A Discourse Concerning Algebra: English Algebra to 1685

A Discourse Concerning Algebra

The original ‘discourse concerning algebra’; A treatise of algebra, both historical and practical by John Wallis (1685).

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship. and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi São Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press. 2002 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2002 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 A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data Stedall, Jacqueline. A discourse concerning algebra: English algebra to 1685/Jacqueline Stedall. Includes bibliographical references and index. 1. Algebra–England–History I. Title. QA151.S74 2002 512.0942-dc21 2002029001 ISBN 0 19 852495 1 (acid-free paper) 10 9 8 7 6 5 4 3 2 1

A Discourse Concerning Algebra ENGLISH ALGEBRA TO 1685

Jacqueline A. Stedall Clifford Norton Student in the History of Science, The Queen's College, Oxford and Open University

Dedication For my father, who understands how the past lives in the present, and in memory of John Fauvel

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Preface I begin with thanks to my husband, a reader of book reviews, who thought that The history of mathematics: a reader, edited by Fauvel and Gray, might interest me. Little did he know what he was setting in motion, but I am deeply grateful for the amicable, efficient and generous way that he has adapted to the subsequent changes in both our lives. My children have spent most of their teenage years with the writing of this book, and know its contents better than anyone. Every author should have such witty and articulate critics and such loyal support. Their unquestioning faith in my ability to complete the book sustained me more than they could know, and I would like them to know how much it was written for them. The research on which this book was based was funded by an Open University Studentship under the supervision of the late John Fauvel. I am proud to be associated with the Open University and the Centre for the History of the Mathematical Sciences, and I would like to thank especially June Barrow-Green, who encouraged me to begin this work and who, with Jeremy Gray, has supported me in every conceivable way throughout. I thank also David Clover, Liz Scarna and Cheryl Manning who have provided me with invaluable technical and administrative support. Amongst historians of mathematics I have found friends and colleagues second to none. Philip Beeley, Henk Bos, Stephen Clucas, David Fowler, Ivor Grattan-Guinness, Jan van Maanen, the late George Molland, Peter Neumann, John North, Steve Russ, Christoph Scriba, Muriel Seltman and Tom Whiteside have all given me generous support, and I hope that each of them will recognize behind the alphabetical list of names the gratitude, respect and affection I feel for them. I am grateful too to the many librarians and archivists in academic libraries, Public Record Offices and church archives who have assisted me. In particular I thank the counter staff of Duke Humfrey's Library, Oxford, who have fetched and carried for me more volumes than they or I would care to count, and who have been endlessly patient and helpful in guiding me amongst the rich but sometimes buried treasures of the Bodleian Library.

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During the final months of completing this book I have been privileged to hold the Clifford Norton Studentship in the History of Science at The Queen's College, Oxford. Once again I hope that an expression of collective gratitude, to the Provost, Fellows and staff, will be read as many individual notes of thanks: my time at Queen's has been not only some of the most productive of my life but also the happiest. It is a source of immense sadness to me that John Fauvel did not live long enough to see the completion of a book he did so much to encourage. He read every page of it in one draft or another, and never failed to respond with stimulating and constructive comments. Many of those comments, whether as to theme and structure, or the correct placing of commas, are incorporated into the text. But I hope that the book captures John's spirit in other ways too: his sense of mathematics as one of the great fields of human endeavour, and the history of mathematics as the story of those who have loved and created the subject. Above all, I would like to think that discernible traces of John's enthusiasm, humour and deep understanding of life and human nature might live on through the pages of this book. The Queen's College, Oxford J.A.S. April 2002

Acknowledgements Chapter 3 is based on a paper originally published in Annals of Science57 (2000), 27–60, see http://www.tandf.co.uk, and Chapter 7 on papers originally published in Notes and Records of the Royal Society54 (2000), 293–316; 317–331. Some of the material in Chapters 4 and 6 was originally published in Archive for History of Exact Sciences54 (2000), 455–497 and 56 (2001), 1–28. Figure 1.2 is reproduced by permission of The Examination Schools, Oxford; Figs 2.1, 2.2, 3.4, 3.5, 3.6, 3.7 by permission of The Bodleian Library, University of Oxford; Fig. 2.5 by permission of Irene Barton; Figs 4.2, 4.3, 4.4, 4.5, 5.1 by permission of The British Library; Fig. 4.6 by permission of Lambeth Palace Library; Fig. 5.3 by permission of Lincoln Cathedral Library; Fig. 6.8 by permission of the Syndics of Cambridge University Library; Fig. 7.1 by permission of The Royal Society.

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Contents 1 A large discourse concerning algebra English mathematics before 1649 A treatise of algebra 2 How algebra was entertained and cultivated in Europe ‘In Arabic it is called al-jabr’ ‘The Italians have given it the name of Regula cosae’ ‘Cardan gives it the name of Ars magna’ ‘What we commonly call algebra is by a Greek name called Analysis’ Wallis's account of early algebra 3 Ariadne's thread: William Oughtred's Clavis The first edition: 1631 The second editions: 1647 and 1648 The third edition: 1652 The fourth edition: 1667 Wallis and the Clavis The final editions: 1693 onwards 4 Rob'd of glories: Thomas Harriot and his Treatise on equations The Treatise on equations Harriot's Will and the writing of the Praxis The contents of the Praxis The Corrector and the Summary

1 5 8 19 35 36 45 50 52 55 59 65 68 73 77 82 88 94 97 100 107

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The fate of Harriot's material after 1631 Wallis's account of Harriot's algebra 5 Moving the Alps: uncovering the mathematics of John Pell An introduction to algebra Pell and Wallis 6 Reading between the lines: John Wallis's Arithmetica infinitorum Squaring the circle Praise and criticism The enlargement of the mathematical empire 7 Catching Proteus: the mathematics of William Brouncker Squaring the hyperbola and the circle The challenges from Fermat 8 ‘Many pretty things worth looking into’ Wallis's perspective on algebra Wallis's perspective on history Notes Bibliographies Primary sources: manuscripts Primary sources: printed books Secondary sources Index

111 117 126 135 139 155 156 165 173 183 185 196 208 212 215 219 261 261 264 276 289

1 A large discourse concerning algebra Without algebra, modern mathematics and science would be, literally, unthinkable. Algebra as a discipline, a tool and a language evolved gradually over many centuries and in different mathematical cultures, but emerged in forms we recognize and use today in western Europe in the sixteenth and seventeenth centuries. In 1685 John Wallis, Savilian Professor of Geometry at Oxford, published the first substantial history of the subject. He described his book as ‘a large discourse concerning algebra’,1 and gave it the title A treatise of algebra, both historical and practical, shewing the original, progress, and advancement thereof, from time to time; and by what steps it hath attained to the heighth at which now it is (Frontispiece). It was soon apparent to its readers that as history Wallis's treatise was neither general nor unbiased. Wallis devoted most of its four hundred pages to English mathematics alone and in particular proclaimed, sometimes to an apparently absurd degree, the achievements of his immediate predecessors and contemporaries. The account could not be seen as anything but partial, in both senses of the word, and soon ceased to be regarded as a serious piece of historical writing. Three centuries later it is possible to view Wallis's text in a new light, and to see what were formerly considered its weaknesses as perhaps also its strengths. It can be read not just as history but as an invaluable compendium of work by seventeenth-century English mathematicians, both well-known and obscure, some of whom made original and important contributions to the development of algebra. In concentrating on English advances Wallis was telling a story that had not been told before and has hardly been addressed since.2 He was writing from first-hand experience, about mathematicians he knew personally and often about mathematics he himself had helped to

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develop, so that the contents of the book provide a fascinating contemporary view of the evolution of algebra in England in its crucial formative period. The present book takes up Wallis's history afresh, with the aim of combining the first-hand quality of his account with the benefits of hindsight. It explores Wallis's treatise from his opening discussion of English mathematics during the medieval period to his inclusion of some of Newton's results on infinite series, then the latest work available, and at every stage investigates his sources and his interpretations. It also supplements Wallis's account with the findings of new research and modern scholarship and, in particular, provides detailed studies of the work and influence of William Oughtred (1573–1660), Thomas Harriot (c. 1560–1621), John Pell (1611–1685), William Brouncker (c. 1620–1684) and of Wallis himself. This book is, in other words, a new ‘discourse concerning algebra’, and a reappraisal of those who helped to bring about the seventeenth-century renaissance of English mathematics. John Wallis (Fig. 1.1) was born on 23 November 1616 at Ashford in Kent.3 His father, a clergyman, died when John was just six years old. Three years later Ashford was afflicted by plague, and John was sent by his mother to a tutor in Tenterden about twelve miles away, where he remained until at the age of fourteen he spent a year at Felsted school in Essex. As was usual at the time, such an education gave him a thorough grounding in Latin (which he wrote as easily as English) and also in Greek and some Hebrew, but he learned elementary arithmetic only from the textbooks and instruction of a younger brother who was preparing to go into trade.4 At Emmanuel College, Cambridge, from 1632 to 1640 he studied among other things some astronomy, but claimed that he pursued mathematics only as ‘a pleasing Diversion’.5 In his autobiography, written sixty years later, he wrote that ‘Mathematicks, (at that time, with us) were scarce looked upon as Academical Studies’,6 but he may have moved in relatively limited circles, for John Pell as a student at Cambridge five years before Wallis, devoted much of his time to mathematics, as did Charles Scarborough and Seth Ward a few years later.7 After his ordination in 1640 Wallis was employed as a private chaplain. Always inclined towards the puritan tendency in the English church, he was appointed in 1644 as one of the secretaries to the Westminster Assembly of Divines, a body set up to resolve the problems of church government arising from the abolition of the episcopacy in 1643. The outbreak of civil war in 1642, however, had already changed Wallis's fortunes in more important ways. In the first year of the war he was shown a ciphered letter in the possession of Sir William Waller, then a colonel in the Parliamentary army.

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Fig. 1.1 John Wallis (1616–1703) by Sir Godfrey Kneller. The portrait was commissioned by Samuel Pepys and now hangs in the Examination Schools in Oxford.

Wallis broke the cipher with ease and such letters continued to come his way not only during the war years but for the rest of his life. In 1653 he was confident about the significance of his work: ‘I do not know that there hath been any [letter] deciphered save those that came to my hands; and I believe that if those had escaped my hands, they had likewise escaped that danger’.8 In 1648 the University of Oxford was purged of its Royalist supporters, including both the Savilian mathematical professors, John Greaves (astronomy) and Peter Turner (geometry). Seth Ward was given the chair in

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astronomy and in 1649 Wallis was rewarded for his loyalty to Parliament with the chair in geometry.9 Self-taught from his brother's arithmetic books and Oughtred's Clavis mathematicae, Wallis knew little enough mathematics when he was appointed, but probably not much less and perhaps rather more than most of his contemporaries. After the disruption of the war years Wallis's first great contribution to English mathematics was the new stability and seriousness he brought to the Oxford post. He helped to create an atmosphere in which mathematics was valued and could flourish, and with dedication and commitment rapidly established himself as one of the leading mathematicians of the day. Wallis did his most creative mathematical work within ten years of his appointment. By 1657 he had published a comprehensive textbook on arithmetic that also includes some historical sections, his Mathesis universalis sive arithmeticum opus integrum (1657); the first systematic algebraic treatment of conics, De sectionibus conicis (1656); and a book that was to be particularly influential, his Arithmetica infinitorum (1656), in which he explored methods of finding areas and volumes by the summation of infinite sequences. These books and some smaller pieces, including his inaugural lecture, were printed together in two volumes of collected works, the Operum mathematicorum (1656–57). Shortly afterwards, Wallis worked on properties of the cycloid and cissoid (Tractatus duo de cycloide et… de cissoide, 1659) and with Brouncker on what later came to be known as number theory (Commercium epistolicum, 1658).10 During the 1650s he enjoyed other fruitful collaborations with Brouncker and also began a lifelong mathematical friendship with John Pell. During the 1660s Wallis's output slowed a little but he published his Mechanica sive de motu tractatus (1669–71) and wrote many shorter pieces in the form of letters or treatises, often in response to particular problems. In the early 1670s he wrote the greater part of A treatise of algebra (though it was not published until 1685) and in his later years he edited a number of Greek mathematical and musical texts from manuscript. He was a tireless letter writer and corresponded from time to time with the finest mathematicians on the continent, from Fermat in the 1650s to Huygens and Leibniz in the 1690s.11 During the 1690s all Wallis's publications and some of his letters were reprinted in three volumes to form a second set of collected works, the Opera mathematica (1693–99). Volume II was published first, in 1693, and included A treatise of algebra now revised and extended and translated into Latin. Long and successful as it was, Wallis's career was not without personal and mathematical controversy. In 1657 he was appointed Custos archivorum

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(Keeper of the University Archives) despite protestations from those who argued that this was incompatible with the Savilian statutes. A year earlier Wallis had become embroiled in what was to become a long-running and bitter dispute with Thomas Hobbes arising from their very different concepts of politics, theology and mathematics.12 In 1658 he engaged in acrimonious discussions with Fermat and Frenicle on number problems and quarrelled with Blaise Pascal over the latter's dismissive treatment of his work on the cycloid,13 and in 1667–68 he went to considerable lengths to contest the work of two otherwise little known French mathematicians Francis Dulaurens14 and Vincent Leotaud.15 Against such behaviour must be set Wallis's generosity and loyalty to many English mathematicians both well-known and obscure, for Wallis defended his friends as ardently as he attacked his enemies. In particular, he was consistent in his concern that worthwhile results should be published and recognized, and nowhere is this more evident than in A treatise of algebra written almost entirely in support of English mathematics and mathematicians. To see Wallis's text only as polemic, however, is to underestimate the strength of his feeling for mathematics as a historical subject. His interest in history was already evident in his inaugural lecture and in his Mathesis universalis of 1657 and was to continue throughout his life: he was still actively pursuing historical research not long before he died in 1703 at the age of 86. During the first thirty years of his professorship, he witnessed nothing less than a revolution in the nature and scope of mathematics as it liberated itself from the constraints of its Classical past and began to take on a life and momentum of its own. A few years before he died Wallis wrote:16 ‘It hath been my lot to live in a time, wherein have been many and great Changes and Alterations’, and though he was speaking of political change his words applied as aptly to mathematics. With his keen sense of the importance of recording both the past and the present, Wallis would have been one of the first to recognize the value of setting down what he knew for the benefit of his contemporaries and posterity.17

English mathematics before 1649 When Wallis gave his inaugural lecture as Savilian Professor in 1649 England was far from the forefront in algebra or any other branch of mathematics. In the immediate past the country had suffered six years of civil strife that had brought normal academic activity almost to a standstill, but the malaise

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in English mathematics went back far beyond that, for England had fallen outside the mainstream of European mathematical activity for almost three centuries. It had not always been so, for during the thirteenth and fourteenth centuries the University of Oxford had boasted a large number of natural philosophers and astronomers who explored a mathematical view of the universe. After about 1400, however, Oxford's pre-eminence faded and by the seventeenth century the names of some of the finest English medieval mathematical scholars were all but forgotten. During the fifteenth and sixteenth centuries England can only be described as a mathematical backwater, particularly in relation to the development of algebra, the subject with which this book is chiefly concerned. Although the first translation of algebra from Arabic to Latin was done by an Englishman, Robert of Chester, in Spain around 1150, Robert's translation was unknown in England until many centuries later. Instead, as will be described more fully later in this book, Arabic algebra was taken up first in Italy and then Germany, where it became known as the Cossick Art, until by the sixteenth century it was known in every country in western Europe. Even England produced a single cossist writer, Robert Recorde (c. 1510–58), whose Whetstone of witte was published in 1557. In England, however, the Whetstone was the first and last new text devoted to algebra to appear until 1631, and up to the closing years of the sixteenth century the advances made by Cardano and others in Italy from about 1540 onwards remained unknown. During the 1590s the French mathematician François Viète (1540–1603) brought together the two major strands of European algebra: the Arabic-cossist ideas that had been in circulation for four centuries, and the revolutionary new methods of equation-solving developed in Italy. Viète's real genius, however, was in seeing how to use those ideas to illuminate the newly available Greek texts of Diophantus and Pappus. For Viete, algebra was the key to understanding and reconstructing Greek thought, the Analytic Art that would open up Greek mathematics. The profound influence of Viète's work can be seen in the achievements of the two greatest French mathematicians of the early seventeenth century, Pierre de Fermat (1601–1665) and René Descartes (1596–1650), both of whom began, as Viète did, with the aim of exploring Classical mathematics through analytic methods. In England, however, analytic or algebraic geometry, so central to continental mathematics, found few serious followers. To an observer at Wallis's inaugural lecture the prospect of England catching up in mathematics with France or the Netherlands must have

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seemed remote. Yet within thirty years a revolution had taken place and English mathematics was among the most advanced in Europe. How did such rapid change come about? Some of the first signs were already discernible in the early years of the seventeenth century. As early as 1600 Harriot had mastered Viète's algebra and had taken it in important new directions, quite different from those followed by Fermat or Descartes, and although Harriot himself never published any of his mathematics his papers circulated privately and some of his algebra was published posthumously in 1631. In that same year Oughtred brought out his Clavis mathematicae, an elementary but influential textbook on arithmetic and algebra, and Oughtred's algebra, like Harriot's, was based on Viète's. Harriot's researches on mathematics, astronomy and optics were carried out under the patronage of Henry Percy, Earl of Northumberland, while Oughtred was a country clergyman, employed by local gentry as tutor to their sons, and particularly encouraged and supported by Thomas Howard, Earl of Arundel. Thus at a time when paid posts in mathematics were almost non-existent the patronage of Percy and Howard did much to enable mathematical studies to flourish. Charles Cavendish (1591–1645) was a third member of the English aristocracy who played a vitally important role in nurturing mathematical activity by collecting, copying and disseminating new ideas.18 Cavendish spent time in France where he became acquainted with the most eminent mathematicians of the day,19 and brought the best of continental mathematics back to England. Oughtred read Viète's De aequationum recognitione in a copy owned by Cavendish,20 and it was also Cavendish who some years later showed Oughtred the ideas of Bonaventura Cavalieri (1598–1647), and so introduced into England another important strand of mathematical thought. English mathematics owed a great deal to this small group of learned aristocrats, Percy, Howard and Cavendish, but at the same time there were moves towards a new professionalization of the subject. Gresham College was founded in 1597 to provide public lectures in seven subjects including geometry and astronomy, and in 1619 Henry Savile (1579–1632) founded chairs of geometry and astronomy at Oxford. Henry Briggs (1561–1631), now remembered for his work on the calculation of logarithms, was appointed as the first Savilian Professor of Geometry and John Bainbridge (1582–1643) as the Professor of Astronomy. Thus during the first half of the seventeenth century several important developments were under way. Mathematics was becoming increasingly

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respected as a university discipline and the work of Harriot and Oughtred gave new impetus to mathematical study, particularly to algebra. Eventually ideas from abroad, passed at first only from one individual to another, began to become more easily available: Cavalieri's theory of indivisibles became more widely known through Torricelli's Opera geometrica of 1644, and in 1646 Francis van Schooten published the first complete edition of the work of Viète, followed by a Latin translation of Descartes' La géométrie in 1649. From the 1650s onwards these factors combined to produce an explosion of creativity in English mathematics, and right from the beginning Wallis was at the heart of it.

A treatise of algebra Wallis first stated his intention of writing a textbook on algebra in 1657 at the end of his Mathesis universalis, where he explained that he had hoped to include the ‘doctrine of analysis, the perfection of arithmetic’ but that he had already written more than he intended. Rather than give too short an account of ‘analysis’, or algebra, he thought it better to devote a separate volume to the subject, which he proposed to do. Given Wallis's prolific output on other topics it is perhaps not surprising that the volume did not materialize, and there was no further mention of it until some ten years later. To understand how A treatise of algebra eventually came to be written in the early 1670s, and why it was not printed until 1685, it is helpful to know something about the state of English mathematical publishing during those years, particularly with respect to algebra. Much of our information on the subject comes from John Collins (1625–1683), government clerk and mathematical enthusiast, and a Fellow of the Royal Society from 1667. Collins not only collected and circulated information about new mathematical texts from abroad, acquiring copies whenever possible, but also did his best to encourage new publication at home. Modest, and untiring in his efforts for others, he is one of the most likeable figures of the seventeenth-century mathematical scene, and has contributed invaluably to our knowledge of the period. In the Netherlands, the publication of Van Schooten's Latin translation of Descartes' La géométrie in 1649 had led to a wave of new results, many of which were included in Van Schooten's second, enlarged edition of 1659–61. The 1660s saw the publication of a number of books on Cartesian algebra by Dutch and French writers, but in England over the same period there was almost nothing new. Instead there were attempts to remedy the dearth

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of good books by reprinting or translating existing texts: Oughtred's Clavis mathematicae, almost forty years old, was republished in a fourth Latin edition in 1667, and Johann Rahn's Teutsche Algebra of 1659 was translated into English and published in 1668 as An introduction to algebra with new material added by John Pell. Adding new material to existing texts served two purposes: it introduced the latest ideas in less time than it took to write a new book, but it also enabled the publication of problems, theorems or short treatises on the back of established texts. Collins' letters are full of suggestions as to what pieces of work might be added to this or that edition, and he was particularly keen to make more and better algebra available to English readers. In 1667, for instance, he suggested that part of Kinckhuysen's Algebra ofte stel-konst (1661) should be appended to the Pell and Rahn Introduction to algebra then in the press.21 When this plan failed Collins put forward a different idea, namely that Kinckhuysen's text should be printed in full along with extracts from another Dutch algebra, Ferguson's Labyrinthus algebrae (1667), and that Wallis might assist in the editing.22 This too came to nothing, as did the next proposal, that Kinckhuysen's text should be published with notes added by Newton.23 The most ambitious of these schemes, again originating with Collins, but also supported by William Brouncker, President of the Royal Society, was to follow up Kersey's Elements with a further volume containing results discovered by Dutch or French mathematicians: Hudde, De Beaune, Bartholin, Dulaurens, Kinckhuysen, Ferguson, Brasser and Verstay.24 Wallis was again invited to advise and assist, though the work of transcribing was to be done for £6 by a friend of Collins, the impoverished Michael Dary. Wallis was willing, though he hoped that ‘we find not a stop at the press, which we meet with too often in mathematical books’.25 Wallis's remark about the press points to a fundamental reason for the shortage of new books, the reluctance of publishers to undertake mathematical texts which incurred high costs and offered low returns. English booksellers had lost heavily even on the works of such respected authors as Wallis and Barrow and were unwilling to take further risks.26 There was talk for a while of getting some of Wallis's works reprinted in the Netherlands,27 but probably the situation was no easier there (and according to Collins the situation in France was even worse).28 Wallis, Barrow and others had adopted the habit of depositing their papers with the Royal Society until an opportunity for publishing arose, and by April 1677 Collins had held some of Wallis's papers for several years.29

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We do not know exactly when Wallis began to write A treatise of algebra. In 1668 he had written that ‘an Introduction to Algebra I have not yet ready’,30 and it seems unlikely that he would have agreed to assist Collins and Dary in 1672 if he was already engaged on a major work on algebra of his own, so we may suppose that the book was not begun in earnest until 1673 and that Wallis continued to work on it until he delivered it to Collins in 1677. The dating 1673 to 1677 is confirmed by a number of mathematical letters that Wallis wrote in response to queries from Collins in 1673 and 1676, and then included in A treatise of algebra.31 Newton's Epistola prior and Epistola posterior were written in June and October 1676, and the final third of A treatise of algebra, in which extracts from Newton's letters are embedded, may have been written in its entirety in the winter of 1676–77. Collins died in 1683 but by then the Royal Society had promised to underwrite the publication of A treatise of algebra and a deal had been negotiated with Richard Davis, an Oxford bookseller, who agreed to handle it if sufficient sales were guaranteed. Down payment on 100 copies seems to have been the necessary level of support,32 and the Royal Society undertook to buy 60 copies at 1½d per sheet and invited further subscriptions at the same rate. A Proposal to publish A treatise of algebra was circulated in 1683 (Fig. 1.2); it invited subscribers to send a deposit of five shillings before December 1683 and promised to print at a rate of two sheets a week from 1 August 1683.33 (The book eventually required four quires, or 96 sheets, of paper, and so cost twelve shillings to subscribers but sixteen shillings or more to later buyers.) The book was printed by John Playford in London, who possessed the necessary range of type,34 and it was eventually completed not in 1684, as had been hoped, but in 1685, some twelve years after Wallis first began to write it.35 The delay between writing and printing gave Wallis plenty of opportunity to add to his text and he continued to do so up to the last possible moment, the final ‘Additions and Emendations’ being inserted in 1684 when much of the book was already printed. The opportunity to publish was too valuable to waste and prompted Wallis to include as much as he could of the work deposited with Collins, whether directly related to algebra or not. Some of it was relegated to appendices, but some was integrated at what Wallis considered appropriate points in the main text. This makes A treatise of algebra read at times like an anthology of results and ideas with little obvious relation to one another. Running through the text, nevertheless, is a clear historical thread. At times it seems in danger of vanishing but Wallis always managed

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Fig. 1.2A Proposal About Printing A treatise of Algebra, circulated in 1683.

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to bring it back into focus: the history that was incidental to his Mathesis universalis thirty years earlier had now become his main theme. A unique combination of circumstances in seventeenth-century Oxford made Wallis's history possible. From the opening of the Bodleian Library in 1602 there had been energetic and wide-ranging efforts to collect and preserve texts from England and abroad,36 and Wallis had access to an unprecedented accumulation of books and manuscripts. First and foremost, as a Savilian professor, he had the use of the Savile collection, originally given by Henry Savile when he founded his two mathematical chairs in 1619. Savile had donated his personal library of mathematical books, notes and manuscripts: sixteenthcentury printed texts, about forty handwritten volumes containing copies of Greek or Latin texts, and a few important volumes of medieval manuscripts. All the seventeenth-century Savilian professors, including eventually Wallis himself, added generously to the Savile Library, making it the best collection in England, or perhaps anywhere, of mathematical texts up to 1700. Wallis knew the library thoroughly and his annotations are to be found frequently in both the books and the manuscripts.37 Another great collection of mathematical manuscripts was given to the Bodleian Library in 1634 by Sir Kenelm Digby (1603–1665), then a naval commander, later a diplomat. Over half of Digby's collection had been bequeathed to him by his former tutor, the Oxford mathematician Thomas Allen (1542–1632), who had rescued some of the mathematical manuscripts that Merton College had disposed of after the reformation, so the Digby collection was (and remains) a particularly rich source of medieval mathematics.38 William Laud (1573–1645), Archbishop of Canterbury and Chancellor of Oxford University from 1629 to 1645, almost doubled the Bodleian Library's manuscript holdings by donating his own collection, which included Greek mathematical texts acquired for him by John Greaves, later Savilian Professor of Astronomy.39 Further manuscripts were given in 1659 by John Selden (1554–1654),40 and in 1683 by Elias Ashmole (1617–1692), the latter reflecting Ashmole's special interest in alchemy and astronomy.41 In addition, the 1650s saw a rapid proliferation of oriental studies in Oxford, with new translations by Edward Pococke and John Greaves.42 Wallis knew his Classical sources thoroughly but also recognized the debt of European mathematics to Indian and Islamic sources, and a subject he explored in depth was the spread of learning from Islamic Spain to northern Europe during the twelfth and thirteenth centuries. He also read widely beyond mathematics, and his longstanding interests in grammar, etymology, cryptanalysis, music,

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astronomy, calendar reform and general history all informed his historical writing. The concentration of such a wealth of material in Oxford both reflected and encouraged new attitudes to historical study. Histories of mathematics before Wallis had generally adopted what may be described as the ‘list of names’ approach, in which chronological lists of writers were presented with little or no reference to culture or historical context. The underlying assumption was that mathematical knowledge derived from divine revelation or ancient authority, so that the history of mathematics was essentially the handing on (traditio) of such knowledge from one generation or culture to the next. Thus the earliest post-Classical histories, those of Isidor of Seville (570–636),43 and Bede (672–735),44 asserted that mathematics was handed from the Babylonians or the Hebrews to the Egyptians and thence to the Greeks. Later medieval accounts became gradually more sophisticated but presented much the same story. By the thirteenth century Roger Bacon (c. 1214–1292) saw the history of science as a process of decline in which ancient knowledge was occasionally recovered only to be lost again.45 Two centuries later, in 1464, Johannes Regiomontanus (1436–1476) wrote a history of mathematics in which the main theme was not change, but the continuity and stability of mathematics as handed from one mathematician to another.46 Sixteenth-century writers continued in the same style: in 1553 Girolamo Cardano (1501–1576) produced a list of twelve great mathematical or scientific writers, in which six Greeks, two Britons, a Roman and three Arabs were ordered by eminence rather than chronology, and without any suggestion of historical development or context. Bernardino Baldi (1553–1617), in his Vite de matematici written almost at the end of the sixteenth century still conveyed mathematics as a continuous tradition running from the Babylonians (Chaldeans) and Egyptians through the Greeks, Romans and Arabs to his own time, in which the greatest achievement was the restoration of Archimedes,47 and his Cronica de matematici listed an unbroken line of mathematicians from 600 BC to 1596.48 A similar list was compiled by Henry Savile in Oxford in 1570 and is preserved in the Savile Library. Savile's list began with the sons of Seth and continued through the Druids and Zoroastrians to Abraham, Joseph, Homer and Pythagoras before reaching the firmer historical ground of Classical Greece.49 Only very gradually did there begin to emerge ideas of mathematical progress, a sense that later mathematicians could add to or even improve upon the existing body of knowledge.50 Writing in the 1640s, the Dutch scholar Gerard John Vossius (1577–1649) still presented

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the various branches of mathematics as existing largely independently of age or culture, so that the detailed tracing of ideas was less important than identifying those who preserved and handed on the tradition. Sometimes, especially in the early chapters of A treatise of algebra, Wallis subscribed to the same approach: thus we see him in his second chapter producing a list of Arab writers, all of them gleaned from Vossius's De scientiis mathematicis (1650) (Fig. 1.3).51 In Wallis's copy of De scientiis mathematicis the pages on Arab astronomy are particularly heavily annotated and it is clear that he read them carefully, but he also extracted Arab writers from other sections of the book and ordered the list more or less chronologically for his own account.52 He did so, however, without giving any information as to when or where any of the writers lived, or what they wrote about. Later Wallis provided a similar indiscriminate list, also compiled largely from Vossius, of medieval English writers. Yet at the same time he began to tease out for himself the story of the origin and spread of the Hindu-Arabic numeral system. Thus we see side by side with the traditional ‘list of names’ approach, the beginning of a modern historiography of mathematics, with careful reading of primary sources and attention to the emergence and spread of new ideas. When it came to his own day, it was quite clear to Wallis that mathematics, far from being a relatively fixed body of knowledge, was taking off in new directions, and that history could no longer be regarded as the handing on of tradition, but as the story of growth and change. Wallis's title itself indicates as much: shewing the original, progress, and advancement thereof, from time to time; and by what steps it hath attained to the heighth at which now it is. There was perhaps nobody better able than Wallis to understand and record the ‘Changes and Alterations’ that were taking place in seventeenthcentury mathematics. The main text of A treatise of algebra runs to just under 400 pages divided into one hundred chapters as follows: Chapters 1 and 2 Chapters 3–12

Chapters 13 and 14 Chapters 15–29 Chapters 30–56 Chapters 57–72 Chapters 73–97 Chapters 98 and 99 Chapters 100

Hints of algebra in Classical and Islamic writers; how such ideas spread to Europe, especially England. The development of the numeral system from Archimedes to the seventeenth century, with particular emphasis (Chapters 3 and 4) on the arrival of the Hindu-Arabic numeral system in northern Europe. The development of algebra from Leonardo of Pisa to Francois Viète. The algebra of William Oughtred, and applications. The algebra of Thomas Harriot, and applications. The algebra of John Pell and miscellaneous related topics. Wallis's Arithmetica infinitorum and work derived from it by Isaac Newton and others. Methods in number theory developed by William Brouncker and Wallis. Conclusion.

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15

Fig. 1.3 Title page of De quatuor artibus popularibus … et scientiis mathematicis by Gerard John Vossius (Amsterdam 1650 and 1660).

