoniam hoc expetunt, sic demonstro. Intelligantur a 2 , e 2 , quadrat! inter se primi, seu in eadem ratione minimi; (quippe vel tales sunt, vel, facta ad communem mensuram applicatione, ad tales reduci possunt, non mutata ratione rectarum AD, DE.) Ergo et a, e, inter se primi, (quippe siquis hos duos metitur numerus, idem et a 2 , e2, metietur.) Ergo et a, e, a — e, item a, e, a + e, singuli singulis inter se primi; (nani siquis vel horuni vel illorum duos metitur numerus, idem et tertium metietur; quippe, qui totuni et ablatum metitur, metitur reliquum; quique ablatum et reliquum metitur, metitur totum.) Cumque a — e sit ad utrumque duorum o, e, primus; primus item erit ad ae. factum ex illis. Similia turn ostendetur, a + e primum esse ad eandem ae quoniam ita est turn ad e turn ad a. Cum itaque ae primus sit turn ad a + e, turn ad a — e; primus item erit ad o2 — e2 ex illis factum. Non sunt igitur a2 — e 2 , et ae, plani similes; quippe similes plani non possunt esse inter se primi; (intellige, nisi sint quadrati; qui casus hie non obtinet.) Quod erat demonstrandum.

237.

JOHN COLLINS to WALLIS 15/[25] August 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-COLLINS 25.VIII/[4.IX]. 1668. In this letter, Collins apparently sent Wallis news of Oldenburg's marriage to Dora

1-2 ego, (1) apud viros tantos, Lemma illud, etiam absque diserta demonstratione (2) Lemma illud, apud tantos viros | quibuscum res agebatur add] admissum 3 quoniam (1) non admittunt, (2) hoc expetunt, 8 siquis (1) eorum (2) vel horum vel illorum 12 primus; (1) erit (2) primus 12 ad jeandem add. and del\ ae. factum 13-14 primum esse (1) ae. (2) ad eandem ae. (3) ad eandem ae, quoniam ita est turn ad e turn ad a. (a) Cumque sit ae primus (b) Cum itaque ae primus sit 16 primi; (1) sed vel alter alterum metietur, vel saltern his similium minimus metietur utrumque. (2) (intellige, nisi sint quadrati; qui casus hie non obtinet.)

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238. WALLIS to COLLINS, 25 August/[4 September] 1668 Katherina Dury, Oldenburg's former ward. He also informed Wallis of the state of printing of his Mechanica sive de motu, which he was seeing through the press. Finally, he reported on Huygens's rule for squaring the hyperbola by means of logarithms, which the Dutch mathematician had recently published at the end of his review of Gregory in the Journal des Scavans (2 July 1668), 56.

238. WALLIS to JOHN COLLINS Oxford, 25 August/[4 September] 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 199r-199av (f. 199V and 199ar blank) (our source). At top off. 199av at 90° to address in Collins's hand: 'About an Hyperbolick Space'. Postmark on f. 199av: 'AU/26'.— printed: RIGAUD, Correspondence of Scientific Men II, 492-3. Reply to: COLLINS-WALLS 15/[25].Vin.l668. Oxford Aug. 25. 1668.

Sir,

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I received yours1036 of the 15th instant. I shall be very glad of any happyness for Mr O.1037
yours: i.e. COLLINS-WALLIS 15/[25].VIII.1668. Mr O.: i.e. Henry Oldenburg. He married Dora Katherina Dury on 13/[23] August 1668. 1038 they . . . Presse: Wallis's Mechanica was typeset by William Godbid for Moses Pitt, who published it in 1670. 1039 rule: i.e. HUYGENS, 'Maniere . . . pour trouver par le moyen des Logarithmes la dimension de 1'espace Hyperbolique compris entre la Courbe & une de ses Asymptotes, & deux lignes paralleles a 1'autre Asymptote, la proportion de ces deux lignes etant 1037

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238. WALLIS to COLLINS, 25 August/[4 September] 1668 it will not be possible that any such Hyperbolick space, can bee lesse then (or so little as) the number answering to the Logarithme 0.36221,56868, continually added. (Unlesse I misapprehend your meaning.) The parallelogramme ahfi (which I suppose you mean by the Parallelogramme of the Hyperbola, as you describe it,) being in all places equal; & therefore its measure 1.00000,00000, being a standing measure.

Pray let them make what dispatch may bee, at the presse: which (do what we can) will be considerably retarded by passing of letters & papers too & fro; which is that which I would have prevented (beside your own trouble) by having it done here if it had been convenient1040. It shal stay with me as little as is possible. I am obliged to you for your care. And rest

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Yours &c John Wallis. [199av] These For Mr John Collins, next to the three Crowns in Bloomsbury-Market. London.

2 to (1) this (2) the donnee en nombres', Journal des Scavans (2 July 1668 n.s.), 56; HUYGENS, (Euvres completes VI, p. 231. 1040 which . . . convenient: cf. OLDENBURG-HuYGENS 31.V/[10.VI].1669, OLDENBURG, Correspondence V, pp. 580-584, 581: 'Les deux volumes de Motu et Mechanice etc. par Monsieur Wallis n'advancent pas si promptement, comme on le souhaite; c'est a cause que 1'autheur est a Oxford, et le livre s'imprime a Londres.'

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239. WALLIS to OLDENBURG, 25 August/[4 September] 1668

239. WALLIS to HENRY OLDENBURG Oxford, 25 August/[4 September] 1668 Transmission:

W Letter sent: LONDON Royal Society Early letters Wl, No. 61, 2pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Accepi d. (1) 27 (2) 26. Aug. resp. d. 28.' Postmark on p. 2: 'AU/26'.—printed: OLDENBURG, Correspondence V, 26-7. Answered by: OLDENBURG-WALLIS 28.VIII/[7.IX].1668.

Oxford Aug. 25. 1668. Sir,

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Amongst this Monthe's Transactions, I am told this is one, Te secundas nuptias iniisse;1041 precor ut sint secundae. As to others, which you mean to publish; I doubt I have burdened you so much allready, that I am on that account rather to beg your pardon, than for not suggesting inough. Onely you have this reserve, that you may lay off as much of your burden as you think fit to be eased of. To the doubt1042 you made in one of your last to mee; I think I did, in one of mine to my Lo. Brounker, give tha satisfaction that may be sufficient. Which, with some others about the same time, I hope hee did receive: though by reason of his imployments of one kind, & yours of another, I have not heard, from either, that they be yet come to hand. The two letters1043 which concern the Hyperbola, I think are clear inough, so that they need not much explication; and so 7 of (1) it (2) your burden as you (a) see (6) think fit to be (ao) omitted. (66) eased of. 13 concern (1) Mr Merc breaks off (2) the 14 not (1) much (2) much 1041

Te secundas nuptias iniisse: Oldenburg married Dora Katherina Dury on 13 August. doubt: expressed in a now missing letter of Oldenburg's and possibly concerning Wallis's review of Mercator's Logarithmotechnia in WALLIS-OLDENBURG 8/[18].VII.1668 or the revised version WALLIS-BROUNCKER VII/VIII.1668. The letter to Brouncker, to which Wallis refers in the following, may be WALLIS-BROUNCKER 5/[15].VIII.1668. That letter continued the demonstration of a method for summing logarithms, which at the end of the review was only intimated. 1043 two letters: i.e. WALLIS-BROUNCKER VII/VIII.1668 and WALLIS-BROUNCKER 5/[15].VIII. 1668, which were both published in Philosophical Transactions No. 38 (17 August 1668). 1042

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240. OLDENBURG to WALLIS, 28 August/[7 September] 1668 short, that I know not well how to contract them. Those two (or three) concerning Dulaurens1044, if in any thing My Lord think they need further addition; there may bee matter inough to adde: Or, if any thing there sayd, bee not to his mind, it may bee omitted safely inough, & yet the remainder be sufficient; or altered to his content. But the last, of both subjects perhaps you may reserve for the next month. And, between this & that there will bee time to suggest any thing that my Lord shall think fit to be further insisted on, or altered. I have little more to adde at present, but to wish you joy, & assure you that I am Your affectionate friend & humble servant. John Wallis.

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[2] These

For my honoured Friend, Mr Henry Oldenburg, at his house in the Old Pelmel near St James's London.

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240.

HENRY OLDENBURG to WALLIS 28 August/[7 September] 1668 Transmission:

Manuscript missing. Existence and date: Indicated by Oldenburg's endorsement to WALLIS-OLDENBURG 25.VIII/[4.IX].1668.

1 (or three) add. 5 sufficient (1) . (2) ; or altered to his content. 1044 two (or three) concerning Dulaurens: i.e. Wallis's animadversions on Dulaurens, contained in WALLIS-OLDENBURG 2/[12].VII.1668 (ii), which was published in Philosophical Transactions No. 38 (17 August 1668), and in WALLIS-OLDENBURG 18/[28].VII. 1668, printed in two parts in Philosophical Transactions No. 39 (21 September 1668) and No. 41 (16 November 1668).

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241. PELL to COLLINS, 29 August/[8 September] 1668

241. JOHN PELL to JOHN COLLINS [Brereton], 29 August/[8 September] 1668 Transmission:

C Copy of (missing) letter sent: LONDON British Library MS. Add. 4278, f. 127r. Reply to: COLLINS-PELL 18/[28].VII. 1668 and COLLINS-PELL 20/[30].VIII. 1668 (LONDON British Library MS. Add. 4278, f. 341r-341v).

For Mr J. C. an accomptant &c at his house next the three Crownes in Bloomsbury market LONDON. Sir 5

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Since my last1045 of July 6, I received two from you1046; to which I am now to return some answere. Mr Brancker thinks it needless to send you a coppy of Mr Merry's paper1047; because you may have a coppy of it from Mr Merry himself, and then you may compare it with Mr Branckers reply. In the paper of 19 faults1048, which he sent you, it seemes h had miss-written one; 40597 for 40599. Till he be more exercised in nice transcribing, he will be subject to such oversights. His first & second Collection hath 135 errata. Dr Wallis his paper1049 hath ten more, which I excerpted & sent to him. Since he received it, he hath beene heere once He seemed resolved to write to Dr Wallis a letter1050 of thanks for taking the paines to examine that long table &c. I have not heard from him, since he went hence just 3 weeks agoe. I feare he is sick. Of the marriage1051 mentioned in your last, heere we know more circumstances. The Bridegroom hath written to my Lord. I know the 1045

last: i.e. PELL-COLLINS 6/[16].VIL1668 (LONDON British Library MS. Add. 4278, f. 126v-127r). 1046 two from you: i.e. COLLINS-PELL 18/[28].VIL1668 and COLLINS-PELL 20/[30].VIII. 1668 (LONDON British Library MS. Add. 4278, f. 341r-341v). 104r paper: cf. COLLINS-PELL 18/[28].VII.1668. 1048 paper of 19 faults: see enclosure to COLLINS-PELL 18/[28].VIL1668. 1049 Dr Wallis his paper: i.e. Wallis's catalogue of errata in Brancker's table of incomposite numbers enclosed in COLLINS-PELL 18/[28].VII. 1668. 1050 letter: No letter from Brancker to Wallis has been found. 1051 marriage: In COLLINS-PELL 20/[30].VIII.1668 Collins reports that Oldenburg had recently married a 15-year-old gentlewoman. The marriage to Dora Katherina Dury took place on 13 August. 566

241. PELL to COLLINS, 29 August/[8 September] 1668 Brides age was but 14 yeares and 3 months. I very well remember that she was born in May Anno 1654; when her Father1052 and I were travelling together in Germany. I thank you for your newes1053 concerning Tacquet, Slusius, Fermat and Anderson. Riccio will not thank those that coupled him1054 with Mercator. It may be it will make him set some-body to write against him: for he will not vouchsafe to doe it himself. I shall not wonder to see some transmarine pen fly at him. If he print not the same things in English, I shall let him alone, till the incomposits and their Logarithms bring him in my way. My L. Br.1055 hath received all your letters. He hath been very busy of late. They say, He hath paid all his Fathers debts. I desire not, that He should be called hence, before he have put his affaires into a faire way for the paying of his owne also. Three months agoe, you named a Deane1056 whom, you said, the King would make a Bishop at the next opportunity. You may now tell his Freinds that the Bishop of Chester1057 is dead & buried. I heard not of it till yesterday. A Dieu. August 29. 18 [Note in Pell's hand:] The Bishop of Chester (George Hall) died at Wigan on Sunday August 23 1668 and was buried there on Wednesday August 26. This I heard August 29 after the Poste was gone.

15 Deane |{—} del. whom 1052

Father: i.e. John Dury (1596-1680). newes: i.e. in COLLINS-PELL 20/[30].VIII. 1668. 1054 those that coupled him: The initiative to print Mercator's Logarithmotechnia with Ricci's Exercitatio geometrica de maximis et minimis apparently came from Collins. See COLLINS-PELL 5/[15].IX.1668 (LONDON British Library MS. Add. 4278, f. 342r-342v). 1055 L. Br.: i.e. Lord Brereton. 1056 Deane: possibly John Wilkins, at the time Dean of Ripon. 1057 Bishop of Chester: i.e. George Hall (1612-68), DNB. His successor was John Wilkins. See COLLINS-PELL 5/[15].IX.1668. 1053

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242. WALLIS to HUYGENS, 31 August/[10 September] 1668

242.

WALLIS to CHRISTIAAN HUYGENS Oxford, 31 August/flO September] 1668 Transmission: W Letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 1659, 4pp. (our source). On p. 4 above address blind drawing by Huygens.—printed: HUYGENS, (Euvres completes VI, 251-7. w Paxt copy of letter sent in scribal hand (with corrections in Oldenburg's hand): LONDON Royal Society Early Letters Wl, No. 62, 4pp. On p. 4 in Oldenburg's hand: 'Dr Wallis's Latin Letter to M. Hugens Aug. 13. 68.' Answered by: HiJYGENS-WALLis [3]/13.XI.1668.

Oxonii, August! 31. 1668.

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Cum harum lator (Vir Nobilissime) D. Theodoras Ryckius1058, Geldrus, (qui per mensem unum aut alterum hie moram posuit, studiorum gratia,) jam recta in Galliam trajecturus1059, a me petierit, ut siquas haberem literas Parisios amandandas ipso latore uterer: Ansam hanc arripui te compellandi: ne feliciter inita pridem amicitia, (quod fore nollem,) longa desuetudine exolescat. Aderat hie, jam ante tres annos, celeberrimus Pater tuus1060: qui me diligenter exquisivit, humanissime invisit, et eleganti ornavit Epigrammate1061: quod amicitiae tuae debeo. (Quern gaudebam adhuc in vivis conspicere, et in ea aetate vegetum valere.) Et jam nuperrime, ab eodem per literas commendatus, consobrinus tuus D. Mauritius Guilielmius1062, qua potui humanitate exceptus est: et quae secum attulit Parentis tui Opera, Bibliothecae Bodleyanae dicata, eidem ingessimus, ornamento futura. Indicavit ille (non potui utique de Hugenio nostro non 2 Geldrus, add. 1058

Ryckius: i.e. Dirk de Rycke (1640-90), who studied at Leiden before becoming a school rector in Kampen. After travelling through Europe 1667-71, he was appointed professor of history and eloquence at the University of Leiden in 1672. 1059 trajecturus: Rycke brought the letter to Huygens on [21J/31 October. See HUYGENS-WALLIS [3]/13.XL1668. 1060 Aderat . . . tuus: Constantijn Huygens was appointed ambassador to England in 1664. 1061 Epigrammate: i.e. CONSTANTIJN HUYGENS-WALLIS 7/[17].IX.1664 and CONSTANTIJN HUYGENS-WALLIS 6/[16].X.1664. 1062 D. Mauritius Guilielmius: i.e. Maurits le Leu de Wilhem (b. 1641). Cf. CONSTANTIJN HUYGENS-WALLIS 21.VI/[1.VII].1668.

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242. WALLIS to HUYGENS, 31 August/[10 September] 1668 inquirere) Te Parisiis adhuc manere; credo, et mansurum. Quod quidem, nisi quod tuis rationibus sic convenire existimem, secus mallem. Maluissem utique ut vel apud Belgas tuos, vel (si ad Exteros transire libuisset) aequis conditionibus nobiscum sedes posuisses; Societatis nostrae Regiae, cujus Tu egregium Membrum es, auxilio futurus porro, et ornamento. Ut autem de communibus studiis nonnihil addam: Vidi ego nuper tuam in Jacobi Gregorii Quadraturas Animadversionem1063, in le Journal de Sgavans insertam: et quid ille respondeat1064, insertum Transactionibus Philosophicis nostratium. Et quidem mallem quod Hugenii nomen abfuisset. Nam (praeterquam quod ad Prop. 10, Exemplum fuerat non appositum; et, ad Prop. 7. fractio m^I^fe non fuerat ad minimos terminos reducta; quae leviora sunt, et summam rei non attingunt;) non video quid haereat. Nam, quantum ego judico, Propositio ll a , quam tu oppugnas est legitime demonstrata: Nempe, quod series convergens, ibidem exposita, non sit (eo sensu quo ille haec intellecta vult) Analytice composita. Quaeque tu ex prop. 7. hue pertrahis, rem non conficiunt: quippe quae, particular! casui accommodata, non erant praeter Authoris mentem alio pertrahenda; nee hie quadrant. Quodque alii jam antehac (atque Tu inter eos) varias per numeros approximandi methodos ad Circuli quadraturam commenti fuerint; non impedit quin et grata sit haec sua, subtilis sane et nova; praesertim cum ita comparata sit, ut Ellipsin etiam et Hyperbolam ex aequo respiciat atque Circulum; et quidem belle. Dimensionem Hyperbolae cum Logarithmorum doctrina connexam esse, (quod varii variis modis ostenderunt,) agnosco rem esse non plane novam: non modo quod Tu forsan hisce non absimilia inveneris, (quod ille, credo, juxta atque ego nesciverat,) sed quod, post editum Gregorii de Sancto Vincentio Opus Geometricum, pro passim cognita habeam. (Vide, si placet, nostrum Commercium Epistolicum, Epist. 37. 39. 40.1065) Sed [2] nee ille, credo, ut rem plane novam proponit (ut sit eo nomine culpandus,)| sed rem alias notam exponit suo modo. Haec autem non ideo dicta putes, 1063

Animadversionem: i.e. HUYGENS, 'Examen de Vera Circuli fe Hyperboles Quadratura, in propria sua proportionis specie inventa fe demonstrata a Jacobo Gregorio Scoto, in 4°. Patavii', Journal des Scavans (2 July 1668 new style), 52-6; HUYGENS, (Euvres completes VI, 228-30 (partly). 1064 respondeat: i.e. GREGORY, 'Mr. Gregories Answer to the Animadversions of Mr. Hugenius upon his Book, De vera Circuli & Hyperbolae Quadratura; as they were publish'd in the Journal des Scavans of July 2. 1668', Philosophical Transactions No. 37 (13 July 1668), 732-5; HUYGENS, (Euvres completes VI, 240-3. 1065 Epist. 37. 39. 40.: i.e. FERMAT-DiGBY [28.III]/7.IV.1658, WALLIS-DIGBY 5/[15].V.1658, and WALLIS-BROUNCKER 11/[21].V.1658.

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242. WALLIS to HUYGENS, 31 August/[10 September] 1668 velim, quasi tibi derogatum earn, ipsi faveam: (est utique ille mihi, pariter atque tibi, credo, ignotus: quern non memini me vidisse unquam, vel allocutum esse, nee quicquam cum eo habuisse negotii; tantum abest ut ipsi debeam, prae te, antiqua jam amicitia cognito, faventior esse:) sed, 5 ut apud amicum, libere quae sentio eloquor. Idque eo potissimum facio, (quippe res ipsa tenuis est momenti,) quoniain jam aliquoties expertus scio, quam ex levibus initiis ad altercationes et aemulationes facile transitur: quod nollem hie factum; vel si fieri sit necesse, nollem Hugenii nomen in partes tralii. Aliud quod monendum duxi, hoc est. Regulam quam de Spatio Hy10 perbolico Logarithmorum ope mensurando Tu tradis; eandem esse quam, inter alia, Barovius1066 noster jamdudum tradiderat1067. Cujus cum ego Apographum apud me habeam, non gravabor hie (mutato tantum Idiomate ex Anglico in Latinum) verbatim transcribere.

1066

Barovius: i.e. Isaac Barrow (1630-77), fellow of Trinity College, Cambridge (164955), professor of Greek at Cambridge (1659-62), and from 1662 professor of geometry at Gresham College. He later became Lucasian professor at Cambridge. DNB. 106r tradiderat: i.e. in BARROW-COLLINS 6/[16].III. 1667/8; RIGAUD, Correspondence of Scientific Men II, 48-56, 48-9. Apparently Collins sent Wallis a copy of this letter. Cf. COLLINS-GREGORY ?.II.1668; RIGAUD, Correspondence of Scientific Men II, 174-9, 175. 570

242. WALLIS to HUYGENS, 31 August/[10 September] 1668 Hyperbolae Asymptotis ZH, ZK, in angulum rectum coeuntibus; sint ZA, ZB, ZC, ZD, &c, continue proportionales; puta, in ratione a ad /?. Ductisque rectis DM, EN', FP, &c, ipsi ZK parallelis; Rectangulisque F/M, F//x, et .FAT, Fz/, &c, completis: Erunt turn omnia Rectangula circumscripta EM, FN, GP, &c, invicem aequalia; turn omnia inscripta Ep,, Fv, GTT, &c. Ideoque EM toties sumptum quot sunt (DE, EF, &c,) partes in DH, superabit spatium Hyperbolicum DHSM: Et E\i toties sumptum, ab eodem deficiet. Est autem partium illarum in DH numerus, aequalis numero partium in ZH demptis partibus in ZD; (hoc est, Logarithmus ipsius ZH, dempto Logarithmo ipsius ZD; quam differentiam appello L.) Et quoniam DE = ZE - ZD = ZD - ZD = ZD: Erit EM = ZD x DM: Adeoque L x ZD xDM majus erit spatio DHSM. Similiter, EN = DM: Ideoque DE x EN = ZD x DM = ZD x DM: Adeoque L x ZD x DM minus erit spatio DHSM. Haec ille. Quam eandem esse cum tua Regula, non est quod moneam. Quippe si intelligantur Rectangula DM vel D/J, tam exigua, ut inscripti a circumscripto differentia, et utriusvis ab interjecto spatio hyperbolico, merito possit negligi: Dicit ille, Si L (differentia Logarithmorum rectarum ZD et ZH, vel his proportionalium HS, DM,) Multiplicetur in EM (rectangulum illud, seu spatium, exiguum,) habebitur ipsum DHSM spatium: Tu vero, Si Logarithmus ipsius L, Addatur Logarithmo ipsius EM, habebitur spatii DHSM Logarithmus. Quod tantundem valet. Sed de his hactenus Me autem quod spectat; miror ego quid Gallis vestris in mentem venerit, quod in me, omni data occasione, (vel non data,) involant. Quid ego certum in illos peccaverim, nescio: nisi forte quod Problemata1068 ab ipsis proposita, solverim1069 aliquoties. Quam male habitus ab ipsis olim fuerim, tu partim nosti; nee repeto. Quod fecit, ut tanquam religiose abstinuerim, ne cum illis quicquam negoti haberem (quamquam aliquoties lacessitus) quibus cum non erat nisi conviciis agendum. Quippe id genus 23 Multiplicetur in (1) D (2) EM 1068

Problemata: i.e. the challenges on number theory issued by Fermat and on the cycloid issued by Pascal. 1069 solverim: i.e. the solutions published in Commercium epistolicum and Tractatus duo.

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arma mihi non placent. Post id temporis, Sorberius (quamquam quod a me benigne exceptus1070 fuerit non diffitetur) non potest suum in Angliam Itinerarium (et quidem satis oscitanter) perscribere, nisi et me perstringat. Cumque nihil habuerit quod vitio verteret,| saltern id sibi ridendum pro- [3] posuit (et quidem satis ridicule) quod fuerim pro nostrorum more pileatus; quodque Academicus fuerim, non Aulicus. Sed contemnenda isthaec. Nuper mihi allatus est1071 Leotaudi Liber, quam Cyclomatheiam vocat, (sed aliquamdiu antea scriptus,) qui dum Aynscomium suum sibi refutandum1072 posuerit; quo volumen compleret, meum de Angulo Contactus Tractatum1073, prolixe satis, sed et satis jejune et parum Mathematice refutandum aggreditur. Sed, quam feliciter, non alium optarem quam te judicem. Cui privatis ad eum literis nuper respondi1074: nescio an publicum responsum mereatur. Tandem Dulaurensius, dum Specimina1075 sua ostentare satagit; quo pompam ornaret, subjungit Appendicem1076; ubi me Authorem fingit Problematis cujusdam leviusculi, omnibus Europae Mathematicis propositi; ut, eo soluto, de me triumphum ageret. Cumque hoc mihi injuriam fuisse conquestus sum1077; furibundus ille novum in me cadit dicterium1078, conviciis plenum; nee me quiescere patietur nisi velim libri sui scribere Refutationem. Quern quidem ego ne refutatione dignum censeo1079. Qualis ille apud suos haberi possit, nescio: At 8 refutandum (1) proposuerit (2) posuerit 1070 exceptus: Samuel Sorbiere (1615-70) reported on his visit to Wallis in Oxford in 1664 in SORBIERE, Relation d'un voyage en Angleterre, ou sont touchees plusieurs choses, qui regardent I'estat des Sciences, de la Religion, & autres matieres curieuses, Paris 1664, 41. 1071 mihi allatus est: cf. WALLIS-LEOTAUD 17/[27].II.1667/8. 1072 Aynscomium suum sibi refutandum: i.e. AYNSCOM, Expositio ac deductio geometrica quadraturarum circuli, Antwerp 1656. 1073 meum ... Tractatum: i.e. WALLIS, De angulo contactus et semicirculi tractatus, Oxford 1656. 1074 respondi: i.e. WALLIS-LEOTAUD 17/[27].II. 1667/8. 1075 Specimina: i.e. DULAURENS, Specimina mathematica, Paris 1667. 1076 Appendicem: i.e. DULAURENS, Specimina mathematica, 249-55. Cf. WALLISOLDENBURG 30.III/[9.IV].1668. 1077 conquestus sum: i.e. WALLIS, 'Another Letter written by the same hand, concerning . . . Specimina mathematica Francisci Du Laurens', Philosophical Transactions No. 34 (13 April 1668), 654-5. See WALLIS-OLDENBURG 30.III/[9.IV].1668. 1078 dicterium: i.e. DULAURENS, Responsio Francisci Dulaurens ad epistolam D. Wallisi ad clarissimum virum Oldenburgium Scriptam, [Paris 1668]. 1079 Quern . . . censeo: Wallis nevertheless wrote a number of replies to Dulaurens: WALLIS, 'Some Animadversions . . . ' , Philosophical Transactions No. 38 (17 August 1668), 744-50; 'A second Letter ...', Philosophical Transactions No. 39 (21 September

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243. WALLIS to BROUNCKER, [August ? 1668] certe, vel ego admodum fallor, vel non is est a quo in Mathematicis eximia sperare debeamus. Ego hujusmodi altercationibus non delector; ut qui bonas horas melius collocari posse judico. Si tamen necesse facient ut in mei defensionem, toties lacessitus, respondere debeam; non debent aegre ferre si cuneum nodo dignum adhibuero. Non tamen alios se laesum iri existiment, qui de me meisque rectius aestimant: non enim ex paucorum petulantia de omnibus judico. Tu vero (Vir Optime) si me audis, fac (quantum poteris) ut altercationum semina quam fieri potest caveantur: saltern nequid inter nostram vestramque Societatem Regiam aemulationis malae suboriatur, sed ut amice potius utroque junctis operis veram promoveat Philosophiam, nee altera alteri derogatum eat, vel jactandi captet occasiones. Denique Tu vale feliciter, et amare pergas,

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Tui amantissimum et observantissimum Joh: Wallis. [4]

Clarissimo Eruditissimoque Viro, D. Christiano Hugenio, de Zulichem; Parisiis.

243. WALLIS to WILLIAM BROUNCKER [Oxford, August ? 1668] Transmission:

W Letter sent: LONDON Royal Society Early Letters W2, No. 65, 10 pp. At top of p. 1 in Wallis's hand: 'Epist. D. Joh. Wallis, ad D. Vicecomitem Brouncker.' This paper largely represents a revised version of Wallis's investigations on Fermat's negative theorem for n = 3 (enclosure to WALLIS-BROUNCKER 3/[13].VIII. 1668) and of Wallis's demonstration of Fermat's other negative theorem (enclosure to WALLISBROUNCKER 8/[18].VIII.1668). The former is contained in parts I and II, the latter in part IV of the present letter. It was presumably written some time after Wallis's letters to Brouncker of 3 and 8 August 1668 and was probably intended for publication in the Philosophical Transactions.

10 promoveat (1) inq breaks off (2) Philosophiam 1668), 775-9, and 'A Continuation of Dr. Wallis his second Letter . . . ' , Philosophical Transactions No. 41 (16 November 1668), 825-32.

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243. WALLIS to BROUNCKER, [August ? 1668] Honoratissime Doctissimeque Doniine,

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Cum experimento aliquoties edoctus fuerim, quam facile certamina sive Philosophica sive Mathematica, utcunque placide inchoata, in iras et contumelias degenerant; fuerimque id expertus ipse in alio de Cycloide nego108 tio, ubi indignis modis me habitum censerim, [et speciatim a D (cui nominando abstineo, quia jam forsan ipse non scriptas malit) turn in literis ad me, turn aliis de me, crassissimis opprobriis onustum, (quae a Viro Nobili baud speraverim,) et quidem indicta causa (quid enim id sit quod eum ita commoveret nee dum scio, sed nee solicitus inquire,) et quibus omnino non respondendum erat, nisi in aperta jurgia et dimicationes velim ut res rediret:] Statui mecum hujusmodi cum Gallis commercium abrumpendum. Adeoque per aliquot annos prorsus silui; et, quanquam aliquoties lacessitus, nihil respond! quicquam, (nisi quod exigua quaedam chartula mea1081, me nescio, et praeter intentionem meam, in eorum manus devenerit,) necdum video cur ab ea sententia recedam. Adeoque D. Fermatii Epistolae (quippe hujus esse videtur, atque totius tenor idem indicare) aut cujuscunque fuerit, typis editae1082, neque publice respondendum censui, neque etiam privatis literis ad eum scriptis; quo manus denuo consererem, aut instaurarem proelium: pacem prae triumpho potiorem habens. Ne tamen vadimonium deseruisse videar; libet Demonstrationes eas quae hue spectant in formam redigere, (quarum materiam jamdudum paratam ipse moveris, utpote cujus magna pars tua est,) atque apud Te deponere; ut siquando contegerit Commercium Epistolicum1083 iterate imprimi, possint et haec adjungi. Interim Prooemio ejus aliquid reponendum videtur; in quo conqueri videtur Vir Clarissimus, quod eo inconsulto Commercium ediderim Epistolicum. Cui saltern duo reponenda habeo. 12 Adeoque (1) jam ab aliquot annis (2) per aliquot annos 16-17 (quippe hujus esse (1) perhibetur,) (2) videtur, atque totius tenor idem indicare) aut cujuscunque fuerit, add. 19 aut instaurarem proelium: \(—} del. add. 1080

aD : presumably Blaise Pascal. chartula mea: not identified. 1082 Epistolae ... editae: presumably FRENICLE-DIGBY ?.II.1657/8. Wallis would appear to be incorrect in ascribing this piece to Fermat rather than to Frenicle. 1083 Commercium Epistolicum: i.e. WALLIS, Commercium epistolicum, Oxford 1658. 1081

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243. WALLIS to BROUNCKER, [August ? 1668] Primum est, quod D. Freniclius (qui una cum D. Fermatio adversas partes contra nos tuebatur) nos hac in re praeoccupavit: edito libello1084 (non nobis tantum sed et aliis quorum id intererat nee consentientibus nee suspicantibus; quod queritur Schotenius, Epist. XXXIII.1085) de his ipsis quaestionibus; in quo non Mini tantum nominatim, sed Genti nostrae sive insultavit sive triumphavit satis liberaliter. (Cujus fere summa hue tandem redit, Solvisse ipsum problema quoddam, priusquam nos de eo quicquam inaudiveramus; Nos autem illud, nobis turn inauditum, non solvisse; ut Epist. XVIII.1086 ostensum est.) Cui omnino aliquid in Gentis nostrae vindicias publice reponendum par erat. Neque simplicius illud aut magis innocenter faciendum videbatur, quam Literas illas quibus res utrinque agebatur palam exhibendo. Siquis itaque hac in re culpendus sit, non nobis certe, sed ei potius qui et nobis praeivit, nobisque hanc necessitatem induxit, res imputanda erit. Alterum est; quod Cl. DD. Fermatius et Freniclus, nihil quicquam ad nos immediate scripserunt, neque nos ad eos; sed ad D. Kenelmum Digby utrique: Quo mediante primum usus est Fermatius me nominatim ad Certamen Provocans; eoque deinceps tanquam communi Mediatore usa est pars utraque. | Ilium itaque consuluimus, nee eo in consulto fecimus, sed et [2] approbante et incitante, (ut Epist. XXVIII. et XXXVI1087, liquet.) Neque alius a nobis consulendus videbatur, quam a quo omnia acceperamus, et quicum solo rem habuimus: Ut neque hie culpandi simus. Sed, missis his, ut ad rem ipsam deveniamus: Postquam Solutiones

1 (qui add.

3 id add. 4 Epist. XXXIII. add. 6 liberaliter. (1) Cui omnino aliquid in Gentis nostrae vindicias reponendum, par erat. Neque simp breaks off (2) Cujus tamen (summa) hue fere {redit,} Solvisse (3) (Cujus 12 utrinque add. 13 sed ei (1) qui (2) potius (3) potius qui 15 ad nos add. 19 utraque. (1) Cum (2) Ilium 20 XXVIII. et add. 21 acceperamus (1) : Ut (2) , et 1084

edito libello: i.e. FRENICLE DE BESSY, Solutio duorum problematum, Paris 1657. Epist. XXXIIL: i.e. SCHOOTEN-WALLIS [8]/18.III.1657/8. 1086 Epist. XVIIL: i.e. WALLIS-DIGBY 26.XII.1657/[5.I.1658]. 1087 Epist. XXVIII. et XXXVI: i.e. WALLIS-DIGBY 15/[25].III.1657/8 and DIGBYWALLIS [24.IV]/4.V.1658. 1085

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nostras aliquas extenuatum itur, ob D. Freniclii librum editum, (de quo abunde dictum est, Epist. XVIII1088; atque ostensum, libelli ejus tradita non ultra se extendere quam nos jam diu ante conspectum ilium pertigeramus:) alias dissimulat; (nam alia solveram suas, quas vocat, Hyperbolas infinitas spectantia earumque centro gravitatis, Epist. IV. V. XII. XVI.1089 Et quae Hyperbolam Apollonii spectant, Epist. XXXVII. XXXIX. XL.1090 Quae et Theoremata generalia sunt, et nullum ab ipsis auxilium habuimus: Ut mirum sit qua ratione dicat, quod neglexerimus omnino ea in quibus ipsi nobis non praeiverint: neque ullas Problematum generalium demonstrationes agnoscat; sed vel negat vel saltern dissimulat:) Quatuor saltern adhuc a nobis exigit, ut omni ex parte victoriam consequamur. Quibus eodem ordine quo proponuntur, respondeo. I.

Theorem^ inquit, praecipuum hoc erat; Dato quovis numero non-quadrato in integris, dantur infiniti quadrati in integris, qui in datum numerum ducti adscita unitate confidant quadratum.1091

15

1 aliquas add. 2 ostensum, (1) libellum ejus (2) libelli ejus tradita 3 diu add. 4-11 alias dissimulat; . . . saltern dissimulat:) add. in margin 4 solveram (1) , Hyperbolas quas vocat (2) suas, quas vocat, Hyperbolas 8 ratione add.

8 quod (1) neglexerim (2) neglexerimus 11 saltern (1) Capit breaks off (2) adhuc 12 Quibus (1) hoc (2) eodem ordine |quo proponuntur, add] respondeo. 14 praecipuum, add. 1088

Epist. XVIII: i.e. WALLis-DiGBY 26.XII.1657/[5.I.1658].

1089

Epist. IV. V. XII. XVI.: i.e. FERMAT-DIGBY [10]/20.IV.1657, WALLIS-DIGBY 6/[16].VL1657, FERMAT-DIGBY [5]/15.VIII.1657, and WALLIS-DIGBY 21.XI/[1.XII]. 1657. 1090 Epist. XXXVII. XXXIX. XL.: i.e. FERMAT-DIGBY [28.III]/7.IV.1658, WALLISDIGBY 5/[15].V.1658, and WALLIS-BROUNCKER 11/[21].V.1658. 1091

Dato quovis ... confidant quadratum: i.e. the equation generally known as Pell's equation. 576

243. WALLIS to BROUNCKER, [August ? 1668] Memini mihi a D. Freniclio, Epist. XXVI1092, vitio datum esse quod hoc Theorema probaverim; Quoniam, inquit, Hoc, nempe quod sint infiniti quadrati qui in 3 vel alium numerum non-quadratum ducti [adscita imitate] faciant quadratum; est expositum Theorema, quod se demonstrasse affirmat Fermatius, et quod affert ad exemplum tantum et ad praeparationem; non est igitur Problema quod solvendum proponitur. Quod autem ille culpat ut -rrapefrfoi' seu a-Kpoabibwaov: exigit alter et culpat ut non factum. Tarn est difficile, utrique placere. Sed utcunque, queritur hie Author (sive id demonstrandum proponeretur sive non,) saltern demonstratum non esse, atque nunc exigi ut demonstretur. Respondeo: jam olim demonstratum erat (Epist. XVI.1093) quo casu Unus aliquis ejusmodi quadratus dari possit, dari posse (illius ope) alios infinitos; Et, qua methodo illud fieret, item ostensum est. Et quidem, Epist. XIV, iterumque Epist. XVII1094, ex dato primo quadrato, dari posse non infinitos tantum sed seriatim omnes possibiles, ostensum est; et, qua methodo id fiat. Sed qua methodo dato quovis numero non quadrato primus ille quadratus (sed et sequentes omnes) haberi possit; ostensum item est, (ut alibi, sic) Epist. XVII et XIX1095: Et quidem tali methodo quam nullus non-quadratus fugiat. Si vero haec pro justa demonstratione non habeantur: Subjungam hanc aliam ita legitime deductam ut nullus sit exception! locus, qua probetur saltern Unum Quadratum cuilibet nonquadrato convenire: et quidem, si unus, infinitos inde haberi posse jam ante demonstratum esse non inficiabuntur ipsi. Exponatur itaque Theorema, cum Problemate annexe, in hac forma. Exposito n numero quovis integro non-quadrato: Dantur

3 3 vel alium add. 9 Author, (1) id dem breaks off (2) (sive 10 non,) (1) salte breaks off (2) saltern 18 quadratus (1) (nedum (2) (sed 19 tali methodo add. 21 Subjungam |hanc add. aliam (1) modo et forma (2) ita legitime (a) due breaks off (b) deductam 1092

Epist. XXVI: i.e. FRENiCLE-DiGBY ?!1.1657/8. Epist. XVI.: i.e. WALLIS-DIGBY 21.XI/[1.XII].1657. 1094 Epist. XIV ... Epist. XVII: i.e. BROUNCKER-WALLIS 22.X/[1.XI].1657 and WALLIS-BROUNCKER 17/[27].XIL1657. 1095 Epist. . . . XIX: i.e. WALLIS-BROUNCKER 20/[30].I.1657/8. 1093

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243. WALLIS to BROUNCKER, [August ? 1668]

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quadrati in integris innumeri, ut a?, qui in expositum nonquadmtum ducti, assumpta imitate, efficiant na2 +1 numerum quadratum. Proponitur Theorema demonstranduni; Et construendum problema, Exposito n; numeros a invenire. Theorematis Investigationem Analyticam utut apponere non teneamus; cum Demonstratio inde deducta (repetitis Analyseos vestigiis) praesenti sufficiat postulate: Ne tamen hanc illis invideamus, haec esto. Investigatio Theorematis. 2

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Si sit na + 1 numerus quadratus integer; erit ^/ : na2 + 1, (ejusdem latus,) numerus item integer; qui major sit quam a^/n, et minor quam a^/n+ (Quippe quadratus ab illo esset na 2 , justo minor; quadratus ab hoc, na2 + 1 + -^^ justo major.) Cumque horum differentia 2al/n sit unitate minor (propter a, n, numeros integros,) oportet ^ : na2 +1, ilium esse integrum qui surdum a\Jn proxime superat; et quidem excessu minore quam Sit m numerus integer surdo ^/n proxime major; hujusque ad ilium complementum p = m — ^/n < 1. Adeoque surdi a-^/n, complementum ad am, erit ap = am — a^/n. Sit I numerus integer surdo a^/n proxime major. Cujus quidem surdi, ad am complementum ap, cum possit esse majus unitate, (utut enim sit p < 1, possit tamen ap unitatem superare;) sit z integer proxime minor quam ap: qui itaque hinc demptus, relinquet surdi a^/n ad I (integrum proxime majorem) complementum ap — z = I — a

17 qui (1) qu breaks off (2) surdum 23 complementum |ad del. ap

578

[3]

243. WALLIS to BROUNCKER, [August ? 1668]

Sit r =

ap — z\ et a2p — az < r; Et

erit itaque ^ —

propterea (tanquam resolvendo aequationem) a

adeoque

Erat autem z < ap- adeoque f < p. Est ergo f < p < Quod omnino possibile esse constat; quoniam ita sumi possunt a, z, integri, ut sit ^ turn minor quam p, turn hinc tamen deficiat differentia auae sit data minor: adeoaue tali ut sit • p. Quod indicio est, Theorema esse possibile quicunque exponatur numerus non-quadratus n. Theorematis Demonstratio.

5

10

Numero surdo yn (utpote radice non-quadrati) sit m numerus integer proxime major: Cujus itaque excessus unitate minor erit, nempe m — \/n < 1. Ponatur p — m — Jn\ et r — TTT-. Sumantur porro integri duo z, a, ita comparati, ut sit | < 15 Quod fieri posse constat. (Certum utique est sumi posse in numeris rationalibus, fractum f, qui tarn prope dericiat ab exposito irrational! p, ut nunc superet Hoc Hoc est, (posito f + ^ = p) ut sit est (demptis utrinque aequalibus) adeoque z + y < 20 2 2 2 V/ : z + 4pr. Hoc est, ut ^/ : z + 4pr : superet ^z majori excessu quam est y; qui poterit esse quantumvis exiguus, seu dato minor.) Sumptis itaque ut dictum est z, a, numeris; Propter f < p; adeoque z < ap; erit ap — z quantitas positiva, seu major quam 0. Item; propter p < %/:z2 ^f r:+z ; erit 2ap < ^ : z2 + 4pr : +z; et 25 lap — z < ^/ : z2 + 4pr. Hoc est (sumptis utrinque quadratis) 4a2p2 4apz + z2 < z2 + 4pr: Adeoque 4o2p2 — 4apz < 4pr. Hoc est, a2p — az < r; •*

v

j^-yTl

2 propterea (1) a < (2) ( jtanquam add] resolvendo 15 ita (1) constituti, ut sit (2) comparati, ut sit (3) comparati 19 (posito (1) ^+y=p(2) ***=p(3) f + £ = p

19 sit (1) ^+y(=P) (2) *¥(=p) (3) I + £(=P) 20 aequalibus) (1) £ + y < (2) ^ < 20 adeoque z + y < \/ : z2 + 4pr. add. in margin 21 ut (1) V''z22l4pr'' superet ^ (2) ^ : z2 + 4pr : superet ^z2

579

243. WALLIS to BROUNCKER, [August ? 1668] et ap - z <

5

10

15

20

seu am -a

Hoc est, ap-z<

Ergo

am-z a^/n. Cum itaque sit am — z numerus integer (propter TO, a, z, integros) major quidem quam a-^/n, sed minor quam a^/n + 2a1/n '• Quadratum exhibebit numerum integrum, majorem quam na 2 (= Q : a^/n;) minorem [4] vero quam na2 + 1 + 4^2 (= Q '• a^/n + 2^7^-) Cumque inter hos duo numeros na2 et na2 +1 + ^2, (propter -^-^ unitate minorem, sunt utique n, a, integri,) nullus intercedat integer praeter na2 + 1, (quern propter n, a, 1, integros, integrum esse constat, et quidem proxime majorem quam no2;) Sequitur, quadratum numeri I = am — z, esse ipsissimum na2 + I quadratum. Constat itaque ejusmodi esse numerum a2 quaesitum. Adeoque et infinites. Uno enim habito, quotlibet alios in infinitum haberi posse, demonstratum est Epist. 14. 16. 17. 19.1096 et alibi. Atque ibidem Problematis constructionem fuse docuimus; quomodo istiusmodi quadrati a2 quotlibet inveniantur, exposito numero quovis nonquadrato n. Demonstravimus igitur Theorema, et Problema construximus. Quae erant facienda. II.

Secundum, inquit, Theorema Negativum hoc erat: Nullus numerus Cubus in duos Cubos rationales dividi potest.1097

25

9 (propter -^-^ unitate minorem, (1) cum sint n, a, (2) sunt utique n, a, integri,) add. in margin 20 et (1) co breaks off (2) Problema 1096

Epist. 14. 16. 17. 19.: i.e. BROUNCKER-WALLIS 22.X/[1.XI].1657, WALLIS-DIGBY 21.XI/[1.XII].1657, WALLIS-BROUNCKER 17/[27].XIL1657, and WALLIS-BROUNCKER 20/[30]. 1.1657/8. 1097 Nullus . . . potest: i.e. Fermat's 'last theorem' for n = 3.

580

243. WALLIS to BROUNCKER, [August ? 1668] Hoc autem ab origine (Problematice propositum) Schotenii fuerat, (atque ab ipso mutuatum videtur.) ut patet ex Epist. XXXIII.1098 Illud vero ego sic Demonstro. A A A Esto,' sifierirpotest,i a4 ^^^^) a + ' pm(= \ aApA ' = a-f^s, p ' in rationalibus. Potest igitur et in Integris idem fieri. Quippe, sublato communi Denominatore, erit in Integris a3/?3 + 63a3 = d3. Sunt autem hi tres Cubi, a3/?3, a363, e?3, vel singuli singulis inter se primi; vel saltern, facta ad communem mensuram applicatione, ad tales reduci possunt. Non potest enim eorum duos aliquos eadem mensura metiri, quin et tertium metietur. Quae enim Totum et Ablatum metitur, metitur Reliquum; Quaeque Ablatum et Reliquum metitur, metitur Totum. Sunto igitur, si fieri potest, ejusmodi tres Cubi, in Integris singuli singulis inter se primi; Puta a3 + b3 — c3. Adeoque a, 6, c, erunt similiter singuli singulis inter se primi. (Nam siquis horum duos mensuret numerus, idem et eorum Cubos mensurabit.) Est autem c major utrovis duorum a, 6, (propter c3 — a3 + 63,) sed simul utroque a + b minor. (Nam si c aequalis sit vel major quam a + b; erit (? aequalis vel major quam a3 + 63 + 3a26 + 3a62: ponitur autem C3 = 03 + b s_) Esto igitur c — a+b—e. Eritque e numerus integer (utpote integrorum differentia,) minor utrovis a, 6, (secus enim non esset c utrolibet a, b, major.) Ponamus b—e = /, et a—e = g. Adeoque c(= a+b—e) = a+f = b+g. Tune, propter a + / = c, erit a3 + / 3 + 3a2/ + 3a/2 = c3 = a3 + ft Adeoque 63 = / 3 + 3a2/ + 3a/2. Cujus Aequationis pars posterior est divisibilis per /; Ergo et prior. Adeoque y = / 2 + 3a2 + 3a/ numerus integer. Cum igitur / dividat 63; idem etiam dividet b: (vel illius saltern aliquota pars major quam 1; de quo casu post dicetur.) Adeoque (propter [5] a, 6, inter se| primes) idem / non dividet a; nisi sit / =1. Esto igitur b = f h ; adeoque b3 = f3h3. Ergo ^^ = /i3/2 - / 2 = 3a2 + 3a/; et h3 - 1 - ^ + % numerus integer. 1-2 Hoc autem . . . Epist. XXXIII. add. 1 fuerat, (1) atque ab ipso mutuatum (2) (atque ab ipso mutuatum videtur. 3 vero add. 13 similiter add. 1098

Epist. XXXIIL: i.e. SCHOOTEN-WALLIS [8]/18.III.1657/8.

581

5

10

15

20

25

30

243. WALLIS to BROUNCKER, [August ? 1668]

5

10

15

20

25

30

35

Est autem / vel divisibilis per 3, vel non est. Si sic; esto / = 3fc. Ergo / 1 3 -1(=^ + ^) = ^J + |. Ergo, 3 dividit a. Quod fieri non potest: quoniam f = 3k diviserat 6; sunt autem a, 6, inter se prirni. Si autem / non sit divisibilis per 3: turn, propter h3 — I — erit ^-^ = j2 + j numerus integer. Ergo / dividit a. Sed et diviserat b. Est igitur (propter a, 6, inter se primos) / =1. Similiter; propter b + g — c, adeoque 63 + g3 + 3b2g + 3bg2 — (? — 3 a + fe3; erit a3 = g3 + 3b2g + 3bg2. Cujus aequationis cum pars posterior sit divisibilis per gr; sic erit et prior: Adeoque ^- = g2 + 362 + 3bg, numerus integer. Cum igitur g dividat a3, idem etiam dividet a: (vel illius saltern aliquota pars, quae major sit quam 1; de quo casu mox dicetur.) Adeoque (propter o, 6, inter se primos) idem g non dividet b: nisi sit g =1. Esto a = gl] adeoque a3 = gr 3 / 3 , et ^^ = fig2 - g2 = 3fe2 + 36#, et I3 - I = 2jr + y, numerus integer. Est autem g vel divisibilis per 3, vel non est. Si sic; esto g = 3m. Eritque / 3 - l ( = y + |) = ^ + A.ErgO; 3 dividit b. Quod fieri non potest; quoniam g = 3m diviserat a: sunt autem a, 6, inter se prirni. Sin g non sit divisibilis per 3: turn (propter fi — 1 = ^- + y,) erit numerus integer. Ergo g dividit 6. Sed et diviserat a. Est ergo (propter a, 6, inter se primos) g — I . Erat autem et / = 1. Ergo b — e = (f = g =)a — e. Adeoque a = b, 3 et a — 63, et a3 + 63 — 2a3 — c3. Quod impossibile est. Non potest enim in rationibus dari (nedum in integris) Cubus cubi duplus. Sin dicatur (quod modo insinuatum est) fieri posse ut / dividat ft3, nee tamen dividat 6; (et similiter g dividat o3 nee tamen dividat a:) Quoniam, licet nullus sit numerus primus (vel etiam ex solis primis semel positis compositus) Cubum dividens, qui non et Latus dividat: fieri tamen potest, ut numerus ex primi quadrate vel cubo compositus, Cubum dividat, nee tamen dividat Latus. Puta, si ponetur / — rs2, et b — rst, adeoque fe3 = r3s3t3: qui quidem Cubus, sed non item ejusdem Latus, ab / = rs2 dividetur. Non tamen minus procedit demonstratio. Quippe si non ipse /; ipsius saltern pars aliquota, quae major sit quam 1, eundem b dividet. Nam, 3 non |non del] potest 582

243. WALLIS to BROUNCKER, [August ? 1668] propter —^J— — 3a2 + 3a/: Positis b — rst, et / — rs2; ipsius / — rs2 pars aliquota major quam 1, nempe rs, dividet b = rst: eritque 3a2 + 3ars2. Adeoque r3s3t3 - r3s6 = 3a2rs2 + 3arV. Cujus cum aequationis pars prior divisibilis sit, non quidem, ut prius, per / 3 — r3s6, sed saltern per hujus aliquotam partem quae major sit quam 1; nempe per r3s3: Erit et posterior pars ita divisibilis: Adeoque t3 - s3 = 3a^2gTs2, numerus integer. Unde ostendetur (ut prius) quod, vel 3 dividet o, qui prius diviserat 6; (nempe si 3 dividat r:) vel (si secus) idem r2s (vel hujus aliquota pars major quam 1) qui prius diviserat b, dividet a. (Quod fieri non potest [6] propter o, 6, inter se primos.)| Unico saltern casu excepto: nempe si r =1 et s — 3. Quippe turn 3 t - s3 = t3 - 27 = a2 + 9a; seu t3 = a2 + 9a + 27. Eritque hoc casu / = rs2 = 9. Atque eadem eodem modo ostendenda sunt si dicatur g dividere o3 sed non et a. Nempe vel casus erit impossibilis, propter a, 6, inter se primos; vel saltern erit g = 9. Jam vero si duorum /, g, uterque dicatur — 9: perinde est atque si sit uterque = 1. Quippe turn, ut prius, b — e = (f = g =)a — e~ adeoque a = b, et a3 — 63, et a3 + b3 — 2a3 — c3, quod est in rationalibus impossibile. Id unicum ostendendum restat; nempe, si duorum /, g, dicatur alter = 9, alter = 1, rem adhuc esse impossibilem. Puta, si / = 9, et g = 1; vel vice versa: utrumvis enim dicatur eodem res recidit. Id si ponamus: erit b — a = 8, (ut ex adjuncto calculo patet;) adeoque b = a + 8, et 63 = a3 + 24a2 + 192a + 512.

5

10

15

20

25

f=b-e=9 g =a-e = l f-g=b-a=8 Sed et, propter r = 1, s = 3, et r3s3t3 = 63, erit ^63 = (t3 =) o + 9a + 27: Adeoque 27o2 + 243o + 729 = 63 = o3 + 24a2 + 192a + 512. Et propterea a3 - 3a2 - 51a - 217 = 0, seu a3 - 3a2 - 51a = 217. ut adjuncto calculo patet. 2

1 ipsius . . . dividet b = rst: add. in margin 4 pars (1) posterior (2) prior 19 ut prius, (1) b-e(- f — g (2) b - e — (f - g-)a - e

583

30

243. WALLIS to BROUNCKER, [August ? 1668] a 3 +24a 2 +192a+512 = 63. subdue. 27a2 +243a +729 = b3. a 3 -3a 2 -51a -217 = 0. a 3 -3o 2 -51a =217. 5

10

Hoc autem fieri non potest, ita ut a sit numerus rationalis integer, prout supponitur. Posito enim o3 — 3o2 — 51a = 217; erit a major quam 10 sed minor quam 11. Quippe, si ponatur a — 10, erit a3 — 3a2 — 51a — 190: Si a = 11; erit a3 - 3a2 - 51a = 407. (Eadem, paucis mutatis, eadem methodo ostenderentur, si, pro / = 2 rs poneretur / = rs3, aut in alia forma simili: ut non sit opus eadem repetere.) Non est igitur, in Integris, a3 + 63 = c3. Adeoque (per ante demonstrata) neque in Rationalibus. Quod erat demonstrandum. Hinc sequitur; Freniclii Problema, Epist. XXXIII.1099 memoratum; Nempe;

15

10 aut in alia forma simili: add. 13-14 demonstrandum. \Idem aliter, et brevius \\ Si potest Unus aliquis Cubus rationalis in duos rationales dividi; potest quilibet sic dividi. Nam si c3 = a3 + 63, erit et (1) a3e3 (2) c3e3 = a3e3 + bse3. Quicunque autem sit cubus c3; potest hie, per cubum aliquem integrum vel fractum (quern e3 dicamus) multiplicatus, quern libet constituere cubum. || Potest igitur (verbi gratia) cubus 8 sic dividi. Puta 8 = a3 + fe3, ut (a) sit (6) sint a3, 63, Cubi in integris, (quod fieri non posse certum est; cum, praeter 1, nullus sit cubus eo minor;) Vel saltern 8 — ^ + j$, ut dividatur in duos Cubos fractos. || Sed neque hoc potest esse. Quippe turn erit 8a3/33 = a3/33 + a363. Qui quidem Cubi tres cum non sint singuli singulis inter se primi, possunt omnes (facta ad communem divisorem applicatione) sic reduci ut ita fiant. (Nam siquis numerus eorum duos metiatur, metietur et tertium: siquis enim numerus Totum et Ablatum metitur, metitur et Reliquum; et siquis metitur Ablatum et Reliquum, metitur totum.) Ergo, propter 8a3/33 et a3/?3 divisibiles per /33, etiam a363 sic erit divisibilis, adeoque %|- numerus integer: et propterea 8a3 = a3 + ^|- |in add. integris. Iterumque, propter 8a3 et ^-p- divisibiles per a3, erit et a3 sic divisibilis, adeoque ^ numerus integer: et propterea 8 — ^ + j% in integris. Quod fieri non posse, ante demonstratum est. || Non potest igitur jcubus add.\ 8 (oa) sic dividi (66) dividi in duos cubos rationales. Ergo (per ante demonstrata) nee Cubus ullus alius. Quod erat demonstrandum. del.\ \\ Hinc sequitur; 15 Epist. XXXIII. (1) propositum; nempe (2) memoratum; 1099

Epist. XXXIII.: i.e. SCHOOTEN-WALLIS [8]/18.III.1657/8. 584

243. WALLIS to BROUNCKER, [August ? 1668] Trouver un nombre triangulaire, dont le sextuple +1 soit nombre cube: impossibile esse. Nam cujusvis numeri triangularis sextuplum +1, est duorum cuborum continue proximorum differentia, quorum minoris latus idem est cum latere istius numeri triangularis. (Quae itaque differentia si cubus esse posset, haberetur Cubus duobus cubis aequalis.) Nam (per Arith. Infin.1100 prop. 172.) Lateris a. numerus Triangularis est |a2 + ^a, cujus itaque sextuplum unitate auctum est 3a2+3a+l; quae est differentia [7] Cuborum lateris a, et lateris a + l.|

III.

5

10

Tertium, inquit, Theorema generale, quod sub forma problematis concipi potest, hoc erat; Datus quivis numerus de duobus Cubis compositus in duos alios cubos est divisibilis. Vel, si problema universale proponendum mavis,

15

Datum numerum ex duobus cubis compositum in duobus alios cubos rationales dividere. lino vero, inquam, Problema primitus propositum erat, Treuver deux nombres cubes dont la summe soit esgale a deux autres nombres cubes. Duos numeros cubos invenire quorum summa sit aequalis duobus aliis numeris cubis. (Non, duobus aliis datis.) Atque hoc est quod Freniclius solvisse dicitur, Epist. X.1101 ubi de Cubis Datis, ne jpv. Sed crescit pensum nostrum, quod tamen absolvendum curabimus. At interim Problemati, ut prinium proposito, quo respondeamus, Cubos exhibebimus binos binis aequales, alios ab eis quos exhibuit Freniclius,

3 cujusvis add. 4 quorum jminoris add.\ latus idem (1) habet (2) est 22 aliis (1) aliis cubis (2) datis 26 primum (1) pos breaks off (2) proposito 1100 1101

Arith. Infin.: i.e. WALLIS, Arithmetica infinitorum, Oxford 1656. Epist. X.: i.e. BROUNCKER-WALLIS 3/[13].X.1657.

585

20

25

243. WALLIS to BROUNCKER, [August ? 1668] Epist. X, et XXVI1102; sed neque eorum multiples aut submultiplos, eisve alias proportionales; ut ne hoc nomine conquerantur.

Cub: 5 + C: 204 + C: 623 + C: 464 + C: 2655 + C: 4580 + C: 2453 + C: 1397 + C: 11208 + C: 16367 + C: 2015 + C: 7700 + C: 31695 + C: 15285 + C: 42464 + C: 4768 + C: 18581 + C: 31229 + C: 9775 + C: 71892 +

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10

15

20

25

Cub: 315. C: 2046. C: 3913. C: 2280. C: 11169. C: 17290. C: 8547. C: 22660. C: 36708. C: 51025. C: 15288. C: 23058. C: 91785. C: 83580. C: 119536. C: 100688. C: 51051. C: 140896. C: 130169. C: 193410.