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Everything from Chapter 15 onwards is about the work of seventeenthcentury English mathematicians, with most of whom Wallis was personally acquainted. The first fourteen chapters were therefore designed to set the scene by tracing the Classical, Islamic and Renaissance precursors to the later English flowering.53 In their content and analysis the early chapters are uneven, containing both the best and the worst of Wallis's historical writing. They have generally been overlooked by later commentators but include much interesting material. The opening chapters prepare the ground not just historically but mathematically: throughout his career Wallis was more at home with arithmetic than with geometry, and for him algebra was generalized arithmetic. It was therefore natural for him to begin a history of algebra with a brief history of the numeral system without which, he argued, algebra could not conveniently be managed.54 It was Wallis's stated aim ‘to consider pure Algebra from its own Principles; abstracted from Geometry and other Accommodations to particular Subjects’,55 and on these grounds he largely avoided any discussion as the book progressed of algebra in relation to geometry, though he could not resist putting in a few examples to demonstrate the applicability of the methods of Oughtred, Harriot or Pell. Only once did he come close to a definition of algebra, as a subject chiefly concerned with solving equations,56 but his history ranged far beyond the limitations of this or any other contemporary definition. By the end of the book Wallis had covered not only the standard subject matter of any algebra text, namely quadratic, cubic and quartic equations, but also indeterminate equations, geometrical interpretations of negative or complex solutions, summation of infinite series,

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the general binomial theorem, early results of the calculus, and problems in number theory. Neglect of ‘Accommodations’, or applications, was one of the few restrictions Wallis maintained; in every other respect he followed wherever his subject led him, so that his text is a mixture of historical survey, mathematical demonstration, textual criticism, commentary and polemic. This has led to its readers interpreting it according to their own circumstances or prejudices: as a textbook on algebra, as a repository of interesting mathematics, or as history, good or bad. In the chapters that follow, I take up Wallis's text anew and, as previous readers have done, engage with it in a variety of ways. Through exploring Wallis's original sources, his annotations, his letters and his mathematical notebooks, one can begin to enter into his own historical and mathematical standpoint. It is also now possible, however, to elucidate or correct Wallis's account in the light of later research and a longer historical perspective, and this I have attempted to do wherever possible and appropriate, so that this book is a new contribution, based on Wallis's foundations, to the study of early modern algebra. The chapter that follows this one pays particular attention to Wallis's historiography. In his discussion of the spread of the Hindu-Arabic numerals in particular Wallis drew his conclusions not from earlier authorities but from direct reading of manuscripts and artefacts, and in this respect he can be considered perhaps the first modern historian of mathematics. Unfortunately his enthusiasm for historical research did not extend to the rise of algebra in Renaissance Europe, which had little bearing on his English theme, and here I have attempted to redress the inadequacies of his history with my own account of European algebra from the medieval period to the end of the sixteenth century. Though often based on the same sources that were available to Wallis had he chosen to use them, I go further than he did in separating and analysing the different strands of algebraic thought that emerged during this important transitional period. The remaining chapters of this book focus on seventeenth-century England, on the work of individual mathematicians, and on the ways in which they influenced and collaborated with one another. The book sheds new light, for instance, on the profound influence of the teaching of Oughtred on Wallis and others, and on the extent of Wallis's lifelong support for Oughtred's Clavis mathematicae. It investigates the seventeenthcentury circulation of Harriot's unpublished manuscripts and reassesses Harriot's important contribution to algebra. It also demonstrates that

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A treatise of algebra contains a considerable amount of material that can be traced to Pell, a surprising and hitherto unsuspected finding, and one that leads to a completely new understanding of the long relationship between Wallis and Pell. The final part of the book is an evaluation of the significance and influence of Wallis's finest work, his Arithmetica infinitorum, which gave rise directly or indirectly to a great deal of new mathematics, culminating in Newton's brilliant discovery of the general binomial theorem. Now less well known, but no less ingenious, is Brouncker's discovery of a sequence of continued fractions for multiples of π: Brouncker never explained his method, and I offer a possible reconstruction of his work using only the relatively limited mathematical tools that he had at his disposal. The chapter on Brouncker also compares the very different mathematical styles of Wallis and Brouncker in the then new field of number theory. Seventeenth-century mathematicians knew that what they were doing was both new and exciting: Harriot, liberated by his notation, took flight in previously unimaginable ways; Wallis was visibly moved as he closed in on his fraction for 4/π; Newton's delight in his infinite series for logarithms spurred him to page after page of calculation; Brouncker threw into the mathematical arena some extraordinary gems of mathematical invention. They would not have considered Oughtred's description of the new mathematics as ‘the discovery of wonders’ any exaggeration,57 and it was exactly this sense of inspiration and progress in English mathematics that Wallis wanted to convey. That he himself had created part of the history he was describing, and that he wrote into it his personal preferences and prejudices, does not invalidate his account but draws us all the more fully into his world. The present book invites its readers to share afresh an inside view of English mathematics through the eyes of one of its foremost practitioners.

2 How algebra was entertained and cultivated in Europe Commentators on Wallis have tended to ignore or pass in haste over the early chapters of A treatise of algebra, those that deal with the earliest hints of algebra in Classical and Islamic cultures, and its reception in Europe from the twelfth century to the sixteenth.1 Yet these chapters give us a unique insight into a seventeenth-century perception of mathematical history. They reveal an evolving sense of mathematics as a historical subject, and the introduction of new historiographical techniques, to both of which areas Wallis made important contributions. Here we see juxtaposed his skills and his shortcomings as a historian: sometimes his sources were few and his use of them was desultory, but on other occasions he warmed to his subject and displayed greater objectivity and a truer sense of the complexities of historical development than in almost anything else he wrote. In his investigations into the origins of the number system in particular, he began to apply new historiographical standards and methods and challenged commonly accepted ideas by teasing out a different story from written and material primary sources. Wallis was concerned from the beginning to set his history of algebra into an English context, and for his account of mathematical activity in England in the medieval period he drew heavily on Vossius's De scientiis mathematicis. Vossius had lived in England from 1629 and had been a canon of Canterbury, but had returned to the Netherlands in 1633 to take up the chair of history in the new University of Amsterdam. Between 1643 and 1646 he met the English mathematician John Pell, then teaching in the Gymnasium at Amsterdam, and it may have been Pell who later suggested De scientiis as a useful source to Wallis. Wallis's copy of De scientiis is preserved in the Bodleian Library and his frequent and detailed annotations show how thoroughly he read it. Vossius classified the mathematical sciences as arithmetic, geometry, logistics, music,

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optics, and so on, and for each branch he wrote a description and history, followed by a chronological list of writers on the subject. For each author Vossius gave, as far as possible, a date and details of extant works, but he was not himself a mathematician and did not discuss mathematical content: ‘I do not teach the science itself, but only write about it’.2 Vossius based his history on numerous facts and details drawn from earlier authorities (whom he cited frequently); his own contribution to scholarship was the collating and ordering of this material.3 For English writers Vossius drew especially on the work of three sixteenth-century English historians: John Leland, John Bale and John Pits. The earliest of these was Leland (c. 1506–1552), who in 1530 became chaplain and library keeper to Henry VIII. Three years later he was commissioned by Henry to search out manuscripts and artefacts in the monasteries and colleges of England, and he spent the best part of the next ten years on the work. On New Year's Day in 1545 he presented the King with an account of his journey,4 and planned a full account of early English writers, but died before it could be completed. His notes, however, were circulated, copied and used by many later historians, and the originals were eventually acquired by the Bodleian Library in 1632.5 Leland's contemporary, John Bale (1495–1563), began his education at the Carmelite monastery in Norwich, but later turned to writing virulent attacks on the Catholic church, earning himself the nickname of ‘bilious Bale’ and seven years in exile in Germany. After his return to England in 1547 he began to keep a detailed notebook of the names, biographical details and works of English writers, drawing freely on the earlier findings of Leland and on his own research. He published two major works: his Summarium in 1548 and his Catalogus in 1557–59,6 and his notebook was eventually acquired by John Selden and given to the Bodleian Library as part of the Selden collection.7 Bale's work was later taken up by John Pits (1560–1616), whose accounts of English writers in his Relationum historicarum de rebus Anglicis of 1619 were closely based on those in Bale's Summarium, though he actually disliked Bale and tried to redress his extreme anti-catholicism. Thus in drawing on Vossius, Wallis was indirectly using the best available evidence of the time, much of it collected during the sixteenth century from the libraries of Oxford, Cambridge, London and Norwich, and from the monasteries at the time of their dissolution. The earliest English writers on mathematics that Wallis could find, all of them gleaned from Vossius, were Aldhelm, Wilfrid of Rippon,

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Bede and Alcuin.8 Aldhelm (640–709), Abbot of Malmesbury and Bishop of Sherborne, was the author of Liber de septenario, a treatise on the number seven, but it was a mystical rather than mathematical work. His reputation for mathematics arose not from this but from the quarrel between the Celtic and Roman churches over the calculation of the date of Easter, in which Aldhelm was a proponent of the Roman method based on the nineteen-year lunar cycle.9 His contemporary, Wilfrid of Ripon (634–709), Archbishop of York, was instrumental in getting the Roman method accepted at the synod of Whitby in 664.10 Bede (672–735) was by far the most prolific scholar of the period.11 Most of his writing was on theology and history, but he also wrote a Computus, a text on the division and measurement of time.12 Bede's work became known on the continent through his pupil Alcuin (or Albinus) (735–804) who became an adviser to Charlemagne.13 He encouraged the study of mathematics and the computus, and is often credited with a set of 53 arithmetic and geometric puzzles, the ‘Propositions for sharpening the minds of youth’.14 Thanks to Alcuin, Bede's influence survived longer in continental Europe than it did in Britain (the best manuscript of Bede's Computus in the Bodleian Library comes not from England but from France).15 In England, Bede's learning was never more than a fragile candle in a vast surrounding darkness, and was all but extinguished in the invasions and instability of the three following centuries. Only early in the twelfth century did scholars in England and elsewhere across Europe become aware of the knowledge that all this time had been accumulating in Islamic Spain, and some of the more adventurous travelled south and brought back texts that were to set intellectual life in northern Europe on a new course. Wallis knew of a handful of scholars of our own Nation, about the twelfth and thirteenth Century, ([who] not satisfied with the Philosophy of the Schoolmen,) were inquisitive into the Arabic Language, and the Mathematical Learning therein contained'.16 The earliest such scholar known to Wallis was Adelard of Bath (c. 1080–1150), who travelled widely in France, Sicily and the eastern Mediterranean, but there is no firm evidence that he ever visited Spain.17 He is best known for the first translations of Euclid from Arabic to Latin, and three different versions are ascribed to him:18 Wallis knew one version in the Savile Library and another in Trinity College, Oxford.19 Wallis also mentioned as collectors and translators of Arabic texts Daniel Morley, William Shelley, John of Salisbury and Roger Infans.20 Daniel Morley (c. 1180), disillusioned with the aridity of teaching in Paris, went south

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to Toledo in search of something better, in particular the contents of the quadrivium, the four classical branches of mathematics. After returning to England with a good collection of books he wrote a treatise entitled Liber de naturis inferiorum et superiorum, with a preface in which he helpfully gave an account of his life and travels.21 Wallis knew of a copy of the preface that had been in Corpus Christi College, Oxford, a few years earlier, but when he came to look for it again he discovered that it was missing (it was not uncommon for readers to extract sections of a manuscript of particular interest).22 The manuscript to which Wallis referred is that now known as MS CCC 95. It includes a copy of the Liber de naturis from which the preface has, as Wallis described, been neatly cut out, but the contents page lists the opening work as Philosophia magistri Daniel de Merlac. Morley's book is followed without a break by the Dragmaticon of Guillaume de Conches, which ends ‘Explicit Will de Conchys’,23 and it must have been this final inscription that led Wallis to connect Morley with Guillaume de Conches, whose name was later anglicized to William Shelley. In fact there is no evidence that De Conches ever left northern France, nor that he was familiar with Arabic language, philosophy or mathematics. John of Salisbury (d. 1180) was one of the best known scholars of the day. He travelled as far as southern Italy, but knew little Greek and no Arabic, and employed an Italian Greek to make translations of Aristotle. He was primarily a theologian and no lover of mathematics, which to him meant astrology: in his Polycraticus he defined mathematicians as those ‘who from the position of the stars and the motion of the planets foretell the future’, and classed mathematics with chiromancy, sortilege and augury as one of the magic arts, and a source of evil.24 Roger Infans was the scholar Roger of Hereford (fl. 1178), but Wallis never used the second, more usual, form of his name. Roger was a natural philosopher, computist and astrologer, with special knowledge of mines and minerals, and was familiar with some Arabic texts, but it is not known whether he made his own translations.25 There is no mention of him in De scientiis and Wallis must have come across the unique occurrence of ‘Infans’ in a manuscript in the Digby collection, where Roger's Tractatus de computo is headed ‘Tractatus Rogeri Infantis’, apparently because Roger himself said that he wrote it while still a young man.26 After Adelard, by far the most important of the Englishmen who travelled to southern Europe in search of new sources of learning was Robert of Chester. There is no mention of Robert in the pages of De scientiis and his inclusion

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in A treatise of algebra is a result of Wallis's own researches among Bodleian manuscripts. At the request of Peter, Abbot of Cluny, Robert made the first Latin translation of the Qur'an, so that the Abbot could better understand and refute the teachings of Islam, and a copy of Robert's translation was acquired by the Bodleian Library as part of the Selden collection.27 In the colophon Robert wrote that he intended to return to his chief interest, mathematics,28 and for Wallis this was reason enough to count him among the English translators of Arab mathematical learning. Little is known of Robert's life. He was in Spain from about 1140 and lived near the river Ebro in the north-east. He worked closely with another translator, Hermann of Carinthia, who appears to have come from the region that borders modern Austria and Slovenia, an indication of trans-European interest in the new learning.29 Over the centuries Robert's surname mutated through the forms Cestrensis, Kestrensis, Ketenensis and Retinensis, leading to confusion that persists to the present day. Robert's name appears at the end of the translation of the Qur'an as Ketenensis, and the Bodleian Library catalogue describes the author as ‘probably of Ketton in Rutland’, whereas Wallis read the looped ‘K’ as an ‘R’ and gave the name as Robert Retinensis or ‘Robert of Reading’.30 As a translator Robert was far more significant and influential than Wallis knew. It is now thought that he may have been the translator of the version of Euclid known as ‘Adelard II’.31 In the late 1140s he translated the astronomical Canons of Arzachel, and Wallis had seen a copy,32 but there Robert's name appeared as Robertus Cestrensis, and Wallis failed to recognize him as also the translator of the Qur'an. Wallis knew nothing, however, of what from his point of view would have been Robert's most important translation, the first Latin rendering of al-Khwārizmī's Al-jabr wa'lmuqābala (c. 825), the seminal text in the evolution of Arabic and European algebra. As far as we know, no copy of Robert's translation reached England until centuries later,33 and Wallis never knew of this early and important English contribution to the history of algebra. The influence of al-Khwārizmī's Al-jabr was not to be felt in England for many years. A much more immediate and revolutionary consequence of the discovery of Arabic mathematical and scientific texts was the introduction of the Hindu–Arabic numerals. Wallis claimed that algebra could hardly have developed without the invention of an efficient numeral system, and devoted three of his opening chapters to exploring the origins and spread of the Hindu–Arabic system. He had already written a brief history of the

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numeral system in his Mathesis universalis of 1657,34 but by the 1670s he could add much that was new, and we shall pause to look at his researches on this subject before returning to algebra itself. Wallis knew of several alphabetic systems that had preceded the Hindu–Arabic numerals, not just the Greek use of letters α, β, γ, δ,… for one, two, three, four,… but similar systems in Hebrew, Arabic, Persian and Turkish.35 Such alphabetic notations were used successfully for hundreds of years both for recording and calculating, but were unwieldy for carrying out large or complex calculations, as was the Roman system based on I, V, X, L, C, D, M. It was not difficult for Wallis to point to examples of the clumsiness of alphabetic systems, for at about the time he was writing A treatise of algebra he was engaged in producing new editions of two works of Archimedes, the Arenarius and the Dimensio circuli.36 The latter included a commentary by Eutocius (c. AD 560) who remarked on the difficulty of Archimedes' calculations with fractions and square roots.37 In this case the difficulty is largely inherent in the calculations themselves but can only have been exacerbated by the limitations of the available notation. Wallis also edited in 1688 a previously unpublished fragment of Book II of the Mathematical collections of Pappus (c. AD 320),38 in which Pappus reproduced the methods of Apollonius (c. BC 225) for multiplying large numbers. The methods rely on memorizing, for example, that 500 × 40 is equivalent to (5 × 4) × 1000, not immediately obvious in an alphabetic system where the former is written with φ and μ (for 500 and 40) and the latter with ε and δ (for 5 and 4). The chief advantage of the Hindu–Arabic numerals was not just the reduction of the number of symbols (the Roman system, after all, required only seven) but in the associated system of place-value, which meant that number facts for 0 to 9 sufficed for calculations of any size. Wallis knew that the numerals by then in common use in Europe had come from Islamic Spain and differed slightly from the Arabic figures used further east. They also arrived by another slightly later route in manuscripts from Constantinople written in Greek: here Wallis cited the ψηφοφορíα, a treatise on calculation written by the Greek monk Maximus Planudes (c. 1255–1310).39 Wallis recognized that the numerals had first arisen, however, in India: the full title of Planudes' treatise is ψηφοφορíα κατ' 'Iνδους (‘Indian calculation’) and the text describes both the figures and the methods of calculation as Indian. Planudes was in fact following a much earlier treatise on Indian figures, that of alKhwārizmī written about AD 825, which no longer survives in Arabic but has been reconstructed from early Latin translations.40

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Al-Khwārizmī's treatise opened with a detailed exposition of the principles of place-value followed by instructions for addition and subtraction, doubling and halving, multiplication and division, all done first for integers then for fractions (common and sexagesimal), and it ended with the extraction of square roots. Later writers all followed a similar plan, though not always covering as much material as al-Khwārizmī had done: Planudes, for example, dealt only with integers. In the west such texts became known as algorisms (a corruption of al-Khwārizmī).41 One of the earliest, well known to Wallis, was in verse, the Carmen de algorismo (Song of algorithm) which, like the ψηϕOϕOpiα described the numeral figures as Indian. The first few lines set out the numerals and explained the principle of place-value:42 Haec Algorismus ars praesens dicitur, in qua Talibus Indorum fruimur bis quinque figuris. 0.9.8.7.6.5.4.3.2.1. Primoque significat unum: duo vera secunda Tertia significat tria: sic procede sinistra Donec ad extremam venias, qua cifra vocatur; Quae nil significat; dat significare sequenti. Quaelibet illarum si primo limite ponas, Simpliciter se significat: si vero secundo, Se decies;… This present art is called ‘algorismus’, in which We make use of twice-five Indian figures: 0.9.8.7.6.5.4.3.2.1. The first signifies one: two the second The third signifies three: thus proceed left Until you come to the end, which is called ‘cifra’; Which signifies nothing; it gives significance to what is behind it. If you put any of these in the first place, It signifies simply itself: if in the second, Itself tenfold…

The Carmen, written by a French Franciscan, Alexandre de Ville Dieu (d. 1240), became immensely popular (Fig. 2.1).43 Little is known about Ville Dieu, but he was sometimes described as Dolensis, which suggests that he came from the region close to Mont Dol and Mont St Michel in northern France, probably from the town now known as Villedieules-Poêles. He also wrote a computus, a text on the reckoning of date and time, also in verse.

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Fig. 2.1 The final verse of the Carmen de algorismo of Alexander Ville Dieu, and a multiplication table with a list of squares and cubes (Bodleian Library MS Savile 17, f. 108v).

Usually known as the Massa computi it is internally dated at 1200,44 so his algorism may be supposed to date from roughly the same period. Even more influential than Ville Dieu's Carmen was the Algorismus of Johannes Sacrobosco (c. 1200–1244 or 1256).45 Almost nothing is known of Sacrobosco's origins but Wallis took him to be English, 46 and knew his Algorismus, composed about 1230, from a copy in the Savile Library.47 Like the Carmen, the Algorismus dealt with the material first set out by al-Khwārizmī: place-value, addition, subtraction, doubling and halving, multiplication and

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division, all for integers. To these Sacrobosco added cube roots and an elementary treatment of arithmetic progression. Sacrobosco's Algorismus became by far the best known of the medieval algorisms. It remained in use as a university text across western Europe for three centuries,48 and set the pattern for all subsequent texts on arithmetic for many years after that. Sacrobosco was not so explicit as Ville Dieu about the Indian origins of the numerals, though he did suggest that the art of computation could be associated with a mythical Indian king, Algus.49 From the appearance of Hindu–Arabic numerals in Planudes' ψηϕOϕOpiα and in the Alphonsine tables created in Spain about 1270, the Jesuit Athanasius Kircher dated the arrival of the numeral figures in northern Europe to the second half of the thirteenth century,50 and Vossius in De scientiis came to much the same conclusion.51 Kircher and Vossius, however, lacked the kind of texts copied and used by working mathematicians in Oxford, and available to Wallis. In the algorismi of Ville Dieu and Sacrobosco, Wallis already had clear examples of the use of the numerals in the early years of the thirteenth century, and his access to Oxford's unique heritage of medieval material was to allow him to carry his argument very much further. Wallis recognized that figures (like diagrams) could be changed in the course of copying from one manuscript to another,52 but he also knew that the word algorism itself was always specifically associated with Hindu–Arabic numerals and could therefore be taken as confirmation of their use.53 Wallis found many further examples of the use of Hindu–Arabic numerals in the first half of the thirteenth century: in the Computus of Robert Grosseteste (d. 1253);54 in the De computo ecclesiastico of Sacrobosco, dated 1235;55 in the Massa computi of Ville Dieu (though Wallis did not know its author);56 in an Algorism by Jordanus who was active c. 1220 (though Wallis thought he lived around 1200);57 and in astrological diagrams dated 1216 (Fig. 2.2).58 Like Kircher, Wallis recognized that one of the earliest ways Hindu–Arabic numerals reached northern Europe was through astronomical tables but, unlike Kircher, he knew that the Alphonsine tables were not the first to spread beyond Spain. Before that, the Toledan tables, originally compiled in the eleventh century, were used throughout Europe and adapted for different cities: Marseilles (c. 1140) and London (1150).59 Wallis knew the London edition, translated by Robert of Chester, from a copy in the Savile Library, and at this point in his research went to some trouble to discover Robert of Chester's identity, but never made the vital connection with Robert the

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Fig. 2.2 Astrological diagrams from 1215 to 1216, on vellum, written with Hindu–Arabic numerals and almost certainly in the hand of Robert Grosseteste (Bodleian Library MS Savile 21, f. 160v).

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translator of the Qur'an.60 Nevertheless in Robert's translation he had clear evidence of the arrival of the numerals in England as early as 1150. Wallis did not confine his researches only to Oxford libraries. His daughter Anne lived in the village of Marston St Lawrence in Northamptonshire, about 30 miles north of Oxford, and in the vicarage in the neighbouring village of Helmdon Wallis found an oak mantelpiece with a date that he read as M133, or 1133 in a mixture of Roman and Arabic numerals (Fig. 2.3).61 His claim for such an early date triggered a controversy that went on well into the nineteenth century. In 1800 Ralph Churton, rector of the neighbouring parish of Middleton Cheney, wrote to the Gentleman's magazine:62 Few of your Antiquarian readers need to be informed how much the inscription on the mantle-tree in the parsonage at Helmdon, in Northamptonshire, has puzzled the learned and curious in such matters ever since the celebrated Dr Wallis gave an account of it in the Philosophical transactions above a century ago.

Churton provided a full size tracing of the inscriptions (Fig. 2.4), considerably more accurate than the drawing published by Wallis in 1685, and concluded: As to the decyphering … having carefully examined the inscription four severall [distinct] times, and copied on thin paper with black lead all the material parts twice as often, I am satisfied, upon the whole, that Dr Wallis gave the true reading, namely, ‘Ano.Doi.Mo. 133’.

Thirty years later, however, George Baker published the first volume of his painstakingly researched History and antiquities of the county of Northampton and after carefully weighing the evidence came to a different conclusion:63 Much disputation and ingenious conjecture have been exercised in decyphering this famous date, and 1133, 1233, 1533, and 1555 have been severally suggested. Some writers have referred the initials W.R.64 following the date to William Renalde or Reynolde, the rector from 1523 to 1560, and the general style of the mantle-piece, its very depressed arch, and the elongated leaves in the spandrils, certainly correspond with that period, and corroborate the supposition: whilst, on the other hand, it must be admitted that the form of the M and the connecting figures strongly favour the interpretation given by Dr Wallis. From a careful examination of the original I am inclined to attribute this singular curiosity to the rector [Reynolde], though it must be confessed his motive for introducing a fictitious date in

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Fig. 2.3 A sketch of the Helmdon mantelpiece, published by Wallis in the Philosophical transactions in 1683 and in A treatise of algebra in 1685.

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Fig. 2.4 Ralph Churton's tracing of the numerals on the Helmdon mantelpiece, published in the Gentleman's magazine in 1800.

rude or arbitrary characters, unless to puzzle future antiquaries seems inexplicable.

Later in the nineteenth century the vicarage was modernized and the mantelpiece, after standing in the porch exposed to weather, was taken into the church for safe-keeping. There it can still be seen (Fig. 2.5), but uncertainty as to its date persists. Architectural experts argue that the carved rosettes are typical of a much later period, and a recent opinion states:65 This is a very nice bressumer but it is certainly not twelfth century! The carving is of provincial quality only, and the rosettes which are the only stylistically datable feature, look to be 1400–50. It is impossible to be more precise than that.

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Fig. 2.5 The Helmdon mantel piece.

A second expert, however, considers that the dragon ‘could easily be twelfth century work’ and admits the possibility that the piece may have originally been a twelfth-century lintel converted to a mantelpiece around 1500.66 None of the claims for a later date, as Baker pointed out, offers any credible alternative reading of the carved date, or takes into account the early form of the 3s. It cannot yet be completely ruled out that Wallis's reading was correct and that the beam was first carved, perhaps as a roof beam or lintel, in 1133. The Crusades from 1095 to 1270 took large numbers of Englishmen to the eastern Mediterranean where some of them must have learned the new numerals, if only for the purposes of bargaining and trading, and it is curious that Wallis never considered the Crusades as important in this respect. The fifteenth-century rosettes, more deeply carved than the numerals, could have been added later: the beam is attractive (the ‘provincial quality’ gives it a pleasing and homely feel) and it is easy to understand why successive generations would have put it to new use rather than see it destroyed. If Wallis was correct, the beam is invaluable evidence of the early use of Arabic numerals in England. The arrival of new ideas and texts, from whatever sources, reinvigorated the intellectual climate of northern Europe. Universities were founded in Oxford, Cambridge and Paris, and Oxford developed a particularly strong tradition of mathematics. Wallis presented a long list of names of

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English medieval mathematical practitioners, almost every one of them associated with Oxford.67 In the thirteenth century Robert Grosseteste, Magister scholarum (1214–1231) and Lector to the Oxford Franciscans (1232–1235), wrote a number of scientific treatises, including a Computus in which he noted the discrepancies between lunar and solar time and suggested appropriate reforms.68 Through Grosseteste's friend and adviser Adam Marsh (c. 1205–1258), himself Lector to the Franciscans in 1247, Roger Bacon (c. 1214–1292) and John Pecham (1230–1292) also entered the Oxford community. Both wrote scientific treatises,69 and Bacon, whose learning became almost legendary, argued for the usefulness of mathematics in every part of intellectual activity.70 His own contribution to the subject, Communia mathematica, was of little consequence, but like Grosseteste he had a good understanding of the shortcomings of the Julian calendar and suggested some practical corrections.71 In the fourteenth century the focus shifted to Merton College, which became renowned for mathematical learning. The Merton names noted by Wallis were those of Thomas Bradwardine (c. 1300–1348), Simon Bredon (c. 1315–1372), John Ashenden (fl. 1347–1357)72 and William Rede (c. 1330–1385), the last of whom founded and endowed Merton's beautiful library.73 Bradwardine was at Merton for over twenty years and was distinguished in theology, philosophy and mathematics. His most important mathematical work was his Tractatus de proportionibus concerning the ratios of speeds and of motive and resistive forces. He was appointed Archbishop of Canterbury in 1349 but died of plague shortly after his consecration at Avignon (then the seat of the papacy).74 Bredon, physician and astronomer, wrote a commentary on Ptolemy's Almagest and left his astrolabe to Merton.75 Ashenden and Rede were both renowned as astronomers: Ashenden is best known for his Summa astrologiae judicialis de accidentibus mundi (‘A summary of the judgements of astrology on the happenings of the world’) eventually printed at Venice in 1489,76 Rede for his astronomical tables based on the Alphonsine tables of 1270. Ashenden and Rede together wrote Prognosticationes eclipseos lunae 1345 and De significatione conjunctionis magne Saturni et Jovis 1365; the preamble to the first describes Rede as having made the calculations and Ashenden the predictions.77 Other Oxford astronomers mentioned by Wallis were William Batecombe (fl. 1348) who compiled astronomical tables for Oxford and an almanack for the year 1348; Nicholas Lynne (fl. 1355) who drew up an astronomical calendar for the latitude of Oxford for 1387–1462, as did John Somer

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(fl. 1380–1395).78 Chaucer later declared his intention of compiling ‘divers tables of longitudes and latitudes and declinations of the sun after the calendars of the reverend clerks John Somer and Nicholas of Lynne’, and his Oxford ‘scienceman’ is thought to have been based on Lynne. John Walter (d. 1412) and John Killingworth (c. 1415–1445) carried the Oxford astro nomical tradition into the early years of the fifteenth century before it finally petered out.79 Among Wallis's list of medieval mathematical practitioners there were also two East Anglian Carmelites, John Baconthorpe (c. 1290–1348) and Richard Lavenham (c. 1380). Neither is now thought of as a mathematician, and their inclusion is an interesting indication of the influence of John Bale, who came from the same East Anglian Carmelite background. Baconthorpe and Lavenham are best known for their theological writings, but both touched on astrology, enough for Bale, and therefore Vossius, and therefore also Wallis, to regard them as mathematicians. If there were some doubtful inclusions, there were also some telling omissions from Wallis's account: Henry Savile in 1570 had classed Richard Swineshead (fl. 1350),80 Roger Bacon and Richard Wallingford (1292–1336) as mathematicians on a par with Archimedes and Ptolemy.81 Wallingford has been described as ‘perhaps the best mathematician and astronomer of the Middle Ages’,82 while Cardano (as Wallis knew) had placed Swineshead (or Suisset) fourth in his list of great scientists (after Archimedes, Aristotle and Euclid but ahead of Apollonius). In England by the time Wallis was writing, however, Swineshead and Wallingford had slipped into oblivion. Swineshead became better known in Italy where his Calculationes was published in three different editions between 1477 and 1520.83 A somewhat similar fate had befallen Bradwardine whose work was first published in Paris in 1495 but who was unknown to Vossius, while Wallis knew of him only through Oxford manuscripts. That the work of Wallingford, Bradwardine and Swineshead was so little appreciated in England is a sign of the low level of interest in things mathematical during the late fifteenth and early sixteenth centuries. The English mathematics considered so far has consisted of algorism, computus, natural philosophy, and astronomy intertwined with astrology. To this there may be added some neo-Pythagorean number relationships learned from Boethius (480–524),84 and some elementary geometry from Euclid, but as yet no algebra. To trace the influence of the Arabic algebra texts we now need to turn aside for a while from England to Italy.

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‘In Arabic it is called al-jabr’

85

The text from which European algebra began to evolve and from which eventually it took its name was Al-kitāb almukhtasar fi hisāb al-jabr wa'l-muqābala of Muhammad ibn Mūsā al-Khwārizmī, a member of the scientific academy in Baghdad around AD 825.86 The title is not easy to translate: al-jabr is completing or restoring, used particularly in the sense of setting broken bones, while al-muqābala is setting in opposition or balancing. In al-Khwārizmī's text al-jabr is used when a positive term is added in order to eliminate a negative quantity, while al-muqābala denotes the balancing of an equation by operating simultaneously on each side.87 It seems most likely that the terms originally arose from the traditional geometric technique of solving a quadratic equation by completing a square (Fig. 2.6): there al-jabr would have been the completing or setting together of the incomplete, or broken, square, while al-muqābala would have been the subsequent balancing that such completion requires. In time the words used to describe the geometrical process would naturally transfer to the equivalent steps in the handling of the equation.88 Fig. 2.6 The geometrical method of solving x2 + 10x = 39. A square of area 25 units is required for the completing, or setting together (al-jabr) of the larger square. Balancing (al-muqābala) is achieved by adding the same 25 units to the quantity on the right.

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The subject of al-Khwārizmī's treatise is the handling of linear and quadratic equations of the following six types:89

There is no symbolic notation in al-Khwārizmī's Al-jabr: each type of equation is described verbally, and the method of solution is given as a recipe or set of instructions for each case. The text gives geometrical demonstrations for some of the three-term equations (a literal ‘completing the square’) and there is then a long section of worked examples. Only the first few problems actually illustrate the six types of equation; the remainder are concerned with mensuration and with Islamic laws of inheritance and division. The same material found its way into a slightly later text, Al-Kitāb fi al-jabr wa'l muqābala of Abū Kāmil (c. 850–930), essentially an extension and commentary on al-Khwārizmī's Al-jabr.90 Abū Kāmil's work was taken up by al-Karajī (c. 1010) and, eventually, in the west by Leonardo of Pisa (c. 1170–1240), also known since the nineteenth century as Fibonacci.

‘The Italians have given it the name of Regula cosae’

91

The Liber abbaci (1202) of Leonardo of Pisa was the first major European mathematical text of any kind,92 compiled from knowledge acquired by Leonardo on his travels through north Africa, Syria, Greece and Sicily. Through numerous practical problems and puzzles it helped to spread and standardize the use of Hindu–Arabic numerals, but it also introduced algebra by presenting problems that gave rise to the six equation types set out by al-Khwārizmī (whom Leonardo referred to as ‘Maumeht’). Leonardo's exact sources are unknown: al-Khwārizmī's text was translated into Latin twice during the twelfth century, by Robert of Chester (c. 1145) and by Gerard of Cremona (c. 1175), and Leonardo could have known either or both, but he took problems also from the later Islamic writers al-Khayyāmī, Abū Kāmil and al-Karajī.93 Leonardo, like al-Khwārizmī, considered only positive solutions, but where two such solutions existed he could produce both. He also treated equations of higher degree that were essentially quadratic in form, for example

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x 8 + 100x4 = 10 000. In describing such equations (his exposition, like all earlier ones, was entirely verbal) he introduced Latin names that were to remain in use for centuries: numerus for a given or known quantity; radix, res, causa or cosa for the unknown root, or thing; quadratus or census (literally ‘wealth’ or ‘excess’) for its square;94cubus for its cube; census de censu for the fourth power and cubus cubi for the sixth power (not, as one might imagine, the ninth). The fifth power was not used and not named. The Liber abbaci gave rise to a number of thirteenth- and fourteenth-century Italian abbacus texts devoted to the explanation and use of Hindu–Arabic numerals and commercial arithmetic, and some also included a little algebra.95 The first known treatment of algebra after Leonardo's, and the first in Italian, is by Jacob of Florence in 1307.96 A few years later, in 1328, came the Libro de ragioni by the Florentine Paolo Gerardi.97 Like the Liber abbaci, the Libro de ragioni was mainly concerned with practical arithmetic, but in the last eight of his seventy folios Gerardi explained the ‘Regolle delle cose’, the rules for handling an unknown ‘thing’ or quantity, and gave al-Khwārizmī's six equations with the rule and an example for each. Gerardi also added nine types of cubic equation. Most, for example, those of the (modern) form ax 3 ± bx 2 ± cx = 0 were essentially quadratics (the solution x = 0 would not have counted); others, of the form ax 3 = bx + c;ax 3 = bx 2 + c or ax 3 = bx 2 + cx+d. were solved wrongly by simply applying the standard rules for quadratics. The fact that the so-called solutions failed to fit the original equations did not prevent the method from being repeated in many later manuscripts up to and including the Trattato d'abaco by Piero della Francesca in 1480.98 The earliest text to be devoted entirely to algebra was the Aliabraa-argibra of 1344 by Dardi of Pisa.99 Where Gerardi had treated fifteen equation types, Dardi treated no fewer than 194, including some with fourth powers, all, however, reducible by a simple substitution to one of al-Khwārizmī's six basic forms. He also gave solutions to two special and easily solved equations, one cubic and one quartic, arising from three or four years of accumulation of compound interest.100 Later authors repeated his rules for these equations as though they applied generally, without stating the special conditions under which they applied. One further manuscript from this period deserves special mention as the first attempt to explore the history of algebra. The Trattato di praticha d'arismetica of 1463 by Benedetto of Florence described how algebra was brought to Italy not only through Leonardo's work but more directly through the Latin translations of al-Khwārizmī's Al-jabr.101 Benedetto questioned the

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generality of some of the existing suggestions for solving cubic equations and so appears to have been a rather better mathematician than some of his predecessors. He also appeared to go further than most towards a systematic notation by writing r, c and b for cosa, census and cubo, then cc for censo di censo and bb for cubo di cubo. This was not, however, as it might appear, a general method of representing multiplication since Benedetto also wrote br for cubo relato cosa (the fifth power), showing that he was using abbreviations rather than a consistent and generalizable symbolism.102 All the early Italian texts, like their Arabic predecessors, were rhetorical rather than symbolic: the common abbreviations co, ce, p and m for cosa, census, plus and minus were of the kind frequently used in manuscripts (and early printed texts) and were only the precursors of a later true symbolism. By the fifteenth century the teaching of algebra had spread to Germany, and it was there that the practice of solving equations came to be described as the cossick art (from the Italian cosa). The abbreviations R, Z and C (in elaborate Gothic script) were introduced for res, census (or zensus) and cubus, and were combined as ZZ for the fourth power, RZZ for the fifth, ZZZ for the sixth, CZZ for the seventh, etc., the first generally used consistent symbolic representation of powers. (Nicholas Chuquet in. his unpublished Triparty of 1484 had used a system closely related to modern superscript notation: his work was taken up and used by Etienne de la Roche, probably his student, but in L'arismetique of 1520 de la Roche reverted to cossist notation and Chuquet's innovations were forgotten.)103 The first printed text to include some algebra was the Summa de arithmetica of Luca Pacioli, published in Venice in 1494 (Fig. 2.7). Pacioli's system of naming powers was based on multiplication rather than addition, so that for him censo de cubo, or ce.cu, denoted a sixth power, not a fifth. In this system a new name or symbol is required for every prime power, and Pacioli called the fifth power primo relato, the seventh secondo relato and so on. In 1525 the German author Christoph Rudolff introduced the symbols ß, bß, cß,… for fifth, seventh, eleventh powers, etc., a system that became standard during the sixteenth century. The precise symbols used varied from author to author: one of the earliest printed texts, the Summa arithmetica of 1521 of Francesco di Ghaligai, used charming but not very practical squares and rectangles to denote powers, but most authors used some variation on the letters R, Z and C for an unknown, its square and its cube. By the early sixteenth century an elementary symbolism was in place, but in other ways there had been little development of content since

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Fig. 2.7 Part of page 149r from Pacioli's Summa (Venice 1494). At the top is a list of types of equations, some marked as ‘impossible’.

Leonardo's Liber abbaci three hundred years earlier, or indeed since al-Khwārizmī's Al-jabr almost four hundred years before that. Throughout the sixteenth century cossist texts continued to repeat much the same teaching. A cossist algebra typically began by introducing the four operations of arithmetic for whole numbers, and might also have included some treatment

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of fractions, powers and surds. Then the author would define the terms cosa, census and cubus, and set out his chosen notation. Just as for numerical arithmetic he would show how to carry out the basic operations of addition, subtraction, multiplication and division, but now using symbols, and, in the more advanced texts, how to cancel simple algebraic fractions or extract easy roots. Next came an explanation of what an equation was and how it could be simplified: by moving terms from one side to the other (as taught by al-Khwārizmī) and by reducing the leading coefficient to I (also taught by al-Khwārizmī). Some authors also instructed the reader to divide out excess powers of the unknown (so that x4 = 25x2, for example, reduced to x2 = 25), to clear fractions by multiplication and surds by squaring. The heart of the matter, however, was the Rule of Algebra or the Rule of Coss: let the unknown quantity be represented by R and form an equation according to the conditions of the problem; the solution to the equation is then the quantity sought. This transition from physical quantity to mathematical symbol (and back again) was seen as the key process, often emphasized on the printed page by a new heading or a special font. Just as with the word al-jabr in the Arabic texts, we see the word algebra used to describe a process of abstraction or generalization. Al-jabr had allowed one to manipulate quantities without resorting to geometric representations; in algebra those quantities could be weights, measures, money or numbers themselves, since all were handled by the same rule, the regula cosae. Most textbooks ended with worked examples to demonstrate the six types of equation set out by al-Khwārizmī. The exact contents and ordering of the material varied somewhat from author to author but what is most remarkable about the sixteenth-century cossist algebras is not their variety but their essential similarity over time and geographical distance. From Valencia to London and from Venice to Lyon, in Latin, German, Italian, Spanish, French and English, a score or more of cossist texts followed much the same blueprint, a remarkable example of a common intellectual tradition. The best and most original was perhaps the Arithmetica integra of Michael Stifel, published in Nuremberg in 1544. In addition to the material outlined above Stifel gave a single rule that covered all cases of quadratic equations: halve the coefficient of the root, square it, add or subtract the given ‘number’, take the square root of the resulting quantity, then add or subtract half the coefficient (leading, in Latin, to the useful mnemonic AMASIAS).104 Stifel also gave rules for working with negative numbers, and began to

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explore fractional and negative powers:105 he observed, for instance, that what we would now write as is , and −3 6 3 that is ; also that the multiplication corresponds to what would now be written 2 × 2 = 2 . Many later writers acknowledged a debt to Stifel, and the output of new books peaked during the following decade, with works by Scheubel, Peletier, Borrell (Buteo) and Ramus published in France, Aurel and Perez de Moya in Spain, Mennher and Nuñez in the Netherlands, Peucer in Germany and Recorde in England.106 Robert Recorde's The whetstone of witte of 1557 was the single English representative of the genre (Fig. 2.8). Recorde had already published an arithmetic entitled The grounde of artes in 1543, so The whetstone was devoted more particularly to algebra. There is delightful wordplay in the title: not only was the whetstone to be found in the previously prepared grounde (‘The grounde of artes did brede this stone’) but the Latin for whetstone is cos. Recorde's text is written in the form of a dialogue between master and pupil and is lively and thoroughly readable. It is perhaps best known for Recorde's introduction of the equals sign: ‘I will lette as I doe often in woorke use, a paire of paralleles … bicause noe .2. thynges can be moare equalle.’ Here is Recorde's teaching on the Rule of Algebra (Fig. 2.9):107 The rule of equation, commonly called Algeber's Rule … But now will I teache you that rule, that is the principall in Cossike woorkes: and for whiche all the other dooe serve. This rule is called the Rule of Algeber, after the name of the inventoure, as some men thinke… But of his use it is rightly called, the rule of equation: bicause that by equation of nombers, it doeth dissolve doubtefull questions: And unfolde intricate ridles. And this is the order of it. The somme of the rule of equation: When any question is propounded apperteinyng to this rule, you shall imagin a name for the nomber, that is to bee soughte, as you remember, that you learned in the rule of false position. And with that nomber shall you procede, accordyng to the question, until you find a Cossicke nomber, equalle to that nomber, that the question expresseth, whiche you shall reduce ever more to the leaste nombers…

The whetstone remained for many years the only English textbook of algebra. Wallis later searched in vain for any other English treatment of the subject but could find only the little algebra taught in the Stratiotocos, essentially

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Fig. 2.8 Title page of Robert Recorde's The whetstone of witte (London 1557).

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Fig. 2.9 The rule of algebra, or rule of equation, from Robert Recorde's The whetstone of witte, emphasized by decorated capitals and a special italic font.

a military manual by Leonard and Thomas Digges, published in 1579. In A treatise of algebra he wrote:108 Leonard Digges (in his Stratioticos, 1579,) and Robert Record, about the Year 1552, (if I be not mis-informed,) and (I think) Robert Norman,

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about 1560, and some other, (whose Names I do not remember,) have written of it in our own language.

The meagreness of the list obviously troubled Wallis for when he sent his draft to Collins in 1677 he wrote: ‘You may mind me also of the names of ancient algebraists of our own before Vieta. Such I have seen, but have forgot their names’.109 Not even Collins could help him: with Recorde and Digges, Wallis had more or less exhausted the sixteenthcentury English contribution to algebra. Robert Norman was an instrument maker and his Safe-guard for sailers, on navigation, was in the Savile Library but he was not by any stretch of the imagination an algebraist.110 Wallis's knowledge even of Recorde's work was unexpectedly scanty, for he could not name The whetstone of witte and his date of 1552 was incorrect.111 Elsewhere in western Europe, cossist algebras continued to appear well into the seventeenth century.112 The later texts, however, lack the freshness and vigour of their mid sixteenth-century predecessors and are often ponderous or confused. Cossist symbolism at first seemed to carry some promise but always carried the seeds of its own decline, for it was little more than a system of abbreviations, incapable either of generalization or of true symbolic manipulation. It was adequate, just, for handling the linear and quadratic equations given by al-Khwārizmī, but the sixteenth-century cossist writers made little or no attempt to extend their work to cubics, and the notation shed no light on the nature or structure of higher equations. Under the weight of new demands such a system was bound to collapse. Before its lingering death in the seventeenth century, however, cossist algebra taught European mathematicians important lessons about operating with symbols instead of numbers. All the cossist writers saw the rules for manipulating symbols as direct parallels to the corresponding rules for numbers (Recorde was not alone in calling his symbols ‘Cossicke numbers’). The notation may have been flawed but cossist algebra was at least the beginning of a generalized symbolic arithmetic. In technique as opposed to symbolism, on the other hand, the cossists made no advance. A student of Leonardo's algebra would have discovered nothing essentially new in any of the later books, and even a reader of al-Khwārizmī would have found much that was still recognizable. The Arab-cossist tradition lasted for eight centuries without extending either its scope or its methods, until eventually it ran into the ground. Fortunately for the future of mathematics, cossist algebra was not

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the only strand in the story: in mid sixteenth-century Italy something entirely new was under way.

‘Cardan gives it the name of Ars magna’

113

When Pacioli claimed in the Summa that it was impossible to solve a cubic equation by the rules that worked for quadratics,114 it was not so much a statement of failure as a rejection of the erroneous methods proposed by Gerardi and others, and a first important step towards a better understanding of cubic equations. Pacioli himself did not find a solution for cubics, though he did solve a particular quartic by treating it as the product of two (identical) quadratics, foreshadowing later more general techniques for quartics.115 The main advances came a few years after his death in 1517. The first mathematician to succeed with equations of the form x3 + px = q was Scipione del Ferro (1465–1526), lecturer in geometry and arithmetic at the University of Bologna. He never revealed his method publicly but taught it to his son-in-law, Annibale della Nave, and to his disciple Mario Fiore. In 1535 Fiore challenged Niccolo Tartaglia (c. 1499–1557) in a public competition in which every problem gave rise to a cubic of the form x3 + px = q, leading Tartaglia to discover for himself del Ferro's solution. Four years later Tartaglia passed the secret on to Girolamo Cardano (1501–1576) on condition, he later complained, that Cardano would not publish it. Cardano, however, discovered del Ferro's priority (through della Nave) and published the result along with many others, and with full attribution to his predecessors, in the Ars magna of 1545.116 The Ars magna is one of the great mathematical texts of all time. Del Ferro's solution for a special class of cubics is only one of a host of important methods and discoveries the book contains. Cardano worked out the solution to every other case of cubic equation and, together with his pupil and son-in-law, Ludovico Ferrari, went on to tackle quartics too. Unlike the cossist texts, however, the Ars magna was more than a manual for solving set forms of equations. It contained profound and far-reaching insights that were not to be fully worked out for fifty years or more, but which laid the foundations of European algebra. The Ars magna is not easy reading even now. It is written in long winding sentences without any use of mathematical notation, and some of the vocabulary is both specialized and obscure, so that even in modern translation the meaning is not always plain. Cardano presented a plethora of

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rules and methods, some of which he illustrated by examples, but rarely gave any explanation of how he found them. He gave occasional geometrical demonstrations but they have the appearance of being appendages rather than integral parts of the text, and they disappear altogether from the later part of the book. The following account of the contents will focus on those aspects that were particularly important in the later development of algebra, and for clarity Cardano's equations and results will be given in modern notation but it should be borne in mind that his own exposition was entirely verbal. The contents of the first chapter alone display the originality and fertility of Cardano's insights. He began with cubic equations of the form x3 + px = q or x3 + q = px (that is, with no term in x2) and by comparing with q for the second type was able to determine not only the number of real roots but how many were ‘true’ (positive) or ‘fictitious’ (negative). (The quantity 4p3 − 27q2 serves as a ‘discriminant’ for cubic equations of the form x3 − px + q = 0 as ‘b2 − 4ac’ does for quadratics.) The recognition of negative roots was in itself an astonishing advance: up to that time, and for another fifty years to come, most writers steadfastly ignored such a possibility. Cardano went on to say more about negative roots later,117 and also came up with the complex roots of x2 − 10x + 40 = 0 (‘Putting aside the mental tortures involved, multiply 5 + √ −15 by 5 − √ −15, making 25 −( − 15) which is + 15. Hence this product is 40’).118 In Chapter I, he analysed three further classes of equation:

Cardano noted that by changing terms from one side to another a ‘fictitious’ root could be changed to a ‘true’ root. For example, the ‘fictitious’ root (−3) of

becomes the ‘true’ root (+3) of

Under later conventions, where negative terms were allowed, ‘changing side’ became equivalent to ‘changing sign’ and Cardano's observations laid the

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foundations of the ‘Rule of Signs’ proposed by Descartes almost a century later for determining the number of positive real roots of an equation. The next few chapters of the Ars magna treated well-known material: equations of the form ax n = bx m and the standard solutions for quadratics, but then in Chapter VI, entitled De modis inveniendi capitula nova ( ‘On methods for solving new cases’), Cardano took off into the unknown. ‘After I had carefully considered all this’, he wrote, ‘it seemed to me that it would be permissible to go still further’,119 and a few paragraphs later came up with what was perhaps the most profound insight of the whole of sixteenth-century mathematics. It arose from the problem of finding two numbers here, denoted by p and q, such that

The substitution q = 8/p leads to(1)

but the substitution p = 8/q leads to(2)

In the Ars magna the equations are both described verbally in terms of an unknown quantity, so it is closer to Cardano's text to express both (1) and (2) as equations in a general unknown:120(1)

(2)

These equations clearly differ in form yet their roots are related in a simple way (by the transformation x → 8/x ). Thus, Cardano saw that equations of type (1), which he could solve, opened up equations of type (2). Similar transformations applied to other equations could, he reasoned, open up a multitude of new possibilities. As he put it: ‘Translate questions that are by some other ingenuity known, to questions that are unknown, and the discovery of rules will have no end’.121 He was not exaggerating: the transformation and solution of equations by change of root was to be a central

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part of the work of Viète and Harriot at the end of the sixteenth century. Two centuries after that, investigations into the structure and solvability of equations were to become one of the starting points of modern abstract algebra. If any one person deserves to be recognized as the father of European algebra, it must be Cardano. By exploring such transformations, Cardano successfully solved thirteen different cases of cubic. In the penultimate chapter of the Ars magna he also expounded the method devised by Ferrari for solving quartic equations, essentially a method of ‘completing the square’ by the introduction of a new quantity (though Cardano had no notation to distinguish between the original unknown and the new one). The condition for a perfect square led to a cubic equation in the second unknown, which in principle could be solved, and the solution led back in turn to the solution of the original quartic.122 The mathematical skill required to carry out this process, without any notation to assist either the conceptualization or the description, is considerable. Little wonder that Cardano wrote of it: ‘And therefore carrying out such operations as these is about the greatest thing to which the perfection of human intellect or, rather, of human imagination, can come’.123 The lack of notation is one of many remarkable features of the Ars magna and raises profound questions as to what algebra is. Most mathematicians from the cossists onwards have regarded symbolism as an essential feature of the subject, yet the cossists with their symbols made few conceptual advances, while Cardano with no notation at all displayed astonishing imagination and inventiveness. The Oxford English dictionary definition of algebra as a ‘branch of mathematics that uses letters etc. to represent numbers and quantities’ was already inadequate in 1545, whereas Cardano's investigation of mathematical structure was in some ways not far removed from the modern understanding of the subject. The unprecedented nature of Cardano's insights, combined with the obscurity of his writing, meant that in its own day and for a long time afterwards the Ars magna was highly regarded but little understood. The cossist writers sometimes mentioned it but continued with their own set pieces. Cardano found no champion until his work was taken up by another Bolognese mathematician, Rafael Bombelli (1526–1572), who recognized the value of Cardano's methods but wanted to make the exposition clearer. In this he succeeded: Book II of his L'algebra written between 1557 and 1560 is a lucid and systematic treatment of quadratic, cubic and quartic equations by Cardano's methods. Bombelli introduced, for the first time in a printed text, a notation

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for exponents, so that the equation x3 = 6x + 40 appeared as 40. He is now perhaps best known, however, for his handling of imaginary quantities, + √ −n and − √ −n, which he called piu di meno and meno di meno, respectively.124 Part III of L'algebra originally contained a series of practical problems chosen to demonstrate the principles of algebra. However, after studying and partly translating a manuscript of Diophantus' Arithmetic in the Vatican Library, Bombelli re-wrote his Part III entirely and included problems of a more abstract kind, many taken directly from Diophantus.125 Through Bombelli's L'algebra published in 1572, and Xylander's translation of the Arithmetic in 1575, the work of Diophantus began to be more widely known and was to play an important role in the later development of algebra and number theory. A second important book that presented Cardano's work more clearly was L'arithmetique … aussi l'algebre of Simon Stevin (1548–1620) who, like Bombelli, was by profession an engineer. Stevin's book was published in Leiden in 1585 and included a brief historical introduction to the subject of solving equations up to Bombelli, ‘a great arithmetician of our time’. Stevin took up Bombelli's notation for exponents (except that he used full instead of half circles) and in theory extended it to fractional powers, but in theory only, because although he described the meaning of encircled ½, ⅔ etc. such symbols never appeared in print, and were not actually needed in a book dealing only with polynomial equations. Bombelli and Stevin both made important advances in notation but arguably their greatest contribution to the development of algebra was in making Cardano's methods accessible to a wider readership. Detailed consideration of the contents of the Ars magna by later historians of mathematics has been rare.126 Modern commentators have tended to focus almost exclusively on Cardano's treatment of cubic and quartic equations and in this respect have regarded his work as less than wholly original.127 There was, of course, nothing new about trying to solve cubic or quartic equations; where Cardano broke new ground was not in attempting such problems but in the methods he devised. In particular his insight into the way equations could be transformed by a change of root, either linear (x′ = x − k) or reciprocal (x′ = k/x ) was unprecedented. Perhaps the most perceptive general reader of the Ars magna was John Pell, who in 1638 singled out and translated into English the key Chapter VI in which Cardano first announced his breakthrough on the transformation, or transmutation,

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of equations.128 Pell's translation survives among his unpublished papers and until recent years was the only known rendering of any part of the Ars magna into English.129 Pell was thoroughly familiar with the mathematical literature of his day, including the algebra of Viète, Harriot and Descartes, and would have recognized, perhaps more clearly than many later writers did, that Cardano's insights were the foundation of all subsequent progress in the theory of equations for almost a century.

‘What we commonly call algebra is by a Greek name called Analysis’ 130

During the 1590s François Viète (1540–1603) took algebra into the third and last of its sixteenth-century manifestations.131 Viète was familiar with earlier sixteenth-century achievements but what separated him most markedly from his predecessors was the new availability of Greek sources, new editions not only of Diophantus but also of Pappus and Apollonius.132 The distinguishing feature of his work was not, as is so often supposed, any major advance in notation or technique, but his recognition that algebra provided a new means of understanding the work of Classical writers. Bombelli had already yoked algebra to the arithmetic problems of Diophantus, but Viete took an important step further. His symbols, or ‘species’, he realized, could represent not just numbers but geometric quantities: lines, planes and solids, and so geometry, previously called upon to justify algebra, now became itself amenable to algebraic treatment. Viète went so far as to identify the algebraic method as the analytic tool by which Classical geometrical theorems had first been discovered, and so for him algebra became synonymous with ‘analysis’ as the means by which known theorems could be verified and new theorems found.133 Algebra was to be known as the ‘analytic art’ or simply ‘analysis’ for much of the seventeenth century. Viète, following Pappus, spoke initially of two types of analysis: poristic, a method of verifying a proposition in relation to known truths, and zetetic, establishing the relationships, or equations, that an unknown quantity must satisfy. Zetetic in this sense was used by Diophantus throughout his Arithmetica and also corresponds closely to the Rule of Algebra of the cossist texts. To these Viète added a third kind of analysis, rhetic or exegetic, the process of solving equations so that the unknown quantity (whether arithmetic or geometric) can be found.

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Viète's algebra was a rich blend of the work of his predecessors. His first book on the subject, In artem analyticen isagoge published in 1591, took up the old cossist tradition but with a new Classical vocabulary.134 The rules for working with symbols were set out at great length but now under the heading of logistice speciosa, meaning calculation in ‘species’, or ‘types’, while the rules taught by Islamic and cossist writers for handling equations became the leges zeteticae. Moving terms from one side to the other was labelled antithesis, dividing out excess powers of the unknown was hypobibasmus, and reducing the leading coefficient to I was parabolismus. To this point Viète identified and named existing procedures but did not add to them. At the end of the Isagoge, however, Viète went far beyond the cossists in his vision of the power and scope of algebra. He had somehow learned (from what sources we do not know) the methods developed by twelfth-century Islamic mathematicians for solving equations numerically,135 and so recognized that it was possible in principle to solve equations of any degree. Thus the Classical problems of doubling a cube or trisecting an angle could be solved numerically, as could any other problem that gave rise to a cubic or quartic equation. Dazzled by such possibilities Viète ended on a note of unbounded optimism. The analytic art, he claimed, could solve the greatest problem of all: Nullum non problema solvere, ‘to leave no problem unsolved’.136 Viète promised nine more books to demonstrate the use and application of the algebraic method in arithmetic, geometry and trigonometry. These were to constitute his Opus restitutae mathematicae analyseos, seu algebra nova, and seven of them were published individually over the next few years, some after Viète himself had died in 1603.137 The most important for the future development of algebra were those on handling equations: De numerosa potestatum ad exegesin resolutione of 1600, the first exposition in Europe of Islamic numerical techniques, and De aequationum recognitione, written by 1593 but not published until 1615, which provided a theoretical treatment of equations. In De aequationum recognitione the language and techniques of the Ars magna were everywhere apparent, particularly in the methods for transforming equations and for solving cubics and quartics: Viète's exposition, like Bombelli's and Stevin's, owed almost everything to Cardano. Unfortunately Viète failed to take up the notational advances made by Bombelli and Stevin but instead introduced Aquadratus and Acubus for A squared and A cubed, using a vowel for each unknown quantity. Viète's notation had the advantage of making the base quantity clear, but it was both unwieldy and impossible to generalize. It was also potentially ambiguous: was

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Aquadrato-cubus to be read as a fifth power or a sixth? His greatest advance was perhaps the use of letters (non-vowels B, C, D,…) for given or known quantities, so that equations could be written in entirely general form, but he retained verbal links such as ‘B in A’ for multiplication, and ‘aequitur’ for equality, so his writing still appears predominantly verbal rather than symbolic. It is rendered additionally cumbersome by Viète's insistence that equations should be dimensionally homogeneous (areas, for instance, could not be compared with lines or solids) so that he introduced artificial devices such as Zplano or Zsolido to indicate two- or three-dimensional quantities. Viète has sometimes been seen as the founder of modern algebra,138 but he is more correctly seen as heir to Cardano. His most important contribution was not in his notation or his methods, but his insight into the way algebra could open up the understanding of other branches of mathematics, especially geometry, in marked contrast to all previous authors who (recall al-Khwārizmī and Cardano) had used geometry to justify algebra. Viète laid the foundations of algebraic geometry and so stood both chronologically and mathematically at the turning point between the sixteenth and seventeenth centuries. By 1600, the Arabic origins of algebra could still be traced in the name, but in every other way algebra was a thoroughly European and sixteenth-century invention.

Wallis's account of early algebra In the opening chapter of A treatise of algebra Wallis had pinpointed, perhaps without realizing it, the main trends of early algebra, by listing the names by which it had been known: analysis, al-jabr, regula cosae and ars magna. Wallis recognized the origins of these terms in Classical Greece, medieval Islam and Renaissance Italy, respectively, but seems to have regarded them as merely different names for the same subject, whereas we have seen that each carries its own meaning and emphasis, none of which is adequately conveyed by the single word algebra. Wallis's account of late medieval and Renaissance algebra was in fact exceedingly sketchy. For the earlier part of the period this was hardly surprising, for Wallis had never seen an Islamic algebra text. Nor had he seen the Liber abbaci, which circulated only further south in Europe: he could not say anything at first hand about its contents and followed Vossius in dating it at 1400, two centuries too late.139 Algebra, as we have already noted, was not a subject taken up by the medieval Oxford scholars, so although Wallis had seen numerous medieval

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manuscripts dealing with arithmetic or astronomy, he had no such sources for algebra. When it came to printed texts the situation was different, for the Savile Library held a superb collection of sixteenthcentury algebra texts, yet Wallis displayed remarkably little knowledge of, or interest in, their mathematical content. The title of his chapter on early Renaissance algebra, ‘Of Leonardus Pisanus, Lucas Paciolus, Cardano, Tartalea, Nunnes, Bombel and other writers of algebra before Vieta’, already shows that he had reverted to the ‘list of names’ approach, and as the chapter unfolds we see Wallis paying more attention to what each writer said about his predecessors than to any advances in algebra. In other words, Wallis, like so many earlier historians, was mainly concerned to trace a ‘line of descent’ from past to present. Thus he reported in some detail on Pacioli's Summa,140 but drew attention to the teachers and writers Pacioli claimed as his sources, including Leonardo, Prosdocimus of Padua, Jordanus and Sacrobosco. Wallis also noted that all the algebra in the Summa was of Arabic origin with no trace as yet of Greek sources. Michael Stifel's Arithmetica integra and Cardano's Ars magna were dealt with rather more briefly than the Summa.141 Of the Arithmetica integra Wallis had little to say except to mention the notation, and to note Stifel's reference to the earlier authors Adam Ries and Christoph Rudolff.142 On the Ars magna there was a little more, but again Wallis's main interest was in Cardano's sources: al-Khwārizmī (whom Cardano called Mahomet Filius Mosis), Leonardo, Pacioli, Ferro, Tartaglia and Ferrari. Of the mathematical content of the Ars magna Wallis mentioned only the rules for solving cubic equations and remarked (as others have done since) that the rules were ‘by him first Published, though not first invented’.143 Elsewhere he complained that he found Cardano's demonstrations ‘intricate and perplexed’,144 and it seems that he never actually read the later chapters, for he believed to the end of his life that Bombelli was the first mathematician to treat quartic equations. Given Cardano's reputation and the easy availability of the Ars magna in Oxford, Wallis's ignorance of his work was inexcusable, and perhaps the most serious lapse of his entire history. Wallis also made brief references to the works of Scheubel, Peletier, Borrell and Nuñez, all of which were available to him in the Savile or Bodleian Library,145 but added information on Recorde only later. Of Bombelli he noted only that he showed how to reduce a quartic equation to a product of two quadratics, and it is not clear that Wallis in the 1670s had actually

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read Bombelli: he made several references to L'algebre in the Latin edition of A treatise of algebra in 1693, but hardly any in the 1685 edition.146 Passing mention of Ramus, Salignac, Stevin, Clavius and Henisch, all except the last in the Savile Library,147 brought Wallis rapidly into the early years of the seventeenth century. Wallis regarded Viète as sufficiently important to deserve a whole chapter, and made much of his notation which, he claimed, made possible a fully generalized or ‘specious’ arithmetic: he suggested that Viète (trained in law) used ‘species’ in the legal sense in which a single named person may be taken as representative of any person in the same position.148 But Wallis overlooked Viète's most important innovation, the application of algebra to geometry, and completely ignored his concept of algebra as the ‘analytic art’. Instead, he referred to Viète's algebra as ‘specious arithmetick’, a name that emphasized the cossist and arithmetic foundation of his work. He saw correctly that many of Viète's precepts were old rules in new notation, but did concede that Viète had added ‘many new Inventions of his own, for the better understanding the Reasons of those Rules, and the more convenient management of them, with many great improvements thereof ’.149 What the inventions and improvements were, however, he declined to say, for the reader could find them for himself in Viète's work, now easily available, or in the work of later authors. In particular, said Wallis, William Oughtred ‘hath contracted much of it into less room’.150 This was the cue Wallis needed for introducing Oughtred and the seventeenth-century English mathematicians: from this point on, Wallis's history of algebra was to be a history of English algebra alone.