= = = = = = = = = = = = = = = = = = = =

Cub: 248 C: 1535 C: 2820 C: 1589 C: 7568 C: 11439 C: 5540 C: 17397 C: 23375 C: 31992 C: 11231 C: 14261 C: 56084 C: 59189 C: 72255 C: 77935 C: 30560 C: 96765 C: 99148 C: 114767

+ + + + + + + + + + + + + + + + + + + +

Cub: 252. C: 1705. C: 3354. C: 1995. C: 9928. C: 15561. C: 7770. C: 18540. C: 33649. C: 47100. C: 12936. C: 21411. C: 85666. C: 72436. C: 112065. C: 81809. C: 48048. C: 124320. C: 107198. C: 182665.

Et, nisi haec sufficiant, plures Cubos, binos binis aequales, facile exhibebimus: Ut Problemati huic ut primitus proposito abunde satisfactum sit. Sed jam exigitur, ut Duo cubi rationales duobus datis aequales exhibeantur: Seu Datum numerum ex duobus cubis compositum in duos alios cubos rationales dividere. Puta, Invenire numeros g, h, quorum cubi simul sunipti sint g3 + h3 — a3 + 63 datis.

30

Illud ego (sive Theorematice sive Problematice propositum) his passibus expedio.

1 eisve |alias add. proportionales (1) . (2) ; ut ne hoc nomine conquerantur. 29 numeros g, h, quorum cubi simul sunipti sint add. in margin 1102

XXVI: i.e. FRENICLE-DIGBY ?.II. 1657/8.

586

243. WALLIS to BROUNCKER, [August ? 1668] 1. Inveniantur c3 — rf3 = o,3 4- fe3 ciatis. Hoc, modo.

Ponamus n + a = c. et Ergo, c3 = n3 + 3rra + 3na2 + a. et, d3 •Et, c3 Ergo Hoc est, Adeoque, n Ergo, c(= n + a) [8] et, d(= fjn - 6) = Vel, Ponamus a Ergo, c3 = a3 et, d3 = n3 - 3n26 + 3n62 - 63 Et. c3 - d3 = Ergo, Hoc est, Adeoque, n = Ergo, c

5

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15

et, d(— n — b)

2. Si ex c, d, sic inventis, sit 2 c3; turn Iveniantur e, /, ita ut sit e3 — / 3 = c3 —
21 per sequentem operationem tertiam: add. 23 Inveniantur \e, f, ita ut sit add. ez - /3 — c3 - d3. (i) datis. (#) Hoc modo, (a) Ponamus, n - b = e. et ^n - a = f. Ergo, e3 = n3 - 3n26 + 3nb2 - b3. f = . Et e3 - /3 = (6) Ponamus

587

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25

243. WALLIS to BROUNCKER, [August ? 1668]

5

10

15

20

Adeoque, n Ergo, et Vel, ponamus, Ergo, et, Et, Ergo, Hoc est Adeoque, Ergo, et. [91 3. Si ex e, /, sic inventis, nondum habeatur 2/3 < e3; iteranda erit haec secunda operatic donee id habeatur. Verum illud rarissime potest contingere: Nam, ut plurimum, ne secunda operatione opus erit, sed in prima provenit 2e?3 < c3. Habito itaque (vel 2e?3 < c3, vel) 2/3 < t? (aliove simili, si iteranda contingat ea operatic;) Inveniantur, g, h, ita ut sit datis. Hoc modo. Ponamus. Ergo, et, Et,

Ergo Hoc est 25

Adeoque, Ergo,g Et,

Vel ponamus, Ergo 30

14 secunda add. 16 itaque (1) 2/3 (2) (vel 18 g, h, ita ut sit add.

588

243. WALLIS to BROUNCKER, [August ? 1668] Et)

Ergo, Hoc est,

Adeoque, Ergo, a

5

et, h Invenimus igitur numeros g, h, quorum cubi simul sumpti g + h = a + 63 datis. Quod erat imperatum. Et qua methodo, ex datis o, 6, inveniuntur (/, h; eadem, ex datis his, inveniendi duo alii; atque, ex illis, duo alii; atque sic porro in infinitum. Atque ad hanc formam, duabus operationibus inventos (omissis e, /,) datis duobus cubis numerorum 1, et 3, praesto habemus duos alios aequales; qui casus est in Epist. XII.1103 propositus. Item, tribus operationibus inventos, duobus cubis numerorum 1, 2, (qui est casus in hac Epistola et Epist. XXXVII.1104 insinuatus,) duos alios aequales. Sed quoniam in longos numeros (fractos) abeunt, non appono. Non dubito interim quin habeant Cl. Viri expeditiorem methodum idem praestandi (et nos fortasse expeditiorem inventuri, si ad quaestiones hujusmodi numericas studia convertere, animumque eo applicare vellemus,) sed utcunque imperatum absolvimus more nostro, et quidem tali methodo qualem non refugerunt Vieta1105, Bachetus1106, Girardus1107 (ne plures nominem) in casibus non absimilibus: et quam Author hie repudi3

7-8 igitur (1) datis duobus jcubis add.\ a3+63 (quicunque demumfuerint) duos aequales
Epist. XII: i.e. FERMAT-DiGBY [5J/15.VIII.1657. Epist. XXXVIL: i.e. FERMAT-DIGBY [28.III]/7.IV.1658. 1105 Vieta: i.e. Francois Viete (1540-1603). 1106 Bachetus: i.e. Claude Caspar Bachet de Meziriac (1581-1638). 1107 Girardus: i.e. Albert Girard (1595-1632). 1104

589

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20

243. WALLIS to BROUNCKER, [August ? 1668] are non debet, cum eandem in Bacheto et Vieta approbet.| IV.

Quartum, inquit, Problema (lege Theorema,) negativum, hoc erat. Nullum in numeris est triangulum rectangulum cujus area sit numerus quadroons.

5

Hoc ego Theorema demonstraveram Epist. XLIV.1108 in hunc modum In exposito Schemate (cujus constructio patet) Trianguli rectanguli BCD latera non possunt esse numeris effabilia, nisi AD, DE, sint inter se ut numeri plani similes, (secus enim, qui ab ipsis fit non erit numerus quadratus, ejusque radix BD effabilis;) hoc est, ut numeri quadrati inter se. Esto ut 2a 2 , 2e2. Erunt igitur CB, CD, BD, ut a2 + e 2 , a2 - e 2 , 2ae. Adeoque CD, \BD, ut a2 + e 2 , ae. Et proinde (cum duorum quadratorum differentia, atque eorundem medius proportionalis, non possint esse plani similes,) qui ab ipsis fit (hoc est area trianguli) non potest esse numerus quadratus. Quod erat demonstrandum.

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Sed hie, inquit, nullam demonstrationem deteximus. Supponit quippe pro media demonstrations Theorema sequens, Differentia duorum quadraiorum atque eorundem medius proportionalis non possunt esse plani similes: quod nihil aliud est quam obscurum per obscurius aut saltern aeque 3 (lege Theorema) add. 4 rectangulum add. 6 ego (1) Pro breaks off (2) Theorema 13 proinde ( (1) qu breaks off (2) cum 1108

Epist. XLIV.: i.e. WALLIS-DIGBY 20/[30].VI.1658.

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243. WALLIS to BROUNCKER, [August ? 1668] obscurum probare. Quod quidem Lemma Verum esse fatetur, queritur tantum quod non sit Demonstratum. Putaram ego Lemma istud, apud tantos Viros quibuscum res agebatur, admissum iri; neque expressa Demonstratione indigere. Id autem, quoniam hoc expetunt, sic demonstro. Nempe

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Differentia duorum quadratorum et eorundem medius proportionalis (puta a2 — e2 et ae) non possunt esse numeri plani similes. Intelligantur a 2 , e 2 , quadrati inter se primi, seu in eadem ratione minimi; (quippe vel tales sunt, vel facta ad communem mensuram applicatione, ad tales reduci possunt, non mutata ratione rectarum AD, DE.) Ergo, et a, e, inter se primi; (quippe siquis hos duos metitur numerus, idem et a 2 , e 2 , metietur.) Ergo, et trium a, e, a — e; item trium a, e, a + e~ singuli singulis inter se primi; (nam siquis vel horum vel illorum duos metitur numerus, idem et tertium metietur; quippe, qui totum et ablatum metitur, metitur reliquum; quique ablatum et reliquum metitur, metitur totum.) Cumque a — e sit ad utrumque duorum a, e, primus; primus item erit ad ae, factum ex illis. Similiter ostendetur, a + e primum esse ad eundem ae, quoniam ita est turn ad a turn ad e. Cum itaque ae primus sit turn ad a + e turn ad a — e; primus item erit ad a2 — e2 ex illis factum. Non sunt igitur a2 — e2 et ae plani similes; quippe similes plani non possunt esse inter se primi; (intellige; nisi uterque sit quadratus; qui casus hie non obtinet.) Quod erat demonstrandum. Absolvimus itaque (demonstratis his quatuor Theorematis) pensum nostrum, quod exigitur, ut ex omni parte victoriam consequamur. Quod faciendum erat. Interim notandum est, ex omnibus quae nos ex adverse proposuimus (excepto unico, quod, obiter et non sibi propositum, solverit Freniclius,)

10 quippe (1) vel (2) vel 13 et |trium add] a, e, a — e; item |trium add. a, e, a + e; 16 metitur, (1) numerus (2) metitur 22 primi; (1) sed vel alter alterum metietur, vel saltern his similium minimus metietur utrumque |saltem nisi 1 habeatur pro numero piano, nempe quadrato, quicum alio quadrato comparetur; qui casus rem praesentem non attingit; nam neque ae neque a 2 — e2 potest esse 1. add] (2) (intellige; nisi uterque sit quadratus; qui casus hie non obtinet.) 24 (demonstratis his quatuor Theorematis) add. 25 quod (1) exigebatur, (2) exigitur,

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245. WALLIS to OLDENBURG, 3/[13] September 1668 neglexerint omnia, et ne attigerint quidem nedum solverint; ut non sit quod querantur de suorum aliquot a nobis neglectis. Atque his quidem praestitis, non est quod te ultra detineat, Tui studiosissimus atque observantissimus, Johannes Wallis.

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244.

HENRY OLDENBURG to WALLIS I/[11] September 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 3/[13].IX.1668. In this letter, Oldenburg apparently enquired about a certain Mr. Watkins in Oxford, who, being somewhat deaf, reportedly used an otacousticon to listen to sermons and lectures. He had been asked to confirm this report by John Beale in a recent letter. See BEALE-OLDENBURG 29.VIII/[8.IX].1668; OLDENBURG, Correspondence V, 28-32, 29.

245.

WALLIS to HENRY OLDENBURG Oxford, 3/[13] September 1668 Transmission: W Letter sent: LONDON Royal Society Early Letters Wl, No. 63, 2 pp. (our source). Postmark on p. 2: 'SE/4'.—printed: OLDENBURG, Correspondence V, 33-5. Reply to: OLDENBURG-WALLis 1/[11].IX.1668.

Oxford Sept. 3. 1668. Sir

To yours1109 of Sept. 1. concerning Mr Watkins. It is true, as is therein mentioned, that there was, (not lately, but about 14 or 16 years agoe,) 1109

yours: i.e. OLDENBURG-WALLIS 1/[11].IX.1668. 592

245. WALLIS to OLDENBURG, 3/[13] September 1668 one Mr Watkins1110, of Christchurch; who, being deafish, did make us of an Otacousticon to help his hearing: whereby he received advantage so as to hear those (sermons, disputations, discourses, &c) with it, which without it he could not. The Bishop of Salisbury1111, Dean Wilkins1112, Dr Wren1113, Dr Willis1114, Dr Pope1115, (all or most of these,) & many others who were then in Oxford, can as well give an account of it as my self. But, that he could hear farther off, or better with it, than others could without it, or so well as they; I do not think. To us who needed it not, I do not remember that it gave any considerable advantage; (at lest, I did not apprehend it, & I do not remember that others gave a contrary account;) otherwise than that it did make the noise seem greater; but, withall, more confused & indistinct. But it's possible that if wee had been habitually accustomed to it, as he by use had been, it might as well have been advantageous to us allso. (For I doubt not but that the ear & eye are capable of habits, as well as hand, foot, or tongue: whereby wee are more able to distinguish nice differences in things to which wee are accustomed, than otherwise wee could: And it is obvious inough to every one, that a voice to which we are accustomed, shall be easyer understood, & at a further distance, than a strange voice.) The Form of the Instrument,

This I think is too long, for the bignesse, & therefore I drew the other shorter.

5 or most add. 9 not, (1 )(!<-> (2) I 11 seem add. 15 tongue: (1) ) (2) whereby 1110

Watkins: probably Richard Watkins (d. 1708), B.A. 1644 and M.A. 1647 at Christ Church, Oxford, who published News from the Dead, Oxford 1651. 1111 Bishop of Salisbury: i.e. Seth Ward. 1112 Dean Wilkins: i.e. John Wilkins, q.v. 1113 Dr Wren: i.e. Christopher Wren, q.v. 1114 Dr Willis: i.e. Thomas Willis. 1115 Dr Pope: i.e. Walter Pope (d. 1714), half-brother to John Wilkins, who succeeded Wren as professor of astronomy at Gresham College in 1661.

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246. PELL to COLLINS, 6/[16] September 1668

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was much like the greater end of a Trumpet; (but lesse than it;) and at the little end of it, was a small turning to put into the ear, while the great end was directed to the speaker, or to the noise which was to be heard. The length (as I remember) about a foot or more; but I cannot nicely determine it. Hee had divers of them made; whereof some proved better, some worse, though as to the eye wee could see little difference, nor could hee give any other account of the reason (nor, I think, the Artist,) but onely as hee found by tryall. They were of Brasse, & made, he sayd, by the Trumpet-makers; but, by whom in particular, I know not. I beleeve the man may be still living; but, where, I cannot tell. I remember I have severall times mett him (I think, in London,) since he left Oxford. I have given a rude draught of it in the margin; but will not undertake that the proportions are exact, having nothing to direct mee but an Idea in my memory of so many years standing. Perhaps others may remember it better. This is all at present from Yours &c John Wallis.

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These For Mr Henry Oldenburgh, in the Palmal near St James's London.

[2]

246. JOHN PELL to JOHN COLLINS Brereton, 6/[16] September 1668 Transmission:

C Copy of (missing) letter sent: LONDON British Library MS. Add. 4278, f. 127r-127v.

1 was |of Brasse, & add. and del] much. 4 foot (1) fe a half, (2) or more; 8 by (1) (—) (2) tryall. They were |of Brasse, & add. made, 10 but, (1) I know (2) where, 10 I (1) some time (2) have 15 better. (1) which (2) This

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246. PELL to COLLINS, 6/[16] September 1668 Reply to: COLLINS-PELL 5/[15].IX.1668. Yours1116 of Sept. 5 desires me to send some objections against the New Logarithmotechnia1117. Its title-page tells us that it had been communicated in writing almost a yeare before it was printed. I desire to know what Dr Wallis saith of it. As for my Objections, I never threatned M. M. to make any, nor promised to send you any. My design being rather to give precepts differing from his, in hope that some Readers will like mine better than his. He deales, all in short Logarithmes: 2, 3010299 the Log of 200. 1,9978230^ the Log of 99^. 2,0021659^ the Log of 100^. Longer Logarithmes would discover the insufficiency of his precepts. In general (saith he, pag 20) divide 43429 by the medium Arithmeticum terminorum rationis datae: Yet perhaps, saith he, it should be 43430; pag 20, lin ultima. I hope I neede not tell you, that without peradventure the Logarithme of HI is 0.00434,29810,1078. Few are good at making numeros solennes and shewing the lawfull use of them. Nor is it safe for every one to meddle with Theoremes that have forte and propemodum in them; or to judge V [127 ] of problemes that pretend onely to prope verum. \ I have not now time to enter into discourse concerning such propositions. Mr Brancker is newly come hither. He intends to write shortly to you about his Composition for the first fruits of his new Parsonage. My L. Br.1118 sayes, you see Mr William Potter1119 often: I pray you tell him that His letter of September 12, was delivered to the hands to which it was directed. He sayes, he dwells on Clerkenwell-green. Which puts me in minde of a book1120 sold by George Sawbridge at his house on that green; It is called Praxis Francisci Clerk. I pray you let your next tell me the bulk & price of it. A Dieu September 16 1116

Yours: i.e. COLLINS-PELL 5/[15].IX.1668 (LONDON British Library MS. Add. 4278 f. 342r-342v). 1117 New Logaxitlimotechnia: i.e. MERCATOR, Logarithmotechnia, London 1668. The ful title is Logarithmotechnia: sive methodus construendi Logarithmos nova, accurata, & facilis; scripto antehac communicata, Anno Sc. 1667. nonis Augusti: Cui nunc accedit. Ve.ro, Quadratura Hyperbolae, & Inventio Summae Logarithmorum. 1118 L. Br.: i.e. William Brereton, q.v. 1119 Potter: i.e. William Potter (fl. 1656), writer on economics, DNB. See WEBSTER, Great Instauration, 449-55. 1120 book: probably CLERKE, Praxis curiae Admiralitatis Angliae, London 1667. The author, Francis Clerke (or Clarke) (fl. 1594) practised civil law and was sometime senior proctor of the University of Oxford, DNB.

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247. WALLIS to COLLINS, 8/[18] September 1668

247.

WALLIS to JOHN COLLINS Oxford, 8/[18] September 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, 200r-200v (our source). On right margin off. 200V at 90° to address in Collins's hand: 'Mercators Explanation of Ratios disliked'. Postmark on f. 200V: 'SE/9'.—printed: RIGAUD, Correspondence of Scientific Men II, 493-7.

Oxford Sept. 8. 1668.

Sir,

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I do not return the sheet1121 sent by the last post; because there is not much to be altered. & therefore I would not charge the postage. But pag. 20. lin. 11. for continuis put continuatis. And pag. 22. at Prop. 16. Instead of Fig. 15. P.I:: 2P. 2P.I:: 3P. nP.I:: mP. Fig. 14. 2L. L^ 3L. 2L. mL. nL. 2PL7 2PL7I. 6PL7 6PL7I. mnPL. mnPL. r/ifa. let it be Fig. 14. 15. P. 2P. 2P. 3P. nP. mP. 2L. L^ 3L. 2L. mL. nL. 2PL7I :: 2PL7l. 6PL7l :: 6PL7l. mnPL. I :: mnPL. I. that is; I would have I :: in three places taken down from the first line into the third; & in the end, put out mn before I. And in the margin, Fig. 14, 15, (supposing that both figures are to be cited, as, I think, here they are.) So at prop. 17. pag. 23. (if the three figures are to be cited) let it b thus. Fig. 16. 17. 19. But I am very suspicious of mistakes in the citations of the figures in the Margin, which made mee desire in my last that you would look over the copy, &; compare it with the draughts of the schemes, to see whether 5 continuatis. (1) at (2) And 13 So at ... Fig. 16. 17. 19. add. 15 suspicious |of add. mistakes 17 with the (1) schem breaks off (2) draughts 1121

sheet: i.e. proof sheet of Wallis's Mechanica sive de Moiu tractatus geometricus.

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247. WALLIS to COLLINS, 8/[18] September 1668 they answere aright to them or no. For I have no means here to compare them. I do not understand why the sign of multiplication1122 x should more trouble the convenient placing of the fractions, than the other signes H— =>

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As to other things in some of your late letters. I find that Mr Hugens his Rule for the measure of an hyperbolick space, is the same with Mr Barrows, as I have signified in a letter1123 to Mr Hugens, which perhaps my Lord Brounker hath shewed you. For whereas one mentions the difference of Logarithms of ZD, ZH, the other of HS, DM, it makes no alteration; for, the proportion being in both the same, the difference of Logarithms will be so too. And I find Hugens's constant Logarithme 0,3622156868 to be the logarithm of 2,3025853 which is the termination of what you call elenientum Logarithmicum. Andersons book1124 you mention, is not yet come to hand. That which I could have wished altered in Mr Mercators Logarithmotechnia, was, his manner of expression of the composition of Rations: For composition be a word used by Euclide1125 sometime for Addition, sometime for Multiplication; And there being in him two compositions of Rations, the one mentioned Def. 14. lib. 5. which is by addition of the exponents, as when | + | = |; the other by multiplication of the exponents, Def. 5. lib. 6. as when | x | — |: which ambiguity hath cau some confusion, especially where the Latter is called an Addition of Ra22 of (1) Fractions (2) Rations 1122

sign of multiplication: cf. WALLIS-COLLINS 21./[31].VII.1668. letter: i.e. WALLIS-HUYGENS 31.VIII/[10.IX].1668. A copy of this letter was kept in the Royal Society. 1124 book: i.e. R. ANDERSON, Stereometrical Propositions variously applicable, but particularly intended for Gageing, London 1668. 1125 Euclide: see Elements V, def. 14 and VI, def. 5. Cf. Heath's notes on these and Elements V, def. 9, 10 in HEATH (ed.), The Thirteen Books of Euclid's Elements. Translated from the text of Heiherg with introduction and commentary, 3 vols, Cambridge 1908, II, 132-5, 189-90. 1123

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tions: Clavius1126, & Gregory St Vincent1127, & divers others, to avoid this inconvenience, have for distinction sake, called the former Composition by Addition, the Latter Composition by Multiplication; with which most writers (who speak distinctly) have used to comply. (Of which I have spoken at large in what I have writ against Meibomius1128; & against Mr Hobs's fourth Dialog:1129 & else where.) Now Mr Mercator resuming the old term of Addition1130 (which I could wish antiquated) for that which is indeed a Multiplication, reduceth us to the same confusion which hath been endeavoured by distinguishing to be avoided. For certainly Sextuplum is as much the Double of Triplum, as 6 is the double of 3. Not an Addition in either place, but a multiplication. And what Euclide1131 meanes by Proportion Duplicate, Triplicate, Quadruplicate, &c; Def. 10. lib. 5. is but what we now call by other names square, cube, squared square, &c. And consequently what in Mr Mercator is called Numerus rationum1132, I should rather have called Exponentem potestatis expositae rationis; & consonantly else where. Which differs onely in expression, not in substance, from what he means all along. And so, though 0 be indeed the exponens potestatis (pro radice quavis expositae) aequalitatis; yet not 0, but 1, is exponens rationis aequalitatis; as 2 is the exponent of Duplicity. Which, if you please, you may shew him, that hee do not 1 [In left margin in Wallis's hand:] -3. -2. -1. 0. 1. 2. 3.

_L3 r

J_ j.2-

1

r-

I

I-

L

I-

ri I •

tL I •

6 Dialog: (1) ) (2) & 6 Mr (1) Gregory (2) Mercator 12 Proportion add. 12 Def. 10. lib. 5. add. 14 what (1) lie me breaks off (2) Mr Mercator calls (3) in Mr Mercator is called 15 Exponentem (1) rati breaks off (2) potestatis 18 quavis add. 1126

Clavius: see Euclidis elementorum libri XV, ed. Clavius, vol. 1, 161V and 186v-189r. St Vincent: see SAINT-VINCENT, Opus geometncum, 872-3. 1128 against Meibomius: i.e. WALLIS, Adversus Marci Meibomii, De proportionibus dialogum, tractatus elenchticus. 1129 against . . . Dialog: i.e. WALLIS, Hobbius heauton-timorumenos, 63-88. 1130 Mercator . . . Addition: see for example MERCATOR, Logarithmotechnia, 1-2. 1131 Euclide: i.e. EUCLID, Elements V, def. 10. 1132 what . . . rationum: see MERCATOR, Logarithmotechnia, 3. 1127

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247. WALLIS to COLLINS, 8/[18] September 1668 misapprehend me. What you say of the french possibly pretending that Mercator's quadrature1133 was grounded on Lalovere; there is no reason of that pretense, there being much more reason to say that Lalovere's (if the same for substance) was grounded1134 on my Arithmetica Infinitorum, where (in prop. 88 & 95) the ground is layd geometrically, & the rest but an Arithmetical calculation deduced from it, as you see in my letter1135 about that quadrature. But that book is a book which the French (though it's like inough they make good use of) do not desire the world should take notice of. That the Rectifying a Parabolical line, & Squaring the Hyperbole, do mutually give one another, is no new thing, but what hath been discoursed between me & Mr Hugens, by letters1136, many years ago. The whole mystery of rectifying curve lines, being layd down in my Scholia1137 ad Prop. 38. Arithm. Infin. & further prosecuted in my Cycloide1138 pag. 90. &c. And I have not yet seen any thing of that kind, but what are but V [200 ] particular instances of what is there delivered in generall. | Mr Mercators Instruments1139 (mentioned in yours1140 about a month ago) I have yet not spoken of to Dr Wren (having but once had opportunity, & then I forgot it;) but hee being then at London, & for a good while after, I suppose you had there opportunity to discourse it with him. As to Mr Gregory; I do admit to bee demonstrated1141, in Prop. 11. 4 to say add. 9 of) | they del. ed.\ do 21 I do (1) acknowledge (2) admit 1133

quadrature: see MERCATOR, Logarithmotechnia, 31-3. grounded: i.e. LALOUBERE, Veterum geometria promota in septem de cydoide libris, Toulouse 1660. Cf. LALOUBERE, Quadrature, circuli et hyperbolae segmentorum, Toulouse 1651. 1135 letter: i.e. Wallis's review of Mercator's Logarithmotechnia, published as a letter to Brouncker, in Philosophical Transactions No. 38 (17 August 1668), 753-9. See WALLISBROUNCKER VII/VIII.1668. 1136 letters: see for example HUYGENS-WALLIS [3]/13.VI.1655 and WALLIS-HUYGENS 21.VI/[1.VII].1655. 1137 Scholia: see WALLIS, Arithmetica infinitorum, 28-31; Opera mathematica I, 380-2. 1138 Cycloide: see WALLIS, Tractatus duo, 90-8; Opera mathematica I, 551-5. 1139 Instruments: It is not clear to what instruments Wallis refers. Mercator's practical abilities are mentioned more than once in the records of the Royal Society. See BIRCH, Htstory of the Royal Society II, 110, 112, 187. 1140 yours: probably the now missing letter COLLINS-WALLIS 15/[25].VIII. 1668. 1141 demonstrated: i.e. in GREGORY, Vera quadratura circuli et hyperbolae, Padua 1667, 25-8. 1134

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that which hee there undertakes, viz. that the sector ABIP, is not (as he speakes) analytically composed of the Triangle ABP & the Trapezium ABFP. Or that the converging series there proposed, is not analytically composed; & therefore not capable of such a determination. But I do not yet find (though I have again sought for it) that Prop. 10, or 11, or anywhere else, hee doth offer any demonstration, that if ABIP be not analytically composed of ABP, & ABFP, it is not possible there can be any other converging series imagined of which it may bee so composed; or, that, in case such a converging series cannot be found, it is not possible that there can be any other way of squaring the circle by Analyticall operations. For though, it is like inough the thing may be true, (& therefore I do not reject it,) yet, I say, I do not see where hee doth offer to demonstrate it: And in his Epistle, pag. 5. he seems to mee to intimate so much; verum est me hanc demonstrationem integram ad phrasem geometricam non reduxisse. And therefore I did think it was not to be exacted of him, to have demonstrated what he did not undertake to demonstrate; which was a sufficient answere to Mr Hugens his exceptions1142. But, if it be any where done, I shall be willing inough to be shewed it. That is, the strength of this consequence demonstrated; If ABIP, be not analytically composed of ABP, & ABFP; then it is not possible there can be any other way, of giving a streight-lined figure equal thereunto by analyticall operations; viz. Addition, Subtraction, Multiplication, Division, & Extraction of Roots. This at present from Sir

yours &c. John Wallis.

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2 analytically add. 4 But (1) , that (2) I 6 that (1) what (2) if 7 analytically add. 8 imagined add. 18 to (1) see it (2) be 20 of add. 21 any (1) otherwise (2) other way 1142

exceptions: i.e. HUYGENS, 'Examen de Vera Circuli & Hyperboles Quadratura . . . ' , Journal des Scavans (2 July 1668), 52-6; HUYGENS, (Euvres completes VI, 228-30. 600

248. WALLIS to COLLINS, 10/[20] September 1668 These For Mr John Collins at Mr Pittes his house, at the White Hart, in Little Brittain. London.

5

248. WALLIS to JOHN COLLINS Oxford, 10/[20] September 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 201r-201v (our source). On right margin off. 201V at 90° to address in Collins's hand: 'Dr Wallis to Lalovera the 10th of September 1668'. Postmark on f. 201V: 'SE/11'.— printed: RlGAUD, Correspondence of Scientific Men II, 498-9. Enclosure: WALLIS-LALOUBERE c.lO/[20].IX.1668.

Oxford. Sept. 10. 1668. Sir,

The business of this is to desire you to do mee the favour to transcribe two or three propositions out of my papers1143, (which, I suppose, are with my Lord Brounker,) to send away with the inclosed letter1144 to Lalovera; they are those about the Cycloide, and are (as I remember) prop. 18. 20. & 21. de calculo centri gravitatis: (which in those papers is called the 5th chapter.) If I mistake the numbers you will however find them to be these. The first of them gives the measures & center of gravity of the Cycloides Plain & the parts thereof, (after which is another, of the Figure of Sines, Right & versed, or the solides thereof; which you may omit.) Then the next, of the Measures & Centers of gravity of the solides of the Cycloide & its parts. The last, of the Surfaces of those solids. These you may please to transcribe (carefully) with these Titles. De Calculo centri gravitatis Prop. 18. (& so of the rest.) They will contain 13 (which . . . chapter.) add. 1143

papers: possibly an earlier draft of Chapter 5 of Mechanica sive tractatus de motu. letter: WALLIS-LALOUBERE c.lO/[20].IX.1668.