3 Ariadne's thread: William Oughtred's Clavis William Oughtred (1573–1660)1 (Fig. 3.1) was renowned throughout his long life as a teacher of mathematics and especially, in his later years, for his Clavis mathematicae, his textbook on arithmetic and algebra. The book was eventually published in five Latin and two English editions, and outlived Oughtred himself by over forty years. Yet in many ways the book was oldfashioned even when it first appeared, and its survival, eventually alongside much better and clearer texts, cannot be attributed to its mathematical content alone. This chapter traces the story of the Clavis, the reasons for its popularity and long life, and its influence on seventeenth-century English mathematics. Oughtred's father, Reverend Benjamin Oughtred, was the registrar at Eton College, and so would certainly have been a ‘scrivener’ and have understood ‘common arithmetique’,2 but was also said to have taught writing and so was probably responsible both for Oughtred's neat and beautifully formed italic hand and for his early interest in mathematics. Oughtred was educated as a King's scholar at Eton and then from the age of fifteen at King's College, Cambridge, where he wrote his first mathematical treatise, A most easie way for the delineation of plaine sun-dials.3 He remained as a Fellow at King's until about 1606, then became rector of Albury, just below the ridge of the North Downs near Guildford in Surrey, where he was to remain for the rest of his life. During his early years at Albury, Oughtred travelled no further than occasionally to London, but he maintained a reputation for his skill in mathematics. Already in 1616 John Hales, formerly an Oxford lecturer

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Fig. 3.1 William Oughtred (1573–1660) from the 1647 English edition of his Clavis mathematicae.

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in Greek and now fellow of Eton, consulted Oughtred on mathematics and wrote:4 Either your facility was great or your pains very much, who could in so short a space discharge yourself of so many queries… Amongst all the solutions which you then sent me, none there was which gave me not full and sufficient satisfaction, (and so I persuade myself would have given to one of deeper skill than myself).

Oughtred became known to Sir Charles Cavendish (1591–1654), who devoted much of his life to collecting mathematical books and manuscripts, and who cultivated contacts with mathematicians in both England and France. It was almost certainly Cavendish who introduced Oughtred to the analytic art of François Viète, for in the British Library we have some rare and important evidence of Oughtred's mathematical reading, margin notes that he made in Cavendish's copy of Viète's De aequationum recognitione (1615).5 In 1628 Oughtred was engaged as tutor to William Howard, the fourteen-year-old son of Thomas Howard, Earl of Arundel and Surrey, whose country home at West Horsley was about five miles from Albury. (Oughtred had earlier been employed in a similar capacity by another local notable, Sir Francis Aungier, later Baron Longford.) Oughtred was already over fifty years old, and had not written anything mathematical since his Cambridge days but, encouraged by Cavendish, he wrote his first textbook, on arithmetic and algebra, in 1630. It was published early the following year under the title Arithmeticae in numeris et speciebus institutio: quae turn logisticae, tum analyticae, atque adeo totius mathematicae quasi clavis est (‘Calculation in numbers and letters: which was the key first to arithmetic, then to analysis and now to the whole of mathematics’) (Fig. 3.2). The title was lengthy but the book itself was both small and concise, a pocket-size volume containing twenty short chapters in just 88 pages. The title was soon abbreviated: the running head in the first edition was Clavis mathematicae, and the book quickly became known simply as the Clavis. Oughtred's book was written at a time when almost no good algebra text was available to English readers. In 1596 William Phillip had made an English translation of Nicolaus Petri's Practique om te leeren reekenen, first published in Amsterdam in 1583; a literal translation of the Dutch title would be The practice of learning calculation, but Phillip called his work The pathway to knowledge (not to be confused with Recorde's 1551 work of the same name which dealt purely with elementary geometry). From Wallis's later account it would seem

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Fig. 3.2 Title page of the first edition of the Clavis mathematicae (London 1631).

that Phillip translated the text as freely as the title, either mistranslating technical terms or simply leaving them in Dutch.6 A revised version of Phillip's book, again called The pathway to knowledge, was published in 1613 by John Tapp, better known as a writer on navigation.7 To Phillip's weak translation Tapp added from other authors: … for finding the nearest root a number not a right square or cubicall I have observed the method used in Gosselin upon Tartaglia,

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and for the rules and questions in the Arte of cossicke numbers it is only a literal translation collected out of Mr Valentine Mennher his Arithmeticke.

Mennher's Arithmetique seconde had first been published in Antwerp in 1556, and Gosselin's edition of Tartaglia's L'arithmetique in Paris in 1578, so Tapp's book, by his own admission, was a compilation of French and Dutch texts, written between thirty and sixty years before his own. This small volume was the only algebra text to be published in England throughout the first thirty years of the seventeenth century, an indication of how little algebra was generally known or taught in England in those years. If we look to foreign textbooks, the situation was hardly better. Most of the elementary algebra books published in Europe during this period were no more than reprinted editions of sixteenth-century works, and the few new texts that were published went no further than the traditional cossist algebras. An increasing number, in any case, were being written in vernacular languages, predominantly in German or Italian, and so were unlikely to be used in England.8 Oughtred's Clavis can therefore be seen as an important attempt to bring the latest advances in algebra and algebraic notation, as Oughtred understood them, to English readers.

The rst edition: 1631 On New Year's Day 1631, Oughtred wrote an introduction to his newly completed text, beginning with remarks addressed to the young William Howard:9 Most illustrious youth, since the time I have served you so devotedly, by order of your father, in expounding the teaching of mathematics, I have wished nothing more, than that I might in the best of faith show this, that is, the Analytic method (of which teaching it is certainly part). And for this reason, I have recalled the demonstrations of Euclid, among other things, in the Analytic form, of which in the 19 chapters of this book, are to be had several examples.

The last sentence indicates that the twentieth chapter (of assorted problems) was added after the preface was written, in which case the ‘demonstrations of Euclid in Analytic form’ were the culmination of the original text. Oughtred clearly understood and endorsed Viete's ‘analytic method’ as a means of re-writing Classical theorems in algebraic notation, but

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he also, like Viète before him, saw it as a way of recovering the methods by which such theorems were originally found. A few lines later he wrote:10 Then I want to extend to students of mathematics, as it were, Ariadne's thread, by the help of which they may be led to the innermost secrets of this knowledge, and directed towards an easier and deeper understanding of the most ancient and favoured authors.

Oughtred's readers would have understood his reference to Ariadne, the Cretan princess who gave her lover, Theseus, the thread by which he was able to find his way out of the Minoan labyrinth. Oughtred's purpose was to use the new method of analysis to understand and recreate the work of Classical writers, so that for him Ariadne's thread was to be a guide not to the future but to the past. Oughtred went on to describe how the investigation of ancient writings was to be done by ‘interpretation, comparison and reduction of equations’, and in symbols that rendered these matters ‘clearer to the eyes’; his work was short, and rightly so, for he had not written for the ‘half-asleep’ but for those who preferred their mathematics concise and brief. His method of teaching, he stated, was by problems and examples, and so that nothing would be lacking from his text he had added the rules of arithmetic, and instruction in decimals, now more useful than the older sexagesimal system. In the early chapters the Clavis differed little in content from its predecessors. Where cossist texts had treated arithmetic first in numbers, and then in letters or ‘species’, Oughtred handled numbers and letters side by side, but covered much the same content: the four operations of addition, subtraction, multiplication and division, followed by proportion and greatest common measure, and the difficult subject of fractions. Oughtred followed Viète in using A and E for unknown quantities, but abbreviated Aquadratus and Acubus to Aq and Ac, replaced A in E by AE or Æ, and introduced × for multiplication. In Chapter 11 he introduced more new notation by defining for two quantities A and E, the sum Z, difference X, product P and quotient R/S, and he explained how, given A or E and any one of Z, X, P or R/S, the others could be found by use of easy formulae. The text continued with a table of powers up to 98 = 43 046 721, and another for powers of (A + E) up to (A + E)10, and Oughtred showed how to use such tables to find square and cube roots by the method outlined by Viète. In Chapter 16 Oughtred reached the treatment of equations, and once again followed the pattern of the cossist texts in explaining the Rule of

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Algebra and the rules for simplifying equations.11 In the next two chapters he extended his notation to include Z, X, Z, X for sums and differences of squares and cubes:

and showed how to write identities such as:12

In Chapter 19, Oughtred at last gave the results from Euclid promised in his preface, writing each of the fourteen propositions of Book II of the Elements in purely algebraic form and without diagrams, a new departure in an elementary text.13 Propositions II.5 and II.6 led to quadratic equations, expressed in Oughtred's notation as:14

Oughtred used the symbol ± to give, again for the first time in print, the solution formula for the roots of the first equation (both positive):15

and for the positive roots of the second and third:

Chapter 20 went on to give twenty further examples of the application of algebra to geometry. The most interesting was the final one in which Oughtred derived algebraic formulae for the trisection and quinquisection of angles,

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and here too he was following Viète.16 He was unable to give geometric constructions but hoped that with the help of his new methods, solutions to this ancient problem might yet be discovered. What did the Clavis contribute to the development of algebra? Charles Hutton, in his article on algebra in A mathematical and philosophical dictionary in 1796, carefully assessing the innovations of each new text, noted how much of Oughtred's material derived from Viète, but credited Oughtred with the introduction of ‘various symbolical marks and abbreviations which are not now used’, ‘the first instance of applying algebra to geometry’, and ‘a good tract on angular sections’.17 Hutton's remark on notation was substantially correct: Oughtred's invention of × for multiplication was masterly, though he rarely used the symbol himself, but his other innovations, his topped and tailed Zs and Xs, were as incapable of generalization as the cossist symbols that had preceded them, and no more than temporarily helpful. Oughtred's notation enabled him to write identities in a variety of ways, but did nothing to reveal the more general relationships that had to be understood in order to move from the known to the unknown, a limitation that could in the end only hinder rather than help the development of true algebraic thinking. On the second point, Hutton was wrong: Viète's ideas about recovering geometric theorems through algebra had been known for forty years and were familiar to Oughtred who, as his preface plainly showed, was also using algebra for the purpose of understanding the works of Classical writers.18 Where Oughtred led the way was not in attempting such problems but in introducing them for the first time into an elementary textbook. The unique achievement of the Clavis was that in one volume it combined the basic rules and methods of cossist algebra with applications of algebra to geometry. In this way it filled, as no other book then did, the gap between elementary texts of poor quality or in foreign languages, and the more sophisticated and not easily available works of Viète and his followers. Had good textbooks been more common, the Clavis might have appeared unremarkable; instead it became a classic, and made Oughtred's name. Two years after publication William Robinson wrote to Oughtred (already using the abbreviated form of the title):19 I shall long exceedingly till I see your Clavis turned into a pick-lock; and I beseech you enlarge it, and explain it what you can, for we shall not need to fear either tautology of superfluity; you are naturally concise and your clear judgement makes you both methodical and pithy.

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By 1635 Franc Derand was writing to Cavendish that the Clavis was ‘in great estimation amongst the mathematicians at Paris’,20 and Mersenne considered that ‘there was more matter comprehended in that little book than in Diophantus, and all the Ancients’.21 By now the French had Albert Girard's Invention nouvelle en l'algebre, a much clearer text than the Clavis, but with none of the algebraic geometry that Oughtred had included. In England the Clavis had no competitor. In 1636 Robinson wrote again, and his letter implies that an English translation was already under way. He also suggested that for some readers the compactness of the book presented difficulties, and begged Oughtred to expand his teaching:22 I will once again earnestly entreat you, that you be rather diffuse in the setting forth of your English mathematical Clavis, than concise, considering that the wisest of men noted of old, and said stultorum infinitus est numerus [the stupid are infinite in number], these arts cannot be made too easy, they are so abstruse of themselves … Brevity may well argue a learned author, that without any excess or redundance, either of matter or words, can give the very substance and essence of the thing treated of; but it seldom makes a learned scholar; and if one be capable twenty are not;

Among those who sought personal instruction from Oughtred were Seth Ward and Charles Scarborough, who in 1642 were both young Fellows at Cambridge. According to Aubrey, they ‘came [to Oughtred] as in Pilgrimage, to see him and admire him’ and ‘to be enformed by him in his Clavis mathematicae, which was then a booke of Aenigmata’.23 Ward and Scarborough afterwards taught their students from the Clavis, and so raised it to the status of a university textbook. Their direct influence in Cambridge was short lived as teaching fell into disarray on the outbreak of civil war, and both Ward and Scarborough were forced to leave, but Ward continued to spend time with Oughtred and stayed with him during 1644. Oughtred also received written requests for help and William Price's letter of 1642 was probably only one of several:24 Sir, Though I am a stranger to your person, yet I am well acquainted with the fame of your singular skill in the mathematics, and thereupon have so far presumed, to intreat your assistance for the geometrical solution of the inclosed diagram, which, to you that have attained the perfection of the analytical art, perhaps will not appear difficult.

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Oughtred's reply indicated that his renown in mathematics had caused some to accuse him of neglect of his parochial duties, and that now, nearing seventy, he was becoming reluctant to take on new work, perhaps explaining why the English translation expected by Robinson in 1636 had never appeared:25 But now being in years and mindful of mine end, and having paid dearly for my former delights both in my health and state, besides the prejudice of such, who not considering what incessant labour may produce, reckon so much wanting unto me in my proper calling, as they think I have acquired in other sciences; by which opinion (not of the vulgar only) I have suffered both disrespect, and also hinderance in some small preferments I have aimed at. I have therefore now learned to spare myself, and am not willing to descend again in arenam, and to serve such ungrateful muses. Yet, sir, at your request I have perused your problem,…

That complaints were indeed made against Oughtred was borne out by Aubrey: ‘I have heard his neighbour ministers say that he was a pittiful preacher; the reason was because he never studyed it, but bent all his thoughts on the mathematiques.’ Aubrey added, however, ‘when he was in danger of being sequestered for a royalist, he fell to the study of divinity, and preacht (they sayd) admirably well, even in his old age’.26 The Civil War was the most difficult time of Oughtred's life, and the danger of sequestration, according to Aubrey, prompted Oughtred not only to renewed zeal in preaching, but to complete the English translation of the Clavis:27 Notwithstanding all that has been sayd of this excellent man, he was in danger to have been sequestered, and … Onslowe that was a great stickler against the royalists and a member of the House of Commons and living not far from him – he translated his Clavis into English and dedicated it to him to clawe with him, and it did soe his businesse and saved him from sequestration.

Oughtred's own account of how the second edition of his book came to be published was a little different:28 I was unwillingly drawne, at this my declined age, to appear unto the world in such a kinde of Subject. But occasion was administred by one Mr Seth Ward, a young man excellently accomplished with all parts of polite Literature, then Fellow of Sidney Colledge in Cambridge, who tooke the pains to seek me out at my house, and by a gentle violence

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induced me to publish again my former Tractate in a manner new moulded and perfected:

There was probably truth in both accounts. Onslow, the ‘stickler against the Royalists’, to whom the English translation of the Clavis was flatteringly dedicated (Fig. 3.3), was Sir Richard Onslow, who had raised his own regiment for the Parliamentary side in 1642 and was appointed one of the sequestrators for Surrey in 1643. Oughtred's successful appeal against sequestration was probably heard in 1646, and the English version of the Clavis was prepared for publication that same year.29 Even if the immediate danger had passed, it might still have seemed expedient, or grateful, to stay on the right side of Onslow, and Ward, who well understood how to bend before the political wind,30 possibly had just such a motive in mind when he persuaded Oughtred to republish. In the revised preface the original tribute to Cavendish, now a prominent Royalist, was quietly dropped, and the new edition carried a portrait of Oughtred in suitably puritan dress (Fig. 3.1).

The second editions: 1647 and 1648 The first English translation of the Clavis was done by Robert Wood (1622–1685), then a student in his early twenties at Merton College, Oxford. It bore the title The key of the mathematics new filed, and was published in 1647 with a preface in which Oughtred answered those who complained that his work was difficult: Which treatise being not written in the usuall syntheticall manner, nor with verbous expressions, but in the inventive way of Analitice, and with symboles or notes of things instead of words, seemed unto many very hard; though indeed it was but their owne diffidence, being scared by the newnesse of the delivery; and not any difficulty in the thing it selfe. For this specious and symbolicall manner, neither racketh the memory with multiplicity of words, nor chargeth the phantasie with comparing and laying things together; but plainly presenteth to the eye the whole course and process of every operation and argumentation.

And in a mixture of metaphors, he restated the purpose of his book: Now my scope and intent in the first Edition of that my Key was, and in this New Filing, or rather forging of it, is, to reach out to the ingenious lovers of these Sciences, as it were Ariadne's thread, to guide them through the intricate Labyrinth of these studies, and to direct them

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Fig. 3.3 The dedication to Sir Richard Onslow in the 1647 English edition of the Clavis.

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for the more easie and full understanding of the best and antientest Authors.

Oughtred chose his words aptly when he described this second edition as a new forging, for there were significant changes from the first edition, many perhaps suggested by Wood or Ward. The opening chapters carried a number of minor additions, with a longer exposition of sexagesimal arithmetic, and with arithmetic, geometric and harmonic progressions added to the chapter on proportion. Oughtred's idiosyncratic notation was introduced rather earlier, at Chapter 11, and Chapter 16 on the formation and reduction of equations was much expanded with a section dealing specifically with quadratic equations, another on the application of this work to the squares and roots of binomials (quantities of the form a ±√b), and some preliminary work on angular sections. Chapter 17 had disappeared and Chapter 18 was reduced, their contents dispersed elsewhere. The most significant extension of the book was to the original Chapter 19 (now renumbered 18) on Euclid II, which was almost completely re-written and entitled ‘The analitical store’.31 Only Propositions II.5–II.10 remained, but to these Oughtred added a collection of what he called ‘analyticall furniture’: identities on squares and cubes from the original Chapter 18 (but not cubic equations); useful formulae for circles, cylinders and spheres; and twenty theorems and nineteen constructions (from Euclid I, III and VI) that the intending analyst should know. The last four propositions II.11–;II.14 of Euclid II (today known as the ‘golden section’ and the ‘cosine rule’) were moved to the final chapter, where they were followed by a new section on arithmetic progressions, and the twenty problems of the original twentieth chapter. The new edition also carried a long appendix on the numerical solution of equations, short pieces on the calculation of interest and the rule of false position, and Oughtred's early treatise on sundials. A parallel Latin edition was published the following year, identical in content to the 1647 English version but without the dedication to Onslow or the preface, and with two further appendices (one a transcription of Euclid X into symbols, the other on regular solids). The result of these changes was not only a longer text (120 pages in English, 112 in Latin), but one that had lost the methodical structure of the original. Elementary as it was, the 1631 version led the reader by a recognizable and well worn route, and introduced new ideas in such a way as to build steadily towards the more difficult final chapters. In the new version the reader came upon new notation at Chapter 11, but it then disappeared

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until put to use in the difficult new material added to Chapter 16, and the final chapters were no longer a gentle introduction to algebraic geometry but an encyclopaedia of miscellaneous results and problems. The changes may have improved the usefulness of the Clavis as a reference book, but they did little to improve its structure. However, from this point on the text was essentially established and was preserved in future editions without any significant alteration.

The third edition: 1652 In 1652 the Clavis was published in its third Latin edition, this time at Oxford. Why was a further edition thought to be necessary so soon? A letter from Oughtred to Ward about the 1652 Oxford edition has survived among Aubrey's papers (Fig. 3.4), and in addition to the now familiar names of Ward, Wood and Scarborough, it also mentions Wallis:32 Worthy Sir, I made bold lately when I sent my book in a letter to Mr Wood to nominate you and Mr Wallis together with him, to whose judgement and discretion I commit all my right and interest for the printing thereof at Oxford. I have sent the Epistle [preface], which, though written long since, yett was soe mislayed and mingled with many other papers, that I thought it lost. Therin I make noe unloving mention of your self and Dr Scarbrough, whose surname [sic] I remember not. I hope neyther of you will take my officiousnesse in evell part. Yett yf anything shall displease, you are intreated of me to alter it or raze it with a blott; but yf in and by your suffrage it maye passe, I would intreat you to supplie the Doctor's surname … So you will be pleased to remember my best respects to Mr Wallis and favourably to pardon this troublesome interruption of him who am Your truly loving friend to my power Willm Oughtred Aldburie April 19 1651

Wallis and Ward had both been appointed as Savilian professors during the political changes of 1649. Wallis had begun his mathematical studies very much later than Ward, but both owed much to Oughtred (the important influence of the Clavis on Wallis's mathematical development will be discussed in more detail later), and probably both saw the value of introducing the Clavis to Oxford in a new edition cleared of errors and misprints and under an

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Fig. 3.4 A letter from William Oughtred to Seth Ward about the publication of the third edition of the Clavis at Oxford (Bodleian Library MS Aubrey 6, f. 40).

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Oxford imprint. The new edition was printed by Leonard Lichfield, one of the official printers to the University,33 and the price of the book, according to a note in Pell's papers, was four shillings for a bound copy.34 The 1652 edition carried a new preface that explained how the book came to be written. William Howard, now Viscount Stafford, was living in exile, but it was a sign of easier political times that the original influence of Cavendish could once again be acknowledged. The preface went on to reiterate the advantages of the analytic method, and repeated the metaphor of Ariadne's thread. Then Oughtred described, as quoted above, how he was persuaded to bring out the second edition of the Clavis by Ward, supported by Scarborough. For removing typographical errors, checking the calculations and generally supervising this new third edition with ‘unbroken assistance and persistent scrutiny’ he thanked Wallis. There were warm thanks too for Wood. The final acknowledgement was to Christopher Wren, then a student at Wadham College ‘from whom we may expect great things’, and whose translation of Oughtred's 1596 treatise on sundials appeared at the end of the appendices. Now nearly eighty, Oughtred must have been grateful for the help of these younger men, and rarely can any textbook have boasted such illustrious support. Oughtred's admirers were now reaching the prime of their careers. Ward and Wallis were Oxford Professors; Scarborough was a Fellow of the Royal College of Physicians, later to be personal physician to Charles II and James II; Jonas Moore, not mentioned in the 1652 preface but acknowledged in 1647, was tutor to the young Duke of York, the captive son of Charles I. These were the first of Oughtred's pupils to attain positions of eminence, but Wren, still only twenty, was representative of a new generation who were to learn their mathematics from the Clavis. Robert Boyle wrote to Samuel Hartlib in 1647 when he too was twenty years old:35 The Englishing of, and additions to Oughtred's Clavis mathematicae does much content me, having formerly spent much study on the original of that algebra, which I have long since esteemed a much more instructive way of logic, than that of Aristotle.

John Locke, born the same year as Wren, was to write as late as 1681 that ‘the best algebra yet extant is Oughtred's’.36 Isaac Newton first read the Clavis (in its 1652 edition) in 1664 and never met Oughtred, but nevertheless spoke of him thirty years later as ‘a man whose judgement (if any man's) may be relyed on’.37 To this long list of seventeenth-century luminaries may be added many others less well known who likewise cut their mathematical teeth on the Clavis.

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Indeed, for nearly twenty years after it first appeared, no alternative elementary textbook was published in England. (Harriot's Praxis, to be discussed in the next chapter, came out in the same year as the Clavis, 1631, but was not a book for beginners.) Not until 1650 did two new texts appear, but neither was a serious rival to Oughtred's. The first was Richard Balam's Algebra, or the doctrine of composing, inferring and resolving an equation, a tiny volume, which can only have sown confusion and despair in the minds of its readers. Balam introduced strange new notation: Dp 4 for −4, Dq4 for +4 and &z.tdfnc;A&z.tdfnc; for the ‘triplicate or cube’ of A (though ‘triplicate’ and ‘cube’ are not the same), and later also A(3) for the cube. But the text was primarily verbal rather than symbolic, and if Balam's notation was obscure, his vocabulary was even more so. A few short examples suffice to give the flavour of this strange little book: All the affirmed nomes in a multinomiall are addends and all the denied nomes are subducends, to be subducted from the sum of the Addends, and they are to be composed in one summe before subduction … Two like inequimultiplicats (or cossick proportionals) are as any two of their homologall factors … The inferring of an equation, or equative inference, is a numeration, which from an equation given and precedent, inferres a consonant or new equation. That which I here call equative inference is not mentioned by this name, in any Algebraicall Writer, which I have seene. But surely this is the thing, of which are meant the 5 rules, which Mr Oughtred gives in the 16th chapter of his Clavis …

Surely too, anyone reading this would be only too pleased to return to Oughtred's text without seeking Richard Balam's help in the matter. However, the book must have had some success, for it was reprinted in 1653.38 A more serious text, also published in 1650, was Jonas Moore's Arithmetick in two books. Here too, Oughtred's influence was apparent, and not surprisingly since Moore had been one of his pupils. In the traditional manner, the first of the ‘two books’ dealt with ‘vulgar arithmetick’ and the second with ‘arithmetick in species’, or algebra. The latter covered much the same ground as the first edition of the Clavis but in a more elementary way; Moore thought it necessary to point out, for instance, that in forming equations one was dealing only with numbers, not with the things to which they referred, such as men or money. The book included the solutions to quadratic equations

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expressed in Oughtred's notation, but elsewhere in the book Moore used the superscript notation a2, a3, introduced by Descartes in 1637, here seen for the first time in an English printed text. Amongst the general acclaim for the Clavis, there were just a few voices of dissent, as recorded by John Collins, who later wrote: ‘I know many that did lightly esteem [Oughtred] when living, some whereof are at rest, as Mr. Foster and Mr. Gibson’.39 Little is known of Thomas Gibson, but a later letter from Collins to James Gregory indicates that he died in 1657 or 1658.40 Gibson's Syntaxis mathematica was published in 1655 and was a radically new kind of textbook, explicitly indebted not to Oughtred but to Harriot and Descartes, as Gibson clearly stated in his preface to the reader: The method here used is the same as in Master Harriot in some places, that is, in such equations as are proposed in numbers. And as in Des Cartes in some other places, that is, in such equations as are solid [cubic], and not in numbers.

Harriot's influence was immediately obvious in Gibson's notation: a and e for unknowns; b, c, d, f, etc. for knowns, and generally aaa (but occasionally a3) for a cube. Gibson also used Harriot's < and > for inequality. As promised in the preface, quadratic and cubic equations were solved numerically using Harriot's method, and in the following theoretical section Gibson showed how a polynomial could be composed as a product of factors, and stated a number of general results: that an equation can have as many real roots as dimensions, never more, but sometimes fewer because of repeated roots; that the numerical term is the product of the roots (including negatives); that positive roots can be changed to negative by changing the sign of every odd power; and Descartes' Rule of Signs for the number of positive roots. Finally he showed how to transform equations by appropriate changes of root, leading to the usual solution for cubic equations. A later section described the application of algebra to geometry, but Gibson made no attempt, as Oughtred did, to give geometrical constructions for his solutions, being content to leave his readers with algebraic solutions. All this was a world away from the cumbersome text and content of the Clavis and would have seemed to render the ideas of Harriot and Descartes accessible for the first time to the ordinary English reader, but Collins, generally well informed about both English and foreign texts, wrote in 1667 that he had never seen the book.41 Gibson was evidently an obscure figure even in his own day, whose reputation was never likely to compare with Oughtred's, and after his death two years later neither he nor his book ever became widely known.

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Samuel Foster, Professor of Astronomy at Gresham College from 1641 until his death in 1652, appears to have been one of the strongest critics of the Clavis, for Collins later wrote that as early as 1649 ‘Mr Foster of Gresham College seldom heard it mentioned but took occasion to utter his dislike of it’.42 Some of Foster's work was translated and published posthumously in 1659 as Miscellanies or mathematical lucubrations by John Twysden, who took the opportunity to add some work of his own, but Twysden was a lawyer and physician, not a mathematician, and the work was of no great weight. Twysden was evidently ignorant of Foster's opinion of Oughtred's Clavis for in his acknowledgements to other authors he wrote: Amongst them all let Mr William Oughtred, of Aeton, be named in the first place, a Person of venerable grey haires, and exemplary piety, who indeed exceeds all praise we can bestow upon him. Who by an easie method, and admirable Key, hath unlocked the hidden things of geometry.

By the time Oughtred died in 1660 the Clavis remained virtually unchallenged as the primary algebra text for aspiring mathematicians, and it was to outlive its author for many years yet.

The fourth edition: 1667 It was Wallis, always supportive of Oxford publishing, who suggested that a fourth edition of the Clavis should be printed following the disastrous loss of books in the Fire of London in 1666. (Ward was by now Bishop of Exeter and no longer actively engaged in mathematics.) By this time the Rahn–Pell Introduction to algebra was in the press, but the only other algebra published since the Foster–Twysden Miscellanies was Dary's The general doctrine of equation (1664), no more than a slim 16-page summary of the known rules for handling equations. The dearth of good alternative texts (apart from Gibson's neglected Syntaxis) remained. After the death of Leonard Lichfield in 1657, ownership of the imprint of the Clavis had passed to his widow, Margaret, and Wallis suggested to Collins that Moses Pitts, bookseller, should negotiate with her for the rights. Pitts had already expressed an interest in publishing some of Wallis's work,43 but he was less enthusiastic about taking on the Clavis on the grounds that the sale of even the best mathematical texts was slow, and that more up-to-date books were now being prepared either in England or abroad.44

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Collins also recognized that the Clavis was no longer held in uncritical regard. In February 1667 he wrote to Wallis to tell him so:45 The said Mr Kersey hath made notes on the Clavis and to say the truth, doth not admire any thing in it, save what concerns the tenth and succeeding Elements of Euclid. Mr Bunning, an aged minister, near Nuneaton in Warwickshire, hath commented on the Clavis, which he left with Mr Leybourne to be printed; but one Mr Anderson, a knowing weaver,46 told Mr Bunning that the Clavis itself, and his comment thereon, were immethodical, and the precepts for educing the roots of an adfected [polynomial] equation maim and insufficient.47

Despite these objections Collins reported that Pitts was nevertheless prepared to consider reprinting, provided he were allowed to increase his potential sales by adding a commentary to the original text.48 Wallis replied three days later in somewhat defensive tone, that he had made the suggestion only in response to Pitts' expressed willingness to publish mathematical works, and in view of the convenience of using an impression already prepared. To the hesitations expressed by Pitts and Collins he gave a reply in which he elevated Oughtred to the status of Euclid or Archimedes:49 Whether the number be too great, or the book not so vendible, the bookseller, who understands his trade, is a more competent judge than I. But for the goodness of the book in itself, it is that (I confess) which I look upon as a very good book, and which doth in as little room deliver as much of the fundamental and useful part of geometry (as well as of arithmetic and 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 mathematics, fashions will daily alter, and that which Mr Oughtred designed by great letters may be now by others be designed by small; but a mathematician will, with the same ease and advantage, understand Ac, and a3 or aaa. Nor will Euclid or Archimedes cease to be classic authors and in request, though some of their considerable propositions be, by Mr Oughtred and others, delivered now in a more advantageous way, according to men's present apprehensions. And the like I judge of Mr Oughtred's Clavis, which I look upon (as those pieces of Vieta who first went that way) as lasting books and classic authors in this kind;

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Wallis himself was reluctant to see the book made larger: For if Mr Oughtred had intended it to be large, he could with more ease have made it much bigger than it is. But it was by him intended, in a small epitome, to give the substance of what is by others delivered in larger volumes.

The negotiations continued. Collins reported back to Wallis that Pitts was now working in partnership with another bookseller, Mr Thompson, that both were convinced of the need for a commentary and that Thompson had one in mind:50 … they both doubt it will not sell without a comment; and Mr Thompson says he was long possessed of Mr Clarke's comment, who would freely have imparted it to any one to print, and presumes he may have it again if he request it, and affirms it is very large, and will make above 20 sheets.