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(I suppose) about two sheets of paper or more. & copy out the scheme which refers to them. (I think, one scheme serves them all.) Onely for the explication of the Symbols, if in the first of these three propositions they be not expressed, but onely with a general reference, Retentis symbolis ut in praecedentibus aliquot propositionibus; you must then there adde this explication, Nempe &c. and take them out of the former proposition. I think, they are these: Diameter Aa — 2R. Peripheria ABaA — P. Arcus BA = a. Ba = a. Chorda BA = c. Ba = x- Sinus rectus BV = s. versus AV = v. Va = 2R — v = y. VC = x. a + s = f . a — s = e. If there be any more, pray adde them. I shall send you by Moor the carrier, one of my bookes1145 de Cycloide to be sent with it; that being too big to come by the Post. If the thing be dispatched time inough, it may bee sent by the same hand that carryes my letter1146 to Mr Hugens, which perhaps my Lord Brounker may have shewed you. One Theodorus Riccius1147, who lyes at Mr Edward Roberts's house, near York-house; & takes in Postletters. This Riccius is very suddenly going directly to Paris, if not gone allready. My sending of these to Lalovera, is to let him see that wee had not our numbers from him, though possibly they may agree with his; which they must do if both bee right. I do not know whether my Lord Brounker have yet had time to examine those parts of my papers where these propositions are contained. If hee see reason not to send these papers; or not yet: I shall therein be guided by him. While I am writing this, I receive Anderson's book of Gaging1148. What the Geometry of it may bee, I cannot tell: But the language is barbarous inough, I see at the first view. Excuse this trouble given you

by

1 suppose) (1) two (2) about 7 Diameter . . . ABaA = P. add. 10 Moor add. 14-17 One . . . gone allready. add. 21 those (1) pap breaks off (2) parts 1145

one . . . bookes: i.e. WALLIS, Tractatus duo. letter: i.e. WALLIS-HUYGENS 31.VIII/[10.IX].1668. 1147 Riccius: i.e. Dirk de Rycke. 1148 Anderson's book of Gaging: i.e. R. ANDERSON, Stereometrical Propositions variously applicable, but particularly intended for Gageing, London 1668. 1146

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249. WALLIS to LALOUBERE, c.lO/[20] September 1668

Sir your loving friend John Wallis. I wrote1149 to you by the last post, directed to Mr Pits's house. [201V] These For Mr John Collins, next house to the three Crowns in Blooms-bury market. London.

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249. WALLIS to ANTOINE DE LALOUBERE c.lO/[20] September 1668 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-COLLINS 10/[20].IX.1668. Enclosure to: WALLIS-COLLINS 10/[20].IX.1668. The circumstances of this letter are described by Collins three years later in COLLINSVERNON 14/[24].XII. 1671 (RlGAUD, Correspondence of Scientific Men I, 176-9, 177): 'I strangely came to know that Lalovera had a book in print, intituled Geometria veterum promota in septem de Cycloide libris, una cum Appendicibus. After much sending, and long expectation, one was procured from France for Dr. Wallis, who, finding the author and himself wonderfully to agree, wrote a letter to Lalovera, and sent it to me to transmit; but it being rumoured he was dead, I never sent it, having it still by me. The Doctor hath a great esteem for Lalovera; and if you please I shall send you a copy of that letter'.

250. JOHN COLLINS to WALLIS 22 September/[2 October] 1668 Transmission:

Manuscript missing. 1149

wrote: i.e. WALLIS-COLLINS 8/[18].IX.1668.

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251. WALLIS to COLLINS, 26 September/[6 October] 1668 Existence and date: Mentioned in and answered by WALLIS-COLLINS 26.IX/[6.X].1668. In this letter, which enclosed a proofsheet of Wallis's Mechanica and of Gregory's Exercitationes geometricae, Collins apparently conveyed a request from Gregory that Wallis provide further support for him in the dispute with Huygens. Collins also sent news on Pell and requested more details of Wallis's criticism of Anderson's Stereometrical Propositions.

251. WALLIS to JOHN COLLINS Oxford, 26 September/[6 October] 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 202r-203v (our source).—printed: RlGAUD, Correspondence of Scientific Men II, 5005. Reply to: COLLINS-WALLIS 22.IX/[2.X].1668. Enclosure: A Methode of making a Table for the Contents of Segments of Spheares & Circles.

Sept. 26. 1668. Oxford.

Sir, When I opened yours1150 of Sept. 22. I thought I had had two new sheets1151; but when I considered them, they were none of what I ex5 pected; but one E (which I had seen before) &; one of Mr Gregories. I have (of my own) B, C, & E, as they be wrought off. But not D. I can hardly be of your opinion, that they (at this rate) have finished 50 sheets, before next Easter Term. I am glad to hear that Dr Pell promiseth better rules for the Loga1152 10 rithms than those of Mr Mercator. But, in all Tabular operations o this kind, wee must be content with a prope verum; for numbers will not admit of an accurate. Mr Gregories peece1153 that you send the first sheet1154 of, deals 1150

yours: i.e. CoLLiNS-WALLis 22.IX/[2.X].1668. two new sheets: i.e. the fourth proofsheet of Wallis's Mechanica and the first of Gregory's Exercitationes geometricae. 1152 Pell ... Logarithms: see PELL-COLLINS 6/[16].IX.1668. 1153 piece: i.e. GREGORY, Exercitationes geometricae, London 1668. 1154 first sheet: This contained the ' Appendicula ad veram circuli fe hyperbolae quadrat1151

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251. WALLIS to COLLINS, 26 September/[6 October] 1668 pretty roundly with Hugenius. I wish yet that he will consider whether there be not a mistake, when he intimates1155 that Hugenius challengeth that quadrature of the Hyperbole to have been his. I confess his words seem to speak roundly so much; but I took his meaning to have been, not to chalenge this quadrature of the Hyperbole to be his, (which is the same with that of the Circle & Ellipse;) to which certainly he can make no pretense, that I know of: But onely, that the connexion of the Hyperbola's quadrature, with the businesse of the Logarithms, was before known to him. Which may be true inough; for it hath been known likewise to a great many more, ever since the Book1156 of Gregory de Sancto Vincentio. And it was in print in my Commercium Epistolicum, Epist. 39, 40, (long before the Royal Assembly1157 was in being, & therefore before Mr Hugens can pretend to have discovered it to the Royal Assembly,) not as a thing then new, or first discovered by mee, but as a thing sufficiently known; & I quote St. Vincent for it. Nor is that it (I suppose) which Mr Gregory (or Mr Mercator) pretend to be new, in what they now publish. (For I suppose neither of them are ignorant, that this was formerly known; nor Monsr. Hugens neither.) And yet I think there is nothing else to which he can make any pretense: at lest as I did favourably understand his words. If he pretend to more of it, I doubt he is the more to blame. What he sayth1158 was by him made known to the Royal Assembly, about the 2 challengeth (1) his (2) that 6 he (1) hath (2) can 8 quadrature add. 10 the Book of add. uram'. See GREGORY, Exercitationes geometricae, 1-8; HUYGENS, (Euvres completes VI, 315-21. Cf. COLLINS-PELL 22.IX/[6.X].1668, British Library, MS Add. 4278, f. 343r: 'I have likewise sent you the first sheete of a |small add.\ Booke (1) printing (2) wrote by Mr Gregorie, the (a) rest I h breaks off (b) whole booke I hope to send you hereafter, the first sheete is against Hugens, the next will be a geometrick Demonstration, of Mercators quadrature'. 1155 intimates: i.e. GREGORY, Exercitationes geometricae, 1. 1156 Book: i.e. SAINT-VINCENT, Opus geometricum, Antwerp 1647. 1157 Royal Assembly: i.e. the Academic Royale des Sciences in Paris. Huygens refers to himself in the third person in his anonymous 'Examen de Vera Circuli & Hyperboles Quadratura': 'M. Hugens qu'on avoit prie d'examiner ce Livre fe d'en faire rapport a PAssemblee, dit qu'il avoit remarque plusieurs defauts dans la demonstration que cet Auteur pretend avoir donnee de 1'impossibilite de la Quadrature analytique du Cercle & de 1'Hyperbole'. The assembly is earlier referred to as TAssemblee qui se tient a la Bibliotheque du Roy'. See Journal des Scavans (2 July 1668 n.s.), 53. 1158 sayth: i.e. in Huygens's 'Examen de Vera Circuli fe Hyperboles Quadratura . . . ' ,

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weight of the Air, with respect to this of the Hyperbola's Quadrature; I know nothing of it, nor do I remember that I ever heard any word spoken [202V] to that purpose; & so can say nothing to it. 1159 There is no hast of sending to Lalovera, till wee hear whether 1160 hee be dead or alive . Yet I could be content I had one of his books de Cycloide1161 for my self, (for that I have, must be returned to Mr Oldenburg,) & of his Tetragonismica1162, (for I have neither of them,) if they be to be had. I do not find but his method (for the main) may be sound, (but perplex inough, & his figures not clear, nor so fitted as to be understood without difficulty; and in some of his calculations there be some mistakes:) But, I beleeve, I shal hardly take the payns to examine his methods so particularly as to make myself master of them; because I think I have methods of my own much more clear & easy: & I beleeve you wil think so, by that time what you have of mine1163 is printed. As for my peeces1164; if those who are concerned in the copies of them will print all or any of them single, I am not averse from it: but I cannot well do it without them. What you ask of the mistakes of Anderson1165; I confesse I did not so narrowly examine it as to note the mistakes as I went; reading it very

1 Air, (1) to be measured by the quadrat breaks off (2) with. 9 his add. 10 difficulty; (1) ) (2) though (3) and 12 as to (1) be Mast breaks off (2) make 18 not add. Journal des Sgavans (2 July 1668 n.s.), 55-6: 'II ajouta que ce qui est dit dans ce Livre touchant la dimension de 1'Hyperbole & le rapport qu'elle a avec les Logaritmes, est fort bien; mais que Messieurs de 1'Assemblee ne le trouveroient pas nouveau, puis qu'ils pouvoient se souvenir qu'il leur a deja propose la mesme chose . . . de 1'Hyperbole.' 1159 sending: i.e. WALLis-LALOUBERE c.lO/[20].IX.1668, enclosed in WALLIS-COLLINS 10/[20].IX.1668. 1160 dead or alive: Laloubere had in fact died on 2 September 1664 (new style). 1161 de Cycloide: i.e. LALOUBERE, De cycloide Galilaei et Toricellii propositiones viginti, Toulouse 1658. 1162 Tetragonismica: i.e. LALOUBERE, Quadratura circuli et hyperbolae segmentorum, Toulouse 1651. 1163 what . . . mine: i.e. the manuscript of Wallis's Mechanica. 1164 my peeces: presumably the tracts 'de sectionibus Angularibus, Algebra, and some Miscellanies', referred to in COLLINS-PELL 23.X/[2.XI].1668. 1165 Anderson: i.e. R. ANDERSON, Stereometrical Propositions. Cf. WALLIS-COLLINS 10/[20].IX.1668. 606

251. WALLIS to COLLINS, 26 September/[6 October] 1668 cursorily over, (at once before I layd it by;) & though in the reading I took notice of some, yet I did neither charge my memory with them, nor write them down. That of the Parallelogram to the Parabola as 4 to 3; (which should have been as 3 to 2;) I remember (now you mention it) that I did take notice of: And of that in the beginning of pag. 69, which is true in parabola's (meaning it, as I suppose he doth, of a segment cut off by a streight line,) but not of the Hyperbola's or Ellipses. Some cases there are in which it may be true of these; but not universally. He might as well have sayd, that Segments of Circles (for a Circle is but one species of an Ellipse, as a square is of a parallelogram,) if on the same base, are as their altitudes; & if of the same altitude, they are as their bases; &, consequently, Segments of Circles (how unlike soever) in proportion compounded of that of their bases, & that of their altitude. But it is much otherwise. As to what Mr Gregory desires; I know not well what to say more than I have done. You know what I have sayd1166 to you: &, I suppose, you saw what I say1167 to Hugenius himself. That, at the 11th Proposition1168 (against which he excepts) as much is demonstrated as is affirmed: & that more was not there to be exspected, &c. I cannot say that it was there proved (for it is not so much as affirmed) that there can be no other way of squaring the circle analytically. Nor, when Mr Hugens makes this [203r] objection,! '1S there any supplement to that purpose in the Reply. The paper1169 which he sent mee, how cogent so ever, is no part of what is either in the Book1170 or in the Reply1171. To the truth of the Proposition, I have

1 over, (1) (be breaks off (2) at 3 of the (1) Parabola (2) Parallelogram 4 (now you mention it) add. 6 in |the del. parabola's 8 universally. (1) And therefore he should either (2) He 9 one add. 12 in |the del.\ proportion 1166

what I have sayd: i.e. WALLis-COLLiNS 8/[18].IX.1668. what I say: i.e. WALLIS-HUYGENS 31.VIII/[10.IX].1668. 1168-Qth pr0pOSition: i.e. GREGORY, Vera Circuli et Hyperbolae Quadrature*,, prop. 11. 1169 paper: this paper by Gregory on the quadrature of the circle, which he sent to Wallis, has not been identified. 1170 Book: i.e. GREGORY, Vera Circuli et Hyperbolae Quadratura. 1171 Reply: i.e. GREGORY, 'Mr. Gregories Answer To the Animadversions of Monsieur Hugenius upon his Book'. 1167

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251. WALLIS to COLLINS, 26 September/[6 October] 1668

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made no exception: the strength of both the Arguments in the last paper, I had considered before he sent it. For I apprehended the substance of it suggested to my thoughts from what is sayd1172 in his preface, pag. 5. Sicut numeri fracti nunquam precedent ex integrorum Additione &c, sed tanquam ex divisione; et numeri inconimensurabiles &c. tantum ex radicum extractione; ita quantitates non analyticas nunquam ex analyticarum additione, subductione, multiplicatione, divisione, radicum extractione, sed ex sexta hac operatione: &c. Whence I easyly foresaw it might be argued, in like manner as, quorum alterum est, alterum non est, cum eodem tertio commensurabile, ea non sunt commensurabilia inter se: so also, quorum alterum est, alterum non est, cum eodem tertio synanalyticum, ea nee ita sunt inter se. Which is the foundation of both his arguments; & may be of more. This because it seemed to mee a reasonable postulatum, so as that I saw not any thing solidly to be objected against; & the consequences from thence such as I knew not how to avoyd: I did (without his deducing the demonstration) say that I saw nothing to the contrary but that the proposition might be true. And of the same mind I am like to be, till either I do discover myself, or somebody else shews mee, somewhat to the contrary that as yet I see not. If to what he is now printing1173, hee think fit to add the contents of his last paper: and if with all hee adde that the strength of it was allready in his preface, in the place to which I refer, (which is, I think, all that is in the whole book to that purpose;) it will bee a good additionall to his former reply. And if Hugenius rest satisfyed with making any further reply; I think Mr Gregory hath done his work, & hath the better end of the staff. After this is done, I think it will not be necessary for others to appear as seconds in the business, (unlesse upon further provocation,) because it may occasion further heats & animosities between the two societies, (that of ours & that of theirs,) than the business doth deserve; & seeking occasions of revenge & disparaging

2 I had (1) before (2) considered 6 non add. 8 argued, (1) as (2) in 26 for (1) the (2) others 26 business, (1) which will (2) (unlesse 27 it (1) will (2) may 1172

sayd: see GREGORY, Ve.ro, Circuli et Hyperbolae Quadrature,, preface. The passage cited is in fact on page 3. 1173 now printing: i.e. GREGORY, Exercitationes geometricae. 608

251. WALLIS to COLLINS, 26 September/[6 October] 1668 [203V] each other: which may prove of ill consequence. | What I have said to M. Hugens myself, may possibly have as good an influence on him, as if I should more reproach him in publike. Yet, if my Lord Brouncker think fit (who in the first testimony is as much or more concerned as I,) that something briefly to this purpose be inserted; That the persons concerned in the former testimony of that booke, notwithstanding the objections1174 made by Monsr. Hugens (to which Mr Gregory hath made his reply) do not see any cause to receed from their former favourable opinion of it: I shall not be averse from it. But I think there will be no need of it, & perhaps the issue may bee as well without it.

About the segments of Circles: it's true I give no other measure of them, then by the versed sine, the right sine, & the arch, & semidiameter. You say that you take nothing as known (besides the Radius, for so I understand you) but the versed sine. But this being given, the Right sine is known allso (s2 — 2vR — t> 2 ,) & then the Arch by the Tables of Sines. But if you would have a Table computed for segments of Spheres & Circles, according to the several proportions of the versed sine: I know of no more expedient way at present then this. Supposing the Radius CA divided into any number of equal parts; suppose, 10, 100, 1000, 10000, or as many as you please, CV, VV, &c. & consequently CV, CV, CV, 12 by the (1) sine (2) versed 13 Radius, (1) ) (2) for 14 being ( 1 ) known (2) given 18 no add. 117Objections: i.e. HUYGENS, 'Examen de Vera Circuli fe Hyperboles Quadratura'. 609

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252. WALLIS to COLLINS, 26 September/[6 October] 1668, enclosure

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&c, are 1, 2, 3, &c of such parts: & therefore squares of VP, VP, VP, &c, (which represent the Circles of the Hemispher,) are R2 — 1, R2 — 4, -R2 — 9, &c. And the aggregate of all them (multiplied into 1 of those equal parts of the radius,) or so many of them as you shall need; represent the whole hemisphere, or such a part thereof as you need, (that is, they are thereunto, as the square of Radius to the circle.) And the square rootes of them, ^ : R2 - 1. ^ : R2 - 4. ^ : R2 - 9 : &c are the lines VP, & the aggregate of them, or so many of them as you need (multiplied by the altitude of one such part) give you the area of the whole quadrant, or such a part as you need. The Table is easyly calculated; wanting, for each place, but one extraction of the square root, & one addition of two numbers, after a subduction of the successive squares out of the square of Radius. And the result of the work will be either greater or lesse then the just, according as you take the inscribed or circumscribed figure: & the midle between both will be truer then either. I adde here a specimen1175.

252.

WALLIS to JOHN COLLINS Oxford, 26 September/[6 October] 1668, enclosure: A Methode of making a Table for the Contents of Segments of Spheares & Circles Transmission: W Draft of paper sent: OXFORD Bodleian Library MS. Don. d. 45, f. 141r-141v. Enclosure to: WALLis-CoLLiNS 26.IX/[6.X].1668.

2 R - I , R - 4, R - 9, corr. ed. 4 need; (1) axe (2) represent 7 lines (1) of such (2) VP 12 of the square of add. 14 take the (1) great breaks off (2) inscribed 1175

specimen: i.e. the enclosed 'Methode of making a Table for the Contents of Segments of Spheares fe Circles'. 610

252. WALLIS to COLLINS, 26 September/[6 October] 1668, enclosure A Methode of making a Table for the Contents of Segments of Spheares & Circles; supposing Radius or altitude of the Hemisphere, or Semicircle divided into equal parts. Sent to Mr Jo: Collins in a letter of Sept. 26. 1668.

Quoniam in circuli Quadrante ACQ, Ordinatim applicata VP — V7 : CPq — CVq. Si dividatur Radius CA in partes aequales quotlibet in punctis V, adeoque omnes CV, CV, &c, arithmetice proportionales, puta a, 2a, 3a, &c. Erunt omnia quadrata VP, ut R2 — a2, R2 — 4a2, _R2 — 9a2, &c. (quibus proportionales sunt circuli, seu Cylindruli, hemisphaerium complentes;) ipsaeque rectae VP (complentes Quadrantem) ^/ : R2 — a2. y : j R 2 - 4 a 2 . ^/:R2-9a2. &c. Intelligatur Radius (7A = -R = 1. dividi in partes, verbi gratia, 10000; quarum quaelibet sit a = ^onoo = 0-0001. Calculus sic inchoabitur.

7 VP = CPq - CVq corr. ed. 9 V, (1) erunt (2) adeoque U-R-l. add.

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252. WALLIS to COLLINS, 26 September/[6 October] 1668, enclosure

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Aggregata pro Hemisphaerio. 1 1.0000,0000 1 2 1.9999,9999 2 3 2.9999,9995 3 4 3.9999,9986 4 5 4.9999,9970 5 6 5.9999,9945 6 7 6.9999,9909 7 8 7.9999,9860 8 9 8.9999,9796 9 10 9.9999,9715 10 11 10.9999,9615 &c

R2 - a2. &c 1.0000,0000 0.9999,9999 0.9999,9996 0.9999,9991 0.9999,9984 0.9999,9975 0.9999,9964 0.9999,9951 0.9999,9936 0.9999,9919 0.9999,9900 fee.

V:R2- a2. &c 1.0000,0000,0 0.9999,9999,5 0.9999,9998,0 0.9999,9995,5 0.9999,9992,0 0.9999,9987,5 0.9999,9982,0 0.9999,9975,5 0.9999,9968,0 0.9999,9959,5 0.9999,9950,0 &c.

Aggregata pro Circuli quadrante 1.0000,0000,0 1.9999,9999,5 2.9999,9997,5 3.9999,9993,0 4.9999,9985,0 5.9999,9972,5 6.9999,9954,5 7.9999,9930,0 8.9999,9898,0 9.9999,9857,5 10.9999,9807,5 &c

Calculus sic inchoatus facile continuatur. Quippe Columna secunda (indicibus marginalibus non computatis) continuatur continua detractione numerorum 1, 3, 5, 7, 9, 11, 13, 15, &c. Columna prima, continua additione numeri in secunda proxime reperti. Columna tertia sunt numerorum in secunda radices quadraticae. Et columna quarta continuatur continua additione numeri in tertia proxime reperti. Commodum autem erit radicis extractionem per locum unum aut alterum continuare ultra quam in Tabula illius usus erit, ne propter multas continuas additiones faciendas ultima nota numeri in Tabella non sit vera. Ubi autem, in his, operationem ad octavum vel nonum locum partium decimalium protraxi; id non opus erit, sed multo pauciores sufficient. Praesertim quoniam quodlibet aggregatorum| ducendum erit in a, altitudinem unius partis; hoc est, [14F] (propter a = 0.0001,) detrudendum erit adhuc per quatuor loca. Sic, verbi gratia, si sumatur CV — 9a: Aggregatum pro segmento Hemisphaerii erit 8.9999,9796; nempe si sumatur figura ex cylindrulis circumscripta, pro ipso segmento: Vel, si inscripta sumatur, sumendum erit aggregatum proxime inferius, sed detracto primo, hoc est 1. integro: nempe 8.9999,9715: Vel, quod adhuc vero propius erit, his intermedium; (quod figuram exhibeat partim inscriptam partim circumscriptam;) puta 8.9999,9756. Hi autem numeri, in a = 0.0001 ducti, erunt, pro 19 quarta add. 22 multas add. 23 numeri (1) Ta breaks off (2) in Tabella non sit (a) ipse debitus. (6) vera. 29 figura (1) ex parallelogrammis (2) ex cylindrulis

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252. WALLIS to COLLINS, 26 September/[6 October] 1668, enclosure

{

circumscripta, 0.0008,9999,9796 intermedia, 0.0008,9999,9756 inscripta, 0.0008,9999,9715 Nempe tales partes decimales, quibus respondens 1. integer, valeat Cylindrum hemisphaerio circumscriptum. figura

Atque eodem modo, pro segmento quadrantis Circuli; habebitur pro 4 Nempe (1) istiusmodi (2) tales

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253. GREGORY to WALLIS, September/October 1668 ( circumscripta, 0.0008,9999,9898,0 figura ex parallelogrammis < intermedia, 0.0008,9999,9877,8 [ inscripta, 0.0008,9999,9857,5 Nempe tales partes decimales, quibus respondens 1. integer, sit quadraturn Radii. 5 Suntque hi numeri sic detrusi, (vel illorum praesertim mediales si ad tot decimalium locos libet procedere ut inde aliquid discriminis oriatur,) illi ipsi sunt qui in Tabellam construandam sunt in gerendi. Ex Tabella vero sic constructa facile erit pro arbitrio deprimere magnitudinem cujusvis segmenti (per altitudinem suam designati) sive quod 10 Centre sive quod Vertici adjacet, vel etiam Zonae cujusvis intermediae. Quippe si in ipsa Tabella habeatur segmentorum alterum, hujus residuum ad integrum sive Hemisphaerium sive circuli quadrantem, erit reliquum; et duorum segmentorum differentia, erit intermedia Zona. Interim monendum erit, Tabellam pro segmentis hemisphaerii mi15 nus esse necessarian!, quoniam id sine ope Tabellae haberi per se poterit quocunque casu proposito. Si enim ex solido quod fit ex CV in quadratum Radii ducto, auferatur Triens Cubi ex CV, habetur correspondens aggregatum. Puta, posito: R = 1. et CV = 0.0009: erit factum 2 0 ex CV in quadratum Radii, 0.0009; unde si auferatur triens cubi ex CV, 0.0000,0000,0243: manebit 0.0008,9999,9757: qui est praecise accuratus numerus isti Tabellae loco debitus. A quo numerus intermedius nonnisi unitate in ultimo loco differt.

253.

JAMES GREGORY to WALLIS September/October 1668 Transmission:

Manuscript missing. Existence and date: Referred to in GREGORY-WALLIS 26.X/[5.XI].1668. 2 ex parallelogrammis add. 4 Nempe (1) istiusmodi (2) tales 6 detrusi, (1) ill breaks off (2) (vel 7 procedere ut (1) id (2) inde 13 sive (1) Q breaks off (2) circuli 22 debitus. (1) atque a (2) A

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254. OLDENBURG to WALLIS, September/early October 1668 As Gregory reports in GREGORY-WALLIS 26.X/[5.XI].1668, he sought in a postscript to this letter to remove obscurities in repect of proposition 11 of his Vera circuli et hyperbolae quadrature Gregory had probably initiated the discussion in August/September after Wallis had suggested that it was Gregory's intention only to assert that the circle would not be analytically squared. See WALLis-BROUNCKER 4/[14].XL1668. Wallis probably replied to the present letter by WALLIS-GREGORY 22.X/[1.XI].1668.

254. HENRY OLDENBURG to WALLIS September/early October 1668 Transmission:

Manuscript missing. Existence and date: Referred to, as one of several missing letters, in and answered by WALLIS-OLDENBURG 6/[16].X.1668. Oldenburg in this letter (or perhaps other missing letters written at the time) sent Wallis recommendations for a number of persons visiting Oxford. He also apparently sought Wallis's opinion on Laloubere's Veterum geometria promota, in septem de cycloide libris, which he had earlier sent to the Savilian professor.

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BIOGRAPHIES OF CORRESPONDENTS

Blandford, Walter (1619-75). — Born at Melbury Abbas, Dorsetshire, son of Walter Blandford. Matriculated at Christ Church, Oxford, in 1635. In 1638 admitted as scholar to Wadham College. B.A. 1639. Fellow of Wadham College 1644. In 1659 elected warden of Wadham. Created D.D. 1660. About this time chaplain to Edward Hyde, the later earl of Clarendon. Through Hyde obtained a prebend in Gloucester and chaplainship to the king. Vice-chancellor of the University of Oxford 1662-4. Consecrated bishop of Oxford in 1665 and nominated dean of the Chapel Royal shortly afterwards. In 1671 translated to the see of Worcester, where he died in July 1675. Boyle, Robert (1627-91). — Natural philosopher and chemist. Independent scholar. Came from a wealthy Irish family. In 1645 settled in the manor at Stalbridge, Dorsetshire. Established close contacts with the pansophic circle around Samuel Hartlib. Moved to Oxford and joined the group around John Wilkins in Wadham College in 1654. Leading representative of experimental philosophy and active member of the Royal Society. Extensive correspondence, particularly with Henry Oldenburg. From 1668 onwards lived with his sister Katherine, Lady Ranelagh, in London. Numerous works, including Sceptical Chymist (1661), and New Experiments Physico-Mechanicall, Touching the Spring of the Air, and its Effects (1660). Died in London in 1691. Buried in St Martin's-in-the-fields. Bramston, John (1611-1700). — Born at Whitechapel, Middlesex, son of Sir John Bramston, later chief justice of the King's bench. Matriculated at Wadham College, Oxford 1627. Called to the bar, Middle Temple 1635. Associate of Edward Hyde, the later earl of Clarendon. Legal practice in London. Elected to the Long Parliament (1640-60) as member for Bodmin. Following death of his wife, moved to the family house in Skreens. As knight of the shire of Essex, supported the move to Restoration after the resignation of Richard Cromwell. Created knight of the bath at the coronation. Member of parliament for Essex 1660 and 1661-79. Elected to parliament for Maldon in 1679 and again in 1685-7. 611

Biographies of correspondents Brancker (or Branker), Thomas (1633-76). — Mathematician. Born in Barnstaple, Devon, son of the schoolmaster Thomas Brancker. Matriculated at Exeter College, Oxford, in 1652. B.A. 1655. Fellow of Exeter College 1655. M.A. 1658. Received instruction in mathematics and chemistry from Peter Staehl of Strasbourg, a private tutor in Oxford. Expelled from Exeter College for refusal to comply with the Act of Uniformity 1662. Given the rectory of Tilston, Cheshire by William Brereton in 1668. Resigned the benefice after a month to become headmaster of the grammar school at Macclesfield. Published a number of books including Doctrinae sphaericae adumbratio una cum usu globorum artificialium (1662) and a translation of Rahn's Teutsche Algebra, to which Pell added notes and Collins gave assistance. The table of incomposite numbers which Brancker added to the translation was partly corrected by Wallis. Brereton, William (1631-80). — Nobleman and natural philosopher. Born in Brereton, Cheshire. Resident in the Low Countries during the Civil Wars. Educated at Breda by John Pell, to whom he later became a close friend. Associate of Hartlib and Boyle. Married Frances Willoughby 1659. Created third baron Brereton of Leighlin in 1664. Founder member of the Royal Society, in which he was particularly active in the 1660s. Thereafter apparently spent much of his time on his estate in Stonebagge, Cheshire. Brouncker, William (16207-84). — Second Viscount Brouncker of Castle Lyons. Irish mathematician. Studied at Oxford. M.D. 1646/7. Settled in London. First president of the Royal Society after its incorporation by Royal Charter in 1662 (deposed 1677). Published translation of Descartes's Musicae compendium into English (1653). His most important mathematical work carried out during the Commonwealth and the Protectorate; especially intensive correspondence with Wallis in connection with Fermat's challenges on number theory. In 1662 appointed chancellor to Queen Catherine and made keeper of the great seal. President of Gresham College 1664-7. Commissioner for the navy 1664-79, 1681-4. Controller of the navy accounts 1668-79. Master of St Catherines's hospital, London, 1681-4. Died unmarried. Browne, Richard (1605-83). — Son of Christopher Browne of Sayes Court, Deptford. Matriculated at Christ Church, Oxford 1623. B.A. at St Alban Hall 1623. Fellow of Merton College 1624. M.A. 1628. Student of Gray's Inn 1627. Clerk of council to Charles I. The king's resident at court of France 1641-60. Clerk to privy council to Charles II. Knighted 618

Biographies of correspondents 1649. Father-in-law to John Evelyn. Retired to Charlton, Kent 1671/2. Died at Sayes Court. Burscough, John (c.l629-c.!707). — Born in Cheshire. Entered Jesus College, Cambridge 1645. B.A. 1648/9. Incorporated at Oxford 1649. Fellow of Brasenose College (and also of New College) 1649. M.A. 1651. Vice-principal of Brasenose 1657. B.D. 1658. Rector of Stoke by Guildford 1662-1707. Carcavi, Pierre de (c.1600-84). — Born in Lyon. In 1636 left for Paris, having paid for the office of member of the Grand conse.il. In 1645 participated in the dispute surrounding Longomontanus's supposed quadrature of the circle, sending a refutation to John Pell. From 1663 onwards, librarian of the Royal Library in Paris. Friend of Huygens, Pascal, and Fermat. Conducted extensive scientific correspondence. Charles II (1630-85). — Son of Charles I and Henrietta Maria. Created Prince of Wales 1639. Appointed commander of the king's western forces in the Civil Wars 1644. Fled to the Isles of Scilly 1646. Assumed title of king of England following the execution of his father in January 1649. In February 1649 proclaimed king of Scotland (crowned January 1651). Led Scottish forces to defeat at Worcester in September 1651. Fled abroad again, taking up court in exile in France, Germany, and the Netherlands (1651-60). Invited to return to England after intervention by General Monck. Signed declaration of Breda in April 1660. Entered London in May 1660 (crowned king on 23 April 1661). Married Catherine of Braganza in May 1662. Granted first charter to Royal Society in July 1662. Under influence of his lord chancellor, Edward Hyde, first earl of Clarendon (after 1661) pursued war policy against the Dutch (1665-7). Subsequently under the influence of the Cabal ministry (1667-73), abandoned the Triple Alliance and signed Treaty of Dover (1670). Forced to accept the Test Act 1673. Sanctioned marriage of his niece, Princess Mary, to William of Orange (1677). Sanctioned parliamentary Test Act 1678. Died at Whitehall in February 1685, having become a Roman Catholic on his deathbed. Chyliriski, Samuel Boguslav (c.1634—68). — Lithuanian scholar and translator. Matriculated at the University of Franeker in the Low Countries, where he studied theology 1653-7. No evidence of his having taken a degree. On account of Russo-Polish wars and occupation of Lithuania, went to England in 1657. Worked on the translation of the Bible into 619

Biographies of correspondents Lithuanian, for which he received support from Boyle and Wallis. In 1661 he took up residence in London. Relieved of his mandate to complete the translation in 1663. Died in 1668. Clendon (or Clendan), John (fl. 1653-68). Nephew ('cousin') of John Wallis. A chandler or lamp maker with premises in the Strand in London. Collins, John (1625—83). — Born at Wood Eaton, Oxfordshire, son of a non-conformist minister. Around 1638 apprenticed to the bookseller Thomas Allen in Oxford. Employed later by the clerk of the kitchen to the Prince of Wales. During 1642-9 served on board of an English merchantman engaged by the Venetian republic. Devoted leisure time to the study of mathematics and merchants' accounts. On leaving the service, worked as a mathematics teacher in London. After the Restoration, appointed successively as accountant to the excise office, accountant in chancery, and secretary to the council of plantations. Exchanged latter for post of manager of the farthing office in 1672. Not long afterwards, he became accountant to the Royal Fishery Company and remained in this post until his death. Elected fellow of the Royal Society in October, 1667. Often advised Oldenburg on mathematical topics. Alongside numerous mathematical publications of his own, he assisted in seeing the works of others through the press, including Brancker's translation of Rahn's Teutsche Algebra and Wallis's A Treatise of Algebra. As a prolific scientific intelligencer, whose correspondents included Barrow, Newton, Leibniz, Wallis, Flamsteed, and Sluse, often styled as the 'English Mersenne'. Died in October 1683 at his lodging on Garlick Hill, London. Cooper, Benjamin (c.1623-1701). — Son of William Cooper of Hallam, Nottinghamshire. Matriculated Merton College, Oxford 1641. B.A. 1647. M.A. 1648. Registrar of the University of Oxford 1659-1701. Digby, Kenelm (1603-65). — Son of Sir Everard Digby, a convert to Roman Catholicism, executed for participating in the Gunpowder Plot in 1606. Brought up as a Catholic. In 1618, entered Gloucester Hall, Oxford, but left in 1620 without taking a degree. Knighted by James I in 1623. After the death of his wife in 1633, settled in France. Meetings with Hobbes, Mersenne, and Descartes. Returned to England in 1639, but was repeatedly forced to flee the country. In 1657-8 served as intermediary in France between Brouncker, Wallis, Frenicle, and Fermat in connection 620

Biographies of correspondents with the latter's challenges on number theory. Wallis's Commercium epistolicum de quaestionibus quibusdam mathematicis nuper habitum (1658), documenting the exchanges, is dedicated to Digby. Active member of the Royal Society in the early years. Numerous works including Two Treatises (1644). Noted collector of books and manuscripts, many of which he presented to the Bodleian Library. Dillingham, Theophilus (1613-78). — Born at Upper Dean, Bedfordshire, son of Thomas Dillingham. Entered Emmanuel College, Cambridge 1629. B.A. 1633/4. M.A. 1637. Elected fellow of Sidney College, Cambridge 1638. B.D. 1644. Master of Clare Hall, 1654-60, 1661-78. D.D. 1655. Vice-chancellor of the University of Cambridge, 1655-6, 1656-7, 1657-8, 1661 (partly). At the Restoration ejected from the mastership, but re-elected by the fellows in 1661. In 1661/2 became prebendary of Ulleskelf, York. In 1667 installed archdeacon of Bedford. Died in Cambridge in November 1678. Dulaurens, Frangois (d. c.1675). — Probably originated from Montpellier. Evidently met Oldenburg, with whom he later corresponded, in Paris. Author of Solutiones aliquot quaestionum (1663) and Specimina mathematica duobus libris comprehensa (1667). Fell, John (1625-86). — Born in Longworth, Berkshire, son of the dean of Christ Church, Samuel Fell. Entered Christ Church 1637. B.A. 1640. M.A. 1643. Created D.D. 1660. Appointed canon of Christ Church in July 1660. From the end of November 1660 until 1686 dean of Christ Church. Chaplain to the king. Master of St Oswald's Hospital, Gloucester. Vicechancellor of the University of Oxford 1666-9. Bishop of Oxford 1675/686. Under his auspices extensive building work at Christ Church and the construction of the Sheldonian Theatre. Gave strong support to the University press. Numerous publications, including Grammatica rationis sive institutiones logicae (1673) and a critical edition of the works of Cyprian (1682). Frenicle de Bessy, Bernard (1605-75). — Accomplished mathematical amateur in Paris. Visited England in 1634. Councillor at the Cours de Monnaies. In 1666 appointed member of the Academic Royale des Sciences. Corresponded with the most important mathematicians of his day, including Descartes, Fermat, Huygens, and Wallis. Shared interest in problems of number theory with Fermat. Also made important contributions to the problem of the construction of magic squares. 621

Biographies of correspondents Gregory, James (1638—75). — Born at Drumoak near Aberdeen, son of the local minister John Gregory. Studied at Marischal College, Aberdeen. In 1663 published Optica promota, a description of the reflecting telescope he had invented in 1661. During 1664-7 pursued mathematical studies at Padua, where his Vera circuli et hyperbolae quadratura (1667) was printed. The same work was reprinted in Padua in 1668, appended by Geometriae pars universalis. Elected fellow of the Royal Society in June 1668, following his return to England. Published his Exercitationes geometricae in London in 1668. Later the same year appointed professor of mathematics in the University of St Andrews. Married Mary Jameson, daughter of the painter George Jameson, in 1669. Elected professor of mathematics in the University of Edinburgh in 1674. Hevelius (or Hewelcke), Johannes (1611—87). — Astronomer in Danzig. Educated in Poland. Between 1630 and 1634 travelled to several European countries. Studied jurisprudence in Leiden. Visited London and and spent about two years in Paris, where he met Gassendi and Boulliau. Built his own observatory in Danzig and equipped it with instruments, including telescopes, many of which he designed and engraved himself. Assisted in his observations by his wife. Published several important astronomical works. The Selenographia (1647) contains a lunar atlas, descriptions of instruments and details of his observations. Conducted an extensive correspondence with the leading scientific figures of his day. Hobbes, Thomas (1588—79). — Born at Malmesbury, Wiltshire, son of Thomas Hobbes, vicar of Charlton and Westport. Entered Magdalen Hall, Oxford, 1602/3. B.A. 1607/8. Tutor to several members of the Cavendish family, including William (15917-1628), the second earl of Devonshire, and William (1617-84), the third earl of Devonshire. During 1629-31 mainly in Paris. Received in the circles of Mersenne and Descartes. Meetings with Galileo around 1636. After returning briefly to England, fled to France 1640. Taught elements of mathematics to the Prince of Wales in exile in Paris 1646-8. Uproar following publication of Leviathan (1651), led him to return to England. In 1653 rejoined the Cavendish household and remained part of it, first in London and then in Derbyshire until his death. Published De corpore in 1655. Attempts to produce solutions to ancient mathematical problems in the wake of the Webster-Ward debate led to long drawn out war with Wallis. Hopkins, William (c.1641—81). — Attorney and coroner to the University of Oxford. Steward to Oriel, Brasenose, and Lincoln College. Buried 622

Biographies of correspondents at St Mary's church. Huygens, Christiaan (1629—95). — Dutch mathematician and scientist, son of the diplomat, poet, and Latin scholar Constantijn Huygens (1596-1687). Born in The Hague, educated largely by his father and private tutors. Tuition in mathematics by Stampioen 1644. Entered University of Leiden in 1645 to study mathematics and law. Private pupil of Frans van Schooten. 1647-9 studied law at Collegium Auriacum in Breda. In 1655 bought doctorate of law in Angers. From 1654 onwards worked on lenses, constructed microscopes and telescopes. Invented pendulum clock 1655. Carried out numerous astronomical investigations, especially on the rings of Saturn. In 1659, published Systerna Saturnium. Obtained important results concerning curves and surfaces (tangents, quadratures, cubatures, rectifications). Applied theoretical mathematics to problems in physics, astronomy and technology. Numerous stays in Paris in 1650s and 1660s, as well as visits to England. Elected fellow of the Royal Society 1663. From 1666 to 1681 almost exclusively in Paris. Salaried member of Academie royale des sciences. Published Horologium oscillatorium in 1673. Final years spent in The Hague. Huygens, Constantijn (1596-87). — Dutch statesman and poet. Born in The Hague, son of Christiaan Huygens, the elder. Early instruction in Latin by Johannes Dedelus. Entered the University of Leiden in 1616. Travelled to England in 1618, accompanying the English ambassador. In 1619 sent on diplomatic service to Venice. Further diplomatic appointments in England in 1621, first for three months and then from December to March 1623. In autumn 1622 knighted by James II. In June 1625 appointed private secretary to the Stadholder of the Netherlands. Married Susanna van Baerle in 1627. Father of Christiaan Huygens (1629-95). In 1630 made privy councillor to the Stadholder, serving Frederik Hendrik, Willem II, and Willem III. Died in March 1687. Buried in St Jacob's Church, The Hague. Jenkins, Llewelyn (or Leoline) (1623-85). — Born in Llantrisant, Glamorganshire, son of Llewelyn Jenkins. Entered Jesus College, Oxford in 1641. During the Civil Wars served in the royalist army in Wales. In 1648 became tutor to the son of Sir John Aubrey of Llantrithyd, Glamorganshire. In 1651 returned to Oxford and for the following four years gave private tuition in a house in the High Street. In 1655-8 retired to the continent, travelling in France, the Low Countries, and Germany. On his return to England, resided for a time in the house of William Whit623

Biographies of correspondents more in Apley, Shropshire. Returned again to Oxford at the Restoration. Elected fellow of Jesus College. Created LL.D. 1660/1. Principal of Jesus College, 1660/1-73. In 1662 made deputy professor of civil law in the University and appointed assessor to the chancellor's court. In November 1664 entered the College of Advocates. Judge of the high court of the Admiralty, 1668-73. Judge of the prerogative court of Canterbury 1668/985. Knighted on returning from diplomatic mission to France 1669/70. Appointed manager of the university press in Oxford 1672. Member of parliament for Hythe 1672/3-8. English representative at the congress of Cologne (1673) and at the congress of Nijmegen (1676-9). Member of parliament for the University of Oxford 1679-85. Secretary of state (northern department) 1680-1. Secretary of state (southern department) 1681-4. Died unmarried in Hammersmith in September 1685. Buried in the chapel of Jesus College. Laloubere (Lalovera), Antoine de (1600-64). — Born in the diocese of Rieux, Languedoc. Admitted S.J. 1620. Professor in the Jesuit college in Toulouse. Submitted a solution to Pascal's prize questions on the cycloid in 1658. Publications include Quadratura circuli et hyperbolae segmentorum (1651), De cycloide Galilaei et Torricellii propositiones viginti (1658), and Veterum geometria promota in septem de cycloide libris (1660). Died in Toulouse in September 1664. Leotaud, Vincent (1595-1672). — Born in La Vallouise. Admitted S.J. on completion of his studies. Taught mathematics for fourteen years at the Jesuit college in Dole, Burgundy. Thereafter sent to the college at Lyons. Publications include Examen circuli quadraturae (1654), a refutation of Gregoire de St Vincent's work on the quadratur of the circle, and Cyclomathia sen multiplex circuli contemplatio (1663), in which he responds to certain disciples of St Vincent. Towards the end of his life he retired to the Jesuit house at Embrun, where he died in 1672. Moray (or Murray), Robert (1608-73). — Son of Sir Mungo Moray of Craigie, Perthshire. Educated at the University of St Andrews and in France. In 1633 served in the Scottish regiment which joined the French army. Returned to Scotland at the beginning of the Civil Wars to fight on the Royalist side. In 1641 admitted to the lodge of Edinburgh. 1643 knighted by Charles I. Returned to the Continent to serve under French command. Captured and imprisoned in Bavaria in 1643 (released on payment of ransom in 1645). Spent Interregnum partly in Scotland, partly on the Continent (especially in Paris and Maastricht). Devoted much of 624

Biographies of correspondents his time in exile to chemical investigations. At the Restoration returned to London, residing at court in Whitehall. Played a decisive role in the creation of the Royal Society 1660 (president at most meetings 1661-2). In 1667 sent to Scotland as member of governing commission under the authority of the king. Withdrew from politics 1670. Died at Whitehall in July 1673. Burried in Westminster Abbey. Oldenburg, Henry (16187—77). — Born in Bremen, son of the schoolmaster Heinrich Oldenburg. Studied theology at the Gymnasium Illustre in Bremen. Matriculated at the University of Utrecht 1641. During the following years apparently worked as a tutor and travelled on the Continent. Returned to Bremen 1652. In 1653 sent to England on diplomatic mission. Around 1655 became tutor to Richard Jones, the son of Lady Ranelagh, the sister of Robert Boyle. Matriculated at the University of Oxford 1656. Devoted time to the study of experimental philosophy. 1657-60 travelled on the Continent with Richard Jones. 1660 returned to London. Founder member of the Royal Society. In April 1663 appointed secretary of the Royal Society (with John Wilkins). Prolific scientific intelligencer with correspondents at home and abroad. Owner and publisher of Philosophical Transactions 1664-77. In June 1667 accused of espionage and imprisoned in the Tower of London for two months. Married twice. Died in Charlton, Kent in September 1677. Pascal, Blaise (1623-62). — French mathematician, physicist, and philosopher of religion. Born in Clermont, son of the deputy Etienne Pascal (1588-1651). Educated by his father. Already as a young man introduced to the scientific gatherings of Roberval, Mersenne, and others in Paris. Published an essay on conic sections in 1640. In 1642 began work leading to development of a calculating machine. In 1646 first religious conversion. From 1646 onwards conducted barometric experiments, and published results in Experiences nouvelles touchant le vide (1647). Later studies on arithmetic, combinatorics, and probability. Strong ties to Port-Royal and to Jansenism, following second religious conversion in 1654. Wrote Lettres a un Provinciale in defence of Arnauld against the Jesuits. His first and second challenges on the properties of the cycloid in June and July 1658 provoked a dispute with Huygens, Wallis, and others. Various works including the Pensees sur la religion and the Traite du triangle arithmetique published posthumously. Pell, John (1611—85). — Born in Southwick, Sussex, son of the local vicar John Pell. Entered Trinity College, Cambridge 1624. B.A. 1628. 625

Biographies of correspondents Proceeded M.A. 1630. Incorporated at Oxford 1631. In 1632 married Ithumaria Reginolles of London. 1638-43 apparently teacher of mathematics in London. In 1643 succeeded Hortensius as professor of mathematics at the Athenaeum Illustre in Amsterdam. Removed to the college at Breda 1646. Publications include Controversiae de vera circuli mensura (1647), based on his dispute with Longomontanus, and An Idea of Mathematics, which appeared as an appendix to Durie's The Reformed LibrarieKeeper (1650). Returned to England 1652. Appointed by Cromwell to lecture on mathematics. During 1654-8 served as Commonwealth agent in Zurich. Took holy orders on return to England. Rector of Fobbing, Essex 1661-85. Vicar of Laindon, Essex 1663-85. Elected fellow of the Royal Society in May 1663. Created D.D. at Lambeth in November 1663. Collaborated with Brancker in producing An Introduction to Algebra (1668), a translation of Rahn's Teutsche Algebra (1659). Resided for some years at Brereton Hall, Cheshire, as guest of his former pupil William Brereton. Financially impoverished towards the end of his life, he died in London in December 1685. Phillips (or Phillippes), Henry (fl. 1648-77). Practitioner of mathematics in London, specializing in navigation, surveying, and gauging. Published his own almanack 1653-7 and took over from Henry Bond in editing Tapp's Seamans kalender. Other publications include The geometrical sea-man (1652) and The advancement of the art of navigation (1657). Lived for many years in a house on London Bridge from where he observed the tides. Appears to have died in 1677. Titus, Silas (1622-1704). — Son of Silas Titus of Bushey, Hertfordshire. Entered Christ Church, Oxford 1637/8. Admitted student of the Middle Temple in 1639. Served in the parliamentary army at the outbreak of the Civil Wars. As a Presbyterian later switched his support to the Royalist side. Representative of the Presbyterians at negotiations between Charles II and the commissioners for Scotland in Breda. Sent by Charles to France to convey his marriage proposals in 1651. At the Restoration became member of parliament for Ludgershall. Appointed keeper of Deal Castle in 1661. Member of parliament for Loswithiel 16708, for Hertfordshire 1678-9, for Huntingdonshire 1679-81, 1681, and for Ludlow 1691-5. Married Catharine Winstanley. Died in December 1704 and buried in Bushey. Wharton, Anne (d. 1692). — Daughter of William Carr of Fernihurst, Roxburghshire, a groom of the bedchamber. Married, first, admi626

Biographies of correspondents ral Edward Popham (16107-51), who sided with parliament during the Civil Wars. By this marriage two children, a daughter, Letitia, and a son, Alexander, born deaf. Following Edward's death, married the former parliamentary commander Philip, fourth Baron Wharton (1613-96) in August 1661. As politician after the Restoration, Baron Wharton rose under William III to the post of privy councillor. Anne Wharton's only son by her second marriage was killed in a duel. She died in August 1692. Wilkins, John (1614-72). — Born at Fawsley, Northamptonshire, son of Walter Wilkins, an Oxford goldsmith. Entered New Inn Hall, Oxford 1627. Migrated to Magdalen Hall. B.A. 1631. M.A. 1634. Took holy orders. Vicar of Fawsley 1637. Resigned benefice and became private chaplain to William Fiennes, first Viscount Saye and Sele. Later chaplain to the prince palatine, Charles Lewis. While in London participated regularly in scientific meetings. Supported parliament during the Civil Wars. Took the covenant. In 1648 made Warden of Wadham College in place of the ejected John Pitt. Created D.D. 1649. Supported by the Independents following his subscription to the engagement. In 1652 named one of five commissioners to carry out the office of chancellor of the University of Oxford. Increased influence following marriage to sister of Cromwell in 1656. At Wadham College he formed a circle of like-minded scholars interested in furthering experimental philosophy. 1659 resigned wardenship of Wadham College to become master of Trinity College, Cambridge. Incorporated D.D. at Cambridge 1659. Deprived of his mastership at the Restoration. Made prebendary of York 1660. Founder member of the Royal Society. Various fields of activity including mathematics and philosophical languages. Was also a skilled instrument maker. Presented by the king to vicarage of St Lawrence Jewry, 1662. 1663 dean of Ripon. In 1668 made bishop of Chester. Died in November 1672 in Chester. Wren, Christopher (1632-1723). — Born at East Knoyle, Wiltshire, son of the local rector Christopher Wren and his wife Mary. Received instruction in mathematics from William Holder, the husband of his father's sister Susan. Attended Westminster School 1641-6. Before entering university worked as assistant to Charles Scarborough. Admitted to Wadham College, Oxford 1648. B.A. 1651. Proceeded M.A. 1653 (incorporated at Cambridge 1664). Fellow of All Souls College 1653-61. Professor of astronomy at Gresham College, London 1657-61. Founder member of the Royal Society. Savilian professor of astronomy at Oxford 1661-73. D.C.L. Oxford 1661. Assistant to Sir John Denham, the surveyor of the royal 627

Biographies of correspondents works 1661-9. 1665 sojourn in Paris. Surveyor-general for the rebuilding of St Paul's Cathedral, the parochial churches, and other public buildings of the city after the Great Fire of London 1666. Architect of the Sheldonian Theatre in Oxford, erected 1664-8 from funds donated by Gilbert Sheldon. Surveyor-general of the royal works 1669-1719. Married Faith Coghill in December 1669. Following the death of his first wife, married Jane Fitzwilliam in 1676. Admitted to Lincoln's Inn 1676. President of the Royal Society 1680-2. Member of parliament for Plympton 1685-7, for Windsor 1689, 1690, and for Weymouth and Melcombe Regis 1701-2. 1698-1723 surveyor-general for the repairs of the collegiate church of St Peter's, Westminster. Last five years of his life resided at his houses in Hampton Court and at Piccadily. Died in London in February 1722/3 and buried in St Paul's Cathedral. Wren, Matthew (1629-72). — Son of the Bishop of Ely, Matthew Wren. Cousin of Christopher Wren (1632-1723), with whom he associated closely from childhood. Matriculated Peterhouse College, Cambridge 1642. Created M.A. Oxford 1661. Secretary to Edward Hyde, the later Earl of Clarendon 1660-7. Member of parliament for St Michael 1661-72. Elected fellow of the Royal Society in May 1663. Author of Considerations of Mr. Harrington's Common-wealth of Oceana (1657). Secretary to James, Duke of York 1667-72. Died in June 1629 and buried alongside his father in Pembroke Hall, Cambridge. Yate, Thomas (1603-81). — Born in Cheshire. Matriculated Brasenose College, Oxford 1619. B.A. 1623. M.A. 1625. Created B.D. 1636. Fellow of Brasenose 1624-48. Created D.D. 1660. Principal of Brasenose 1660-81 (though elected already in 1648). Rector of Middleton Cheney, Northamptonshire 1633-40, 1642, ejected 1646, 1660-78. As law practitioner often employed in University afiairs in London. Married Elizabeth Bartlet. Died in April 1681. Buried in Brasenose College chapel.

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LIST OF MANUSCRIPTS

CAMBRIDGE Cambridge University Library MS Add. 9597/13/6, f. 184r-190v, f. 192V-203V: Letters 98, 100, 121, 123, 137, 139-42, 182, 186, 222, 238, 247, 248, 251. CHATSWORTH The Devonshire Collection Hobbes MSS, letters 51, 85: Letters 14, 15. THE HAGUE Koninklijke Bibliotheek KA XLIIIc, f. 27r-27v: Letters 74, 77. KA XLIV, 618, pp. 618-19: Letter 200. DUBLIN Marsh's Library Z3.4.24: Letter 57. KEW The National Archives SP29/33, No. 66: Letter 11. SP29/221, No. 41: Letter 149. SP29/222, Nos. 31, 311: Letters 155, 156. SP29/233, Nos. 4, 41, 111, 1111, 111II: Letters 168-71. SP29/234, No. 145: Letter 178. SP29/441, No. 34: Letter 157. LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, Nos. 722, 735, 736, 754, 759, 779, 971, 972, 1659: Letters 1-4, 6, 8, 16-18, 242. LONDON British Library Add. MS 4278, f. 81r-82v, f. 87r-87v, f. 127r-127v, f. 340r-340v: Letters 219, 223, 224, 241, 246. Add. MS 4398, f. 148r, f. 175r: Letters 210, 220. Add. MS 32499, f. 15r: Letters 93, 94. LONDON Royal Society Boyle Letters 5, f. 170r-173v: Letters 29, 91. Boyle Letters 6, f. llr-12v: Letter 25. 629

List of manuscripts Boyle Papers 44, f. 23r-27v: Letter 45. Classified Papers 24, No. 11: Letter 172. Early Letters Wl, Nos. 3-11, 13-20, between 20 and 21, 21-44, 46-61, 63, 65, 110: Letters 24, 46, 50, 51, 54, 56, 57, 59, 75, 80, 86, 87, 102-4, 106, 107, 109, 110, 113-16, 120, 124, 126, 134, 136, 143, 146, 160, 163, 165, 174, 176, 177, 189-91, 193, 194, 196, 198, 203-5, 207-9, 213, 216, 217, 221, 229, 233, 234, 239, 245. Early Letters W2, No. 65: Letter 243. Early Letters O2, No. 177: Letter 111. Early Letters OB, No. 63: Letter 148. Letter Book Original 1, between pp. 222 and 223, p. 308, between pp. 330 and 331: Letters 87, 106, 113. OXFORD Bodleian Library MS Add. D. 105, f. 12r-21v, f. 25r-27v, f. 29r-32v, f. 130ar-130bv: Letters 32-5, 41, 82, 84, 88, 147, 164-6, 174, 179. MS Don. d. 45, f. 116r-117r, f. 136r-138r, f. 141r-141v: Letters 231, 236, 252. MS Savile a. 11, f. 89r-89v: Letters 225, 226. Savile A 4 (iv): Letters 212, 215. OXFORD Oxford University Archives WP/3/10/17, f. 13r-18v, f. 21r-24v: Letters 60, 61, 63, 65-70. WP^/15/19/30: Letter 10. WP/3/R/7b/6: Letter 12. WP7/16/1, f. 70r-71cv, f. 76r-96v: Letters 9, 13, 37, 130, 131, 144, 153, 158, 161, 167, 173, 180. PARIS Bibliotheque Nationale Nouv. acq. latines 1641, f. 110r, f. HT-llF: Letters 39, 47.