In the matter of the commentary, Pitts lost and Wallis won. When the Clavis was reprinted later the same year it was not for Pitts and Thompson, but for booksellers John Crosley and Amos Curteyne, and there were no new additions to the text.51 Gilbert Clark's commentary, mentioned more than once during the negotiations over the fourth edition of the Clavis,52 finally appeared in 1682 under the title Oughtredus explicatus, sive commentarius in clavem mathematicam Oughtredi (‘Oughtred explained, or a commentary on Oughtred's key to mathematics’) but did no more than expand and explain the easier sections of Oughtred's original text. An important letter from Collins on the Clavis was probably addressed to Clark,53 and is quoted here at length for Collins' careful appraisal of the value of Clavis by the early 1670s. Worthy sir, I have yours in answer to what was objected against the Clavis. It was not my intent to disparage the author, … I do not search atramentum in nive [blackness in the snow], but my design was to acquaint you with the argument of certain books, whereby the Author might be improved … Nor is the Author or any man blamed for making a collection of things already known. Collection, translation, and illustration of matters scarce, exotic, and obscure, cannot but have its encouragement. You grant the author is brief, and therefore obscure, and I say it is but a collection, which, if himself knew, he had done well to have quoted his authors, whereto the reader might have repaired.

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Collins went on to cite a number of authors who had taken up Viète's notation before Oughtred did.54 He carefully refrained from blaming Oughtred for omitting the names of his precursors, but wanted them mentioned so that readers could return, if they wished, to the original sources. When it came to Oughtred's notation, however, his criticism was more direct: And as for Mr Oughtred's method of symbols, this I say to it; it may be proper for you as a commentator to follow it, but divers I know, men of inferior rank that have good skill in algebra, that neither use nor approve it. One Anderson, a weaver, … Mr Dary, the tobacco cutter, … Wadley, a lighterman, … and [I] may acquiesce in these men's judgments, or at least in Dr Pell's, who hath said it is unworthy to the present age to continue it, as rendering easy matters obscure. Is not A5 sooner writ than Aqc? Let A be 2. The cube of 2 is 8, which squared is 64: one of the questions between Maghet[,] Grisio55 and Gloriosus is whether 64 = Acc or Aqc. The Cartesian method tells you it is A6, and decides the doubt.

Collins continued: As to the third objection, about the defect of argument, and fourth about the improvement of the general method, they cannot properly concern the author, nor is he to be blamed for not publishing what probably he knew not, which yet, in good part, was then extant in Gerrard56 and Vieta de Recognitione et Emendatione Aequationum,57 but those works of Vieta came out piecemeal, most of them at his own dispose, and thence became almost unknown and unprocurable.

Thanks to John Pell, we now know that Oughtred in fact read Viète's De aequationum recognitione et emendatione with great care.58 Collins recognized, however, that Oughtred was writing within the limitations of his time, and what he now wanted was not that the Clavis should be abandoned, but that it should be revised and updated for modern readers: The aim of those objections was not to disparage the author, but to incline you to supply the defect of him, that his book, together with yours, might be of the more durable esteem, and not be undervalued (as that author now is by Mr Hooke59 and Dr Croone,) as wanting the most material parts of algebra. I agree with you. the author is not to be rejected; he was, without doubt, a very learned divine and mathematician, and one that did much good in his generation. I know no man that would willingly

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be without his book, and certainly it had been a great detriment to learning to have wanted it.

Collins' reasonable and carefully stated opinion that the Clavis had been invaluable in its time, but was now capable of improvement, was only what any objective person might have argued about a book that had been in circulation for some forty years, and his view was echoed by Henry Oldenburg who remarked in 1668 that many English mathematicians now preferred the method of Descartes and Pell which ‘seemes to them more facile and compendious [concise] than that of Oughtred’.60 It must have seemed, even to most of its admirers, that the Clavis had served its purpose and could without dishonour be set aside, but Wallis was not yet ready to let it lapse into oblivion.

Wallis and the Clavis Wallis's admiration of the Clavis has already been touched on, but at this point deserves further study, for the Clavis played a crucial role in Wallis's mathematical development and he in turn became the book's most ardent and lasting supporter. By his own account, Wallis read the 1631 edition of the Clavis ‘with great delight’ in 1647 or early 1648.61 There are three copies of the first edition annotated by him in the Savile Library.62 Many of the annotations are beginner's notes: rules for the four operations on negative numbers, lists of relationships between A, E, Z and X, additional diagrams for the geometrical problems, and so on (Fig. 3.5). Wallis's first use of Oughtred's text was to take the identities involving cubes from Chapter 18:

and to use them to find a solution for cubic equations, by a method similar to Oughtred's for quadratics. His solution was identical to that of Cardano a century earlier, but then unknown to him, and he sent it off to John Smith, who had been a slightly younger contemporary of his at Emmanuel College, and was now lecturing in mathematics at Queens' College, Cambridge. Wallis referred to his 1648 correspondence with Smith on several occasions,63 but unfortunately no copies of the letters survive (Smith died in 1652). Wallis published his solution to cubic equations in 1657 in the preface to his Adversus Meibomi,64 and wrote it out again for Collins in 1673.

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Fig. 3.5 Wallis's annotations on positive and negative numbers in a copy of the first edition of the Clavis (Bodleian Library Savile Z.19).

To Collins he wrote:65 I was not displeased at this my good success upon the first attempts of a young algebraist; and the rather because I did not know but that I was the first that had made this discovery, though since I find that Cardan had been before me. … I was well content with my success so far, and proceeded, for my further exercise, where Mr Oughtred ends his Clavis, to the business of angular sections.

The ‘business of angular sections’, taken up by Wallis from the final section of the Clavis, became the subject of a small discourse, which he also sent to Smith.66 This early work was somewhat laboured, and the results were not new (though Wallis did not know it at the time), but the value to Wallis was in what he learned from the attempt:67 And this speculation was then the more pleasing to me, because from hence I discovered the necessity, of what I did before suspect: that, in superior Equations there might be more than Two Roots; though

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I had not found, in Mr Oughtred, any mention at all of Negative Roots; nor, of more than Two affirmatives in any Equation. 'Tis true that Harriot, and (after him) Des Cartes, do expressly declare it; and I find that Vieta, was also aware of it … But I had then seen none of these; knowing then no more of Algebra than what is in Oughtred's Clavis, (from whence I had newly learned it,) and what my own thoughts did suggest from thence.

Wallis's study of the Clavis not only gave him his first crucial mathematical insights, but helped to launch his career. When the Savilian professorship fell vacant in 1648 Wallis was unknown as a mathematician except through his correspondence with Smith, and his results on cubic equations and angular sections were his only credentials.68 Wallis probably met or corresponded with Oughtred shortly after he first read the Clavis, and by 1651 he was involved in revising and correcting the text for the third, Oxford, edition. Aubrey, who had no liking for Wallis, suggested that he even made his own self-serving contribution to the preface:69 When Mr Oughtred's Clavis was printed at Oxford (edition tertia with additions) the author W.O. in his Preface gives worthy characters of several young mathematicians that he enformed and amongst others of Dr Wallis who would be so kind to Mr O. to take the pains to correct the Presse, which the old gentleman doth with approval also acknowledge, and after he hath enumerated his titles: “Viri ingenui, pii, industrii, in omni reconditiore literatura versatissimi, in rebus mathematicis ad modum perspicasis, et in enodatione explicationeque Scriptorum intricatissimis ‘Zipherarum’ involucris occultatorum (quod ingenii subtilissimi argumentum est) ad miraculum faelicis.” This last of the cyphers was added by Dr Wallis himself which when the book being printed the old gentleman saw he was much he vexed at it and said that he thought he had given him sufficient praise with which he might have rested contented.

The Latin eulogy reads: ‘Of a man talented, pious and industrious, most able in all abstruse literature, sharp sighted as to the method in the things of mathematics, and wonderfully fortunate in the analysis and explanation of secret coded writings most intricately entangled, (a sign of his great subtlety and skill)’. Whether Aubrey's story was true or only half true, it makes a refreshing antidote to the more generally held view that Wallis and Oughtred always held each other in mutual esteem.

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Wallis's role in promoting the 1667 edition of the Clavis, despite the reservations of Collins and the booksellers, has already been noted. His greatest tribute to Oughtred, however, was to come in 1685 when he published A treatise of algebra and devoted fifteen of its 100 chapters entirely to the Clavis.70 Wallis, always interested in notation, began by noting the advantages of Oughtred's over Viète's (Fig. 3.6):71 Thus what Vieta would have written

, equal to FG plane would with him be thus expressed

… He doth also (to very great advantage) make use of several Ligatures, or Compendious Notes, to signify the Sums, Differences and Rectangles [products] of several Quantities. And by this means … he hath in his Clavis, a great deal of very good Geometry brought into a very narrow room; and you shall hardly find in any who have written before him so much of it delivered with so much clearness in so few words.

Wallis recognized that there are those who find fault with his Clavis, as too obscure’ but argued that the content once apprehended is much more easily retained, than it were expressed with the prolixity of some other writers; where a Reader must first be at the pains to weed out a great deal of superfluous Language, that he may have a short prospect of what is material.

Having claimed here, and on other occasions, that the beauty of the Clavis was its brevity, Wallis now proceeded to double the length of it by quoting extensive sections and adding long explanations of his own. For example, he discussed in detail the rules for the multiplication of negative numbers that he had long ago written into his own copy of the Clavis. In further chapters he expanded Oughtred's treatment of fractions and proportion, and of arithmetic and geometric progressions, and also set out Oughtred's method (adapted from Viète) for finding numerical solutions to polynomial equations. Next Wallis dealt with Oughtred's algebraic method of solving quadratic equations (excusing the fact that the Clavis dealt only with quadratic equations, and only with positive roots, on the grounds that the work was meant as an introduction for, he was sure, ‘Mr Oughtred could not be ignorant’ that an equation of higher degree would have more roots). Finally, he gave examples, quoted verbatim, of Oughtred's application of algebra to geometry, ending, as

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Fig. 3.6 Wallis's notes on Viete's Zeteticorum libri quinque in Oughtred's notation. This and three similar sheets are to be found between the pages of Viete's Opera mathematica (Leiden 1646) in the Bodleian Library (Savile N.6).

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Oughtred himself did, with the work on angular sections. In this way Wallis essentially republished the entire content of the Clavis, generally putting the material into better order than it appeared in the Clavis itself from the second edition onwards, with substantial commentary and explanation of his own, despite his criticisms in the opening paragraphs of ‘the prolixity of some other writers’.

The nal editions: 1693 onwards There was to be one further Latin edition of the Clavis, and once again Wallis was behind it. An invitation to subscribe to a new fifth edition survives among the manuscripts in the Savile Library.72 It is dated April 1692 and purports to be signed by Leonard Lichfield (the younger) but the handwriting is Wallis's (Fig. 3.7): Whereas several Learned Persons have taken Notice, That Mr Oughtred's Clavis Mathematicae etc has too long lain out of Print; and do complain of the many Typographical Mistakes in the last Edition. Therefore Leonard Lichfield of Oxon Printer (in whom the Propriety of the said Copy now remains) having by the Favour and Assistance of Dr Wallis, promised a Correction of the Errors and Mistakes in the former editions of the said Clavis Mathematicae, does now Propose to Re-print the former. And himself Offers to those Gentlemen who shall be Pleased to Assist and Incourage this New Edition: 1. To Print it on Good Paper and in new Characters. 2. To Finish and Deliver them in Michaelmas Term, next. 3. To Deliver them well Bound (notwithstanding the present Dearness of Paper) at 3s per Book, or in Sheets at 2s–6d per Book. 4. That he may be the better Enabled to produce a good Edition of the Said Book, he Prays, That some money may be paid upon Subscription, the other upon Delivery of the Book. Leonard Lichfield We do not know who the ‘several learned persons’ were who complained about the previous edition, apart from Wallis himself. Wallis sent the above letter to David Gregory, recently appointed Savilian Professor of Astronomy, for his endorsement, and Gregory's signature appears in the bottom left-hand corner of the draft. With it, Wallis sent a covering letter in which he repeated

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Fig. 3.7 A letter purporting to be signed by Leonard Lichfield, but in Wallis's handwriting, about a proposed fifth edition of the Clavis. David Gregory's endorsement is in the lower left-hand corner (Bodleian Library MS Savile 101, f. 14).

his praise of the Clavis and his offer of assistance:73 Sir, I understand from Leon Lichfield that you are willing to incourage and assist him in re-printing Mr Oughtred's Clavis, by getting subscriptions for taking off a number of copies, at a moderate rate, when they shall be printed. Wherein I think you do very well. For the book is certainly a very good book, and the first that brought Algebra (the subtlest piece of Mathematics) into considerable reputation and practice. It hath been several times printed and revised with good approbation, by those who understand and apply themselves to that kind of study; and hath done a great deal of honour to our nation: But it is now quite out of print. I do not know any book which doth in so small a bulk boast

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so much of sound mathematicks. And it is not for the credit of our own nation, that foreigners, who have learned from him, should vent those notions under other names, without acknowledging whence they had them. The book will now fairly be acceptable both at home and abroad. But because mathematicks is not so universal a study it is not to be expected that such books should by as speedy a sale (without some other assistance) encourage a printer as common pamphlets do, which are every body's money; (and for such reason many a good book is lost.) I have promised him my assistance in correcting the editions, to free it from divers typographical faults, which in some former editions have escaped. Which to the reader will be no small advantage. And if I can be otherwise assistant I shall be willing to it. I am sir, Yours to serve you, John Wallis

This letter reveals not only Wallis's active involvement in promoting the fifth and final Latin edition, but also some of his motives for doing so. The Clavis was no longer needed for its mathematical content, for it was by now not only ‘quite out of print’ but long out of date: its notation had fallen into disuse and its contents were inadequate for even the most elementary understanding of algebra. Wallis, however, was concerned less with the relevance of the text than with the honour of Oughtred and the nation. Isaac Newton himself was persuaded to support the efforts of Wallis and Gregory, and Newton's hopes for ‘the honour of our nation and advantage of Mathematicks’ echo Wallis's sentiments so closely as to hint that Wallis himself probably suggested the wording:74 Mr Oughtred Clavis being one of ye best as well as one of ye first Essays for reviving ye Art of Geometricall Resolution & Composition I agree wth ye Oxford Professors that a correct edition thereof to make it more usefull & bring it into more hands will be both for ye honour of or nation & advantage of Mathematicks. Is. N.

Wallis's efforts bore fruit, and the new edition was published by Leonard Lichfield in 1693. The typographical corrections were done not by Wallis himself but by Thomas Cook, a young fellow of New College, perhaps a protégé of Wallis's, and the last in a long line of young mathematicians who had helped to correct the text of the Clavis.

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The fifth edition was followed in 1694 by a new English translation with explanatory notes, mostly very brief, at the end of each chapter. Neither translator nor commentator was named,75 but the book carried a recommendation from Edmund Halley, then Secretary to the Royal Society: The Clavis Mathematicae of Mr William Oughtred is a book of so established a reputation, that it were needless to say any thing thereof. It was formerly translated by Dr Wood into English; but from an edition which has been since much bettered and augmented; and besides, the concise brevity of the Author is such, as in many places to need an explication, to render it intelligible to the less knowing in mathematical matters. This translation is new and from the fullest edition, and may be of good use to all beginners in the Analytical Art. And especially to such, who tho they be ignorant of the Latin tongue, may yet be desirous to inform themselves in Geometry: and to all such I recommend it as a very useful Treatise. E. Halley Those ‘ignorant of the Latin tongue’ yet ‘desirous to inform themselves’ are listed on the title page as ‘gagers, surveyors, gunners, military officers, mariners etc’ (Fig. 3.8), for whom some of the arithmetic in the book might have been useful but for whom the algebraic content almost certainly was not. These readers were not quite those students of the mysteries of Classical mathematics to whom Oughtred had first offered his Ariadne's thread. The role of the Clavis was finally coming to an end, but it survived just a few more years: the fifth Latin edition was reprinted in 1698 with a specific acknowledgement to Wallis (now over eighty) for his revisions, and the second English edition was reprinted for the last time in 1702, the year before Wallis died. By then, Ariadne's thread had wound its way through English mathematics for just over seventy years. The reasons for the popularity of the Clavis changed over its long life. When first published, it not only satisfied an urgent need for a new elementary textbook in algebra, but it brought the beginnings of algebraic geometry into common circulation for the first time. Small and concise and lacking any serious competitors, it became the primary textbook for a whole generation of young mathematicians, some of whom were also taught personally by Oughtred and were to remember him with respect and gratitude all their lives. Two of these men, Ward and Wallis, gave the book a new lease of life during the political changes of the late 1640s and early 1650s.

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Fig. 3.8 Title page of the 1694 English edition of the Clavis.

Wallis continued to promote the book for the next fifty years both by encouraging new editions and by commenting on it extensively in his own work. As early as the 1650s, and certainly over the subsequent decade, others began to see the Clavis as outdated, but the opinions of men like Foster, Gibson or Anderson would have carried little weight against those of the Oxford Savilian Professor of Geometry. Moreover, until the end of the 1660s good alternative texts by English writers were still not available, Gibson's

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Syntaxis remaining unaccountably unknown. It is harder to explain the final republishing of the book in both Latin and English in the 1690s, by which time it had long outlived its usefulness, but by this time the Clavis and Wallis himself were both so established in English mathematics that perhaps no one could seriously question the standing of either. The long use of the Clavis as a standard textbook can in some ways be seen as detrimental, for it kept ideas and notation that belonged to the closing years of the sixteenth century in circulation until the turn of the next century and beyond. Oughtred's clumsy notation had to be abandoned before progress could be made, as many mathematicians in the second half of the seventeenth century clearly saw. Nevertheless, the Clavis profoundly influenced the development of mathematics in England, not so much by the value of its content but because it came into existence at a time when mathematical teaching in England was at a nadir, and almost alone began the revival of serious mathematical learning. Almost every seventeenth-century English mathematician or scientist of note learned his early skills from it. One can only speculate on the subsequent course of English mathematics if Moore, Ward, Wallis, Wren, Boyle, Hooke, Newton and many lesser figures had not had the Clavis to set them on their way. Oughtred had offered the book as ‘Ariadne's thread’ to lead aspiring mathematicians into the mysteries of Classical writings, and in this it succeeded, well beyond the circle of Oughtred's personal pupils. But the real value of the Clavis in the end was not as a guide to the past but as an inspiration for the future; Oughtred's key was to open the way to mathematics that Oughtred himself could have hardly imagined.

4 Rob'd of glories: Thomas Harriot and his Treatise on equations The year that sawhe year that saw the first publication of Oughtred's Clavis also saw the publication of an algebra of a completely different kind, the Artis analyticae praxis (1631), compiled posthumously from the papers of Thomas Harriot (c. 1560–1621) (Fig. 4.1). Because Harriot never published any of his scientific or mathematical findings in his lifetime it has been difficult to establish his true place in the intellectual history of the period or to judge the extent of his mathematical influence on those who came after him. Wallis thought that Harriot should have been more acclaimed than he was, and devoted no less than a quarter of A treatise of algebra to extolling him. Furthermore, he repeatedly accused Descartes of having made use of Harriot's algebra without acknowledgement, and thereby inflamed a controversy that has not been satisfactorily settled since. Jean Etienne Montucla, in 1799, called Wallis's account inexcusable, and a century after that Moritz Cantor dismissed it as nationalistic polemic.1 In the twentieth century, Wallis fared hardly any better: one Harriot scholar wrote him off as a small-minded joker with a bad memory, another questioned his moral character and trustworthiness, while his own biographer described him as pompous and guilty of gross partiality.2 It would seem that not much could be salvaged from such derision, but this chapter looks afresh at Wallis's account, and argues that it was better founded than has been supposed. It also looks in detail at Harriot's algebra and attempts to locate him more precisely than has hitherto been possible within the mathematical developments of his time. Nothing is known of Harriot's early background.3 The appearance of his name in the Oxford University Register in 1577 suggests that he was born about 1560, and the entry implies that he already lived in Oxford, and that he took up residence in St Mary's Hall, affiliated to Oriel College. Some time after

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Fig. 4.1 Thomas Harriot (c. 1560–1621).

Harriot's graduation around 1580 he entered the service of Walter Ralegh, and was employed by him as navigator and scientist on a voyage to north America from 1585 to 1586.4 Harriot's report of the expedition, A briefe and true report of the new found land of Virginia (1588), was the only book he ever published.5 Harriot's reputation as a mathematician became established in the years after his return from America. In 1593 Gabriel Harvey called him a ‘profounde Mathematician’ (alongside Thomas Digges and John Dee), and in 1594 Robert Hues announced in his Tractatus de globis that a further treatise could be expected from the ‘mathematician and philosopher, Thomas Harriot’.6 From the early 1590s Harriot had a lifelong patron and benefactor in Sir Henry Percy, the ninth earl of Northumberland. The Earl was imprisoned in the Tower from 1605 to 1621 on suspicion of involvement in the Gunpowder Plot (his cousin Thomas Percy was one of the ringleaders) but continued to maintain Harriot at his London home, Syon House at Isleworth in Middlesex. Most of Harriot's mathematical and scientific companions were in some way connected with the Earl's household:7 Walter Warner (c. 1557–1643) was keeper of Percy's library and scientific instruments, while Robert Hues

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(1553–1632), a friend of Harriot's since his time at Oxford, became tutor from 1615 to the Earl's sons. Harriot discussed mathematics, optics and astronomy with William Lower (1570–1615), keeper of Percy's Welsh Estates, and made astronomical observations with Lower's successor John Protheroe (1582–1624). Through Protheroe, Harriot also became acquainted with Thomas Aylesbury (1580–1657). Another intellectual companion was Nathaniel Torporley (1564–1632), who had graduated from Oxford four years after Harriot. It is not known when Harriot and Torporley first met but both were interested in mathematics. Torporley wrote to Harriot from Paris on the eve of his first meeting with Viète, the self-styled ‘French Apollonius’:8 I am gathering up my ruined wittes, the better to encounter that French Apollon: if it fortune that either his courtsie or my boldnes effecte our conference; tomorrow beinge the daye, when I am appoynted by his Printer, as litle Zacheus to climbe the tree, to gayne a view of that renoumned analist. What after followes in [his] presence I hope shortly to relate …

What followed was that Torporley became Viète's amanuensis,9 and it was almost certainly through Torporley that Harriot acquired his detailed knowledge of Viète's mathematics. Among Harriot's manuscripts is a sheet headed: ‘A proposition of Vietas delivered by Mr. Thorperly’,10 and there are many other notes on problems from Viète: there are, for example, numerous references to Viète's Zetetica of 1593, and to the Apollonius Gallus of 1600.11 In particular, Viète's algebra became the foundation of Harriot's algebra, but Harriot made important changes to Viète's notation. Harriot's most important innovation in notation was undoubtedly his use of ab to represent a multiplied by b, and consequently aa, aaa for what is now written a2, a3, etc. To denote equality he adopted a variation of the = sign devised by Robert Recorde in The whetstone of witte in 1557.12 In Harriot's manuscripts it always appears with two short cross strokes between the horizontals, but these disappeared when his algebra was put into print. Harriot also introduced two new symbols < and > to represent inequality: in manuscript these are longer and more curved than the modern printed versions, with two short cross strokes across the wider ends (Fig. 4.2).13 Harriot used all three signs <, =, > either horizontally or vertically as occasion demanded, enabling him to work across or down the page at will and also to write circular equations connecting four or more quantities at once.

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Fig. 4.2 Harriot's notation. Note his equals and inequality signs, his use of commas to indicate multiplication by numbers and square brackets for the multiplication of algebraic expressions (British Library Add MS 6783, f. 107).

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Harriot indicated parenthesis by a single comma, as in: √,bb − cc or b −, c − d = b − c + d. He also used commas to denote the multiplication of a letter by a number, as in 2,a. Multiplication of compound quantities was shown by enclosing them in a right-angled bracket:

with the steps of the multiplication carried out exactly as in a long multiplication in arithmetic. Division was represented in the usual way by writing the dividend and divisor above and below a horizontal line, as in fractions.14 Harriot always used the by then well established (but by no means universal) symbols + and − to denote addition and subtraction,15 but also introduced two new symbols: ± and ± and ∓ to denote alternative possibilities.16 This enabled him to handle two, or even four, equations at a time, a useful saving when every combination of signs had to be dealt with separately. Harriot wrote a four-page introduction to his notation, entitled Operationes logisticae in notis (‘Operations of arithmetic in letters’), in which he demonstrated the use of lower case letters in examples of addition, subtraction, multiplication and division, and in the standard rules for simplifying equations. The examples on the first and second sheets are Harriot's own, but all of those on the third sheet, on fractions, are taken directly from Viète's Isagoge of 1591 (Fig. 4.3).17 A comparison of corresponding statements from Viète and Harriot demonstrates immediately the conciseness of Harriot's notation:18 Viète:

If to

there should be added

Harriot: Viète:

If B times G should be divided by both magnitudes having been multiplied by D, the result will be

Harriot:

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Fig. 4.3 The third page of Harriot's Operationes logisticae in notis showing how to handle fractions. The same material is to be found in Viète's Isagoge and on page 10 of the Praxis (British Library Add MS 6784, f. 324).

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(Note how Harriot kept Viète's dimensional homogeneity by replacing A plane by the two-dimensional quantity ac. This is evident throughout Harriot's work where we might find, for example, a + b or aa + bc but never a + bc.) On the fourth sheet of the Operationes logisticae, Harriot turned to the standard rules for simplifying equations: moving terms from one side to another, reducing the leading coefficient to one, and dividing out excess powers of the unknown.19 Viète had given these three rules the names of antithesis, hypobibasmus and parabolismus, respectively,20 and Harriot retained both the names and Viète's examples:21 Viète:

Harriot:

A squared minus D plane is supposed equal to G squared minus B times A. I say that A squared plus B times A is equal to G squared plus D plane and that by this transposition and under opposite signs of conjunction the equation is not changed. Suppose aa − dc = gg − ba I say that aa + ba = gg + dc by antithesis.

Not only is Harriot's notation concise and easy to use, but it is transparent: it reveals algebraic structure and aids thinking in a way that Viète's verbal descriptions do not. The most important influence on Harriot's algebra after the Isagoge was Viète's De numerosa potestatum de exegesin resolutione of 1600, the book in which Viète taught the solution of equations by numerical methods. Harriot studied De potestatum resolutione in detail and, as he had done with the Isagoge, he re-wrote much of the material in his own notation. More importantly, however, he went on to explore the theoretical underpinning of Viète's method, and so developed his own treatment of the structure and solution of equations. The re-written material from Viète, and Harriot's ensuing theoretical treatment, together formed a self-contained treatise. It is undated, but springs so directly from the De potestatum resolutione that it was almost certainly written not long after that book appeared in 1600. It is also untitled, but for convenience I have called it the Treatise on equations.

The Treatise on equations The Treatise on equations was written in six sections, and since the last five of these were lettered by Harriot as (b) to (f), the first (unlettered) section will be denoted from here on by (a). Sections (a) to (c) deal with the numerical solution of equations and are closely based on the examples given in Viète's

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De potestatum resolutione. In Section (c), however, Harriot began to demonstrate a more theoretical approach, and gave examples of what he called canonical equations, representative forms in which it was clear how the roots related to the coefficients. For example, the canonical equation bc = ba + ca − aa always has two positive roots (a = b and a = c) whereas bc = − ba+ca+aa has only one (a = b). In Section (d) Harriot showed how such canonical forms arose, and in doing so made his most profound and far-reaching contribution to the understanding of polynomial equations: where Viète had analysed equations in terms of ratios, Harriot saw instead the possibility of writing polynomials as products of factors.22 Beginning with the multiplication (a − b)(a − c), he built up a series of quadratic, cubic and quartic polynomials and noted the equations that arose from setting such polynomials to zero. Harriot used only factors of the form (a ± b) and (aa ± df) or, occasionally, (aaa ± dfg), and in the resulting equations the relationships between the roots and the coefficients are immediately clear, especially in Harriot's layout where terms containing the same power are grouped vertically. Harriot realized, for instance, that if certain conditions hold between the roots then one or more powers will vanish: in the equation aa − ba + ca − bc = 0, for example, the linear terms disappear if b = c, and the equation reduces to aa − bb = o (which, like the original, has a positive root, a = b). The most often repeated criticism of Harriot's work has been that he restricted himself to positive real roots.23 Two things need to be said on this point: first, Harriot's work on equations arose directly from his study of Viète's algorithm, an algorithm that had never been used to produce anything but positive roots, and so it was natural that he should take the search for positive roots as his starting point; second, it is clear that as his work progressed, Harriot began to recognize the importance of both negative and complex roots. There are many examples of the former, and a few of the latter, in Sections (e) and (f) of his treatise, but Harriot also began to see how and where they should occur in Section (d). Section (e) is a systematic treatment of cubic equations. Harriot began, as the sixteenth-century Italians had done, with equations lacking a square term and discussed three possible cases:

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Harriot dealt at length with the second case, aaa − 3baa = 2ccc, and distinguished three subcases c > b, c = b and c < b, which he labelled hyperbolic, parabolic and elliptic, respectively.24 Hyperbolic equations have one positive root and a pair of complex roots, while elliptic equations have one positive and two negative roots. It is the elliptic form that gives rise to the ‘irreducible case’ in which the solution formula leads to square roots of negative numbers, but the solution is in fact real because the imaginary terms cancel out. Harriot showed that in principle such an equation, of the form aaa − 3bba = 2ccc, could be solved from its conjugate equation, 3bba − aaa = 2ccc, which has the same numerical roots with different signs. The final part of Section (e) showed how to remove the square term from any cubic equation, so that the reduced equation could then be solved by the methods outlined previously. Section (f) is a similar systematic treatment of quartic equations. Harriot showed how to solve a quartic equation lacking the cube term, by the method devised by Ferrari and Cardano and systematized by Bombelli, Stevin and Viète (Harriot was familiar with the work of all these earlier writers). He also gave detailed instructions on how to remove the cube term from any quartic by a simple linear transformation. This was new: the removal of the cube term from a quartic by this method is not found in the published work of any of Harriot's predecessors.25 Thus, amongst the disarray of Harriot's manuscript sheets we have the scattered pages of an invaluable treatise, a treatise that considerably extended the contemporary understanding of polynomial equations. When William Lower in 1610 urged Harriot to publish some of his mathematical and scientific findings, this work on equations must have been part of what he had in mind:26 Doe you not here startle, to see every day some of your inventions taken from you: for I remember long since you told me as much (as Kepler has just published) that the motions of the planets were not perfect circles. So you taught me the curious way to observe weight in Water, and within a while after Ghetaldi comes out with it in print. A little before Vieta prevented you of the Gharland for the greate Invention of Algebra. al these were your deues and manie others that I could mention; and yet too great reservednesse hath rob'd you of these glories … Onlie let this remember you, that it is possible by to much procrastination to be prevented in the honor of some of your rarest inventions and speculations.

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Unfortunately, Lower's admonitions went unheeded, and Harriot's work on equations, as on all other subjects, remained unknown outside his circle of friends until after his death.

Harriot's Will and the writing of the Praxis A few days before he died in 1621 Harriot dictated a Will, in which he gave special attention to his mathematical papers.27 The relevant instructions were as follows: … I Thomas Harriots of Syon in the County of Midd Gentleman being troubled in my body with infirmities. But of perfect mind and memory Laude and prayse be given to Almightie God for the same do make and ordayne this my last will and testament… I ordaine and Constitute the aforesaid NATHANIEL THORPERLEY first to be Overseer of my Mathematical Writings to be received of my Executors to peruse and order and to separate the chief of them from my waste papers, to the end that after he doth understand them he may make use in penning such doctrine that belongs unto them for public uses as it shall be thought Convenient by my Executors and him selfe And if it happen that some manner of Notations or writings of the said papers shall not be understood by him then my desire is that it will please him to Conferre with Mr Warner or Mr Hughes Attendants on the aforesaid Earle Concerning the aforesaid doubt. And if he be not resolved by either of them That then he Confer with the aforesaid JOHN PROTHEROE Esquior or the aforesaid THOMAS ALESBURY Esquior. (I hoping that some or other of the aforesaid four last nominated can resolve him) And when he hath had the use of the said papers so long as my Executors and he have agreed for the use afore said That then he deliver them again unto my Executors to be put into a Convenient Trunk with a lock and key and to be placed in my Lord of Northumberlands Library and the key thereof to be delivered into his Lordships hands.

It is clear that Harriot thought Torporley the person best fitted to understand, transcribe and edit his mathematical papers. From 1608 Torporley was the rector of Salwarpe in Worcestershire and so may not have been acquainted with Harriot's later work: in this case Harriot suggested that Warner and Hues would be in a position to assist, with Protheroe and Aylesbury (also Executors of the Will) as the final arbiters.

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Preparations for carrying out Harriot's wishes were put in train: in 1622 Torporley resigned his position and probably moved to one of the Earl of Northumberland's residences, Syon House in Middlesex, or Petworth House in Sussex. Protheroe paid him a pension, and instructed his wife to continue the payments after his death (in 1624).28 Later, Torporley was probably supported by Percy: the Earl's household papers show a payment to Torporley in 1626, and he was resident at Petworth in 1627. The mathematical papers handed over to Torporley were carefully listed by Aylesbury, and the list was endorsed by both Protheroe and Torporley.29 It was headed: Copyed from Mr Protheroe A note of the papers and bookes in Mr Harriot's trunke delivered to Mr Torporley

There were sixty items (plus nineteen more added later) and the first few alone give some idea of the overwhelming task faced by the Executors: Analytiques in 16 bundells De Centro gravitatis 3 b.

(b. bundells)

De Jovialibus planetis The spots in the sun The faces of the moon all in one great b. Of the observations of the moon, 1 great b more Eratosthenes Batavus de quadrilatero in circulo, de parabola Silo princeps fecit, diluvium Noachi, generatio maris et feminae with some other papers of genealogies 3 b. On Vietaes zetetiques, with a few miscellaneous papers de Inclinationibus & porismatis (All these bound up in a pack thred together) Of the errors in observations by Instruments which cannot be made exactly ad minutum, 1 b. Certaine observations in a great b. most cleane paper

Torporley was faced with the same daunting quantity of material as every potential editor since, but among his surviving manuscripts is a work now known as the Congestor (‘Compilation’) in which he began the task of presenting some of Harriot's work.30 After a lengthy preamble he identified the

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topics that he proposed to deal with:31

The Congestor contains the first two sections of this scheme: first, several pages on the identification of prime numbers, and related problems, ending with a nine-page table of prime factors for numbers up to 20399;32 second, a section headed ‘Thomas Hariotus, examinatur Stifelius de numeris diagonalibus’, a copy of Harriot's work on Pythagorean triples.33 A fair copy of the Congestor was dedicated to the Earl of Northumberland at Petworth in October 1627. Torporley did not complete the remainder of his plan, and it would seem that at some stage Harriot's Executors relieved him of his duties. We do not know why this happened, but Torporley later complained that ‘my enemies accuse me to the Master of Petworth as being, among other things, an ignorant logic-chopper’,34 so perhaps the Executors were concerned about his intellectual ability to complete the task. For whatever reason, Harriot's work on equations was eventually edited not by Torporley, as Harriot had intended, but by others, foremost of whom were Warner and Aylesbury. Torporley later remarked that one of the editors had been ‘lifted to heaven’,35 and this would have been Protheroe who died in 1624, so perhaps Warner, Aylesbury and Protheroe began to work together on the papers quite soon after Harriot's death. A mistaken identification of handwriting has clouded the issue of Torporley's later involvement. A draft of the closing paragraph of the Praxis, which advised the reader of further work to follow, exists in Warner's hand,36 but an unknown writer has made some changes and added the endorsement: ‘This will do well in this form. And I leave it to Mr Warner's discretion, whether he thinks it fit to give this monition or no, because he seemed to doubt of it.’ Stephen Peter Rigaud, researching Harriot's papers in the early

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nineteenth century, identified Torporley as the author of the imprimatur,37 but the handwriting lacks the distinctive features of Torporley's.38 If Torporley is ruled out, then the most likely alternative is Aylesbury; the writing is certainly very similar (though not identical in every respect) to his.39 The posthumous edition of Harriot's algebra was eventually published in 1631 as the Artis analyticae praxis. There was no editor's name on the title page, perhaps because the Executors knew that in replacing Torporley they had strictly contravened the terms of Harriot's Will. A letter from Aylesbury to the Earl of Northumberland, however, makes it clear that it was Warner who did most to see the book through the press.40 Warner was not previously known as a mathematician, and unfortunately never understood Harriot's work as well as Torporley did.41 Instead of editing the manuscripts as they stood, he chose to select and reorder the material in a manner that will be explained in detail below, and in doing so not only destroyed the coherence of Harriot's treatise but made it appear considerably less sophisticated than in fact it was.

The contents of the Praxis The main text of the Praxis is in two parts: the first, in six sub-sections, deals with the theory of equations; the second teaches practical methods of solving equations numerically. The book opens, however, with a preface and eighteen preliminary definitions. The preface is so close in content to the first few pages of the Congestor as to suggest that Torporley may have had some hand in it. It shows a clear understanding of Viète and what he was aiming to achieve: a restoration of the analytic art that, it was thought, the Greeks had known but which had since been lost:42 Francis Vieta, a Frenchman, a most distinguished man, and on account of his remarkable skill in Mathematical Science the honour of the French nation, first of all with singular genius and with industry hitherto unattempted undertook the restoration of the analytic art, of which subject we are here treating, which after the learned age of the Greeks for a long time had become antiquated and remained uncultivated;…

The writer recognized, however, that Viète had moved far beyond the old methods, as had Harriot: But while [Viète] seriously laboured at the restoration of the old Analysis, which he had proposed to himself, he seems not so much

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to have transmitted to us a restoration of that science, as a new and original method, worked out and illustrated by his own discoveries. This having been enunciated… it may be more clear, what was afterwards performed by our very learned author Thomas Harriot, who followed him in these analytical investigations.

The preface then went on (as did the Congestor) to follow the course of Greek learning, through Diophantus and the Arabs, to Cardano, Tartaglia and Stevin to Viète himself. The mathematics proper began in Section 1 which dealt first with elementary preparatory work: the four operations of arithmetic in ‘species’, or letters, for both whole numbers and fractions, and the standard rules for simplifying equations. This was Harriot's Operationes logisticae, unchanged apart from some slight reordering. The only small, though important, addition was a note on the new signs < and > for inequality.43 At this point, however, the close similarity between the Praxis and the manuscripts came to an end. Warner proceeded directly to the contents of Section (d) of Harriot's Treatise on equations, entitled De generatione aequationum canonicarum (‘The generation of canonical equations’), where Harriot began to develop his insight into the multiplicative structure of polynomials. Here is one of Harriot's first examples of his method, from the sheet he himself numbered d.1) (Fig. 4.4). Here and throughout, a is the unknown quantity:44

Fig. 4.4 The first page of Section (d) of Harriot's Treatise on equations, where he begins to build up polynomials by the multiplication of linear factors (British Library Add MS 6783, f. 183).

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and later:(1)

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The results of the initial multiplication, (b − a)(c + a), are given with the numerical term bc first, then the terms in a listed vertically, and finally the single term in aa. The same layout is useful in longer multiplications where it demonstrates clearly the relationship of each coefficient to the roots. Note also Harriot's use of oo to preserve the homogeneity of the terms. It has been observed many times that the Praxis consistently ignored negative roots, and so it is of interest to note that here, at this early stage, so did Harriot himself. Underneath ‘Let a = b’ in the first line of (1) he appears to have written but then crossed out ‘and let a = − c’ (see Fig. 4.4). Not until some pages later in the treatise did Harriot specifically write a negative root.45 Harriot's method here and throughout Section (d) was a model of clarity, but for reasons we can never know, Warner chose not to follow it. Instead, he based his exposition on some of the definitions from the Introduction to the Praxis, in particular those numbered 14, 15 and 16. It is not clear where these Definitions came from: they may have been written by Harriot but are not found in the surviving manuscripts, which leaves open the possibility that they were written by Warner himself. Definitions 14,15 and 16 are all concerned with canonical equations. Definition 14 shows how polynomials can be constructed as products of linear factors: Definition 14: Originals of canonical equations

If any factor is equal to zero, then so is the resulting polynomial, and it may therefore be rearranged with all the unknown terms on the left and a single known term on the right. Definiton 15 goes on to describe such equations as primary canonicals.

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Definition 15: Primary canonicals established by derivation from the originals

Definition 16 gives examples of secondary canonicals, which arise from primary canonicals when certain relationships between the coefficients cause one (or more) of the powers to vanish. Definition 16: Secondary canonicals established by reduction from primary canonicals

The Material in Part I of the Praxis is arranged strictly in accordance with these definitions: the first half of Section 2 lists a number of multiplications of the type given in Definition 14,46 while the second half of Section 2 shows how to form primary canonicals as in Definition 15.47 Section 3 is concerned with the reduction of primary canonicals to secondary canonicals as in Definition 16.48 Section 4 then lists the roots of each equation.49 Thus material relevant to each equation is found in three or four separate places in the Praxis. To make matters worse. Warner was inconsistent in his ordering of the equations from one section to another so that it is no easy matter to follow any given equation through the book. This was very different from Harriot's handling of his material: all his work on the generation of equations (both primary and secondary canonicals) was gathered in Section (d), where each equation was treated just once in a unified way. Every detail of Harriot's manuscripts points to a confident mastery of his subject, but unhappily the same cannot be said of Warner. Harriot, dealing specifically with positive roots, had omitted cases such as (a + b)(a + c) which

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could not be turned into primary canonicals without admitting negatives. Warner, lacking Harriot's clear purpose, introduced such cases but was then forced to drop them again, so that he began Section 2 with thirty-two equations but ended with only eighteen. One particular set of equations that gave Warner trouble must be noted here. Section 3, like Definition 16, was based on the fact that some terms of a polynomial equation will disappear if certain relationships hold between the roots (there will be no term in the second highest power, for instance, if the sum of the roots is zero). The whole of Section 3 was devoted to listing such relationships and their effects, seven examples for cubics and fourteen for quartics. Warner continued valiantly to write out the conditions and the working in full each time until he faced defeat in the final three problems: 19, 20 and 21. These required the elimination of not just one term, but two, and therefore needed two independent relationships between the roots. Warner gave only one relationship for each, omitted any working, and added an apologetic note: ‘The reductions of these equations, since they are delivered more obscurely in manuscript, must be referred to a better enquiry’.50 Harriot's working was indeed a little obscure: in each case the second condition was taken straight into the working and is not immediately obvious. Furthermore, tired of writing out full solutions to quadratic equations, Harriot had begun to write such things as

leaving the reader to fill in the empty space under the square root sign (Fig. 4.5).51 This was not mere idleness on Harriot's part: he knew that the terms ‘−√’ and ‘+√’ would cancel each other out as soon as he added the two solutions a line or two later. Poor Warner, however, was left with an ungainly square root to fill in. Even worse, he would have seen that the expression under the square root sign was negative: Harriot was dealing here with what we would now call complex conjugates.52 Harriot may have been at ease, but Warner was decidedly not, and he chose to avoid the pitfalls of problems 19 to 21. Both Torporley and Wallis, for different reasons, were to take up Problem 19 later, and we shall return to it. Having dispersed Harriot's Section (d) over Sections 2, 3 and 4 of the Praxis, Warner continued to treat Harriot's treatise in piecemeal fashion in

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Fig. 4.5 Harriot's sheet d.11). Warner complained that the problem was ‘delivered more obscurely in manuscript’ and abandoned any attempt to reproduce it in the Praxis. Harriot wrote +√ and −√ without filling in the expression after the square root; the missing quantity (the same in each case) is negative, so Harriot was here dealing with complex conjugates (British Library Add MS 6783, f. 173).

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Sections 5 and 6. Section 5 of the Praxis contains selected material from some of the sheets of Harriot's Section (e), but falls a long way short of Harriot's careful and systematic treatment of cubics. Section 6 consists of a further material from Section (e), and much of Section (f). The most important part of Section (f), however, the method of solving quartics lacking a cube term, is missing, so the purpose of over thirty examples illustrating removal of the cube term is never made clear. Finally came the long Part II (about one-third of the book) on the numerical solution of equations, the kind of material that Harriot had covered in Sections (a), (b) and (c), but the examples in the Praxis are different from those in the Treatise on equations. On the contents page of the Praxis, and again at the end of Part I, it was stated that such numerical solution was the ‘principal skill’ of the analytic art and that the aim of Part I was to prepare the way for Part II, but it was never made very clear to the reader how it was supposed to do so. Harriot himself appears to have worked in quite the opposite direction, from numerical solutions to theoretical treatment.

The Corrector and the Summary When the Praxis appeared, Torporley attacked its editors bitterly in a piece entitled Corrector analyticus artis posthumae Thomae Harrioti (‘An analytic correction of the posthumous work of Thomas Harriot’).53 Such a document, by Harriot's closest mathematical colleague, deserves to be read with some care, but it still awaits full translation and in the meantime its readers must grapple with Torporley's tortuous Latin. It is headed: Corrector Analyticus artis posthumae Thomae Harrioti

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Torporley then launched into a long and rambling diatribe, the essence of his complaint being that the editors had so misunderstood and altered Harriot's work that scarcely a trace of the original remained. He argued that Harriot himself had selected the material he intended for publication, but that in the Praxis it had been dismembered and scattered, and copied with little understanding. Torporley set out in detail what he thought the editors should have done, and mentioned many specific manuscript sheets, but the task of identifying those sheets amongst the hundreds that have survived has until now seemed daunting. As a result, the important information contained in the Corrector has been badly neglected. At some stage, though, Torporley also put together his own scheme for the material he thought should have been included in the posthumous edition of Harriot's algebra, and using Harriot's original pagination he listed all the relevant sheets together with an abbreviated version of the contents of each.54 This document is untitled but I will refer to it from now on as Torporley's Summary (Fig. 4.6). It contains the Operationes logisticae in notis, a great deal of material on surds, or radicals, and all the sheets of the Treatise on equations together with a few additional examples. The Summary was preserved at Sion College (along with the Congestor and the Corrector) and is a unique and invaluable document in that it brings together details of over two hundred manuscript sheets that were later dispersed, and is now the best guide we have to Harriot's own intentions.55 The Summary is undated but it contains several oblique references to Warner's edition of the Praxis: for example, ‘Prob 16 et 17 mutatis signis W et 18’ (‘Problems 16 and 17, with changed signs in Warner, and 18’) and ‘omissa W’ ('missed by Warner).56 If, as it appears, these notes were written at the same time as the rest of the document, Torporley must have compiled it after Warner had completed the Praxis. This is of some significance as it means that Torporley still had access to Harriot's manuscripts after the Praxis was written. The mathematical content of the Summary makes possible precise identifications of the relevant manuscript sheets,57 and the Corrector and the Summary taken together leave us in no doubt as to how Torporley thought Harriot's material should be ordered. His plan was as follows:

(i) Operations of arithmetic in letters Harriot's Operationes logisticae in notis.

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Fig. 4.6 The first page of Torporley's Summary, containing in condensed form the first three pages of Harriot's Operationes logisticae in notis (see also Fig. 4.3). The same material is to be found in the Praxis, pages 7–10. (Sion College MS Arc L.40.2/L.40, F. 35)

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(ii) Radicals In the Corrector, Torporley argued that Harriot's work on equations should be preceded by a ‘carefully prepared treatise on surds or, as he called them, radicals’.58 In the Summary he listed and summarised over seventy sheets on surds and binomials (numbers of the form √a ±√b).59

(iii) The Treatise on equations On the theory of equations, said Torporley, Harriot was writing three parts. Torporley's detailed description in the Corrector of the contents of those three parts is so important to the reconstruction of the Treatise on equations that it is given here in full:60 On the theory of the analytic art itself he was writing the contents in three divisions, the first part of those thus: On the generation of canonical equations, 21 sheets on that theme, paginated together under paragraph d), with two short appendices on the multiplication of roots. The second part moreover under the title: On solving equations by reduction has 29 sheets, as paragraph e). Under the same heading, fα) 7 sheets; fβ) also 7 sheets; and succeeding these in the numeration of the sheets, fγ) to sheet f18γ) with a short appendix with two lemmas that should not be disparaged, omitted by them. Then fδ) 8 sheets; fɛ) 4 sheets; fζ) also 4 sheets. Next, separately, nine sheets containing old reductions recovered by Harriot's method. But the third part (thus I am not eager for disagreement) he was writing like Viète.61On solving equations in numbers, and rightly and deservedly. Not nearly all is Viète's in each example. And in the paragraph supposed a),62 and in 13 sheets are three examples of quadratics, of which the first is his, the other two are Viète's, and five cubics, all Viète's apart from the first. And five quartics of which the fourth is his, the rest Viète's. And these, according to the method of Viète, are all of affirmative equations. The other part of it,63 as paragraph b) in 12 sheets, has as Viète has, analysis of powers negatively affected: quadratics in b1), b2), b3), cubics in b4) to b10), quartics in b10), b11), b12). The third part of this.64 as paragraph c), has 18 sheets, and treats the analysis of avulsed powers,65 as Viète, where there are multiple roots, and the limits of each are demonstrated. The examples of this are two quadratics, four cubics, two quartics.

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It seems strange that Torporley chose to place those sections lettered (d), (e) and (f) ahead of those lettered (a), (b) and (c). Despite that, we have in this extract from the Corrector a complete and almost entirely accurate description of Harriot's Treatise on equations. In the Summary Torporley set out exactly the material described here, and in the same order, that is, Sections (d), (e) and (f) followed by (a), (b) and (c). With the help of the Corrector and the Summary together it has now been possible to identify the sheets belonging to all six sections among the surviving manuscripts, and apart from some minor discrepancies, they are just as Torporley described them.

The fate of Harriot's material after 1631 Under the terms of Harriot's Will, his mathematical papers should have been returned after Torporley had finished with them to the Earl of Northumberland. Certainly all the sheets mentioned by Torporley can be found amongst the surviving manuscripts discovered at Petworth in 1784, but whether they were returned in Torporley's lifetime, or immediately after his death in 1632, or some time later, it is impossible to say. There is evidence, to be discussed below, that at least some of Harriot's papers remained in circulation for up to thirty years, and copies of parts of them for even longer, ample time for their contents to become known outside Harriot's immediate circle. Someone who took a particular interest in Harriot's work was Charles Cavendish, who copied many sheets of Harriot's mathematics.66 The importance of Cavendish as a disseminator of mathematical ideas can hardly be underestimated: he was responsible for bringing the ideas of Viète and Cavalieri to England, and it is not unreasonable to suppose that he also carried some of Harriot's ideas from England to his mathematical acquaintances on the continent. Whether or not Descartes in the Netherlands had learned of Harriot's ideas before he wrote La géométrie became a subject of controversy as soon as that book appeared and has remained so since. Jean Beaugrand, who had published some of Viète's work in 1631, thought he detected the influence of both Viète and Harriot in La géométrie,67 but Descartes denied that he had read either.68 Descartes had lived in Paris from 1626 to 1628 and had remained in touch with the French mathematical community through Mersenne, and it is hard to believe that in 1637 he was unfamiliar with ideas that had by then been in the open for twenty to thirty years. Certainly his notation in La géométrie was strikingly similar to Harriot's, especially in his use of lower case letters, and xy for x multiplied by y. Descartes introduced

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x3, x4, etc. for cubes and higher powers, but retained xx as Harriot did for x squared.69 Descartes also used a vertical layout very similar to Harriot's to demonstrate the removal of a cube term from a quartic. Compare, for instance, the following examples from Harriot's manuscripts and La géométrie: the equations are different but the procedure and layout is the same in each:70

More importantly, Descartes introduced without comment the factorization of polynomials, yet this idea has previously appeared nowhere but in the work of Harriot. Descartes may have discovered factorization independently but his contemporaries could be forgiven for thinking that he had some hint of it, even if indirectly, from Harriot. According to a story later repeated by Wallis, Cavendish himself thought that Descartes had learned from Harriot. Wallis related the following discussion, supposed to have taken place between Cavendish and Roberval after the publication of La géométrie:71 I admire (saith M. Roberval) that notion in Des Cartes of putting over the whole equation to one side, making it equal to Nothing, and how he lighted upon it. The reason why you admire it (saith Sir Charles) is because you are a French-man; for if you were an English-man, you would not admire it. Why so? (saith M. Roberval) Because (saith Sir Charles) we in England know where he had it; namely from Harriot's algebra. What Book is that? (saith M. Roberval,) I never saw it. Next time you come to my Chamber (saith Sir Charles) I will shew it you. Which after a while, he did: And upon perusal of it, M. Roberval exclaimed with admiration (Il l'a veu! Il l'a veu!) He had seen it! He had seen it! Finding all that in Harriot which he had before admired in Des Cartes; and not doubting but that Des Cartes had it from thence.

A tale told forty years after the event is unlikely to be accurate in detail, but the fact that it was repeated at all demonstrates the suspicions of Harriot's English supporters. The source of Wallis's story was John Pell, who in the

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late 1630s was acquainted not only with Cavendish himself but also with Aylesbury and Warner. Aylesbury and Warner always hoped to publish more of Harriot's work,72 and we know from Samuel Hartlib that in 1639 Pell too was working on some of Harriot's problems: ‘[Pell] hase finished those Problemes of Hariot which Warner should have perfected. Sir T. A. promised to let them have Hariots papers but hee did solve them without them’.73 Twelve years later, Pell and Aylesbury were still discussing Harriot's mathematics, for in 1651 Cavendish in Antwerp wrote to Pell in Breda:74 Sr. Th. Aleyburie remembers him to you and desires to knowe if you would be pleased to shew the use of Mr. Hariots doctrine of triangulare numbers; which if you will doe he will send you the originall; I confess I was so farr in love with it that I coppied it out; though I doute I understand it not all;75

As Aylesbury, Warner and Pell were all actively interested in Harriot's mathematics long after his death, the fate of the papers of those three is of some interest, and I have searched especially for evidence that any of them retained any of Harriot's Treatise on equations. Of Aylesbury's papers we know little except that many of his books and manuscripts were destroyed when he was cashiered as a Royalist in 1642.76 From Cavendish's letter to Pell we know that Aylesbury still held some of Harriot's papers in 1651, but after that we hear no more of them. Aylesbury's daughter, Frances, later married Edward Hyde, Earl of Clarendon, and the Clarendon family papers were searched on behalf of the Royal Society for any surviving Harriot manuscripts, but without success.77 Of Warner's papers we know rather more, and the story of their dispersal and circulation can be pieced together from a number of contemporary references. A year after Warner's death in 1643, Pell wrote to Cavendish that he feared Warner's papers were lost:78 And first for Mr Warner's analogickes, of which you desire to know whether they be printed. You remember that his papers were given to his kinsman, a merchant in London, who sent his partner to bury the old man: himselfe being hindred by a politicke gout, which made him keepe out of their sight that urged him to contribute to the parliament's assistance, … Since my comming over [to Amsterdam], the English merchants heere tell me that both he and his partner are broken, and now they both keepe out of sight, not as malignants, but as bankrupts. …In the meane time I am

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not a little afraid that all Mr Warner's papers, and no small share of my labour therein, are seazed upon, and most unmathematically divided between the sequestrators and creditors, who will, no doubt, determine once in their lives to become figure-casters, and so vote them all to be throwen into the fire, if some good body doe not reprieve them for pye-bottoms, for which purposes you know analogicall numbers are incomparably apt, if they be accurately calculated.

Fortunately, some of Warner's papers escaped both fire and pie dish, and were inherited by Nathaniel Tovey, his nephew by marriage. Tovey passed them on to Herbert Thorndike, a prebendary of Westminster, who was interested in mathematics and who did what he could to get the papers seen by those who might understand them. In 1650 Hartlib remarked that Seth Ward, a friend and admirer of Thorndike, was ‘to set out the mathematical and other workes of Warner conc[erning] coyne etc.’.79 At the time, Ward shared the Savilian Professors' study with Wallis, but thirty years later, in a letter to Aubrey, Wallis was vague about whether Ward had also held any of Harriot's original papers:80 I have formerly heard that they [Harriot's papers] had been, at some other time, in Mr Hobbes' hands. That they had been at some other time, in Dr Pell's hands. And that some time they had been in the hands of the present Bishop of Salisbury [Ward]. But it is many years since I heard any thing of certainty where they are: and feared they might have perished.

John Collins later reported that in 1650 the mathematician Thomas Gibson had also borrowed some of Warner's papers from Thorndike.81 Gibson's Syntaxis published in 1655 drew explicitly on the published work of both Harriot and Descartes, but there is no evidence that he used any of Harriot's manuscript material. In 1652 Thorndike sent the papers on ‘analogics’, or antilogarithms, to Pell, but a month later Pell told Thorndike ‘that I feared I must change my resolution of putting out Mr. Warner's writings, because they were so incomplete’.82 Some of Warner's papers were evidently dispersed to other owners too, for in 1653 Hartlib recorded that Sir Justinian Isham (of Lamport Hall, Northamptonshire, and a collector of books and manuscripts) ‘hath gotten all the MS Mathematical of Warner and Tin as himself shewed them Mr Pell’.83 The papers were acquired for Isham by Charles Thynne, a sometime acquaintance of Warner, and in recent years have come to light among

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the Isham papers now held in the Northamptonshire Record Office.84 Most are on chemical topics, mechanics and coinage, but there is also a bundle of twenty-three sheets of mathematics, which includes copies in Warner's hand of some of Harriot's work on geometery and optics.85 There is nothing relating to Harriot's Treatise on equations, but there is a neat method of solving the quadratic equation aa + ba = bc by ‘completing the square’:86

Wallis later remarked that Harriot had ‘a peculiar way of his own’ for solving quadratic equations, and described exactly the method outlined here;87 the method was clearly known to Warner also, but is not to be found in the Praxis. Other Warner papers were given by Thorndike to Collins for safe keeping in 1667, and Collins' possession of them was confirmed in a letter to James Gregory written early in 1668:88 I have some papers of Mr Warner deceased, wherein he proves if parallels be drawn to an asymptote, so as to divide the other into equal parts, the spaces between them, the hyperbola, and asymptote, are in musical progression, the which, if desired, I may communicate.

The inventory of the papers handed over by Thorndike, however, reveals nothing relating to Harriot's algebra (unless contained in a bundle mysteriously entitled ‘Mr Protheroe’).89 Collins, for all his information on contemporary mathematical writing, knew little more of Harriot's papers than anyone else:90 The Lord Brouncker has about two sheets of Harriot de Motu et Collisione Corporum, and more of his I know not of: there is nothing of Harriot's extant but that piece that Mons. Garibal hath.91

All attempts from 1662 onwards to trace the papers met with failure. By 1660 the only surviving Executor was Robert Sidney, Viscount Lisle, son-in-law of Henry Percy, who was only twenty-six when Harriot died, but after 1632 he lived first abroad and then in semi-retirement, and no one seems to have thought of making enquiries of him. Meanwhile, both

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the Will and the papers became the subject of speculation and hearsay. A 1653 entry in Hartlib's Ephemerides, for example, suggested that Sir Robert Naunton, who had married William Lower's widow, Penelope, had acquired some of Harriot's papers from Protheroe, but the report was suspect since it also described Protheroe as one of Harriot's last surviving friends when in fact he was the first to die after Harriot himself.92 Even the Royal Society, for all its prestige and influence, failed to discover anything new: requests for surviving papers were made on behalf of the Society in 1662 to the Earl of Clarendon, the son-in-law of Aylesbury, and in 1669 to John Vaughan, Earl of Carbery, brother-in-law of Protheroe, in neither case to any avail.93 After Thorndike's death in 1672 the Warner papers still in his possession passed to another prebendary of Westminster, Richard Busby, headmaster of Westminster School. Busby's ownership of the papers was well known, and was mentioned by both Anthony Wood and by Wallis, who wrote that Pell had ‘seen and perused them’.94 A final search among the Earl of Clarendon's papers was noted by Wallis and Aubrey in 1683, but Wallis was forced to conclude that: ‘concerning those papers of Mr Harriot's which were supposed to be in his hands. He … doth assure us [he] hath them not. So that, I guess, there are no other of them to be found’.95 Pell's name has now been mentioned many times in connection with Harriot's papers, from his early collaboration with Warner in 1639, his correspondence with Cavendish and Thorndike and his perusal of Isham's papers in the 1650s, to his inspection of the papers in Busby's possession in the 1670s. After Pell's death in 1685, his papers too were left with Busby, and in 1755 the papers of both Pell and Warner were rescued from Westminster School by Thomas Birch, Secretary of the Royal Society, and deposited in the British Museum.96 Neither set now contains anything significant in relation to Harriot's Treatise on equations except that amongst Warner's papers there is a single page in his handwriting containing material from Harriot's sheets d.4) and e.10), on conjugate equations.97 Amongst the hundreds of sheets left by Pell there is little relating to Harriot except for some scattered notes on the Praxis.98 There is some evidence, however, that during the early 1670s Pell discussed Harriot with Wallis, for Leibniz, who met Pell on his visit to London in 1673, appears to have been familiar with the opinions of both of them on the subject of Harriot's algebra.99 Then in 1685, Wallis published his Treatise of algebra, and devoted no fewer than twenty-six of its one hundred chapters

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to Harriot's algebra. It is to this lengthy and controversial account of Wallis's that we now turn.

Wallis's account of Harriot's algebra Wallis made his first public reference to Harriot in 1656 in the Dedication of his De sectionibus conicis. There he introduced the theme he was to take up so strongly later:100 In the symbols used, we have followed in part our own Dr Oughtred, in part Dr Descartes (unless a better contender is the name of our own Dr Harriot, who went before Dr Descartes on almost the same path), sometimes both.

By the time Wallis came to write A treatise of algebra twenty years later, acrimonious quarrels with Fermat, Pascal and Dulaurens had considerably hardened his attitude to the French. Wallis's account of Harriot's algebra was shot through with denigration of Descartes and accusations that he had taken ideas from Harriot without acknowledgement, criticisms that have clouded all later assessments of both Wallis and Harriot. Leaving aside, for the moment, Wallis's polemics, let us consider his description of the algebra itself. Wallis began by pointing out the advantages of Harriot's notation: his use of lower case letters, and the fact that his notation was unambiguous, clear and free from geometrical considerations.101 Wallis recognized, however, that Harriot's notation was only a first step towards a deeper understanding of polynomial equations and their structure:102 Beside those conveniences in the Notation mentioned in the former Chapter, (which are things less considerable:) Mr Harriot, as to the Nature of Equations, (wherein lyes the main Mystery of Algebra;) hath made much more improvement. Discovering the true Rise of Compound Equations; and Reducing them to the Originals from whence they arise.

Up to this point, Wallis's assessment of Harriot's achievements was both correct and astute. But then he made the first of the comments that has brought his account into such disrepute: And here first, Beside the Positive or Affirmative Roots, (which he doth, through his whole Treatise, more especially pursue, as the principal and most considerable:) He takes in also the Negative or Privative Roots; which by some are neglected.

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At first this seems a very strange remark for, as has been observed many times, there are no negative roots in the Praxis. Indeed, their absence was so noticeable that one reader after another was forced to question Wallis's credibility. But note that Wallis mentioned something else, a ‘Treatise’, in which Harriot especially pursued positive roots. This is an accurate description of Harriot's Treatise on equations where, just as Wallis described, Harriot did concentrate to begin with on positive roots only. There is already a strong hint here that Wallis was working from more than the Praxis and that he knew at least something of the contents of Harriot's unpublished Treatise on equations. Wallis's use of the word ‘here’ was not only vague but confusing, for he was ascribing to the Praxis something that was evident only in the manuscripts. A few pages later there was a similarly misleading remark about imaginary roots:103 And of such imaginary Roots, we find Mr Harriot particularly to take notice (in the Solution of Cubick Equations) in his 13th Example of his 6th Section; pag. 100.