630

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LIST OF LETTERS

BROUNCKER to WALLIS (1657/XI/l), 547, 549, 577, 580 BROUNCKER to WALLIS (1668/VIII/13), 554, 554 BROUNCKER: Solution to Dulaurens's Problem (1667/XII-1668/I), 394, 397, 398, 400, 402 BROWNE to CHYLINSKI (1662/X/13), 74, 75, 76 BURSCOUGH to WALLIS (1664/XII/4), 168, 168

AUZOUT to OLDENBURG (1665/early I), 174 BALL to HUYGENS (1659/VI/27), 12 BARROW to COLLINS (1668/111/16), 570 BEALE to OLDENBURG (1668/IX/8), 592 BELLAIR to HUYGENS (1659/VII/4), 6 BLANDFORD to WALLIS (1664/VI/26), 142-144, 144 BLANDFORD to WALLIS (1664/VII/l), 145, 146, 148 BLANDFORD to WALLIS (1664/VII/7), 149, 154, 155 BOYLE to OLDENBURG (c.!664/X/12), 162 BOYLE to OLDENBURG (c.!667/IX/25), 326 BOYLE to OLDENBURG (1668/1/8), 398 BOYLE to OLDENBURG (1668/11/11), 394, 398 BOYLE to WALLIS (1662/1/14 or 15), 50, 51 BOYLE to WALLIS (1662/III/8), 50, 51, 51 BOYLE to WALLIS (1662/IV/15), 68 BOYLE to WALLIS (1663/VIII-IX), 86, 87 BOYLE to WALLIS (early 1665), 182, 183 BRANCKER to COLLINS (c.!668/VII/26), 526 BRANCKER to PELL (1668/VIII/4), 533, 534 BROUNCKER to OLDENBURG (1668/III?), 454 BROUNCKER to WALLIS (1657/X/13), 585, 586

CARCAVI to HUYGENS (1659/IX/13), 1, 5, 9, 17 CARCAVI to HUYGENS (1660/111/06), 1, 7-9, 18 CARCAVI to HUYGENS (1660/VI/25), 7, 13, 13, 20, 28 CARCAVI to HUYGENS (1662/1/1), 45 CARCAVI to WALLIS (1660/VI?), 13, 18, 20, 28 CAVENDISH to MERSENNE (1640/IX/l), 527 CAVENDISH to PELL (1648/VIII/2), 303 CHYLINSKI to BROWNE (1662/X/14), 75, 75, 76 CHYLINSKI to WALLIS (1662/X/14), 76, 166 CHYLINSKI to WALLIS (1664/XI/29), 166, 166 CoLEPRESSEto OLDENBURG (1667/111/16), 322, 323 CoLEPRESSEto OLDENBURG (1667/VIII/5), 328 CoLEPRESSEto OLDENBURG (1668/1/17), 437

651

List of letters COLEPRESSEto OLDENBURG (1668/III/3), 437 COLLINS to BRERETON (1668/VII/21), 469, 491, 530, 532 COLLINS to GREGORY (1668/II/?), 433, 570 COLLINS to GREGORY (1669/1/10), 433 COLLINS to PELL (1666/IX/7), 189 COLLINS to PELL (1667/IV/19), 526 COLLINS to PELL (1668/V/l), 533 COLLINS to PELL (1668/VII/28), 469, 491, 525, 527, 534, 566 COLLINS to PELL (1668/VII/28, enclosure), 469, 525, 527, 534, 566 COLLINS to PELL (1668/VIII/30), 566, 567 COLLINS to PELL (1668/IX/15), 567, 595 COLLINS to PELL (1668/X/6), 604 COLLINS to PELL (1668/XI/2), 606 COLLINS to VERNON (1671/XII/24), 599, 603 COLLINS to WALLIS (1666/1/12), 188, 191 COLLINS to WALLIS (1666/111/10), 191, 191 COLLINS to WALLIS (1666/VIII/12), 275, 278, 279 COLLINS to WALLIS (1667/1?), 298, 301, 302, 309, 317 COLLINS to WALLIS (1667/11/12), 301, 301, 307, 309, 311, 313 COLLINS to WALLIS (1667/11/12, end.), 302, 307 COLLINS to WALLIS (c.1667/11/20), 309, 311 COLLINS to WALLIS (1668/III/6), 409, 429, 429, 434, 475 COLLINS to WALLIS (1668/VII/24), 469, 495, 531 COLLINS to WALLIS (1668/VIII/25), 561, 562, 599 COLLINS to WALLIS (1668/X/2), 603, 604 COLLINS to WALLIS (1671/111/31), 526

652

COOPER to WALLIS (1661/V/3), 37 DESCARTES to MERSENNE (1641/1/28), 303 DIGBY to WALLIS (1658/V/4), 575 DlLLINGHAM to WALLIS (1661/IV/ll), 33, 36 DlLLINGHAM to WALLIS (1662/XI/6), 77,

77, 335 DULAURENS to OLDENBURG (1668/11/12), 406 DULAURENS to OLDENBURG (1668/111/10), 461 DULAURENS to OLDENBURG (1668/V/23), 447, 473

FELL to WALLIS (1666/X-XI(i)), 288 FELL to WALLIS (1666/X-XI(ii)), 288 FELL to WALLIS (1666/XI/28), 288, 288 FELL to WALLIS (16667/XII/15), 290 FELL to WALLIS (16677/III/1), 321 FELL to YATE (16617/11/15), 30 FERMAT to CARCAVI (1659/VIII), 3, 9 FERMAT to CARCAVI (1659/LX), 4, 17 FERMAT to CARCAVI (1660/11), 9 FERMAT to CARCAVI (1660/VI/?), 16,17 FERMAT to CARCAVI (1661/XII?), 45, 47 FERMAT to DIGBY (1657/IV/20), 576 FERMAT to DIGBY (1657/VIII/15), 589 FERMAT to DIGBY (1658/IV/7), 560, 569, 576, 589 FRENICLE to DIGBY (1658/II/?), 574, 577, 586 FRENICLE to DIGBY (1661/XII/20), 45, 47 FRENICLE to WALLIS (1661/X-XI?), 45, 45 FRENICLE to WALLIS (1661/XII/20), 45, 46 GREGORY to OLDENBURG (1668/VII/23), 491 GREGORY to WALLIS (1668/IX-X), 614 GREGORY to WALLIS (1668/XI/5), 614,

List of letters 615

491, 568 HUYGENS for WALLIS (late 1670), 12 HUYGENS, CONSTANTIJN to CHRISTIAAN HUYGENS (1655/XI/6), 222 HUYGENS, CONSTANTIJN for WALLIS (1664/IX/17), 159, 163, 568 HUYGENS, CONSTANTIJN for WALLIS (1664/X/16), 159, 163, 568 HUYGENS, CONSTANTIJN to WALLIS (1668/VII/l), 467, 568 HYDE to BLANDFORD (1664/VII/5), 156

HAAK to MERSENNE (1647/VIII/16), 304 HEVELIUS to OLDENBURG (1664/1/4), 100, 101, 103 HEVELIUS to OLDENBURG (1664/IX/10), 158 HEVELIUS to OLDENBURG (1665/VI/l), 187 HEVELIUS to OLDENBURG (1665/IX/12), 187 HEVELIUS to OLDENBURG (1666/IX/15), 285, 292 HEVELIUS to WALLIS (1664/1/4), 79, 82, 100, 103, 108 HEVELIUS to WALLIS (c.!664/IX/10), 103, 158, 165 HOBBES to WALLIS (end of 1662), 79 HOBBES: La duplication du cube (1661/

JENKINS to WALLIS (1664/VI/25), 142, 144 JENKINS to WALLIS (1664/VII/l), 147, 150 JENKINS to WALLIS (1664/VII/earlymid), 157, 158 JENKINS to WALLIS (1667/XI/5), 332, 332, 333 JENKINS to WALLIS (1667/XI/10), 330, 332, 337, 340 JENKINS to WALLIS (1667/XI/15), 334, 340, 352 JENKINS to WALLIS (1667/XI/26), 353,

VI?), 38, 42

HUYGENS to CARCAVI (1659/1/16), 4 HUYGENS to CARCAVI (1659/V/22), 2 HUYGENS to CARCAVI (1659/IX/04), 2 HUYGENS to CARCAVI (1660/11/26), 1, 7, 17 HUYGENS to CARCAVI (1660/111/27), 7, 13 HUYGENS to CARCAVI (1660/VII/15), 13 HUYGENS to MORAY (1661/VI/24), 12 HUYGENS to MORAY (1661/XI/4), 42 HUYGENS to SCHOOTEN (1656/VI/28), 12 HUYGENS to SCHOOTEN (1660/111/19), 12 HUYGENS to WALLIS (1655/VI/13), 222, 599 HUYGENS to WALLIS (1659/XII-1660/ II), 1 HUYGENS to WALLIS (1660/111/31), 2, 11, 21, 27, 30 HUYGENS to WALLIS (1660/VII/15), 2, 13, 20, 27, 28 HUYGENS to WALLIS (1668/XI/13), 433,

359

JENKINS to WALLIS? (1668/1?), 378 JENKINS to WALLIS (1668/II/9), 382, 395 JENKINS to WALLIS (1668/11/23), 405, 407, 408 JUSTEL to OLDENBURG (1666/V/26), 239, 244 JUSTEL to OLDENBURG (1667/XI/16), 366 JUSTEL to OLDENBURG (1667/XII/21?), 378 JUSTEL to OLDENBURG (1667/XII/28), 376 JUSTEL to OLDENBURG (1668/1/4), 366 JUSTEL to OLDENBURG (1668/11/29), 407 JUSTEL to OLDENBURG (1668/V/2), 546 JUSTEL to OLDENBURG (1668/V/5), 413

653

List of letters JUSTEL to 447 JUSTEL to 447 JUSTEL to 479, JUSTEL to 525,

OLDENBURG (1668/V/23),

OLDENBURG to BOYLE (1667/IX/22), 330 OLDENBURG to BOYLE (1667/IX/27), 325 OLDENBURG to BOYLE (1667/XII/5), 360 OLDENBURG to BOYLE (1667/XII/13), 365 OLDENBURG to BOYLE (1667/XII/27), 326 OLDENBURG to BOYLE (1668/1/3), 376, 377 OLDENBURG to BOYLE (1668/II/7), 394, 398 OLDENBURG to BOYLE (1668/11/21), 407 OLDENBURG to BOYLE (1668/III/6), 428, 429 OLDENBURG to BOYLE (1668/111/13), 432, 437, 443 OLDENBURG to BOYLE (1668/111/27), 440, 454 OLDENBURG to BOYLE (1668/VIII/6), 542 OLDENBURG to HEVELIUS (1664/V/21), 103 OLDENBURG to HEVELIUS (1664/VI/6), 84, 103 OLDENBURG to HEVELIUS (1666/IX/8), 285 OLDENBURG to HUYGENS (1669/VI/10), 563 OLDENBURG to MORAY (1665/X/17), 187 OLDENBURG to PALMER (1667/XII/13), 377 OLDENBURG to PALMER (1667/XII/31), 377 OLDENBURG to POWLE (1666/IX/22), 293 OLDENBURG to SLUSE (1668/III/5), 366 OLDENBURG to WALLIS (1664/V/l), 106, 111, 111, 112 OLDENBURG to WALLIS (1664/V/14), 111, 114, 114, 115, 118 OLDENBURG to WALLIS (1664/X/9), 160,

OLDENBURG (1668/VI/6), OLDENBURG (1668/VII/7), 480 OLDENBURG (1668/VII/14), 531

L., H. to WALLIS (1665/IV/20), 179 LACOSTE to OLDENBURG (1669/11/10), 299 MORAY to HUYGENS (1662/II/3), 47 MORAY to HUYGENS (1662/II/9), 47 MORAY to OLDENBURG (1665/X/29), 187 MORAY to OLDENBURG (1665/XI/8-9), 187 MORAY to WALLIS (1662/IV-V), 69, 71, 71 MORAY? to WALLIS (1668/II/?), 432, 432 MORAY? et al. to WALLIS (1662/V/18), 69, 72 OLDENBURG to BOYLE (1664/X/9), 163 OLDENBURG to BOYLE (1664/XI/l), 164 OLDENBURG to BOYLE (1665/VIII/20), 319 OLDENBURG to BOYLE (1665/IX/28), 187 OLDENBURG to BOYLE (1665/X/8), 187 OLDENBURG to BOYLE (1665/X/20), 187 OLDENBURG to BOYLE (1665/X/27), 187 OLDENBURG to BOYLE (1665/XII/29), 187 OLDENBURG to BOYLE (1666/1/9), 187 OLDENBURG to BOYLE (1666/111/27), 193 OLDENBURG to BOYLE (1666/VI/18), 249 OLDENBURG to BOYLE (1666/VII/18),

238, 242

654

List of letters 162 OLDENBURG to WALLIS (1664/X/23), 164, 164 OLDENBURG to WALLIS (1664/X/31), 164, 165 OLDENBURG to WALLIS (1664/mid-XII), 170, 171 OLDENBURG to WALLIS (1665/X/8), 186 OLDENBURG to WALLIS (1666/1/5), 187 OLDENBURG to WALLIS (1666/1/9), 187 OLDENBURG to WALLIS (early 1666), 195, 195 OLDENBURG to WALLIS (1666/V/15), 201, 223, 223 OLDENBURG to WALLIS (1666/V/27), 201, 231, 232, 237, 250 OLDENBURG to WALLIS (1666/VI/8), 201, 232, 238, 240, 241, 246, 250 OLDENBURG to WALLIS (1666/VI/10), 201, 232, 238-240, 240, 241, 246, 249, 250 OLDENBURG to WALLIS (1666/VIII/5), 270, 271, 285 OLDENBURG to WALLIS (1666/VIII/10), 270, 271, 285 OLDENBURG to WALLIS (1666/VIII/14), 278, 281, 282 OLDENBURG to WALLIS (1666/mid-VIII), 281, 283, 284, 285 OLDENBURG to WALLIS (c.1667/1/18), 291, 292 OLDENBURG to WALLIS (1667/1/25), 291, 292 OLDENBURG to WALLIS (1667/II/3), 295, 296 OLDENBURG to WALLIS (1667/111/29), 322, 322, 323 OLDENBURG to WALLIS (1667/XII/6), 360, 361 OLDENBURG to WALLIS (1667/XII/20), 365, 368, 397, 401, 430 OLDENBURG to WALLIS (1668/1/3), 201, 252, 264, 361, 368, 375, 397 OLDENBURG to WALLIS (1668/11/14),

397, 400, 400 OLDENBURG to WALLIS (1668/11/21), 400, 406, 408, 437, 442 OLDENBURG to WALLIS (1668/III/6), 428, 432 OLDENBURG to WALLIS (1668/111/20), 436, 440, 440 OLDENBURG to WALLIS (1668/IV/5), 446, 447, 472 OLDENBURG to WALLIS (1668/VII/10), 447, 469, 470, 472, 479 OLDENBURG to WALLIS (1668/VII/14?), 479, 479, 480 OLDENBURG to WALLIS (1668/VII/23), 492, 497 OLDENBURG to WALLIS (1668/VII/28), 525, 530, 531 OLDENBURG to WALLIS (1668/VII-VIII), 542, 542 OLDENBURG to WALLIS (1668/IX/7), 564, 565 OLDENBURG to WALLIS (1668/IX/ll), 592, 592 OLDENBURG to WALLIS (1668/IX-early X), 615 OLDENBURG to WALLIS (1668/eaxly XI), 499 OLDENBURG: Objections against Dr Wallis's Hypothesis of Tides (1666/VI?), 232, 239, 250, 252, 258 PAISEN to OLDENBURG (1669/1/11), 299 PALMER to OLDENBURG (1667/XII/22), 376 PELL to BRANCKER (1668/VII/31), 470, 527, 533, 535 PELL to CAVENDISH (1645/11/18), 303 PELL to COLLINS (1668/VII/16), 491, 566 PELL to COLLINS (1668/IX/8), 491, 525527, 566 PELL to COLLINS (1668/IX/16), 491, 594, 604 PELL to COLLINS (1668/XI/7), 527 655

List of letters 228, 240, 242, 249, 252, 264, 270272, 377, 397, 436, 442 WALLIS to BRAMSTON (1668/VIII/6), 536, 538 WALLIS to BROUNCKER (1656/X/26), 29 WALLIS to BROUNCKER (1656/X/30), 29 WALLIS to BROUNCKER (1656/XII/15),

PHILIPS to WALLIS (1668/1V/16), 462 POWLE to OLDENBURG (1666/late IX), 293 SCHOOTEN to HUYGENS (1660/1/17), 6 SCHOOTEN to WALLIS (1658/111/18), 575, 581, 584 SLUSE to OLDENBURG (1667/XI/24), 360, 364 VERDUS to MERSENNE 1643?), 303

(end

28, 306 WALLIS to BROUNCKER (1657/XII/27), 547, 549, 577, 580 WALLIS to BROUNCKER (1658/1/30), 547, 549, 577, 580 WALLIS to BROUNCKER (1658/V/21), 449, 543, 569, 576 WALLIS to BROUNCKER? (1661/VII/3), 38, 41 WALLIS to BROUNCKER ? (1664/III/? (i)), 102, 102, 107 WALLIS to BROUNCKER ? (1664/III/? (ii)), 102, 107 WALLIS to BROUNCKER (1668/VIIVIII), 482, 541, 544, 555, 564, 599 WALLIS to BROUNCKER (1668/VIII/13), 542, 545, 545, 558 WALLIS to BROUNCKER (1668/VIII/13, enclosure), 545, 545, 558, 573 WALLIS to BROUNCKER (1668/VIII/15), 482, 541, 554, 554, 558, 564 WALLIS to BROUNCKER (1668/VIII/16), 545, 554, 558 WALLIS to BROUNCKER (1668/VIII/18), 559, 559 WALLIS to BROUNCKER (1668/VIII/18, enclosure), 559, 559 WALLIS to BROUNCKER (1668/VIII?), 546, 559, 573 WALLIS to BROUNCKER (1668/XI/14), 432, 433, 542, 615 WALLIS to BURSCOUGH (1664/XI/?), 168, 168, 169 WALLIS to CARCAVI (1659?), 13, 18 WALLIS to CHYLINSKI (c.!664/XI/27), 166, 166 WALLIS to CLENDON (1667/V/15), 325

1641-

WALLIS to ? (1701/IX/14), 536, 538 WALLIS to BLANDFORD (1664/VI/23), 139, 142, 144 WALLIS to BLANDFORD (1664/VI/25?), 144, 144 WALLIS to BLANDFORD (1664/VI/28), 145, 146, 148, 149 WALLIS to BLANDFORD (1664/VI/30), 145, 146, 148, 152 WALLIS to BLANDFORD (1664/VII/3), 149, 154 WALLIS to BLANDFORD (1664/VII/8), 154, 155 WALLIS to BLANDFORD (1664/VII/28), 157, 157 WALLIS to BOYLE (1662/1/9), 47, 51 WALLIS to BOYLE (1662/III/2), 50 WALLIS to BOYLE (1662/111/24 (i)), 50, 51, 51, 60, 69 WALLIS to BOYLE (1662/111/24 (ii)), 63, 76, 166 WALLIS to BOYLE (1663/IV/6), 80, 398 WALLIS to BOYLE (1663/IX/20), 86, 87, 88, 91, 228 WALLIS to BOYLE (1663/IX/20, enclosure^)), 87, 88, 88 WALLIS to BOYLE (1663/IX/20, enclosure^!)), 87, 91 WALLIS to BOYLE (1665/V/4), 182, 183 WALLIS to BOYLE (1665/V/9), 182, 182 WALLIS to BOYLE (1666/V/5), 200, 223,

656

List of letters WALLIS to COLLINS (1666/I-II), 191, 191 WALLIS to COLLINS (1666/VIII/17), 275, 276, 278 WALLIS to COLLINS (1667/1?), 298, 301, 302, 309 WALLIS to COLLINS (1667/11/15), 302, 309, 311, 409 WALLIS to COLLINS (early 1668?), 409, 410 WALLIS to COLLINS (1668/11/25), 409, 409, 413 WALLIS to COLLINS (1668/III/8), 429, 429, 433 WALLIS to COLLINS (1668/early VII), 469, 491, 495, 527, 531, 532 WALLIS to COLLINS (1668/VII/31), 469, 495, 531, 597 WALLIS to COLLINS (1668/IX/4), 561, 562 WALLIS to COLLINS (1668/IX/18), 533, 596, 603, 607 WALLIS to COLLINS (1668/IX/20), 601, 603, 606 WALLIS to COLLINS (1668/X/6), 604, 604, 610 WALLIS to COLLINS (1668/X/6, enclosure), 604, 610 WALLIS to DIGBY (1657/VI/16), 576 WALLIS to DIGBY (1657/XII/l), 476, 478, 547, 549, 576, 577, 580 WALLIS to DIGBY (1658/1/5), 575, 576 WALLIS to DIGBY (1658/111/14), 25 WALLIS to DIGBY (1658/111/25), 575 WALLIS to DIGBY (1658/V/15), 543, 569, 576 WALLIS to DIGBY (1658/VI/30), 560, 590 WALLIS to DIGBY (1660/IX/3), 22, 204 WALLIS to DILLINGHAM (1661/111/26), 32, 36 WALLIS to DILLINGHAM (1662/X/31), 77, 78 WALLIS to FRENICLE (1661/XI-XII?), 45, 45, 47

WALLIS to GASSENDI (1655/IX/10), 29 WALLIS to GREGORY (1668/XI/l), 615 WALLIS to HEVELIUS (1663/11/19), 79, 100 WALLIS to HEVELIUS (1663/IV/9), 82, 100, 101, 103, 111, 113 WALLIS to HEVELIUS (1664/IV/15), 82, 84, 103, 106-108, 113, 158 WALLIS to HOPKINS (1667/XI/ll), 339, 343 WALLIS to HUYGENS (1655/VII/l), 599 WALLIS to HUYGENS (1659/1/1), 12 WALLIS to HUYGENS (1659/XII/4), 2, 8, 11, 12, 27 WALLIS to HUYGENS (1660/IX/10), 2,11, 20, 27 WALLIS to HUYGENS (1668/IX/10), 159, 163, 410, 413, 467, 468, 568, 597, 602, 607 WALLIS to JENKINS (1667/XI/5), 330, 337, 379, 383 WALLIS to JENKINS (1667/XI/6), 332, 332, 337 WALLIS to JENKINS (1667/XI/10), 334, 352 WALLIS to JENKINS (1667/XI/12), 331, 337, 339, 339, 346, 352 WALLIS to JENKINS (1667/XI/12, enclosure), 340, 346 WALLIS to JENKINS (1667/XI/21), 353, 359 WALLIS to JENKINS (1668/1/31), 379, 380-382 WALLIS to JENKINS (1668/1/31, enclosure), 31, 331, 359, 379, 380, 405 WALLIS to JENKINS (1668/II/7), 378, 382, 385, 395, 396 WALLIS to JENKINS (1668/II/7, enclosure), 32, 290, 340, 344, 379, 381383, 384, 396 WALLIS to JENKINS (1668/11/21), 405, 408 WALLIS to LALOUBERE (c.!668/IX/20), 601, 603, 606 WALLIS to LEOTAUD (1668/11/27), 412,

657

List of letters 430, 572 WALLIS to MORAY (1662/IV/17), 69, 71, 72 WALLIS to MORAY (1662/V/16), 71, 71, 72 WALLIS to MORAY? (1668/II/?), 432, 432 WALLIS to MORAY (1668/VII/24), 494, 542 WALLIS to OLDENBURG (1664/1V/16), 103, 104, 106, 110, 111, 160, 221, 225, 441 WALLIS to OLDENBURG (1664/1 V/17), 110 WALLIS to OLDENBURG (1664/V/10), 82, 111, 111, 114, 225, 251, 441 WALLIS to OLDENBURG (1664/V/17), 114, 114, 118 WALLIS to OLDENBURG (1664/V/24), 114, 118, 137 WALLIS to OLDENBURG? (1664/V/26), 137 WALLIS to OLDENBURG (1664/VI/4), 118, 137 WALLIS to OLDENBURG (1664/X/l), 160, 162, 165, 221, 225, 441 WALLIS to OLDENBURG (1664/XI/8), 158, 164, 164, 171 WALLIS to OLDENBURG? (1665/1/3), 165, 170, 171 WALLIS to OLDENBURG (1665/1/31), 172, 174 WALLIS to OLDENBURG? (1665/V/18), 184 WALLIS to OLDENBURG (1666/IV/13), 195, 195, 235, 286, 294, 407 WALLIS to OLDENBURG (1666/V/3), 197, 235 WALLIS to OLDENBURG (1666/V/17), 223, 223 WALLIS to OLDENBURG (1666/V/22), 226, 284 WALLIS to OLDENBURG (1666/V/29), 201, 231, 232, 241, 246, 250, 252 WALLIS to OLDENBURG (1666/VI/3),

235, 295 WALLIS to OLDENBURG (1666/VI/12), 201, 232, 238, 240, 240, 246, 250, 252, 255-257, 262, 274 WALLIS to OLDENBURG (1666/VI/18), 232, 240, 241, 246, 250, 252, 259 WALLIS to OLDENBURG (1666/VII/28), 201, 232, 241, 246, 250, 251, 270272, 397 WALLIS to OLDENBURG (1666/VIII/3), 201, 252, 263, 264, 270-273, 397 WALLIS to OLDENBURG (1666/VIII/12), 251, 264, 266, 270, 271, 284 WALLIS to OLDENBURG (1666/VIII/21), 278, 281, 283, 284, 292 WALLIS to OLDENBURG (1666/VIII/28), 283, 284, 293 WALLIS to OLDENBURG? (1666/XI/2), 287, 289, 293 WALLIS to OLDENBURG (1667/1/29), 291, 291, 437 WALLIS to OLDENBURG (1667/11/10), 295, 296, 319 WALLIS to OLDENBURG (1667/11/22), 318, 399 WALLIS to OLDENBURG (1667/111/31), 322, 322, 355 WALLIS to OLDENBURG (1667/XI/26), 354, 361, 367 WALLIS to OLDENBURG (1667/XII/10), 360, 361, 376 WALLIS to OLDENBURG (1667/XII/23), 365, 368, 376, 403 WALLIS to OLDENBURG (1668/11/11), 201, 252, 264, 376, 396, 400, 401 WALLIS to OLDENBURG (1668/11/18), 201, 252, 264, 400, 400, 402, 406 WALLIS to OLDENBURG (1668/111/10?), 428, 432 WALLIS to OLDENBURG (1668/111/17 (i)), 434, 436, 440 WALLIS to OLDENBURG (1668/111/17 (ii)), 293, 434, 435, 440, 442, 443 WALLIS to OLDENBURG (1668/111/27 (i)), 436, 440, 440, 442

658

List of letters WALLIS to OLDENBURG (1668/111/27 (ii)), 436, 440, 441 WALLIS? to OLDENBURG (1668/IIIIV?), 461 WALLIS to OLDENBURG (1668/IV/9), 365, 430, 446, 446, 471-473, 475, 477, 499, 507, 572 WALLIS to OLDENBURG (1668/VII/12 (i)), 469, 470, 471, 477, 479 WALLIS to OLDENBURG (1668/VII/12 (ii)), 446, 470, 471, 477, 479, 492, 497, 525, 530, 565 WALLIS to OLDENBURG (1668/VII/14), 472, 477, 480, 499, 507 WALLIS to OLDENBURG (1668/VII/16), 469, 472, 479, 479, 531 WALLIS to OLDENBURG (1668/VII/18), 481, 489, 490, 541, 544, 554, 555, 564 WALLIS to OLDENBURG (1668/VII/21), 489, 532 WALLIS to OLDENBURG (1668/VII/26), 492, 497 WALLIS to OLDENBURG (1668/VII/28), 498, 530, 543, 565 WALLIS to OLDENBURG (1668/VII/30), 498, 499, 525, 530 WALLIS to OLDENBURG (1668/VIII/13), 482, 541, 542, 542, 545 WALLIS to OLDENBURG (1668/IX/4), 564, 565 WALLIS to OLDENBURG (1668/IX/13), 592, 592

659

WALLIS to OLDENBURG (1668/X/16), 615 WALLIS to OLDENBURG (1668/XI/13), 467 WALLIS to OLDENBURG (1668/XI/17), 499 WALLIS to OLDENBURG (1673/X/14), 20, 22, 30 WALLIS for the ROYAL SOCIETY (1663/ X/4), 99 WALLIS to TITUS (1662/VI/22), 73 WALLIS to WILKINS (1668/VII/26), 492, 493, 496 WALLIS to MATTHEW WREN (1665/VI/ 14), 185, 186 WALLIS: Humble Petition to Charles II (1661/III?), 35 WALLIS: Notes on MartivelPs case (1667?), 331, 345, 350 WALLIS: Solution to Dulaurens's Problem (1668/11/18), 368, 400, 402 WALLIS: Note on letter to Boyle of 14/[24].III.1661/2 (i) (1670/ VII/21), 60 WALLIS and CHRISTOPHER WREN to the ESSEX COMMISSIONERS (1668/ VIII/6), 536, 537, 538 WHARTON to WALLIS (1662/X/10), 73 WILKINS to WALLIS (1668/VII/23), 492, 496 WREN, MATTHEW to WALLIS (1665/ VI/9), 185, 186

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INDEX: PERSONS AND SUBJECTS

ANDERSON, ROBERT (d. 1710), 316, 567 identified, 304 Anderson, Robert: Stereometrical Propositions, 597, 602, 604, 606 angle, 410-412, 414, 417-420, 422, 423, 425, 426, 431, 452, 453, 506 angle of contact, 266, 267, 410, 414, 417423, 425-428, 431 angular sections, 269, 431 animals, 94, 95 Antwerp, 303, 328 APELLES of Kolophon (2nd half of 4th century BC), 265 APOLLONIUS of Perga (c.260-c.!90), 421, 576 apothecaries, 31, 96 Aquitaine, 236 ARCHER, JOHN (1598-1682), 148 identified, 140 ARCHIMEDES of Syracus (2877-212), 116, 265, 310 screw, 280 Archimedes: De dimensione circuli, 411, 423 Archimedes: De quadratura parabolae, 423 Archimedes: De sphaera et cylindro, 411, 423 arithmetic, 55, 265, 270, 309, 377, 523 armillary sphere, 97 ARUNDEL, THOMAS, EARL OF (15861646) identified, 196 ARZET, ANDREAS (1604-75) identified, 315 Arzet: Clavis mathematica, 315

Aberdeen, 377 Abulfeda: Geography, 81 Academic Royale des Sciences, 573, 605, 608 Africa, 437, 444 agriculture, 377 air compression, 116 pressure, 94 Aleppo, 377 ALEXANDER, ANDREAS (b. c.1475) identified, 276 Alexander: Mathemalogium, 276 algebra, 48, 55, 275, 303, 309, 317, 475 symbols, 309, 447, 474, 533, 597 algebraists, 430, 448, 452, 461 ALLEYN, THOMAS (d. 1690), 65 identified, 65 Alost (Aalst), 329 America, 437, 444, 446 Amman: Abhandlung von der Sprache, 51, 60 Amsterdam, 6 anatomists Paris, 326 anatomy, 105, 204 dissections, 326 experiments, 327 ANDERSON, ALEXANDER (1582-1619?),

315 identified, 317 Anderson, Alexander: Exercitationum mathematicarum deces prima, 317

661

Index: persons and subjects astronomers, 84, 101, 103, 211, 218, 220, 221 continental, 85 English, 79, 80, 83, 85, 286, 296, 319 astronomy, 101, 104-106, 109, 110 instruments, 12, 30, 165, 176, 293, 296 observations, 12, 79, 80, 82-86, 101, 103, 104, 109, 111, 113, 161, 162, 165, 170-178, 188, 201, 209, 210, 220, 238, 243, 244, 249, 261, 286, 291, 293, 296, 297, 319, 377 tables, 83, 84, 103, 109, 172 AUZOUT, ADRIEN (1622-91), 174, 188, 286, 293, 319 Auzout: L'Ephemeride, 174 AYNSCOM, FRANgois-XAviER (1624-60), 412, 414, 415, 421, 423, 424 Aynscom: Expositio ac deductio geometrica quadraturarum circuli, 417, 421, 572

identified, 316 Barlaam: Logistica, 316 barometer observations, 281-283 BARROW, ISAAC (1630-77), 597 identified, 570 Barrow: Euclidis elementorum libri XV, 302 BARTHOLIN, ERASMUS (RASMUS BERTHELSEN) (1625-98), 279 identified, 275 Bartholin: Dioristice, 275, 279 BARTLET, ? (Oxford carrier) (17th century), 183, 282, 284, 330, 332, 337 BATHURST, RALPH (1620-1704) identified, 74 BAYER, JOHANN (1572-1625) identified, 85 Bayer: Uranometria, 85 BEALE, JOHN (1603?-c.l683), 592 identified, 328 BEAUGRAND, JEAN (1595-1640) identified, 315 Beaugrand: Ars analytica, 315 BECKER, JOHANN JOACHIM (1635-82) identified, 6 Belgium, Belgian, 569 Berkshire, 344 Bermuda, 254, 377 BERNARD, ? (17th century), 175 BERNARD, EDWARD (1638-96) identified, 398 Bible, 61 in Lithuanian, 63-68, 75, 76, 166

BACHET DE MEZIRIAC, CLAUDEGASPARD (1580-1639), 589 Bachet de Meziriac (ed.): Diophanti Alexandrini arithmeticorum libri sex ... Accessit ... inventum novum collectum ex . . . Fermat epistolis, 303 Bago, 256 BALIANI, GIOVANNI BATISTA (15861666), 208 identified, 208 Baliani: De motu naturali solidorum et liquidorum, 208 BALL, SIR PETER (d. 1680) identified, 12 BALL, WILLIAM (c.1627-90) identified, 12 Baltic Sea, 444 Bandung, 377 Barbados, 377 BARLAAM (12907-1348)

BlLLINGSLEY, HENRY (fl. 1551-1606)

identified, 316 Billingsley: Elements of Geometry, 316 BILLY, JACQUES DE (1602-79), 360, 361, 367 identified, 354 Billy: Extrait d'une lettre, 354, 367 BIRCHENSHA, JOHN (fl. 1664-72), 114, 115, 131, 136

662

Index: persons and subjects identified, 115 Black Sea, 444 BLANDFORD, WALTER (1619-75) (B), 139, 142, 144-146, 148, 149, 154, 155, 157 BOETHIUS, ANICIUS MANLIUS SEVERINUS (480-524), 125, 138 Boethius: De institutions musica libri quinque, 138 BOISMORAND, ? (fl. 1659/60), 1 identified, 5 BOOKER, JOHN (17th century) identified, 463 booksellers and printers, 310 ALLESTREE, JAMES (d. 1670) (London, at the 'Bell', at the 'Rose fe Crown'), 30 BEE, CORNELIUS (1636-72) (London) identified, 237 BROWN, SAMUEL (d. 1665) (The Hague) identified, 12 DICAS, THOMAS (d. 1669) (London, at the 'BelP, at the 'Hen & Chickens') identified, 30 GODBID, WILLIAM (fl. 1656-77) (London), 562 LEYBOURNE, ROBERT (fl. 1645) (London) identified, 304 LICHFIELD, ANN (d. 1672) (Oxford), 143, 180, 288, 309, 310, 312, 313 identified, 180 LICHFIELD, LEONARD, the elder (d. 1657) (Oxford), 312 identified, 180 LICHFIELD, LEONARD, the younger (d. 1686) (Oxford), 187 identified, 180 MARTIN, JOHN (fl. 1649-80) (London, at the 'Bell')

663

identified, 30 MEURS, JACOB (17th century) (Antwerp) identified, 303 PITT, MOSES (fl. 1654-96) (London, at the 'White Hart', at the 'Angel'), 299, 301, 302, 304, 307, 309-312, 318, 562, 601, 603 identified, 298 SAWBRIDGE, GEORGE (fl. 1647-81) (London), 595 STEPHENS, ROBERT (17th century) (London) identified, 304 THOMPSON, SAMUEL (d. 1668) (London, at the 'White Horse', at the 'Bishop's Head'), 30, 112, 311-313, 433 identified, 2 UNDERBILL, THOMAS (d. 1660) (London, at the 'Bible', at the 'Anchor and Bible') identified, 30 BORELLI, GIOVANNI ALFONSO (1608-79) identified, 303, 542 Borelli: De m percussionis, 366, 401, 494, 495, 542 BOULLIAU, ISMAEL (1605-94), 20 BOYLE, ROBERT (1627-91) (B), 12, 47, 50, 51, 60, 63, 68, 80, 86-88, 91, 163, 165, 167, 169, 171, 178, 182, 200, 283, 287, 324, 325, 376-378, 398, 407, 437 Boyle: New Experiments PhysicoMechanicall, 50, 183 Boyle: Origine of formes and qualities, 328 Boyle: Usefulnesse of Experimental Natural Philosophy, 86, 87, 91-99 BRADFORD, EDMUND (17th century), 139, 143, 144, 147, 149 BRAMSTON, JOHN, the younger (16111700) (B), 536 BRANCKER, THOMAS (1633-76) (B), 298, 470, 526, 527, 533, 534, 566, 595

Index: persons and subjects 344, 346-348, 380, 536, 537, 539 Clare Hall, 33 Gonville and Caius College, 196 vice-chancellor, 36, 77, 346 Canterbury, 237 Caravaggio: Geometria applicationum, 188 CARCAVI, PIERRE DE (c. 1600-84) (B), 1, 7, 13, 20, 21, 28, 29, 45, 378, 398 CARDANO, GERONIMO (1501-76), 315 Cardano: Artis magnae sive de regulis algebraicis liber unus, 315 CARDINAEL, SYBRANDT HANSZ (15781647)

Brancker: Table of Incomposit Numbers, 469, 491, 496, 525, 527-529, 532535, 566 BRERETON, WILLIAM (1631-80) (B), 299, 308, 491, 526, 566, 567, 595 BRIDGEMAN, ORLANDO (16067-74), 143, 149, 151-153, 155, 156 identified, 141 BRIGGS, HENRY (1561-1630), 190 identified, 193 Briggs: Arithmetica Logarithmica, 302 Broscius: De numeris perfectis, 194

BROUNCKER, WILLIAM (16207-84) (B), 21, 23, 28, 30, 41, 69, 102, 106, 107, 111, 113, 164, 175, 178, 224, 234, 239, 249, 251, 279, 287, 297, 306, 361, 365, 368, 376, 394, 397, 398, 400, 402, 406, 430, 433, 441, 449, 478, 541, 542, 544, 545, 554, 558, 559, 564, 565, 573, 597, 599, 601, 602, 609

identified, 316 Cardinael: Arithmetica, 316 Cardinael: Tractatus geometricus, 316 Caspian Sea, 444 Cassini: Martis circa axem proprium revolubilis observationes, 249 catoptrics, 422

Brouncker: The Squaring of the Hyperbola, 454

centre of gravity (math.), 556, 576, 601 centre of gravity (phys.), 201, 211, 212, 219-221, 224, 225, 233, 243, 250252, 254, 260 CHACE, THOMAS (d. 1449), 379, 381, 382, 384, 396 identified, 331 CHAMPANTE, FRANCIS (17th century), 535 Channel, 240, 249, 438, 445 CHARLES II (1630-85, king of England), 35, 61, 65, 67-69, 75, 76, 101, 105, 143, 217, 399, 536, 539, 540, 567 coronation, 80 CHAVEAU, JEAN-BAPTISTE (17th century) identified, 303 chemical processes, 87 chemistry, 105 chemists, 95, 96 Cheshire, 299 Brereton, 535 Middlewich, 534, 535

BROWNE, RICHARD (1605-83) (B), 74, 75, 167 BROWNE, SAMUEL (d. 1668), 383, 396 identified, 345 BUNNING, 7 (17th century), 304, 313 BURSCOUGH, JOHN (c.l629-c.!707) (B), 168 Caen Academic de Physique, 326 Caithness, 377 calendar computing, 354-358, 360-364, 367, 376 Cambodia, 256 Cambridge (city), 333, 339, 343, 345-348, 353 alehouses, 36 butchers, 36 victuallers, 36 Cambridge, University of, 33, 34, 36, 77,

664

Index: persons and subjects China, Chinese, 54 CHYLINSKI, SAMUEL BOGUSLAV (c.163468) (B), 74-76, 166 publication of the Bible in Lithuanian, 63-68, 75, 76, 166 CICERO, MARCUS TULLIUS (106-43 BC), 91, 266, 273, 274 Cicero: Orationes Philippicae, 204 circle, 265, 267, 317, 366, 367, 369, 371, 394, 402, 406, 411, 416, 421, 425, 426 quadrature, 25, 26, 268, 299, 431, 433, 491, 544, 569, 600, 605, 607, 615 segment of, 277, 607, 609-614 circular wedge, 69, 70, 72 cissoid, 8, 11, 12, 29 quadrature, 492 CLARENDON, FIRST EARL OF, see HYDE, EDWARD CLARK (CLERKE), GILBERT (1626-97?), 305, 312 identified, 304 Clark: Oughtredus explicatus, 304, 312 CLARKE, SAMUEL (1625-69), 398 identified, 81 CLAVIUS, CHRISTOPH (1537-1612), 314, 413, 414, 417, 420-422, 424, 426, 430 identified, 266, 267 Clavius: Algebra, 314 Clavius (ed.): Euclidis elementorum lib. XV, 410, 414, 598 CLEMENT IX, POPE (1600-69, pope from 1667 onwards), 329 CLENDON, JOHN (fl. 1653-68) (B), 35, 38, 143, 145, 149, 154, 325 CLERKE (CLARKE), FRANCIS (fl. 1594) identified, 595 Clerke: Praxis curiae AdmiraMtatis Angliae, 595 COCKER, EDWARD (1631-75) identified, 305 Coke: The fourth part of the institutes of

665

the laws of England, 340 COLEPRESSE, SAMUEL (d. 1669), 322, 328, 437, 443 identified, 323 COLLINS, JOHN (1625-83) (B), 188, 191, 275, 278, 298, 301, 307, 309, 311, 367, 368, 376, 406, 409, 413, 429, 433, 434, 469, 475, 482, 491, 495, 525, 527, 531, 534, 561, 562, 566, 594, 596, 601, 603, 604, 610 Collins: Geometricall Dyalling, 194 Collins: Introduction to Merchants Accounts, 194, 275, 279 Collins: Mariners Plain Scale, 193, 275 Collins: Method For finding the Number of the Julian Period, 367 Collins: Sector on a Quadrant, 193 COLWALL, DANIEL (d. 1690) identified, 313 comets, 170-178, 188, 286, 293, 296, 297 COMMANDING, FEDERIGO (1509-75), 116, 117 Commandino (ed.): Apollonii Pergaei conicorum Libri quattuor, 116 Commandino (ed.): Pappus, Mathematicae Collectiones, 116 Commandino (ed.): Sereni Antinensis Philosophi Libri Duo, 116 commensurability, 515-523, 608 conchoid, 300, 301 quadrature, 492 cone, 276, 277, 281, 453 conic sections, 315, 421, 452 conoids, pyramids, spheroids, 4, 9, 12, 17, 317 convocation, 31, 37, 143 COOPER, BENJAMIN (c.1623-1701) (B), 31, 37 COPERNICUS, NICOLAUS (1473-1543), 85, 249, 260, 262, 431 Cornwall, 234, 297, 438, 439 Land's End, 254 tin mines, 328 COTTON, JOHN (1621-1701), 295

Index: persons and subjects identified, 237 COTTON, ROBERT BRUCE (1571-1631), 237 identified, 196 Court of Common Pleas, 139-141, 331, 340, 350 CRABTREE, WILLIAM (1610-44?), 161, 165, 171, 175, 221 identified, 161 CROKE, UNTON (fl. 1658) identified, 290 CROMWELL, OLIVER (1599-1658), 79 CROONE (CROUNE), WILLIAM (1633-84) identified, 313 CROSSE, JOHN (17th century), 180 identified, 169 CROSSE, MRS. (17th century), 169 CROUCH, THOMAS (1610-79), 359 identified, 343 cube, duplication of, 38-44, 50, 267 curves, 4, 189, 416 cycloid, 9, 16, 19, 20, 28, 571, 574, 601 rectification, 8, 11 CYRIAQUE DE MANGIN, CLEMENT (C.1570-C.1642), 315 identified, 314 Cyriaque de Mangin: Collection, ou, Recueil, 315 Cyriaque de Mangin: Cursus mathemati-

Deinze, 329 DENIS, JEAN-BAPTISTE (c.1640-1704), 324, 376 DESCARTES, RENE (1596-1650), 14, 305 Descartes: Geometria (ed. Schooten) 1649, 14, 314

Descartes: Geometria (ed. Schooten) 1659, 23, 279, 314, 316, 430, 447, 475 Descartes: La Geometric, 14, 314 Descartes: Lettres (ed. Clerselier), 303 Devonshire, 297, 438, 439, 443, 445, 446 tin mines, 328 DIGBY, KENELM (1603-65), 21, 22, 29, 45, 47, 575 DlLLINGHAM,

THEOPHILUS

(1613-78)

Cyriaque de Mangin: Supplementum cursus mathematici, 314

(B), 32, 36, 77 divinity, 430 DOHNA, KRISTOFFER DELFICUS (162868) identified, 329 DOLBEN, JOHN (1625-86) identified, 289 DORCHESTER, FIRST MARQUIS OF, see PIERREPONT, HENRY Dover, 234, 254 Dublin, 118 DUCKWORTH, RICHARD (fl. 1649-92) identified, 169 DULAURENS, FRANgois (d. c.1675), 366, 368, 376, 394, 397, 398, 400-402, 406, 407, 429, 450, 461, 471, 475-

Danzig, 102, 103, 116, 187 DARY, MICHAEL (1613-79) identified, 305 Dary: General Doctrine of Equation, 305 deaf-mutes instruction in speech, 48-63, 69, 7274 DEBEAUNE, FLORIMOND (1601-52), 279 deciphering, 57, 185, 186 DEE, JOHN (1527-1609), 316

Dulaurens: Specimina mathematica, 365, 366, 369, 401, 407, 429, 446-454, 472, 475, 480, 499, 507, 525, 572 DURET, NICOLAS (17th century) identified, 315 Duret: L'algebre, effections geometriques ...,315 DURY, DORA KATHERINA, see OLDEN-

cus, 314

480, 524, 525, 531, 543, 565 Dulaurens: Responsio ad epistolam D. Wallisii, 446, 469-477, 492, 497525, 572

666

Index: persons and subjects BURG, DORA KATHERINA DURY, JOHN (1596-1680), 567 identified, 308 Dury: The Reformed-School, 308 DYBVAD, CHRISTOFFER (15787-1622) identified, 314 Dybvad: In arithmeticam irrationalium Eudidis . . . demonstratio, 314 Dybvad: In arithmeticam rationalium Eudidis ... demonstratio, 314 Dybvad: In geometriam Eudidis ... demonstratio, 314 Dybvad: Problema de arcuum descriptione, 314

identified, 251 equations, 377 biquadratic, 306, 360, 364, 365, 431 cubic, 306, 431 higher, 431 limits of, 188 quadratic, 453 roots of, 304-308, 453 solution of, 307, 308, 431 Essex, 297, 536, 537 Commissioners, 537-540 Lees Priory, 283 Little Hayes, 536, 538 Rettendon, 536, 538 EUCLID of Alexandria (c.325 BC), 116, 265, 266, 273, 274, 310, 314, 412, 420-423, 426, 452, 453, 473 Eudid: Elements, 269, 270, 304, 410, 411, 415-418, 421-427, 448, 503-506, 509, 511, 514, 527, 597, 598 Eudid: Sectio canonis, 115, 119, 120, 129, 133 Europe, 438, 444, 446, 448, 449, 467, 473, 572 Exchequer, Court of, 77, 78, 153, 333, 335, 336, 339, 340, 344-348, 351, 352, 380, 384, 396 experiments, 105, 120, 238, 242, 243, 305, 322, 324, 327, 376, 462, 494, 495, 542

Earth, 211, 221, 233, 243, 244, 253, 261, 262, 514 motion, 202, 206-208, 210, 211, 216, 219, 220, 223-225, 242, 247, 251, 254, 258-261, 437, 438, 444-446 earthquake, 407 East Indies, 239, 242, 251, 255 eclipse lunar, 85, 377 EDGAR (944-75, king of England from 959 onwards), 329, 330 Edinburgh, 377 education, 336 EDWARD I (1239-1307, king of England from 1272 onwards), 236 EDWARD II (1284-1327, king of England from 1307 onwards), 236 ELIZABETH (1596-1662, queen of Bohemia), 61 ellipse, 430, 446, 448-452, 607 quadrature, 431, 433, 569, 605 engines, 280 England, English, 1, 7, 9, 11, 19, 21, 22, 30, 48, 63, 76, 105, 157, 170, 203, 217, 236, 294, 303, 324, 328, 344, 378, 388, 398, 433, 449, 450, 473, 572 ENT, GEORGE (1604-80), 239

Fabri: Opusculum geometricum, 188 Fabri: Synopsis optica, 430, 434 FARINGTON, WILLIAM (d. 1420) identified, 381 Faulhaber: AcademAa algebrae, 316 FELL, JOHN (1625-86) (B), 30, 282, 288, 321, 331, 333, 345, 380, 383, 396, 405, 409 FERMAT, PIERRE DE (1607/8-65), 2-4, 9, 10, 13, 18, 21, 23, 26, 45, 303, 476, 478, 546, 567, 571, 574, 575 as geometer, 9 Fermat's 'last theorem', 545, 549-

667

Index: persons and subjects 553, 558, 573, 580-584 negative theorems, 545, 546, 549553, 558-561, 573, 580-584 Fermat: De linearum curvarum cum lineis rectis comparatione, 13, 2123, 29 Fermat: De solutione problematum geometricorum, 14 FERNE, HENRY (1602-62) identified, 36 fixed stars, 82-84, 101, 104, 107, 178 Flanders, 328, 329 Florence, 303 Flussaies: Ele.me.nta, geometrica, 421 FOSTER, SAMUEL (c.1600-52), 85, 194, 313 identified, 83 Foster: Art of Dialling, 194 FRACASSATI, CARLO (1630-72), 327 France, French, 11, 27, 28, 42, 49, 54, 189, 195, 322, 324, 328, 397, 400, 402, 406, 450, 473, 525, 531, 546, 568, 603 Franche Comte, 378 Franeker, 85 University of, 63, 64 FREDERICK V (1596-1632, elector Palatine), 61 FRENICLE DE BESSY, BERNARD (160575) (B), 25, 45, 46, 473, 476, 478, 574, 575, 577, 584, 585, 591 Frenicle: Solutio duorum problematum, 575, 576

tagma, 272 Gassendi: Syntagma philosophicum, 252, 259, 260, 272 Geldern, 568 geographers, 81 geometers, 5, 19, 115, 264, 266, 274, 275, 318, 415, 420, 424, 461 ancient, 420 Greek, 421 modern, 420, 421 geometrical problems, 461, 462 geometry, 55,106, 193, 265, 270, 309, 315, 317, 422, 423, 426, 430, 448, 450, 523, 602 Cartesian, 279 Germany, German, 299, 567 GHETALDI, MARINO (1568-1626) identified, 314 Ghetaldi: De resolutione & compositione mathematica, 314 GIBSON, THOMAS (ft. 1655), 316 Gibson: Syntaxis mathematica, 316 GIRARD, ALBERT (1595-1632), 315, 589 Girard: Invention nouvelle en I'algebre, 315 glass, 96 globe, celestial, 172, 173, 176, 178 Gloucestershire, 537, 540 GODDARD, JONATHAN (1616-75), 239, 242, 251, 284 identified, 113 GOEDART, JAN (1620-68), 326 Goedart: Metamorphosis et historia naturalis insectorum, 326 Goedart: Metamorphosis naturalis, ofte historische beschryvinghe, 326

GALILEO GALILEI (1564-1642), 5, 202, 203, 214, 258-260, 324 Trattato delle Meccaniche (manuscript), 276, 279 Galileo: Dialogo sopra i due massimi sistemi del mondo, 202, 206-208, 214, 220, 247, 259 GALLOIS, JEAN (1632-1707), 491 GASSENDI, PIERRE (1592-1655), 28, 272 Gassendi: Philosophiae Epicuri syn-

GOUSSAINVILLE, PlERRE DE (c.1621-83),

195 grammar, 266, 274, 450 gravity, 202, 211, 215, 495 GREAVES, EDWARD (1608-80) identified, 399 GREAVES, JOHN (1602-52)

668

Index: persons and subjects HERIGONE, PIERRE, see CYRIAQUE DE MANGIN, CLEMENT HEURAET, HENDRIK VAN (1633-60?), 11, 21-23, 204, 430 Heuraet: Epistola de transmutatione curvarum linearum, 21, 23, 430 HEVELIUS, JOHANNES (1611-87) (B), 79, 82, 100, 103, 107, 108, 113, 116, 158, 165, 174, 188, 285, 286, 293, 296, 297 controversy with Auzout, 286, 293 election to Royal Society, 84 Hevelius: Cometographia, 101, 480 Hevelius: Descriptio Cometae anno 1665, 278, 282-286, 292, 293, 296-298, 319 Hevelius: Machina Coelestis, 101 Hevelius: Mercurius in Sole visus, 79, 80, 83, 104, 109 Hevelius: Prodromus cometicus, 187, 296 Hevelius: Selenographia, 211, 318, 320, 481 HOBBES, THOMAS (1588-1679) (B), 38, 48, 52, 79, 116, 264-270, 272-274 as mathematician, 267, 268 Hobbes: De principiis et ratiocinatione Geometrarum, 264-270, 272-274, 397 Hobbes: Dialogus physicus, 38, 41, 50, 52, 116 Hobbes: Examinatio et emendatio mathematicae hodiernae, 50, 265-267, 269 Hobbes: Leviathan, 79 Hobbes: Mr Hobbs considered, 79 Hobbes: Problemata physica, 38, 42 Hobbes: Seven philosophical problems, 38 HOLDER, WILLIAM (1616-98) identified, 74 Holder: Elements of Speech, 74 Holland, Dutch, 21-23, 159, 217, 234, 254, 286, 328 ambassadors, 328 HOLLOWAY, CHARLES (d. 1679), 142,

identified, 399 GREAVES, THOMAS (1612-76) identified, 399 Greenwich, 234, 254, 285 GREENWOOD, DANIEL (16057-73), 169 identified, 168 GREENWOOD, DANIEL (d. 1679) identified, 169 GREGORY, JAMES (1638-75) (B), 412, 432, 433, 470, 491, 495, 533, 543, 569, 604, 605, 607-609, 614 Gregory: Answer to the Animadversions of Monsieur Hugenius, 491, 530, 532, 542, 543, 569, 607, 609 Gregory: Exercitationes geometricae, 604, 605, 608 Gregory: Vera circuli et hyperbolae quadratura, 429, 431-433, 454, 491, 496, 530, 542-544, 562, 569, 599, 607, 608, 615 Gresham society, 72 GUISONY, PIERRE (17th century) identified, 12 GUNTER, EDMUND (1581-1626), 193 identified, 190 hailstones, 137 HALE, MATTHEW (1609-76), 335 identified, 78 HALL, GEORGE (1612-68) identified, 567 Hamburg, 299 Hampshire, 143, 147, 297 HARRIOT, THOMAS (1560-1621), 474 identified, 314 Harriot: Artis analyticae praxis, 314 HARTLIB, SAMUEL (c.1600-62), 449 HARVEY, THOMAS (fl. 1657-63), 193 HENRION, DENIS, see CYRIAQUE DE MANGIN, CLEMENT HENRY II (1133-89, King of England from 1154), 236, 237 Heraklion, 329

669

Index: persons and subjects 152, 359, 408 identified, 140 HOLMES, ROBERT (1622-92) identified, 285 HOLWARDA, JAN FOKKES (JOHANNES PHOCYLIDES) (1618-51), 85 Holwarda: HavaeXr/vcr^ . . . , 85 HONDIUS, JODOCUS (1563-1611), 178 HOOD, PAUL (c.1585-1668), 31, 32, 34 identified, 33 HOOKE, ROBERT (1635-1703), 187, 204, 209, 210, 238, 242, 297, 319 Hooka: A Spot in one of the Belts, 210 Hooka: Some New observations, 209 Hooke: The Particulars of those observations, 209 HOPKINS, JOHN (17th century) identified, 142 HOPKINS, WILLIAM (c.1641-81) (B), 331, 332, 339, 341-345 Horace: Ars poetica, 473 Horace: Satirae, 303 HORROX, JEREMIAH (16177-41), 106112, 114, 160-162, 164, 165, 171, 174, 175, 182, 184, 220, 222, 225, 441 identified, 80 Horrox: Opera posthuma (ed. Wallis), 108, 160, 161, 221 Horrox: Venus in Sole visa, 80, 83, 109, 165 HUDDE, JAN (1628-1704), 303, 526 Hudde: Epistola prima de reductione aequationum, 303, 447 HUET, PIERRE-DANIEL (1630-1721), 326 HUIPS, FRANS VAN DER (fl. 1654) identified, 316 Huips: Algebra, 316 HUME, JAMES (fl. 1636), 317 identified, 315 Hume: Algebre de Viete, 315

HUYGENS, CHRISTIAAN (1629-95) (B), 1, 7, 11, 13, 17, 20, 23, 27, 47, 80, 159, 222, 285, 286, 433, 480, 491, 496, 525, 533, 543, 544, 562, 568, 597, 599, 602, 604, 605, 607-609 visit to London, 80 Huygens: Brevis assertio systematis Saturnii sui, 12 Huygens: De circuli magnitudine inventa, 269 Huygens: Examen de Vera Circuli & Hyperboles Quadratura, 433, 491, 530, 532, 542, 543, 562, 569, 600, 605, 609 Huygens: Horologium oscillatorium, 23, 430 Huygens: Maniere pour trouver ... la dimension de I'espace Hyperbolique, 562 Huygens: Systema saturnium, 2, 12, 21 HUYGENS, CONSTANTIJN (1596-1687) (B), 159, 163, 467 visit to Oxford, 159, 163, 467, 468, 568 HYDE, EDWARD (1609-74), 140,141,143, 148-153, 155, 156, 185, 186, 290, 537 identified, 140 HYDE, THOMAS (1636-1703), 104, 107, 112, 114, 116, 378, 398 identified, 81 hyperbola, 430, 492, 607 infinite, 576 quadrature, 5, 14, 431, 433, 454460, 482-491, 531, 532, 543, 544, 562, 564, 569-571, 597, 599, 604, 606 impetus, 206, 207, 215, 255 India, 84 instruments, 95, 244, 274, 593, 599 Ireland, 234, 254 Irish Sea, 438, 445 iron, 96, 233, 252, 253 IRONSIDE, GILBERT, the younger (1632-

HUNTINGTON, ROBERT (1637-1701)

identified, 398 670

Index: persons and subjects 138 KERSEY, JOHN (1616-77), 304 identified, 303 Kersey: Algebra, 303 KlNCKHUYSEN, GERARD (d. 1661), 316 Kinckhuysen: Algebra ofte stelkonst, 316 Kinckhuysen: De grondt der meetkonst, 316 King's Bench, 32, 334, 335, 381, 382, 384,

1701) identified, 536 ISHAM, JUSTINIAN (1611-75) identified, 305 Italy, 432 Izmir, 377 Jamaica, 377 JAMES II (1633-1701, duke of York, king of England, Scotland, and Ireland 1685-88), 61, 187, 329 JENKINS, LLEWELYN (1623-85) (B), 139, 142, 147, 148, 150, 153, 154, 157, 158, 289, 330, 332, 334, 337, 339, 346, 350, 352, 353, 359, 378-380, 382, 395, 401, 405, 408 Jesuites, 413 Jews, 334 Journal des Scavans, 491, 562, 569 JOYLIFFE, GEORGE (1621-58) identified, 204 Jupiter, 48, 209, 210, 221, 222, 233, 253, 259, 260 Great Red Spot, 204 satellites, 209, 210, 221, 233, 238, 253, 259, 260 jurists, 105 JUSTEL, HENRI (1620-93), 195, 244, 326, 328, 366, 401, 407, 413, 428

392 KOOPMAN, CATERINA ELISABETHA (17th century) identified, 101 KOOPMAN, JOHANNES (17th century), 100, 106 identified, 101 KOPRULI), ACHMED (1635-76) identified, 329 KRAINSKI, JAN (1625-85), 76 identified, 65 L., H. (17th century) (B), 179 LA COSTE, BERTRAND DE (fl. 1663-76) identified, 299 La Coste: Klarer Bewiis van het Quadraet des Circels, 299 LALOUBERE (LALOVERA), ANTOINE DE (1600-64) (B), 599, 601-603, 606 Laloubere: De cydoide, 606 Laloubere: Quadratura circuli et hyperbolae, 599, 606 Laloubere: Veterum geometria promota,

Kattegat, 444 Kent, 85, 280, 294, 297, 399, 436-438, 442, 443, 445, 537, 540 Ashford, 88, 288 Chatham, 234, 250, 253, 254, 399 Deal, 254 Gravesend, 441 Hythe, 287 Romney Marsh, 205, 216, 234, 254, 272, 287, 445 KEPLER, JOHANNES (1571-1630), 118, 137, 138 Kepler: Astronomia nova, 221 Kepler: Harmonices Mundi Libri V, 137,

599, 603, 615 LAMPLUGH, THOMAS (1615-91) identified, 37 Lancashire, 165 Wigan, 567 languages, 48, 49, 52-59, 62, 63, 481 Arabic, 52, 378, 398 Chinese, 54 English, 52, 56, 58, 59, 62, 92, 95 French, 95 Greek, 48, 52, 95

671

Index: persons and subjects Irish, 52 Latin, 52, 59, 62, 91, 94 Lithuanian, 64 oriental, 378 Persian, 112, 378 Polish, 62 Turkish, 378 universal, 55, 479, 481, 493, 494, 496 Welsh, 52 LANSBERGE, PHILIP VAN (1561-1632), 109 identified, 109 Lansberge: Tabulae Goelestium Motuum perpetuum, 109 LANTZ, JOHANNES (1564-1638) identified, 315 Lantz: Institutionum arithmeticarum libri quatuor, 315 LA TOUR D'AUVERGNE, HENRI DE, Vi-

LICHFIELD, SOLODELL (d. 1671), 335 identified, 78 lightning, 226, 227 Limousin (French province), 236 Lincolnshire, 297 line, 265-267, 416, 424, 452, 453, 503, 505, 506, 511, 519 liquids, 94, 95 Lithuania, 63, 66, 67 Calvinist church, 63-68 loadstone, 233, 252, 253 logarithms, 190, 192, 460, 483, 488, 531, 532, 543, 562, 563, 567, 569-571, 595, 597, 604, 605 sums of, 482, 488, 532, 554-557, 564 London, 2, 27, 30, 48, 57, 64-66, 69, 80, 81, 95, 96, 100, 105-107, 112, 154, 172, 175, 176, 182, 187, 190, 196, 217, 234, 236-238, 250, 254, 274, 280, 284, 285, 289, 292, 293, 297, 311, 312, 325, 341, 377, 399, 401, 413, 430, 439, 441, 442, 450, 463465, 467, 494, 496, 563, 594, 599 Billingsgate, 441 Clerkenwell Green, 595 Court of Aldermen, 65 Drury Lane, 289 Excise Office, 190 Exeter House, 401, 407 Great Fire, 289, 309, 311 Great Plague, 188, 189, 193 Gresham College, 83, 85, 113, 313, 450 King Street, 217 Little Britain, 298, 311 London Bridge, 466 lord mayor, 65 Market Place, 217 ministers, 65 Palace Yard, 217 St Paul's Churchyard, 311 Strand, 217 Tower, 326, 343 Westminster, 78, 147, 150, 156, 217,