What we actually find on page 100 of the Praxis is a statement that the equation aaa − 3bba = 2ccc is ‘impossible’ because it requires as part of its solution the term √ − dddddd, which is ‘inexplicable’. In other words, the writer of the Praxis ‘particularly took notice’ of imaginary roots only to dismiss them as rapidly as possible. There is no other reference to imaginary roots in the Praxis. Harriot's manuscripts, on the other hand, supply ample evidence that he himself worked comfortably with imaginary solutions. Thus it seems that Wallis was not making it at all clear where his information came from. This was so uncharacteristic of Wallis, who was well used to public argument, and to the accurate quotation of chapter, verse and line number, that one is forced to ask whether he was deliberately throwing a smoke-screen around his true sources. Wallis's specific examples of Harriot's method confirm the suspicion that he was working from more than the printed text. The first example he gave was the following, which corresponds to that in Harriot's sheet d.1), given above at (1):104

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And then Adding or Subducting bc to each side [Harriot] deduceth such as these, aa − ba + ca = bc: Which he calls Canonical Equations. (2)

After some discussion to clarify that a − b = 0 and a + c = 0 could not both hold at the same time, Wallis went on to show, as Harriot did in sheet d.1), that the canonical equation thus formed is indeed satisfied by a = + b: If for a, we put every where + b; we shall have

Thus Wallis's exposition of the material, though interspersed with his own commentary, is much closer to that in the manuscript sheet d.1) than to the scattered version that appears in the Praxis. Note also Wallis's use of 00 to indicate multiplication of two zeros, a notation required by the principles of homogeneity, and often used by Harriot in the manuscripts, but which never appears in the Praxis. The correspondence between Wallis's example (2) and the manuscript version (I) is close enough to suggest that Wallis did indeed have inside knowledge of Harriot's Treatise on equations.105 Further evidence that this was the case comes from a letter Wallis wrote to Samuel Morland in 1689, in which he compared Harriot's and Descartes' work in some detail.106 As examples of Harriot's use of negative and imaginary roots, Wallis gave a = −f and a = ±√ − df, and page numbers (14 and 15) of the Praxis where the relevant equations appeared. But once again his citations were less than transparent: on pages 14 and 15 of the Praxis the equations were indeed listed, but without their roots; the negative and imaginary roots cited by Wallis are to be found only in the manuscripts, in sheets d.7.2°) and d.13.2°), respectively.107 We have also seen that Wallis knew Harriot's method of solving quadratic equations by completing the square, though this too is not to be found in the Praxis. Thus, though Wallis wrote to Aubrey in 1683 that it was many years since he had heard anything of Harriot's original papers, he displayed a remarkably intimate knowledge of their contents. Wallis himself made several apparently contradictory statements as to whether he had seen Harriot's manuscripts or not. In March 1677, shortly after he had completed the first draft of A treatise of algebra, he wrote a long preamble in the Savile Library copy of the Praxis, in which he stated quite clearly that he had seen some of Harriot's unpublished work:108 There were many other very worthy pieces of Mr Harriots doing, left behind him, & well worth the publishing: as appears by Mr Warners

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Preface, & Title-page & some of them I have seen. But in who's hands they now are, or whether they be since perished, I cannot tell.

In 1683, however, he wrote to Aubrey, in a letter already quoted above, that it was many years since he had heard anything of the papers, and continued:109 I have never read any of his things, but that only of his Algebra; which hath had the good hap to be published by Mr W.W., as a prodromus to some other of his works; which at the same time, he gave hope of publishing, but hath not done it.

And in 1685 in A treatise of algebra itself, he lamented that the disappearance of Harriot's papers prevented him from detailing any further applications of his algebra:110 What uses [Harriot] hath made of Algebra in order to other parts of mathematical knowledge, in his other Treatises, I cannot say; because they are not publick: nor do I know in whose hands they are; if extant; nor whether they are ever like to see the light.

Wallis's 1683 letter to Aubrey appears to contradict the claim he made in 1677, that he had seen some of Harriot's pieces in manuscript, but there is perhaps a distinction to be made between having seen something and having read it, and there may also have been a difference between what Wallis would write in a private library book and what he would wish to convey to Aubrey. It is also possible, however, to interpret the 1683 passage further. The text can be read as: ‘I have never read any of his things in manuscript, but that only of his Algebra, which hath had the good hap to be published by Mr. W.W.’ In other words, Wallis was implying, that he had read some of the originals of Harriot's algebra (though not any of his other work), but that the content was no more than had been published by Warner. Wallis's failure to distinguish clearly between the content of the manuscripts and the content of the Praxis was, as we have seen, a recurrent feature of his description of Harriot's algebra. All the evidence suggests that when Wallis wrote A treatise of algebra he had first-hand knowledge of at least some of Harriot's Treatise on equations.111 How did this come about? Wallis's earliest opportunity to study Harriot's manuscripts, or Warner's, would have been around 1650 when he shared a study with Ward, but his letter to Aubrey implied that he had no clear recollection of having seen the papers at that period. Wallis may have been unwilling to share all he knew with Aubrey, and it is just possible that he saw and copied some of

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Harriot's papers in the early 1650s. He had a more immediate and later source, however, in Pell. When Wallis wrote about Harriot in A treatise of algebra his only mention of Pell was as the source of the story about Cavendish and Roberval. Some years later, however, after Pell had died, Wallis was more explicit about his involvement. For the 1693 Latin translation of A treatise of algebra he wrote a new piece entitled De Harrioto addenda (‘What must be added concerning Harriot’) which contains the following important paragraph:112 Certainly, no one at all has judged this to have been taught before Harriot. Which Pell himself recognised, and I some time later came to understand; so that he urged me all the more often in this; and from his words I have written everything I have said; and after it was written I showed it to him (to be examined, altered, corrected as he decided, or as he preferred said otherwise) before it went to press, and everything which was published was said with Pell's assent and approval.

Such close collaboration between Pell and Wallis adds a new dimension to Wallis's account. In particular it helps to explain Wallis's criticisms of Descartes, for Pell had known Descartes personally and, unlike Wallis, had some reason to dislike him. Pell's coolness towards Descartes seems to have arisen from his first meeting with him in 1646, when Descartes was unenthusiastic about Pell's efforts to edit Apollonius and Diophantus.113 Later, Pell appears to have been disgruntled by Descartes' reception of his Idea (his plan to compile a catalogue of mathematics and mathematicians, and a library of their most important works).114 Collins, with whom Pell lodged for a time, knew something of this affair, for he reported it to Leibniz:115 The said Doctor [Pell], being censorious of others, and incommunicative, himself declining discourses about his methods, was at last censured by Des Cartes for those assertions, concerning whom the Doctor never had any extraordinary esteem. And those letters or censures of Des Cartes, one Mr Haak, … hath a copy of, but, upon the account of his friendship to Dr Pell will not impart.

Theodore Haak was finally persuaded to publish the letters in the Philosophical collections in 1682, and it turned out that Descartes‘ attitude was not so much one of antagonism as of indifference, but to Pell it probably amounted to much the same thing. ’I inspected the Mathematical idea only incidentally',

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wrote Descartes, ‘and now only recollect that there was nothing with which I greatly disagreed’.116 It was unfortunate that Pell's dislike of Descartes found its way into A treatise of algebra, for in other ways Pell had helped Wallis to a deeper understanding of Harriot's algebra than any of his contemporaries. Wallis's grasp of the implications of Harriot's work is evident throughout his account, and he took some pains to point out features that he considered obvious from Harriot's work, but that were not overtly stated: the clear relationships between the coefficients and the roots of a polynomial equation, for instance, or the fact that the degree of an equation could be decreased by division, just as it could be increased by multiplication. 117 Generally Wallis was clear that these results were not explicitly stated in the Praxis but could be inferred from its contents, as the following quotations show (my italics):118 Whence it follows, That all Cubick Equations have (Real or Imaginary) Three Roots, (all Affirmative or all Negative or partly the one, partly the other.)… 'Tis manifest also, that as Compound Equations are made up of others more simple, by Multiplication; So they may by like Divisions, be reduced into those Simples again.… 'Tis also manifest (from these Compositions,) not only, how many Roots (Real or Imaginary,) every Equation contains, (viz. so many as there are the Dimensions of the Highest Term:) But Likewise, of what Members each of the Coefficients are made up. Which appears, without further trouble, by a bare inspection of the Composition. … And whereas Mr Harriot gives Rules to determine how many Affirmative Roots there are in any Equation proposed; the same Rules (by this means) serve as well to determine, how many Negatives are therein real. … And these [improvements] are either explicitly delivered by him, in express words; or be obvious Remarks, upon the bare inspection of what he delivers. … How he [Descartes] came by that Rule, he doth no where tell us; nor give us any Demonstration of it. … But from Harriot's Principles, It follows naturally.

This approach was typical of Wallis's account: over nineteen chapters he described or quoted the entire content of the Praxis, much of it verbatim,

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but repeatedly filled in what was missing, or what he considered the obvious consequences of Harriot's work. Thus he supplied those equations with only negative roots that Warner had so conspicuously left out,119 and at Problems 19, 20 and 21 of Section 3, where Torporley had stumbled and Warner had fallen, Wallis restored the missing conditions without difficulty.120 Wallis saw as clearly as Pell the true magnitude of what Harriot had done, and was concerned, with Pell's encouragement, to put the record straight. His description of Harriot's algebra was intended not just as straight rendering of content but as an assessment of its place in the history of the subject, and of its potential for development. In this he can only be said to have succeeded: his exposition of what could be built on Harriot's foundations was essentially correct, and his list of twenty-five ‘Improvements of Algebra to be found in Harriot’ was a fair summary of what could be found in Harriot's work or easily deduced from it.121 Wallis's account was for over 300 years the most thorough and detailed analysis of Harriot's algebra available and, had he not been so aggressive towards Descartes, or so secretive about his sources, might have stood as the fine testament to Harriot that he intended it to be. What should we now consider to be the ‘Improvements of Algebra to be found in Harriot’? The first and most obvious must be his notation, the use of lower case letters, with repetition to indicate multiplication. The only significant difference between modern notation and Harriot's is in the use of superscripts for exponents. Terms such as aaaabb seemed to beg for some kind of abbreviation, and Torporley in copying Harriot's manuscripts did indeed sometimes write aI, aII, aIII, aIV, where Harriot had written a, aa, aaa, or aaaa.122 Clearly it was only a matter of time before such conventions became general. The lack of abbreviation may have been tedious for the writer, but does nothing to hinder the reader, and Harriot's work is clear and easy on the eye. Oughtred later took great pains to devise an algebraic notation that ‘plainly presenteth to the eye the whole course and processe of every operation and argumentation’, but never succeeded to the extent that Harriot had done more than twenty years before him. Harriot's notation made possible his second great achievement, the handling of equations at a purely symbolic level. His interest in the structure and manipulation of equations was firmly in the tradition of Cardano, Bombelli and Viète, but Harriot was the first to have a notation that actually helped rather than hindered his investigations. His third outstanding achievement was his crucial insight into the way polynomials could be built up as products of linear or quadratic factors and to see that such composition was in turn

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a powerful analytic tool. In Harriot's clear layout, results about the number and kind of roots, and the relations between the roots and the coefficients became immediately obvious. Wallis pointed this out repeatedly but most modern commentators on the Praxis have been too distracted by the absence of negative roots to notice it.123 It was, however, evident to Hutton who, a century after Wallis, said the same thing in more measured tones:124 [Harriot] shewed the universal generation of all the compound or affected equations, by the continual multiplication of so many simple ones, or binomial roots; thereby plainly exhibiting to the eye the whole circumstances of the nature, mystery and number of the roots of equations; with the composition and relations of the coefficients of the terms; and from which many of the most important properties have since been deduced.

Harriot was not in the habit of describing his mathematics verbally, but if he failed to state such results explicitly it did not mean he was not aware of them, for his entire work on equations was based on the way coefficients were composed from roots. Such relationships between roots and coefficients were to become the foundation of all subsequent work on polynomial equations, and were to help to lead eventually to the development of modern abstract algebra. These three achievements alone—the invention of clear notation, symbolic handling of equations, and an understanding of the structure of polynomials—were enough to place Harriot among the first rank of early algebraists, on a par with Cardano, Viète and Descartes. Furthermore, Harriot was fully aware of the use of algebra as an analytic tool for studying geometrical problems, as numerous examples in his notes testify. For his time, he was, as Pell so rightly said, ‘so learned, that had he published all he knew in algebra, he would have left little of the chief mysteries of that art unhandled’.125 The tragedy was that Harriot published nothing, and those who afterwards tried to do so failed to do him justice. Instead, from 1637 onwards, Descartes' La géométrie became the foundation and inspiration for continental mathematicians, and eventually became the dominant influence in England too. No wonder that Wallis, already deeply suspicious of the French, became bitter as he came to know Harriot's algebra better at what he, like Pell, saw as Descartes' usurpation of Harriot's rightful place. Wallis, like Torporley and Warner before him, did his best to ensure that Harriot was given the recognition he deserved, but later readers of Wallis's account, lacking the

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supporting evidence of the manuscripts, saw only his apparent exaggeration and polemic, and dismissed it. Harriot's procrastination had cost him dear, and those who came after him failed, through incompetence, incomprehension or misjudgement, to reward him with the ‘Gharland of the greate Invention of Algebra’.

5 Moving the Alps: uncovering the mathematics of John Pell John Pell (1611–1685) has long been considered the most enigmatic of the seventeenth-century mathematicians.1 There is no question of his lifelong devotion to mathematics: at the age of seventeen he was already discussing mathematics with Henry Briggs, first Gresham Professor of geometry, and he spent the first twenty years of his working life teaching mathematics in schools and colleges. During the 1650s he participated in mathematical meetings with Christopher Wren, William Brouncker, Charles Scarborough and others, became an early member of the Royal Society, and was invited by Brouncker to be its Vice-President in 1676.2 He was well read in both Classical and contemporary mathematics,3 and there is no doubt that he was held in esteem. John Collins, a loyal friend to Pell, wrote of him: ‘I take him to be a very learned man. More knowing in algebra, in some respects, than any other …’4 When we try to discover, however, just what Pell's mathematical reputation was based on, the picture is strangely unclear. His name is linked with the equation Np2 ± 1 = q2 (for N, p, q integers), universally known as ‘Pell's equation’ but neither proposed nor solved by Pell.5 His mathematical publications were few and far between:6 the book for which he is best remembered is An introduction to algebra published in 1668, but other books expected of him failed to appear. There were always hints that he was developing further ideas, but he could never be persuaded to share them. And there is the nub of the problem. Here is Collins again:7 To incite him to publish anything seems to be as vain an endeavour, as to think of grasping the Italian Alps, in order to their removal. He hath been a man accounted incommunicable; the [Royal] Society (not to mention myself) have found him so.

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Pell may have been reluctant to publish but for almost sixty years he read widely and wrote prolifically and accumulated vast quantities of papers, and after his death his books and manuscripts were acquired by his friend Richard Busby, headmaster of Westminster School.8 The books are still at Westminster, but the manuscripts are now held in the British Library, where they fill thirty-three large volumes.9 The papers contain calculations, tables, worked problems, and notes on the books Pell had read, and are chronologically and thematically in complete disorder. Pell's correspondence, with Cavendish, Mersenne, Hartlib, Collins and others, fills a further three volumes.10 The papers present the researcher with a daunting task but contain a wealth of material, from which a new picture is at last beginning to emerge of Pell's mathematics and his mathematical influence. Pell was born on 1 March 1611 in Southwick in Sussex. His father, a schoolmaster, died when John was just five years old and, according to Aubrey, left a substantial library. As a boy, Pell went to the newly founded Grammar School at Steyning, just a few miles inland over the downs, and at the age of thirteen, young even for those days, to Trinity College, Cambridge. Aubrey, who knew Pell personally, wrote of him admiringly but perhaps with some exaggeration, that: At 13 yeares and a quarter old he went as good a scholar to Cambridge, to Trinity Colledge, as most Masters of Arts in the University (he understood Latin, Greek and Hebrew) so that he played not much (one must imagine) with his school fellowes, for, when they had play-dayes, or after schoole-time, he spent his time in the Library.

By 1627, aged 16 and still at Cambridge, Pell had taken to mathematics and had started writing small booklets on mathematical subjects. There are many of these, in Pell's small neat hand, among his surviving papers.11 The early booklets demonstrate two things that were to be characteristic of Pell for the rest of his life. The first is a fascination with tables and tablemaking. One of the earliest, dated August 1628, is on the construction and use of multiplication tables (up to 9 × 100),12 but Pell soon moved on to more difficult things. By October that year he had written to Henry Briggs with questions about antilogarithms, and interpolating tables of sines,13 and it seems that Briggs already knew him well, for he signed himself: ‘Yr very lovinge frende, Henrie Briggs’. Pell was to go on making tables for the rest of his life: sheet after sheet of his surviving manuscripts is filled with tables, and calculations for tables: tables of squares, sums of squares,

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primes and composites, constant differences, logarithms, antilogarithms, trigonometrical functions, and so on. Only one set was ever published: tables of the first 10000 square numbers, which were printed in 1672 (Fig. 5.1). The early booklets also give the first hint of another enduring feature of Pell's personality: a strange desire for anonymity. Many of the booklets have a carefully designed title page with a date but no author. One, for instance, is entitled Linea proportionata but in the neatly ruled space where the name of the author should appear is the single word By followed by a blank space.14 Another copy of the same text has the obvious pseudonym By A. FALE, Gent.15 This urge for secrecy was to be evident throughout Pell's life: he was always extremely reluctant to reveal his name in print. It is a stranger and more obsessional quality than mere modesty, and one of the things that has made Pell such a difficult subject of study for later historians. It was probably in Cambridge in 1628 or 1629 that Pell met Samuel Hartlib, newly arrived as a protestant exile from Poland. Hartlib was to work tirelessly both for the European Protestant cause and for educational and social reform, and in 1630 he set up a school in Chichester in Sussex that would realize his educational ideals and provide employment for exiled scholars and others. Pell was employed to teach mathematics. The school lasted only a few months, but Pell remained teaching in Sussex for some years, and in 1632 married Ithumaria Reginalds with whom he eventually had eight children. Hartlib and his friends did their best to find Pell a post in which he could make the best use of his talents, but he was not an easy man to help. Aubrey described him as ‘naturally averse from suing or stooping much for what he was worthy of ’,16 but the fact of the matter was probably that Pell was particular about what he would or would not do. Aubrey reported that Theodore Haak, for instance, introduced Pell to John Williams, Bishop of Lincoln, who offered Pell a benefice, but that Pell turned it down on the grounds that he would rather devote himself wholly to mathematics, ‘for the great publick need and usefullness thereof ’.17 Pell shared Hartlib's aim of strengthening Protestantism through public service and the collection and exchange of knowledge,18 and it was for Hartlib in 1638 that he composed a tract entitled An idea of mathematics. It was published in both English and Latin that same year, but Pell's name did not appear on it, and only later did he admit to being its author (Fig. 5.2).19

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Fig. 5.1 Title page of A table often thousand square numbers (London 1672). Pell's name does not appear in print but has been added in handwriting later. (British Library 528.n.20.)

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Fig. 5.2 The first page of Pell's Idea of mathematics from John Dury's The reformed librarie keeper (London 1650).

Pell began with an admonition: As long as men want will, wit, means or leisure to attend those studies. it is no marvail if they make no great progress in them.

He went on to propose the establishment of three things: first, a catalogue of mathematicians and mathematical texts; second, a library containing those texts; and third, to save time and labour for future generations, a concise summary of all mathematical discoveries so far. These were practical projects that Pell began to work on, though he never completed any of them.20 Of greater interest to us now, however, is his perception of how mathematics

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itself might advance. It should be possible, he argued, to set out the method or process of mathematical argument in such a way that it would be possible to deduce from first principles: … not onely all that ever is to bee found in our Antecessor's writing, and whatsoever they may seeme to have thought on, but also all the Mathematicall inventions, Theoremes, Problemes and Precepts, that it is possible for the working wits of our successors to light upon and that in one certain, unchanged order, from the first seeds of Mathematics, to their highest and noblest applications.

There is something in this of Viète's Nullum non problema solvere, except that Pell projected the power of mathematics into the future as well as the past. His strong sense of an inherent logical structure by means of which all mathematical theorems could be deduced sets Pell apart from most other seventeenth-century English mathematicians, who generally took a much more pragmatic approach to their subject, and brings him in some ways closer to the twentieth century, when it was again believed that mathematics could be established and developed on purely logical foundations. By 1639 Pell had made his way into other circles of acquaintance than those surrounding Hartlib, for he had become known also to Charles Cavendish, and to Thomas Aylesbury and Walter Warner, the editors of Harriot's mathematical papers. Hartlib noted that Aylesbury was prepared to lend Harriot's papers to Pell,21 and Pell's subsequent reputation in algebra probably rested in great part on this early and direct acquaintance with Harriot's work. With Warner, Pell embarked on an ambitious project, the construction of tables of antilogarithms, something he had already discussed with Briggs some ten years earlier. Aubrey knew about these tables but was puzzled as to the use of them:22 Mr. Walter Warner made an Inverted Logarithmicall Table, i.e. whereas Briggs' table fills his Margin with Numbers encreasing by Unites, and over-against them setts their Logarithms … Mr. Warner (like a Dictionary of the Latine before the English) fills the Margin with Logarithmes encreasing by Unites, and setts to every one of them so many continuall meane proportionalls between one and 10 … These, which, before Mr. John Pell grew acquainted with Mr. Warner, were ten thousand, and at Mr. Warner's request were by Mr. Pell's hands, or direction, made a hundred-thousand. Quaere Dr Pell, what is the

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use of those Inverted Logarithmes? For W. Warner would not doe such a thing in vaine.

Unfortunately Warner died in 1643 before the calculations were finished and Pell was by then occupied with other things. Years later, at the end of 1652, Herbert Thorndike handed some of Warner's papers over to Pell, but Pell complained that they were too incomplete for him to continue with them, and the tables were abandoned.23 By the end of 1643, the efforts of Pell's friends to find him a suitable post had finally met with success: through Theodore Haak and Sir William Boswell, Ambassador to The Hague,24 Pell was offered a teaching post at the Gymnasium in Amsterdam and left England in December 1643. During his three years in Amsterdam, much of his time and energy went into refuting a quadrature of the circle published by the Danish mathematician Longomontanus in Amsterdam in 1644. Pell saw it as his duty to argue against Longomontanus, and engaged the help of Cavendish, now living in exile in Hamburg and Paris, to rally support from mathematicians throughout Europe.25 His efforts led him to the discovery of the double-angle tangent formula, but diverted him from what might have been much more productive studies. Cavendish had been hoping since at least 1641 that Pell would publish what he knew on ‘Analiticks’, or algebra, but Pell just as often procrastinated:26 I have thought nothing elaborate enough to be printed, till it were so complete that no man could better it, and did therefore so long keepe my name out of the presse:

Cavendish also urged him to complete and publish two other projects: an edition of Books V to VII of the Conics of Apollonius, the text of which then existed only in manuscript,27 and an edition of the Arithmetic of Diophantus. Early in 1646, however, Pell received a visit from Descartes and afterwards reported to Cavendish that he was disinclined to continue with these classical texts.28 Whether Descartes' discouragement was real or perceived we cannot know, but Pell's editions were never published and no draft of either has been found. In 1646 William, Prince of Orange, founded the Illustre School in Breda,29 and personally invited Pell to be its professor of both mathematics and philosophy. Pell accepted but later, rather to his relief, was required to teach only mathematics (Fig. 5.3). He gave his inaugural lecture on 9 September 1646,30 and we know a little about some of his pupils and acquaintances in the following years. The exiled English king, Charles II, spent some weeks

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Fig. 5.3 A poster advertising the first term's lectures at the Illustre School in Breda. The only known original of the poster is now held in the Archives of Lincoln Cathedral (Bs. 5).

in Breda in 1649 and there is reason to believe that Pell became friends and discussed mathematics with one of his young courtiers, Silas Titus:31 the mathematical friendship between Pell and Titus was to continue for many years, and we shall see more of it later. Another of Pell's pupils was

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William Brereton, then aged seventeen, who had been forced to flee his home, Brereton Hall in Cheshire, in the early stages of the civil war and was sent to Breda for his education. He too became a lifelong friend to Pell, an example of the sort of loyalty Pell could inspire. The same cannot be said, however, of another pupil, the young Christiaan Huygens, who studied in Breda from 1647 to 1649 and visited Pell on several occasions, but who complained that Pell refused to lend or discuss Gregory St Vincent's Opus geometricum (1647) and taught him nothing.32 Pell returned to England in 1652 and, no doubt through Hartlib's intervention, was granted a small salary by Cromwell in return for unspecified teaching duties. He stayed in London only two years, but long enough to make the acquaintance of English mathematicians and to attend the meetings at Gresham that foreshadowed the later formation of the Royal Society. In 1654 he was persuaded by Haak to accept the post of envoy to the Protestant Cantons of Switzerland, and spent the next four years as Cromwell's agent in Zurich, a period of his life that is relatively well known and documented from the many letters that passed between him and John Thurloe, head of Cromwell's intelligence service.33 Pell finally returned to England in 1658, just three weeks before Cromwell's death, and a letter written to him soon after his return shows that he was familiar and respected amongst the London mathematical cognoscenti:34 Mr. Pell, – There is this day a meeting to bee in ye Moore Feilds of some Mathematicall freinds (as you know ye custom hath beene). There will bee Mr. Rook and Mr. Wrenn my Lord Brunkerd Sir Paul Neale Dr Goddard Dr Scarburow &c. I had notice ye last night of your being in towne, from some of ye gentellmen now named, and of there desire to injoy your company, their will bee no such number as you usually have seene at such meetings 12 is ye number invited.

Although his mathematical reputation was by now firmly established, Pell held no post, and it is not clear how he earned his living over the next two or three years, until in 1661 he did what he had refused to do thirteen years earlier and entered the church.35 Pell's first living was at Fobbing in Essex, a church that was in the direct gift of Charles II. This seems a generous reward to someone who had served Cromwell so faithfully, and possibly Pell's friend Titus, by now a Member of the King's Bedchamber, brought some influence to bear in the matter. Two years later, Pell received a second living at Laindon, four miles from Fobbing, from Gilbert Sheldon, Archbishop of Canterbury.

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Both Fobbing and Laindon in the low-lying Essex marshes were notoriously unhealthy and Aubrey tells us that Pell once complained to Sheldon. ‘I doe not intend that you shall live there’, the Archbishop remarked, ‘No, sayd Pell, but your Grace does intend that I shall die there.’ In 1663 Pell was made a Doctor of Divinity by Sheldon at Lambeth. By now he was also a Fellow of the Royal Society, and looked set to enjoy a comfortable middle age, but it was not to be. He became increasingly impecunious and was rescued over and over again by the kindness of his friends. He lodged for a time with the ever patient Collins, and then in 1665 escaped the plague in London by moving with part of his family to Brereton Hall in Cheshire, at the invitation of William Brereton who had been his pupil in Breda.36

An introduction to algebra While in Cheshire, Pell was engaged on the mathematical work for which he is now best known, An introduction to algebra. The book was partly a translation from a German text, the Teutsche algebra, of Johann Rahn who had been Pell's pupil in Zurich in the 1650s.37 There is ample evidence that Pell had made a considerable input into the original German edition,38 and indeed Rahn claimed in his Foreword that his methods were learned from a high and very learned person who would not allow his name to be published: this can only have been Pell. When the book was translated into English by Thomas Brancker during the 1660s, Pell made extensive additions and corrections to it. We have some of the correspondence that passed between Pell and Brancker over this matter, and can only admire the tact and patience that Brancker displayed in his dealings with Pell. It is no surprise by now to learn that a major problem for Pell was whether to allow his name to be added to the book or not. A letter drafted but never sent describes his dilemma exactly; he was reluctant to put his own name to the book but equally reluctant to let others claim credit that was rightly his:39 I know not how my mind may alter but for the present, I think it best not to name mee at all in the title or preface: and yet you may be more ingenuous than Rahn was and not vent all for you owne devices. You may say, that the alterations and additions etc. were made by the advice of one of good reputation in those studies.

When the book finally appeared, the title page carried a compromise: ‘Much Altered and Augmented by D.P.’ (Fig. 5.4). In the British Library we have the

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Fig. 5.4 Title page of An introduction to algebra (London 1668).

original draft of the title page in Pell's own hand, and there ‘D.P.’ is crossed out and then reinstated, as though even in the final stages Pell was not at all sure how much to give away.40 It was well enough known, however, that D.P. was Doctor Pell, and indeed Brancker's preface tells us exactly which pages were Pell's (late) contributions. The opening pages of An introduction to algebra explain the notation and the rules for handling and simplifying equations. Collins, who saw the book through the press, thought the introductory material inadequate, and said

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as much when he wrote to Wallis in 1667:41 As concerning the book of Dr. Pell's scholar, I think the Dr. did little concern himself in it till the introduction was past, and to speak plainly, I account that introduction much worse than the Principia Matheseos Universalis [of Bartholin].42

Collins raised his concerns about the introductory material with Pell while the book was still in the press, but Pell was disinclined to change anything, and Collins, with remarkable loyalty, defended Pell in public, despite his private doubts:43 I know none that account the Introduction a bad one, but divers that think it might have been more plain, and ought to have been more large than it is. This is the judgement of divers of the virtuosi and of some teachers of the mathematics here, who all love and honour the Doctor [Pell]; and I hope I shall do no less as long as I live, albeit I am of their mind, nor do I endeavour to make others of the same opinion, but say to them the Doctor did not much concern himself therein, but lets it come out as his scholar left it;

After the opening pages, the remaining and greater part of the book is devoted to ‘The Resolution of divers Arithmetical and Geometrical Problemes’. There is nothing remarkable in that subheading, but there are one or two features that distinguish An introduction to algebra from any other algebra text of the time. The first is Pell's very strong sense of order and logic, which may be seen on almost every page in his three-column layout (Fig. 5.5). The working out of each problem is contained in a wide right-hand column, while the left-hand column lists a sequence of instructions and the narrow middle column carries line numbers, very much as in a modern computer program. Only the first few lines of each problem are different: there the unknown quantities are listed on the left, and the given or known conditions on the right (see Fig. 5.6). If the solution can be uniquely determined, then the number of unknowns is exactly matched by the number of conditions, and the working can proceed logically to its conclusion. Pell invented a number of new symbols that enabled him to keep each of his instructions on the left on a single line. The only one to have remained in use, and for exactly the reason Pell invented it, is the division sign, ÷, which allows division to be written without resorting to two-line fractions. Pell was not always consistent in his use of this symbol: on page 59 (Fig. 5.5), for example, ‘20 ÷ 21’ means ‘divide line 20 by line 21’, while ‘28 ÷ 4’ means

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Fig. 5.5 Page 59 from An introduction to algebra showing Pell's three-column layout, his use of ÷ and other symbols, and his explanation of Harriot's inequality signs.

‘divide line 28 by 4’, but elsewhere he introduced a short bar to distinguish numbers that were part of the calculation from line numbers. Pell used his three-column method also for geometric theorems, for which he provided algebraic proofs (using, for example, □ AB for the square of AB). The second unusual feature of An introduction to algebra is that Pell showed how to adapt his method to indeterminate problems, those in which there are more unknown quantities than given conditions. As Pell observed,

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‘whensoever the number of required equations is greater than the number of given ones; the question is capable of innumerable answers’.44 Pell's Problem XXIV, for example (taken directly from the Arithmetic of Diophantus), was to find two numbers, either of which being subtracted from the square of their sum, will leave a square number (Fig. 5.6).45 Pell called the leftover square numbers dd and ee, so he had five unknown quantities: a, b, c, d and e, but only three conditions: (1) a + b = c, (2) cc − a = dd and (3) cc − b = ee. Places for two further, but as yet non-existent, conditions were marked by asterisks, which at a later stage could be replaced by arbitrary conditions to aid the solution. In Pell's solution to Problem XXIV the first asterisk was replaced at line 8 by the condition e = 2d, and the second at line 13 by the condition c = 3d (essentially the same conditions as introduced by Diophantus). The discussion of indeterminate problems is one of the most interesting features of An introduction to algebra, and almost certainly arose from Pell's unfulfilled plans to edit Diophantus during his years in Amsterdam. An introduction to algebra has long been regarded as the only significant publication of Pell's mathematics in his later years, but recent research rather changes that view. It has now become apparent that more of Pell's mathematics did appear in print, but in a rather curious way, and to see how it did so, we need to look more closely at Pell's relationship with Wallis.

Pell and Wallis It is not clear when Pell and Wallis first met. When Wallis began his mathematical career in 1649 Pell was abroad, but it is quite likely that they became acquainted when Pell returned to England in 1652. We do know that in 1655 Pell in Zurich received a printed sheet giving news of Wallis's forthcoming Arithmetica infinitorum. Pell was often meticulous about noting the dates when letters arrived, or books or papers changed hands, and we have a scrap of paper in his handwriting recording the following exchange:46 Apr 27. 1655 In ye meantime accept of Dr Wallis Quadratum Circuli here adjoined, which I intreate you to handle soundly. For hee makes himselfe beleeve you will doe no great matters in Mathematicall studys. May 26. 1655 Sir. I thanke you for yours of April 27 with that printed paper inscribed to Mr Oughtred. If his great age have not made him unwilling to looke upon things of that nature, perhaps he will make some reply.