COMTE DE TURENNE (1611-75)

identified, 329 laws of motion, 202, 204, 206 LE TENNEUR, JACQUES-ALEXANDRE (fl. 1639-48), 527 Le Te.nne.ur: Traite des quantites incommensurables, 527 LEAKE, JOHN (fl. 1650-86) identified, 317 Leake: Elements of Geometry, 317 LEOTAUD, VINCENT (1595-1672) (B), 315, 410-412, 429-431 Leotaud: Cydomathia, 410, 412-428, 572 Leotaud: Ele.me.nta geometriae practicae, 315 Leotaud: Examen circuli quadraturae, 412 Levant, 378 LEYBOURNE, WILLIAM (1626-1716), 305 identified, 304 LIBAVIUS, ANDREAS (15407-1616), 544 LICHFIELD, MARGARET (17th century), 288, 313 identified, 311

672

Index: persons and subjects 289 Westminster Stairs, 407 Whitehall Palace, 37, 61, 75, 167 Whitehall Stairs, 407 longitudes and latitudes, 81-83, 101, 103, 190, 192 Louis XIV (1638-1715, king of France from 1643 onwards), 328, 329 LOWER, RICHARD (1631-91), 228, 230, 322, 324 identified, 204 Lumleian library, 196, 237 LUMLEY, JOHN, BARON (15347-1609), 237 identified, 196 lymphatic vessels, 204 LYNE, FISH (17th century), 330, 336, 344, 378, 383, 405, 408

English, 29, 430, 449, 575 European, 448, 449, 473, 572 foreign, 532 French, 450, 571, 574, 599 modern, 411 mathematics, 48, 55, 59, 105, 106, 115, 266, 274, 309, 414, 431, 449, 474, 476, 478, 499, 521, 573 challenges, 448-450, 473, 571, 572 disputes, 10, 23, 574, 575 instruments, 189 matter, 95 MAYNARD, JOHN (1602-90), 140, 141 identified, 140 mechanics, 105, 202, 206, 280 MEDICI, LEOPOLD DE' (1617-75), 5 medicine, 88, 89, 91, 96 Mediterranean Sea, 444 Medway, river, 436, 442 MEIBOM, MARCUS (1630-1711), 28, 413, 414 Mengoli: Geometriae speciosae elementa, 188 MENNHER DE KEMPTEN, VALENTIN (fl. 1550) identified, 315 Mennher: Arithmetique seconde, 315 Mennher: Pratique briesve, 315 Mennher: Pratique pour briesvement apprendre ..., 315 MERCATOR, GERHARD (1512-94), 189, 191, 192, 300 MERCATOR, NICOLAUS (1620-87), 544, 555, 595, 596, 598, 599, 604, 605 Mercator: Logarithmotechnia, 481-491, 526, 531, 564, 567, 595, 597-599 merchants, 105 Mercury, 80, 83, 164, 165, 209, 244 MERRY, THOMAS (d. 1683), 534, 566 identified, 526 MERSENNE, MARIN (1588-1648), 137, 138 Mersenne: Harmonic universelle, 138

Madrid, 238, 274 MAGINI, GIOVANNI ANTONIO (15551617) identified, 85 magnetic forces, 202, 211, 233, 252 Mainz, 6 MALPIGHI, MARCELLO (1628-94), 327 Malpighi: Teiras anatomicarum epistolarum, 327 Manchester, 161 Margate, 234, 254 mariners, 192, 248, 257, 294, 297 MARKE, JOHN (fl. 1665-79), 275, 281, 299, 302 identified, 189 Mars, 12, 209, 243, 244, 261, 262 MARSH, NARCISSUS (1638-1713) identified, 398 MASSUE, HENRI DE (1618-89) identified, 328 mathematicians, 48, 56, 58, 96, 105, 117, 266, 267, 274, 309, 450, 452, 480 ancient, 411 Dutch, 29

673

Index: persons and subjects Mersenne: La verite des sciences, 138 Me.rse.nne.: Traitez des consonances, 138 meteors, 95 methods, 8 analytical, 461, 462 approximation, 25, 280, 433, 569 exhaustion, 423, 426 music theory, 120, 131, 136, 138 of Archimedes, 25 quadratures, 431, 454 rectification, 10 solution of equations, 305-308, 360, 364 summation of logarithms, 554-557, 564 synthetical, 461, 462 Milan, 188 MILLINGTON, THOMAS (1628-1704), 230, 284, 285, 481 identified, 228 mirrors, 412, 422 MOHAMMED IV, SULTAN (1641-92), 329 MONCONYS, BALTHASAR DE (1611-65) identified, 95 Monconys: Journal des voyages, 95 Monmouthshire Chepstow Bridge, 437, 443 MONTAGU, EDWARD, FIRST EARL OF SANDWICH (1625-72), 175, 238, 274 identified, 244 MONTFERT, JEAN DE, see PASCAL, BLAISE Moon, 211, 221, 233, 243, 244, 253, 261, 262, 377, 514 motion, 202, 208, 210-212, 221, 224, 225, 246, 251, 254, 258-261 MOORE, ? (Oxford carrier) (17th century), 149, 200, 236, 238, 282, 284, 319, 477, 479, 602 MOORE, JONAS (1617-79), 114 identified, 109 MORAY, ROBERT (1608-73) (B), 47, 49, 69, 71, 72, 178, 184, 287, 319, 432, 674

494 MORLAND, SAMUEL (1625-95), 318 Morland: A New Method of Cryptography, 318 Moss, RICHARD (17th century), 187 motion, theory of, 322, 323 experiments, 494, 495, 542 music, theory of, 114-116, 118-139 ancient authors, 115, 120 modern authors, 115, 118, 131, 136138 musicians, 115, 131, 136, 138

NAPIER, JOHN (1550-1617), 190 natural philosophy, 105, 202 nautics, 105, 106 NEILE (NEALE), PAUL (1613-86), 23, 30, 231, 250, 287 identified, 12 NEILE (NEALE), WILLIAM (1637-70), 21-23, 204, 322, 323, 430 nerves, 98 NICHOLAS, EDWARD (1593-1669) identified, 37 Norfolk, 297 Normandy, 236 Northampton, 60, 305, 376, 377, 537, 540 Ecton, 84 Norway, 444 NOWELL, JOHN (d. 1670/1) identified, 90 number theory, 571 numbers, 267, 505, 506, 508-510, 519, 520 incomposite, 469, 491, 496, 525-529, 532-535, 567 rational, 397, 398, 400, 403, 404 triangular, 585 OEDIPUS, 502, 518 OLDENBURG, DORA KATHERINA (165477), 562, 564, 566 OLDENBURG, HENRY (16187-77) (B), 60, 74, 82, 84, 103, 106,110, 111, 114,

Index: persons and subjects 118, 137, 160, 162, 164, 170, 171, 174, 184, 186, 187, 195, 197, 201, 223, 226, 231, 232, 235, 238, 240, 246, 250, 251, 263, 270, 271, 278, 281, 283, 284, 287, 291, 295, 296, 318, 322, 325, 354, 360, 361, 365, 368, 375, 396, 398, 400, 406, 408, 413, 428, 432, 434, 435, 440, 441, 446, 454, 461, 462, 469-471, 475, 477, 479, 481, 489, 492, 494, 497, 498, 525, 530, 532, 542, 554, 558, 564, 565, 568, 592, 606, 615 imprisonment, 326 marriage, 561, 562, 564, 566 optics, 105, 411 otacousticon, 592-594 OUGHTRED, WILLIAM (1575-1660), 309, 310, 533 Oughtred: Clavis mathematicae, 299, 301, 304, 305, 309, 310, 312, 314, 318, 447, 474 Oxford (city, diocese), 27, 30, 33, 34, 64, 95, 99, 100, 105-107, 110, 137, 145, 149, 150, 159, 163, 181, 187, 188, 238, 283, 288, 289, 292, 330, 331, 333-336, 338, 340, 341, 351, 378, 382, 383, 385-388, 390-393, 398, 407, 439, 450, 468, 563, 592594, 615 alehouse keepers, 34 alehouses, 33 bailiffs, 33, 37, 341, 342, 381 butchers, 34 Catte Street, 34, 74 charters, 343 inns, 33 market, 31, 37, 336, 382, 389, 390, 392 mayor, 33, 341, 342 Serjeants, 336 St Mary the Virgin, 49, 389 taverns, 33 victuallers, 33, 34 Oxford, University of, 31-34, 37, 77, 140142, 146, 148-150, 154, 155, 288,

289, 321, 330-333, 348, 350-352, 382, 385-392, 536-539 Act, 99 archives, 353, 380, 384 Balliol College, 145, 147, 195, 196, 199, 235 Bodleian Library, 82, 107, 112, 195199, 235, 236, 276, 378, 398, 407, 428, 468, 469, 568 Brasenose College, 143, 145, 154 chancellor, 33, 157, 333-335, 338, 340-343, 346, 349-352, 380, 381 charters, 31, 32, 146, 148, 150, 153, 157, 290, 330, 332, 334-336, 338-340, 342-345, 359, 379383, 385, 386, 388, 389, 392, 393 Christ Church, 35, 49, 282, 593 college lands, 539 college libraries, 195 colleges and halls, 540 curators, 197 Exeter College, 299, 398 Magdalen College, 398 Merton College, 195-197, 199, 236, 398 Oriel College, 196 printers, 180 privileges, 139, 150, 157, 290, 331, 334-337, 340-343, 360, 380 rents from college lands, 538-541 Savilian lands, 536-541 Savilian professor of astronomy, 187, 536 Savilian professor of geometry, 35, 187, 536 Savilian professors rent from estates, 536-541 St John's College, 398 vice-chancellor, 34, 153, 197, 282, 335, 336, 381 vice-chancellor's court, 31, 330, 353 Wadham College, 227, 536

675

Index: persons and subjects Oxfordshire, 344 Banbury, 185 Fiddler's Island, 227 Medley, 227 Port Meadow, 227

PELETIER, JACQUES (1517-82), 414, 420, 421 identified, 266 PELL, JOHN (1611-85) (B), 73, 299, 302, 305, 306, 308, 469, 491, 525, 527, 533, 534, 566, 594, 604 Pell's equation, 545-549, 576-580 Pell: Idea of Mathematicks, 308 pendulum, 215, 238, 242, 495, 542 pendulum clock, 5, 10, 11 perpetuum mobile, 1, 6 PERRAULT, CLAUDE (1613-88) identified, 326 Persia, 407 PETER OF BLOIS, see PETRUS BLESEN-

Padua, 433 PAINTER, THOMAS (17th century), 31, 359, 379, 381, 382, 392, 405 painters, 265 PALMER, JOHN (1612-79), 82-84, 104, 376, 377 identified, 83 Palmer: Catholique Planisphaer, 83-86, 113 parabola, 2, 4, 14-16, 300, 430, 607 rectification, 599 paraboloid, 14, 16 PARACELSUS, THEOPHRASTUS BOMBASTUS VON HOHENHEIM (14931541), 97 Paris, 6, 8, 18, 22, 29, 30, 38, 189, 239, 244, 249, 326, 328, 361, 364, 365, 376, 394, 407, 413, 428, 568, 569, 602 Bibliotheque Royale, 378, 605 parliament, 33, 146, 331, 335, 340, 341, 388 acts, 32, 150, 157, 335, 341-343, 360, 392, 393, 539 PASCAL, BLAISE (1623-62), 3, 8, 9, 18, 19, 28, 29, 450, 571, 574 arithmetical machine, 1, 6 De Montfert's problem, 449-451, 473 Pascal: Historia trochoidis, 9 Pascal: Lettres de A. Dettonville, 1, 2, 8, 18, 28 Pascal: Recit de I'examen, 11, 27 Pascal: Traite du triangle arithmetique, 189 Pegu, see Bago PEIRCE, EDWARD (fl. 1635-9) identified, 339

sis petrifaction, 95 PETRUS BLESENSIS (c.H35-c.l204) identified, 195 manuscripts in English libraries, 195-200, 235-237, 239, 240, 244, 286, 294, 407 Petrus Blesensis: Opera omnia, 195, 407, 428 PETT, PETER (1610-70?), 69 identified, 69 PHILIPS, HENRY (fl. 1648-77) (B), 462 Philips: A Letter ... containing his Observations ..., 462 philosophers, 202, 424 Philosophical Transactions, 60, 187, 201, 209, 210, 224, 251, 270-272, 274, 284-287, 361, 367, 377, 397, 400, 401, 407, 470, 482, 492, 497-499, 524, 525, 530, 541, 542, 546, 554, 559, 564, 569, 573 philosophy, 202, 247, 258, 305, 323, 573 physicians, 92, 95, 96, 105, 376 PIERREPONT, HENRY (1606-80) identified, 276 pigeons, 99, 100 plane, 416, 452, 453 planets, 209, 210, 220, 244, 262 Plato: Timaeus, 95 676

Index: persons and subjects Plymouth, 323, 437, 443 POCOCKE, EDWARD (1604-91), 378, 398 identified, 81 point, 265, 266, 273, 274, 424, 500, 502, 505, 509 Poland, 329 pope, 430, 431 POPE, WALTER (d. 1714) identified, 593 POPHAM, ALEXANDER (17th century), 73 identified, 62 POPHAM, EDWARD (16107-51), 73 Portsmouth, 172, 175 Potter, Charles: Tlv@a~[6pa<; MfTefi^vxo 201, 252 POTTER, WILLIAM (fl. 1656) identified, 595 POWLE, HENRY (16307-92), 297, 437, 443 identified, 293 PRINCE, DANIEL (17th century) identified, 335 privy council, 66-68, 75, 76 problems in number theory, 545-553, 558, 573-592 proportionals mean, 39, 40, 42, 44, 269, 479, 480, 531 proportions, 267, 597, 598 in music theory, 115, 118-136, 138, 139 PTOLEMY, CLAUDIUS (857-1657), 84,

of the conchoid, 492 of the ellipse, 431, 433, 569, 605 of the hyperbola, 5, 431, 433, 454460, 482-491, 531, 532, 543, 544, 562, 564, 569-571, 597, 599, 604, 606 quantities heterogeneous, 414, 416, 417, 424, 431, 504, 506, 511 homogeneous, 410, 411, 414, 416, 417, 424, 425, 427, 431, 453, 504-506, 511, 514 infinite, 415, 416 RAHN, JOHANN HEINRICH (1622-76), 305 Rahn: Introduction to Algebra, translated . . . by T. Brancker ... Much altered and augmented by D[r]. Pfell]., 299, 302, 470, 526, 533 RAWLINSON, RICHARD (d. 1668) identified, 449 rectification, 10, 29, 430, 599 of the cycloid, 8, 11 of the parabola, 599 of the semi-cubic parabola, 21, 22, 204 REEVES, RICHARD (fl. 1641-79) identified, 80 Regula Falsi, 377 religion, 414 RENALDINI, CARLO (1615-98), 303 Renaldini: Ars analytica, 303 Restoration, 105, 217, 336, 536, 539, 540 RICCI, MICHELANGELO (1619-82), 567 Ricci: Exercitatio geometrica, 482, 526, 567 Riccioli: Almagestum novum, 208 RICHARD DE BELMEIS II (fl. 1152) identified, 236 ROBERTS, EDWARD (17th century), 602

116, 172, 249

Ptolemy: Harmonicorum, sen de musica lib. Ill, 115, 136

PYTHAGORAS of Samos (5807-5007), 115, 120, 121, 124, 125, 138 quadrants, 193, 281, 293 quadrature, 265, 269, 414, 599 of the circle, 25, 26, 268, 431, 433, 491, 544, 569, 600, 605, 607, 615 of the cissoid, 492

ROBERVAL, GlLLES PERSONNE DE (1602-

75), 9, 18, 20, 28, 29, 38, 189 ROOKE, LAWRENCE (1622-62), 38, 42, 677

Index: persons and subjects 50, 267 identified, 267 Royal Society of London, 50, 61, 72, 82, 84, 99, 101-108, 110-115, 117, 120, 137, 160, 162, 164, 165, 170, 171, 174, 175, 183, 184, 193, 200, 201, 209, 210, 221, 224-226, 231, 232, 238-243, 246, 249, 250, 252, 282, 284, 285, 287, 288, 292, 293, 296, 297, 323, 324, 327, 328, 354, 360, 366, 367, 376, 394, 432-437, 442, 454, 462, 481, 482, 490, 494, 495, 542, 569, 597, 599, 608 building for, 401 clerk, 201 council, 105, 112, 114, 162, 164, 184, 401, 493 fellows, 105, 319 foundation, 104, 105 president, 101, 105-107, 161, 164, 249, 287, 296, 297, 361 printers, 102, 107 secretary, 162, 195 RUPERT, PRINCE (1619-82), 187 identified, 61 RUVIGNY ET RAINEVAL, MARQUIS DE, see MASSUE, HENRI DE

Samarkand, 84, 103 SANDWICH, FIRST EARL OF, see MONTAGU, EDWARD Saturn, 12, 209, 210, 222, 233, 253 satellite, 209, 210, 222, 233, 253 SAVAGE, HENRY (16047-72), 147 identified, 145 SAVILE, HENRY (1549-1622), 415, 421, 536, 538 Savile: Praelectiones, 421 SAYE, ROBERT (d. 1691) identified, 197 SCHEUBEL (SCHEYBL), JOHANN (14941570) identified, 313 Seheubel: Algebrae eompendiosa faeilisque descriptio^ 313 SCHOOTEN, FRANS VAN (1615-60), 10, 12, 14, 23, 279, 575, 581 death, 21, 29 Schooten: Additamentum, 316 Sehooten: Appendix de cubicarum aequationum resolutione, 447, 474, 475 Schooten: Exercitationum mathematicarum libri quinque, 10, 299 Schooten: Prineipia matheseos universalis, 279, 306, 447, 474, 475 Scotland, 297, 494 SCOTT, ? (17th century), 407 sea charts, 189-193, 300, 318 semi-cubic parabola, 23 rectification, 21, 22, 204 Seneca: De otio, 92 series converging, 544, 569, 600 harmonic, 455 infinite, 299, 456, 457 Severn, river, 293, 297, 437, 438, 443, 445 sextants, 293 SHERLEY (SHIRLEY), THOMAS (1638-78) identified, 326 SINCLAIR, GEORGE (d. 1696) identified, 276

RUYTER, MlCHIEL ADRIAANSZOON DE

(1607-76) identified, 271 RYCKE, DIRK DE (1640-90), 602 identified, 568 St Andrews, 432 St Helens, 254 St John's wort, 535 SAINT-VINCENT, GREGOIRE DE (15841667), 413-415, 421, 423, 424 Saint-Vincent: Opus geometricum, 421, 543, 556, 569, 598, 605 Saintogne (French province), 236 SALUSBURY, THOMAS (17th century) identified, 276 Salusbury: Mathematical collections, 276

678

Index: persons and subjects Stubbe: The Remainder of the Observations, 470 Suetonius: Caesar, 169 sulphur, 327 Sun, 80, 164, 165, 175, 209, 210, 241, 243, 260-262 surds, 25, 26, 314 surgeons, 230 Sussex, 297 SUTTON, HENRY (d. 1665), 189, 193, 302 identified, 189

SLUSE, RENE FaANgois DE (1622-85), 2, 29, 360, 364, 430, 567 Sluse: Mesolabum, 2, 29, 430 SMITH, THOMAS (1638-1710) identified, 398 SMYTH, JEREMY (17th century) identified, 272 SOCRATES (c.469-399), 169 solid, 267, 416, 452, 453 SOMNER, WILLIAM (1598-1669) identified, 237 Somner: Dictionarium Saxonico-LatinoAnglicum, 237 soot, 95, 96 SORBIERE, SAMUEL (1615-70), 572 Sorbiere: Relation d'un voyage en Angleterre, 572 soul, 95 immortality, 94 Spain, 238 Spanish Provinces, 328 speech, 48, 49, 53, 58 organs of, 48, 49, 53, 54, 56-58 sphere, 317 segment of, 277, 280, 281, 609-614 spirals, 2, 4, 9, 17 SPITTY, HENRY (17th century), 536-538 Sprat: History of the Royal Society, 104 STADE, JAN (1527-79) identified, 85 statics, 105, 106, 202, 211, 247, 259-261 steel, 96 STENSEN, NIELS (1638-86) identified, 326 Stensen: Elementorum myologiae specimen, 326, 366, 401 STEVIN, SIMON (1548-1620) identified, 314 Stevin: Appendice algebraique, 314 Stevin: L'arithmetique, 314 STIFEL, MICHAEL (1487-1567) identified, 314 Stifel: Arithmetica integra, 314

SWAMMERDAM, JAN (1637-80), 327

Sweden, 329, 444 SWEIT, GILES (15867-1672) identified, 359 TACQUET, ANDRE (1612-60), 303, 414, 415, 421, 422, 567 Tacquet: Elementa geometriae planae ac solidae, 421 Tacquet: Opera mathematica, 303 TAMERLANE (1336-1405), 82, 84 tangents, 420 Tangier, 376 telescopes, 12, 30, 165, 172, 293 tendons, 98 Terence: Eunuchus, 473 THACKWELL, WILLIAM (17th century), 139, 143, 145, 147, 156 Thames, river, 234, 250, 253, 254, 436, 442 theologians, 105 theology, 422 THOMPSON, ANTHONY (d. 1665) identified, 189 Thraso, 473 thunder, 226, 227 tides, 200-223, 231-234, 238-244, 246262, 272, 297, 322, 323, 399, 401, 435-439, 441, 462-467 history, 203, 220, 241, 247, 255, 258, 259 observations, 204, 205, 216, 217,

679

Index: persons and subjects 234, 249, 250, 254, 258, 284, 285, 287, 288, 293, 294, 297, 322, 323, 407, 437, 439, 442446 periods of, 203, 205, 208, 210, 216, 219, 247, 248, 258-260, 286, 438 time, 416, 424, 452, 503, 505, 520 tin mines, 328 TITUS, SILAS (1622-1704) (B), 73, 287 TORRICELLI,

EVANGELISTA

Venus, 80, 83, 109, 209, 244 Vergil: Aeneid, 145 VIETE, FRANCOIS (1540-1603), 310, 317, 415, 421, 589 Viete: Apollonius gallus, 317 Viete: In artem analyticam isagoge, 310, 315, 447, 474 Viete: Opera, mathematica (ed. Schooten), 314, 447 Viete: Variorum de rebus mathematicis responsorum liber VIII, 421 VLACQ, ADRIEN (16007-67) identified, 12 Vossius, ISAAC (1618-89), 232, 246 identified, 241 Vossius: De motu marram et ventorum liber, 239, 241, 246, 252, 258, 259

(1608-47),

19, 29, 202, 324 transfusion, 204, 322, 324, 330, 376, 544 triangle, 45, 46, 461 TWYSDEN, JOHN (1607-88) identified, 85 Twysden, Roger: Historiae Anglicanae Scriptores X, 237 TYCHO BRAKE (1546-1601), 84, 85, 172, 173, 175, 211

WALLIS, ANNE (b. 1656), 37 WALLIS, ELIZABETH (1658-1700), 37 WALLIS, HENRY (c. 1620-66), 88, 288 WALLIS, JOHN (1616-1703), 1, 2, 8-10, 18, 19, 31, 250, 330, 448, 491, 492, 526, 535, 566, 595 astronomical observations, 171-174, 176-178 attendance at meetings of the Royal Society, 288, 293, 437, 442 barometrical observations, 282, 283 burglary in his house, 292, 299 edition of Horrox's astronomical papers, 106, 108-112, 114, 160162, 164, 165, 182, 184, 225, 441 hypothesis of the tides, 200-223, 231-234, 238-244, 246-262, 272, 285, 286, 288, 293, 294, 323, 377, 397, 399, 436, 437, 440, 442 instruction of deaf-mutes in speech, 48-63, 69, 72, 73 visit to his brother Henry, 90 visit to Kent, 288, 289 visit to London, 81, 193, 250, 280,

UBALDINI, CARLO (17th century), 325, 399 identified, 319 ULUG BEG (1394-1449), 83,101, 111, 113 identified, 82 astronomical tables, 83, 103 catalogue of fixed stars, 84, 101, 103, 104, 107, 108 translation and printing of his astronomical institutions, 102, 103, 106-108, 111, 112, 114, 116, 117, 165 United Provinces, 328 universities, 312 VALERIC, LUCA (1552-1618) identified, 276 Valeria: Subtilium indagationum liber primus, 276, 280 VAULEZARD, JEAN Louis DE (fl. 1630) identified, 315 Vaulezard: Introduction en I'art analytique, 315

680

Index: persons and subjects 285, 289, 325 work as a decipherer, 185, 186 Wallis: A Continuation of . . . his second Letter, 498, 565, 572 Wallis: A Letter . . . concerning the said Doctor's Essay of Teaching a person Dumb and Deaf, 51, 60 Wallis: A Letter . . . concerning the Variety of the Annual High-Tydes, 436 Wallis: A Relation of an Accident by Thunder and Lightning, 226 Wallis: A second Letter . . . on the same printed Paper of ... Du Laurens, 498, 565, 572 Wallis: Adversus Marci Meibomii de proportionibus dialogum tractatus elenchticus, 28, 413, 454, 598 Wallis: An Appendix . . . Being an Answer to some Objections, 241, 251, 270-272, 397 Wallis: An Essay ... exhibiting his Hypothesis about the Flux and Reflux, 223, 251, 270-272, 285, 377, 397, 436, 442 Wallis: Animadversions ... upon Mr Hobs's late Book, 263, 270-274, 397 Wallis: Another Letter ... concerning some Mistakes, 446, 472, 572 Wallis: Arithmetica infinitorum, 18, 25, 26, 28, 308, 430, 447, 455, 474, 475, 485, 555, 556, 585, 599 Wallis: Autobiography, 48, 73, 104 Wallis (ed.): Claudii Ptolemaei harmonicorum libri tres, 115, 136 Wallis: Commercium epistolicum, 7, 449, 476, 478, 543, 547, 549, 560, 569, 571, 574, 605 Wallis: Cono-cuneus, 69 Wallis: De angulo contactus, 410, 411, 413-415, 427, 572 Wallis: De loquela, 58, 62, 63 Wallis: Defence of the Royal Society, 48, 50, 51, 68, 72, 74

Wallis: Defense of the Treatise of the Angle of Contact, 412, 413 Wallis: Discourse of Combinations, 470, 533 Wallis: Elenchus geometriae Hobbianae, 18, 28 Wallis: Exercitationes tres, 354 Wallis: Grammatica linguae anglicanae, 58, 62, 63, 429 Wallis: Hobbius Heauton-timorumenos, 38, 42, 47, 48, 50, 52, 79, 265, 267-269, 598 Wallis: Logarithmotechnia Nicholai Mercatoris: discoursed of in a letter, 482, 541, 599 Wallis: Mechanica I, 533, 562, 563, 596, 601, 604, 606 Wallis: Mechanica II, 28, 313 Wallis: Mechanica III, 12 Wallis: Primam inventionem et demonstrationem, 23 Wallis: Some Animadversions . . . on a printed Paper, 471, 492, 565, 572 Wallis: Tractatus duo, 8, 9, 12, 13, 18, 21, 23, 27, 28, 430, 450, 571, 599, 602 Wallis: Treatise of Algebra, 21, 69, 73, 526, 606 Wallis: Treatise of Angular Sections, 606 WALLIS, JOHN JR (b. 1650), 37 WALLIS, REBECCA (d. 1677), 87-89, 91 WALLIS, SUSANNA, nee CLYDE (162287), 143, 170, 180, 181, 289 Walton (ed.): B^blia Polyglotta, 398 war Anglo-Dutch, 240, 249, 270-272, 285, 287 Russo-Polish, 64 WARD, SETH (1617-89), 82, 178, 187, 296, 593 WARWICK, MARY, COUNTESS OF, nee BOYLE (1624-78), 283 Warwickshire Nuneaton, 304

681

Index: persons and subjects WASSENAER, JACOB VAN (16077-40?),

WILKINS, TIMOTHY (1617-71), 335 identified, 78 WILLIS, THOMAS (1621-75), 90,176,180, 181, 204, 228, 230, 593 identified, 88

316 Wassenaer: Den on-wissen wiskonste-

naer, 316 WATKINS, RICHARD (d. 1708), 592, 594 identified, 593 Watkins: News from the Dead, 593 weight, 416, 424, 452, 506, 520 WHALLEY, DANIEL (17th century), 4850, 56, 58-62, 69, 72 WHALLEY, PETER (d. 1656) identified, 60 WHARTON, ANNE (d. 1692) (B), 62, 73 WHITE, SAMPSON (17th century), 31 WILHEM, MAURITS LE LEU DE (b. 1641), 467, 568 identified, 468 WILKINS, JOHN (1614-72) (B), 78, 103,

Witt: Elementa curvarum linearum, 7, 10 WORTHINGTON, JOHN (1618-71), 165

identified, 161 WREN, CHRISTOPHER (1632-1723) 8, 11, 23, 108, 109, 112, 114, 178, 244, 274, 284, 319, 430, 451, 536, 538, 593, 599 WREN, MATTHEW (1629-72) (B), 149, 156, 185, 186 WRIGHT, EDWARD (1558-1615), 191, 192

(B), 175, 449141, 189,

YATE, THOMAS (1603-81) (B), 30, 37, 139, 143, 145, 150, 154-156 YATES, PETER (17th century), 534, 535

287, 479, 481, 492, 496, 567, 593 Wilkins, John: Essay towards a Real Character, 481, 493

682