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Fig. 5.6 Page 100 from An introduction to algebra showing how Pell handled indeterminate equations by introducing arbitrary conditions, marked by (*).

When it comes to your hand, I pray you to send it to mee. As also if the Author expresse himself more fully heere-after. …

Pell copied the first letter without naming the recipient, so it is not clear whether it was directed to him or to an unknown third person. Wallis's low

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opinion of the recipient suggests that he may have intended the ‘printed paper’ for Thomas Hobbes, but the mathematical comments that follow in the rest of the reply (to be discussed in Chapter 6) are not the same as those later made by Hobbes, and I suggest that they were Pell's own response to the proposed contents of the Arithmetica infinitorum. If so, this scrap of paper is the earliest known mathematical interaction between Pell and Wallis. If Wallis in 1655 held Pell in little regard, his opinion must have changed when Pell returned to England again three years later, because some of Wallis's notes from that period are in Pell's three-column layout.47 Then in 1662, we find from Pell's manuscripts that he and Wallis were discussing a problem that now seems of little interest, but on which both spent a great deal of time: given integers l, m and n, find numbers a, b and c such that

There are numerous pages amongst Pell's papers devoted to this problem,48 and he wrote that he had first come across it in Breda in 1649:49 Mr. William Brereton of Breda anno 1649 brought me an example of this question aa + bc = 16, bb + ac = 17, cc + ab = 22…

Pell then wrote that ‘as triall of logisticall skill’ he transformed it to:

To which I gave this answer:but the manner of investigation I did not shew him. Neither do I now at all remember what course I tooke … but I will heere endeavour to show a way …

Reconstructing the solution was evidently not as easy as Pell supposed; he was unable to fill in the answer and his working rapidly petered out. Silas Titus, who had been in Breda in 1649, and who may have discussed the problem with Pell then, was evidently interested in his renewed efforts, for

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Pell noted that on 13 December 1662 he had left a note about it for Titus at his house.50 Ten days later he left Titus another note, and this time it suggested that a third person was also involved:51 He sayes that x … is greater than

and less than

I say that … x hath four values.

From the general solution that emerged four months later we may surmise that ‘he’ was Wallis. The solution is amongst Pell's papers and in his handwriting, but a note at the end makes it clear that he had it from Wallis:52 I wrote this out from a copy in Wallis's hand, April 14 1663, which I delivered to Captain Titus, November 14 1663.

In Pell's three-column layout Wallis's solution ran to 35 lines. He had shown that the problem gave rise to a twelfthdegree equation in a, which then reduced to an eighth-degree equation in e, where e2 = 2a2. For the values l = 16, m = 17 and n = 18, the equation was

On another page of Pell's papers we have the note ‘Therefore DIW [Dr John Wallis] and MIP [Mr John Pell] agree in all the coefficients of the equation’,53 and Wallis and Pell went to great lengths to solve the above quartic in e2 to produce values of a, b and c to fifteen decimal places. We have a fair copy in Pell's hand of the general solution followed by the calculation of a, b and c, the whole now running to 132 lines.54 The problem surfaced again in the early 1670s, for in 1672 Collins sent James Gregory a copy of the ‘Breretonian problem’,55 and in October 1677, almost thirty years after Brereton first proposed it, Hooke recorded in his diary that he ‘Borrowd of Collins at Rainbow [coffee house], Mr. Bakers solution of problem aa + bc = x.bb + ac = y.cc + ab = z’.56 As mentioned in Chapter 1, several items that were raised in correspondence with Collins during the early 1670s found their way into A treatise of algebra, and this problem was one of them: Wallis included the full solution exactly as he and Pell had arrived at it in 1663.57 The opening paragraph of Wallis's lengthy exposition, however, is rather strange, because Wallis managed to introduce the problem without actually attributing anything to Pell. Although the running title head in this part of A treatise of algebra is ‘Of Dr. Pell's Algebra’, Wallis's chapter

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heading is vague: it is not, as one might expect, ‘An Example of Pell's’, but merely ‘Another Example in Imitation of His’. Then Wallis wrote that he was presenting the problem to demonstrate the use of Pell's three-column method, but as the originator of the problem he named not Pell but Titus:58 I here subjoin another Question, … proposed to my self, long since, by Colonel Silas Titus (then of his Majesties Bed-Chamber;) a very Ingenious Person, and well skilled in affairs Civil and Military, and very well accomplished in Mathematical and other Learning.

Titus had certainly been kept informed of progress on the problem in 1663, but Pell's papers make it clear that it was Pell himself who first took it up in 1649 and continued to pursue it later, so Wallis's failure to acknowledge Pell's interest seems at first either incomprehensible or dishonest. I suggest, however, that Wallis was not, as one might suppose, trying to claim all the credit for himself, but that he was colluding with Pell's obsessional desire for anonymity, and that he did so by deliberately deflecting attention from Pell to Titus.59 There are several other places in A treatise of algebra where Wallis threw a cloak of anonymity around Pell, and did it so effectively that the full extent of Pell's contribution to the book has not until now been recognized. A few further examples will serve to show where more of Pell's work is to be found. In Chapter 11 of A treatise of algebra Wallis sought fractional equivalents for the number we now know as π the ratio between the perimeter and diameter of a circle.60 He began with the well known , proceeded to the less familiar , and ended pages later with . Wallis never suggested that anyone but he had worked on it, but amongst Pell's 61 papers is a sheet that begins like this: July 2. 1636 Of the proportion of ye Periphery to ye Diameter of a circle Archimedes [about 210 years before Christ] determined it as 22+ to 7. Ludolph van Ceulen [A0 Christi 1599] determined it to be as 314159 26535 89793 to 1 at 15 cyphers. Lansberg [A0 Christi 16..] determined it to be as 314159 26535 89793 23846 26433 833 to 1 at 28 cyphers.

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Mr. Henry Briggs [A0 Christi 16..] determined it, to a radius of 1, at 40 cyphers. So that to seeke a greater proportion is meerely needlesse, it is better to seeke to bring it to some smaller one for common use.

1. Some doe by cutting off as many cyphers as they thinke Goode from ye end, so Mr Oughtred saise 3|1416 and Mr Gunter 3142 2. Some seeke ye ratio in some other Diameter, hoping by yt means to find it rationall, so determined it to be as 355 to 113. The missing name in the last sentence was that of Adriaen Anthonisz Metius, a Dutch engineer who discovered the ratio 355 to 113 in 1584. It was published by his son, Adriaen, in his Arithmeticae et geometriae practica of 1611,62 and it is a sign of Pell's wide reading that he knew of it. He went on to wonder how accurate the ratio was, and how Metius had found it: Concerning which we will enquire:

1. How farre this proportion of 355 to 113 will hold? 2. How he found this proportion in small numbers? Pell went on to confirm Metius's figure and to find further ratios of his own, for example repeated long division, essentially the Euclidean algorithm starting from a known value of π.

, by a process of

Now compare the way Wallis introduced the problem:63 The proportion of the Diameter to the Perimeter of a Circle, is by Archimedes shewed to be (very near) as 7 to 22, in small numbers; and nearer than so, in numbers which shall not be greater, it cannot be expressed. … And of later times, Van Culen, Snellius and others, have prosecuted the same to greater exactness, in large numbers, extending to six and thirty places or more.

Then Wallis went on: Amongst others Metius hath pursued the same Inquiry, and gives us the proportion of 113 to 355; which is nearer than that of Archimedes, but in Greater numbers; yet not vastly great like those of Van Culen, but convenient enough for use; and the nearest Proportion which can be assigned in Numbers not greater than such.

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I find some have been wondering by what means Metius came to light upon those Numbers …

That was just what Pell had wondered, in 1636. In fact Wallis's opening is really just an updated version of Pell's 1636 introduction. Furthermore, Wallis went on to tackle the problem by exactly the method that Pell had described then, essentially an application of the Euclidean algorithm, except that where Pell had used fifteen decimal places Wallis calculated from thirty-five. Wallis's introduction and method are so similar to Pell's as to leave little doubt that he was working from Pell's script, yet there is no mention of Pell anywhere. His account of how the problem arose instead introduces another person, Edward Davenant:64 [The work] was occasion'd by a Problem sent to me (as I remember) about the Year 1663, or 1664, by Dr Lamplugh the present Bishop of Exeter, from (his Wives Father) Dr Davenant, then one of the Prebends Residentiaries of the Church of Salisbury, a very worthy Person, of great Learning and Modesty, as I since understand from persons well acquainted with him, and by divers Writings of his which I have seen, though I never had the opportunity of being personally acquainted with him,…

According to Aubrey, Edward Davenant spent a great deal of time on mathematics.65 Aubrey himself had been taught by him and later wrote that Christopher Wren considered him ‘the best Mathematician in the world about 30 or 35+ years agoe’. Davenant was certainly interested in the problem of finding fractional equivalents for π, for Collins in 1676 associated his name with the problem, and Wallis suggested that he had found some but not all of the solutions.66 Wallis's eulogy to Davenant in Chapter 11, however, is strikingly similar to that to Titus in Chapter 60: Titus was ‘very well accomplished in Mathematical and other Learning’, while of Davenant, Wallis said that ‘amongst his other Learning, he was very well skilled in the Mathematicks, and a diligent Proficient therein’. I suggest that the short but effusive tribute to Davenant was written for the same purpose as that to Titus: to divert attention away from Pell to a relatively unknown third person. When Wallis wrote ‘I find some have been wondring by what means Metius came to light upon those Numbers’, he immediately followed it with ‘I guess that somewhat of this nature did first put Dr Davenant upon this inquiry’. We may substitute ‘Pell’ for ‘Davenant’ to

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obtain a version which, from the manuscript evidence, is rather closer to the truth. There is another example of Wallis hiding Pell in Chapter 56 of A treatise of algebra, where Wallis presented Descartes' rule for solving a quartic equation as a product of two quadratics.67 Descartes had given the rule in 1637 but, Wallis complained, ‘How he came by that Rule, he doth no where tell us, nor give us any Demonstration of it.’ To complete what he considered Descartes had left unfinished, Wallis proceeded to show how the factorization could be carried out: first remove the term in x3 (always possible by a simple linear transformation) and suppose that the quartic can be written (in modern notation) as:

Equating coefficients of x2, x1 and x0 then leads to equations for y, b and d in terms of p, q and r (and the equation for y is the same cubic equation that results from the Ferrari–Cardano method of 1545). Wallis later said that he had discovered this method of solving quartics in 1648 and had communicated it to his Cambridge contemporary John Smith,68 but unfortunately his correspondence with Smith has not survived, and there is some reason to doubt his claim. In his Treatise on angular sections begun in 1648, Wallis dealt only with special cases of quartics, those containing fourth powers, squares and constants (and therefore essentially quadratics). There is no evidence that he was yet familiar with the concept of factorizing polynomials, nor with any notation that would enable him to carry out such a process. Pell, on the other hand, had explored the factorization of quartics in 1646. He and Cavendish had discussed Descartes' method that year,69 and there is a sheet of solutions in Pell's handwriting amongst Cavendish's mathematical papers,70 written out forty years before Wallis published the same material in A treatise of algebra. That Pell was regarded as something of an authority on equations is confirmed elsewhere: in 1675 Collins passed on to him a letter from Dary with a query on just the subject under discussion here, that is, factorizing a biquadratic, or quartic, as a product of two quadratics. Pell made a copy of the letter and added, as he so often did, a note in red ink of how and when it was received:71 Mr Collins I have been lately Trying to break Biquadratique Aequations into two Quadratique ones and I have effected my purpose in a great many,

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some by the Aliquote parts and some by the Cubicall Maul, But this soure Crabb, I can not deale with by no method etc. Your sert

Mich: Dary

Tower the 8th Febr. 1674/5 The Aequation is this + yyyy + 8yyy − 24 yy + 104y − 676 = 0 William Lord Brereton gave a copy of this to Doctor Pell on Monday about Noone Febr. 22. 1674|75

Unfortunately we have no record of Pell's reply, but clearly Collins and Brereton thought he was the best person to ask about the matter. Wallis wrote in A treatise of algebra that not only had Descartes himself neglected to explain his rule but that ‘all his commentators have been so kind as not to give us any account of the grounds of it’.72 This was not quite accurate, for there was one algebra textbook that did explore Descartes' treatment of quartics, the Algebra ofte stel-konst of the Dutch writer Gerard Kinckhuysen, published in Haarlem in 1661 (Fig. 5.7).73 Kinckhuysen's approach was very similar to Pell's, raising the question of whether they ever discussed it together. It was not impossible: Kinckhuysen lived only a few miles from Amsterdam, and though he was only eighteen years old when Pell arrived in the Netherlands, he had already published a book on the use of quadrants (a subject on which Pell had also written in his own student days), and might well have come to know Pell.74 There is another striking similarity between Pell's algebra and Kinckhuysen's. We have already observed how Pell handled indeterminate equations by introducing arbitrary conditions, and Kinckhuysen too explained that in indeterminate problems, arbitrary values could be introduced at will, leading to innumerable solutions.75 Neither indeterminate equations nor quartics were usually treated in elementary algebra textbooks, so it seems more than coincidence that both Pell and Kinckhuysen treated both topics, and in such similar ways. There is one further clue that points to an association between them: we know that by 1648 Kinckhuysen was familiar with Book V of the Conics of Apollonius, which then existed only in manuscript,76 and we know that Pell was one of the few people who had access to such a manuscript. Thus, though the evidence is entirely circumstantial, it suggests quite strongly that Kinckhuysen and Pell worked together in Amsterdam sometime between 1643 and 1646. If there was indeed such a connection between Pell and Kinckhuysen it is significant, because Kinckhuysen's Algebra ofte stel-konst later came to be

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Fig. 5.7 Title page of Gerard Kinckhuysen's Algebra ofte stel-konst (Haarlem 1661).

of some importance in English mathematics. Between 1666 and 1668 there were discussions between Collins, Pell and Brancker about including some of the material from the Stel-konst in the forthcoming edition of An introduction to algebra, then in the press.77 For reasons unknown, this scheme came to nothing, but with the support of William Brouncker, President of the Royal Society, Collins arranged for Nicolaus Mercator to make a full Latin translation of the Stel-konst, and invited Isaac Newton to check it for

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mathematical errors or omissions; Newton did so, and eventually provided substantial annotations to the text.78 At about the same time, a second book by Kinckhuysen, his Geometria ofte meet-konst (1663), was also being translated into Latin, by someone whose name we do not know but who was described by Collins as a ‘German gunner’.79 What is less well known is that Brancker was meanwhile translating into English yet a third book by Kinckhuysen, De grondt der meetkonst (1660), a treatment of conics by the methods of analytical geometry. Pell knew about all three translations and sent information about them in a note to Henry Oldenburg, Secretary of the Royal Society, in 1672:80 Introductio Kinckhuyseni translated into Latin and enlarged by Mr Newton's notes, to serve as an introduction to his general method of Analyticall quadratures. As soon as Newton's papers about Analyticall quadrature and 20 Dioptic Lectures come to town, that also of Kinckhuysen will be printed. Kinckhuysen's last book, of Geometricall problems, was transcribed into Latin by a German gunner blown up in trying experiments of firework, but the translation in the hands of M. Bernard is fitted for the press. And Kinckhuysen's Anal … nicks are translated into English, and put into Dr Pell's method by Brancker the publisher of Rhonius his Algebra.

There is a hole in the paper in the final sentence as though something has been scratched out, but the partly erased words may be completed as ‘Analytick Conicks’, that is, the Grondt der meet-konst of 1660. A few days later Oldenburg turned Pell's note into better English and passed the information on to Huygens, but with some telling omissions:81 As for Kinckhuysen, his introduction has been translated into Latin and will be enlarged with Mr. Newton's notes, to serve as an introduction to his general method of analytic quadratures. When this arrives in London to be printed the aforesaid introduction of Kinckhuysen will also be printed. Moreover the last book by the said Kinckhuysen, on geometrical problems, has also been translated into Latin; this translation is at the moment in the hands of Mr. Bernard, Professor of Astronomy at Oxford, who is editing it for the press.

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The unhappy fate of the German gunner was left out but so, significantly, was any mention of Pell, or of Brancker's translation of Kinckhuysen's ‘Analytic conicks’. Clearly Brancker and Pell had done a good deal of work on the Grondt der meet-konst, and had not only translated it into English but rewritten it in Pell's three-column layout, or ‘method’. Why the translation was never printed is therefore something of a mystery. There are hints of it, though, and of problems surrounding it, in letters from Collins to Pell in the winter of 1667–68. In December 1667, Collins wrote to Pell that he had received ‘M. Brankers Translation of Kinckhuysen’,82 but in early February he wrote again:83 Let not Mr. Branker feare that I shall be instrumentall or assenting to the Printing of Kinchuysens Conicks without your and his leave.

Whether it was Brancker or Pell himself who prevented the printing of Kinckhuysen's ‘Conicks’, we do not know, nor for what reason, but the translation was never published, and joined the list of all the other works that Pell might have given the world but did not. Further research might reveal more about the nature and extent of Pell's relationship with Kinckhuysen, but it seems more than coincidence that Pell and Brancker were so closely involved with the translation not only of the text by Rahn, who was certainly Pell's pupil, but also of a text by Kinckhuysen who, it now seems, might well have been Pell's pupil also. There is one further section of Wallis's A treatise of algebra that contains work that can possibly be linked to Pell: four chapters on the geometrical representation of complex numbers.84 Wallis suggested that just as a real number, a, may be thought of as a geometric mean between two positive numbers b and c (that is, a = √bc), so an imaginary number may be considered as a geometric mean between a positive number b and a negative number – c (that is, a = √−bc), and thus the geometric construction of the former can be adapted to give a corresponding construction of the latter.85 This idea enabled Wallis to interpret roots of positive and negative squares as sines and tangents, respectively. In further constructions he showed that if real roots were represented by points on a circle, imaginary roots could be represented on a related hyperbola.86 There is nothing in Wallis's text that overtly links any of this with Pell, but in his final summary Wallis explained that the value of these constructions was to indicate just how far complex or ‘impossible’ solutions deviated from the real

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or ‘possible’:87 [The constructions] while declaring the case in Rigor to be impossible, shew the measure of the impossibility; which if removed, the case will become possible.

Compare these words with a paragraph from a letter written by Collins in 1674:88 These impossible roots, saith Dr. Pell, ought as well to be given in number as the negative and affirmative roots, their use being to shew how much the data must be mended to make the roots possible.

It seems, therefore, that both Wallis and Pell in the early 1670s were interested in what Wallis calls ‘the measure of the impossibility’ of an equation, and whether one could somehow adjust an ‘impossible’ equation to make it ‘possible’. Here again there seems to be evidence of greater collaboration between Pell and Wallis than meets the eye in the pages of A treatise of algebra. It is perhaps no coincidence that Wallis inserted his work on complex numbers immediately after his other references to Pell. In what may be a related matter, Pell repeatedly asserted in conversation that he had his own method for finding numerical solutions for polynomial equations, and claimed that Viète's method, by comparison, was ‘work unfit for a Christian, and more proper to one that can undertake to remove the Italian Alps into England’89 (a statement that perhaps put Collins in mind of the same metaphor in connection with Pell himself). Pell, however, could never be persuaded to explain his method. All we can surmise is that it involved the use of tables, and that it seems to have depended on systematically changing the parameters of the equation and noting the point where ‘possible’ roots became ‘impossible’ or vice versa (observe Collins' and Wallis's words above).90 Collins tried often and valiantly to describe the method to others (sometimes mentioning tables of sines, at other times logarithms),91 but just as often he despaired at Pell's refusal to communicate it for himself: ‘We have been fed with vain hopes from Dr. Pell about twenty or thirty years’, or, ‘Dr. Pell communicates nothing’.92 We do have one sheet on Pell's method, written in 1676, and enclosed in a letter from Wallis to Collins.93 Unfortunately the published version is incomplete and the original has been inaccessible for many years,94 but its existence is yet another indication of close communication between Pell and Wallis during the early to mid 1670s. It is by now apparent that substantial sections of A treatise of algebra were influenced by Pell, a feature of the book that has not previously

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been recognized. The chapters directly or indirectly concerned with Pell's mathematics, together with those on Harriot, which were also encouraged by Pell, constitute well over one-third of Wallis's book. This discovery of Pell's input raises a number of new questions, foremost of which is why was Pell's name was so conspicuously absent? The more cynical reader might think that Wallis hoped to pass off Pell's mathematics as his own, and Wallis was indeed censured by some for too readily publishing other people's work,95 but his motive in doing so was generally to ensure that others were properly acknowledged. Given what we know of Pell's difficult and uncommunicative nature it seems much more likely that it was Pell himself who insisted on discretion, and that Wallis simply colluded with his wishes. Then, we must ask, was it Wallis or Pell who suggested including so much of Pell's work? Or was it Collins who saw A treatise of algebra as an ideal opportunity to get some of Pell's mathematics finally into print? Collins hinted to Gregory in 1675 that Pell was ‘engaged to publish his papers’,96 and was likely to have known more of the matter than anyone else. Much of the material in A treatise of algebra is outlined in correspondence that passed between Wallis and Collins between 1673 and 1676, and it is very likely that Collins both encouraged Wallis in writing his book and made suggestions as to what he might include in it. The uncovering of so much of Pell's mathematics in A treatise of algebra inevitably leads to speculation about other ways in which Pell may have influenced Wallis. Was it, for instance, Pell who persuaded Wallis to write A treatise of algebra as a history, instead of the straightforward textbook that Wallis had previously intended? The historical framework suited the purposes of both: it allowed Wallis to extol Oughtred, Pell to promote the reputation of Harriot, and for both to include long examples of their own work. Was it also Pell who suggested his friend Vossius as such a useful source for the early chapters? Could it also have been Pell who persuaded Wallis to write in English? In 1646 Pell wrote a note about Mercator's Logarithmotechnia: he referred the mathematics in the book to Wallis, but criticized Mercator's use of Latin rather than English:97 In his title page, he says his Logarithmotechnia had beene communicated in writing in August 1667. He says not to whom. Not to Dr. Wallis, I beleeve. I desire to know what Hee saith of it. Howsoever (as I wrote before) I look yt some transmarine pen should fly at him. Englishmen, perhaps, will let him alone till he print the same crudities in English.

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We have also seen from the Rahn and Kinckhuysen translations that Pell chose English before Latin, and indeed ignored Collins‘ suggestion that An introduction to algebra should include a Latin preface ’to explain the symbols, and to signify that the greatest part of the book may be understood by others, ignorant of our tongue'.98 Perhaps Pell persuaded Wallis that his book too should be written in English. If so, it was not the most sensible of suggestions, for the entire text had later to be translated into Latin for inclusion in Wallis's collected works. Many questions about Wallis and Pell will probably remain forever unanswered. During the 1660s and early 1670s Pell was based in London or Cheshire, while Wallis lived in Oxford, so one would imagine that letters must have passed between them, but if so they have vanished without trace. Though we have scores of letters written or received by each of them, there is not one that links them directly. The relationship between Pell and Wallis, like that between Pell and Kinckhuysen, has remained largely unexplored, because the evidence is written not in English or Latin or Dutch, but only in a language a little harder to penetrate, mathematics. We are now perhaps in a stronger position to judge whether Pell's contemporary reputation for mathematics was justified. From his surviving papers and the work to be found in A treatise of algebra, he emerges as a competent but not a brilliant mathematician, one who worked on specific problems that arose in the course of his reading, or that caught his imagination, but who was not himself an innovator. He wrote no great work, held no post of any great importance, and no significant invention or discovery apart from the double-angle tangent formula and the division sign can be attributed to him. In his own eyes his most important achievement was probably his numerical method for solving equations, but he was so secretive about it that to this day no one knows exactly what it entailed. He also understood algebraic methods of solving equations (as learned from Viète, Harriot and Descartes), and his facility in algebra obviously impressed his contemporaries, but he applied his skills only within a relatively limited context, leading Collins to write in a letter already partly quoted at the beginning of this chapter:99 As to his knowledge, I take him to be a very learned man, more knowing in algebra, in some respects, (which I think I can guess at,) than any other, and they in other respects than he; but as to other parts of the mathematics, I grossly mistake if divers of them do not parasangis bene multis surpass him;

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Nevertheless, if the history of mathematics is to be not just about great names but about those who love, practise and teach the subject, then it must take account of men like Pell, who devoted most of his life to mathematics. Widely read, Pell probably knew more about mathematics than almost any of his contemporaries: he knew its history and he knew many of its practitioners, both in England and on the continent. And where he was perhaps unusual for his time was in his sense of a deep structural logic in mathematics, most clearly evident in his three-column ‘method’ which conveys a sense of precision and rigour that contrasts with the more descriptive and ad hoc styles of argument used by Wallis and others. Pell's mathematical vision went beyond the solution of individual problems to encompass mathematics itself, with the claim put forward in his Idea that it should be possible to derive all mathematical knowledge, past, present and future, by an ordered process of logical reasoning. Later, Hobbes in England and Descartes on the continent were similarly concerned with general deductive method, not only in mathematics but in philosophy and science.100 Pell also perhaps prefigured the later ideas of Leibniz who hoped to create an encyclopaedia of all knowledge and a scientia generalis by means of which all truth might be discovered.101 Most English mathematicians, however, troubled themselves little, if at all, with such things. Indeed, Collins feared that Pell's notions were no more than ‘improbable presumptions’.102 By the end of the seventeenth century questions about structure and methodology in mathematics were largely pushed aside by a flood of new results and ideas, and by the time mathematicians once again came to think about the logical underpinning of their subject, Pell's name was almost forgotten. It was a loss to Pell's contemporaries (as to us) that he so steadfastly refused to publish, but at the time it seems to have done his reputation no harm. Paradoxically, and unlike most other mathematicians, he may have achieved his standing by not publishing, for he gave no grounds for anyone to question his mathematical ability. Intelligent, widely read, and possessed of a quick, sharp wit, Pell managed to create an impression of erudition without ever allowing it to be subjected to public scrutiny. Only now has it become apparent that some of his work was published but behind such a veil of discretion that his name has remained hidden for over three centuries. Even Pell could hardly have asked for more.

6 Reading between the lines: John Wallis's Arithmetica innitorum Right at the beginning of his mathematical career John Wallis embarked on the work that was to be published in 1656 as the Arithmetica infinitorum (Fig. 6.1).1 The book was his masterpiece and over the following ten or twenty years was to have a profound influence on the course of English mathematics. It contains the result for which Wallis is now best remembered, his infinite fraction for 4/π, but to Wallis and his contemporaries this was not the book's only, or most remarkable, feature: its real importance lay in the new methods Wallis devised to solve age-old problems of quadrature and cubature. In the Arithmetica infinitorum Wallis took up the best new ideas from the first half of the seventeenth century, Cavalieri's indivisibles and Descartes' algebraic geometry, and applied new techniques of his own on the arithmetic of infinite series. In doing so he not only revolutionized the traditional approach to quadrature but ensured that the seventeenth-century trend from geometric to algebraic thinking became irreversible. The ideas in the Arithmetica infinitorum drew the attention of all the leading mathematicians of the day and in the hands of Newton in particular were to be of far-reaching significance. Wallis was well aware of the importance of his work and later devoted the final quarter of A treatise of algebra to describing the contents and implications of the Arithmetica infinitorum, as developed in the book itself and by Newton and others in the years following its publication.2 From a much longer perspective this chapter revisits the Arithmetica infinitorum and reviews its significance afresh.3

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Fig. 6.1 Title page of John Wallis's Arithmetica infinitorum (Oxford 1656).

Squaring the circle As Wallis himself was the first to acknowledge, the chief inspiration behind the Arithmetica infinitorum was Bonaventura Cavalieri's theory of indivisibles.

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Cavalieri had first published his ideas in his Geometria indivisibilibus continuorum in 1635, and in revised form in Exercitationes geometricae sex in 1647, both published in Bologna, but Wallis in 1650 had not read either: he later reported that he searched the bookshops in vain for a copy of the Geometria.4 Instead, like many other mathematicians, he learned of Cavalieri's ideas through Evangelista Torricelli's Opera geometrica published in Florence in 1644. Torricelli's version of the theory, however, was rather less sophisticated than Cavalieri's. Cavalieri had regarded plane (or solid) figures as being generated by a series of parallel lines (or planes) which he called ‘indivisibles’, and had constructed an elaborate theory to show how areas (or volumes) might be compared by comparing the generating lines (or planes).5 Torricelli offered a simpler and cruder version of the theory, in which a plane (or solid) figure was simply supposed equal to its collection of lines (or planes or surfaces).6 Treating plane or solid figures as the sum of an infinite number of infinitesimal parts was an essential idea in the later development of the calculus, but even in the 1640s it was recognized that Cavalieri had opened up a rich new seam of mathematics. Torricelli supposed that Cavalieri's methods were those by which the Greeks had found their results, and that by reintroducing them not only would ancient methods be made clear, but new results might be discovered (exactly the hopes Viete had for his ‘analytic art’). In 1645 William Oughtred in England wrote of7 … a geometrical-analytical art or practice found out by one Cavalieri, an Italian, of which about three years since I received information by a letter from Paris, wherein was praelibated only a small taste therof, yet so that I divine great enlargement of the bounds of the mathematical empire will ensue.

Wallis, like Torricelli, recognized that Cavalieri's ideas opened up a new way of exploring and developing ancient mathematics, and in particular of approaching the classical problem of squaring the circle. His crucial insight was to see how Cavalieri's ideas could be treated arithmetically. He began by writing out a few easy ratios:

and concluded:8 If there is taken a series of quantities in arithmetic proportion (as the natural sequence of numbers) continually increasing, starting

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from a point or 0, either finite or infinite (for there will be no reason to discriminate), its ratio to a series of the greatest taken the same number of times will be as 1 to 2.

Wallis slid over the problem of what the ‘greatest’ or ‘the same number of times’ might mean for an infinite series: he was satisfied that the ratio of ½ would hold for any number of terms. From this result, and the concept of a plane (or solid) as a sum of parallel lines (or planes), there followed immediately two geometrical corollaries:9 Proposition 3: Therefore, the area of a triangle to that of a parallelogram (on the same base and of equal height) is as 1 to 2. Proposition 4: In the same way, a pyramid or parabolic conoid, (whether vertical or inclined) to a prism or cylinder (on the same base and of equal height) is as 1 to 2.

Several further corollaries followed on the properties of the Archimedean spiral (in modern notation, r = aθ). Wallis was already successfully finding areas and volumes by applying simple arithmetic, and he saw that by summing not just arithmetic progressions, but sequences of squares, cubes and higher powers, he could repeat his success for a wide range of curves. Thus Cavalieri's geometry of indivisibles, or geometria indivisibilium, became for Wallis the arithmetic of infinite sums, the arithmetica infinitorum.10 To handle his infinite sums Wallis introduced a principle he called ‘induction’, not mathematical induction in its modern formal sense, but the argument that a pattern or procedure once established could be continued indefinitely. A few examples were enough, according to Wallis, to show that the same result would hold for all subsequent cases, and using his principle of induction, he went on to find results for sums of squares, cubes and higher powers. He introduced the word ‘index’ for the number denoting the power, though he never wrote his indices in modern superscript notation.11 From the well-known property np × nq = np+q for positive whole number indices, Wallis extended his results by analogy to both fractional and negative indices,12 and at every stage he produced geometrical corollaries to demonstrate the validity of his method. There are numerous examples of quadrature and cubature throughout the first half of the Arithmetica infinitorum, and eventually Wallis arrived (as Torricelli had before him) at plane or solid figures infinite in extent but finite in area or volume.13 He also took the first steps towards something previously considered impossible: rectifying, or finding the length of a parabola.14

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Much of this Wallis had hinted at in another book, also published in 1656, his De sectionibus conicis, the first systematic algebraic treatment of conics. The De sectionibus conicis begins, exactly as the Arithmetica infinitorum does, with a discussion of the method of indivisibles, and ends by promising results on areas and volumes bounded by curves, a promise taken up and developed in the Arithmetica infinitorum.15 In both De sectionibus conicis and the Arithmetica infinitorum Wallis was concerned with what he considered to be the essential properties of curves, properties of proportion and progression:16 This I have here shewed, (briefly and clearly,) by taking [conic sections] out of the Cone, and considering them abstractly as Figures in piano, without the embranglings of the Cone … And this Abstractio Mathematica is of great use in all kind of Mathematical considerations, whereby we separate what is the proper Subject of Inquiry, and upon which the Process proceeds, from the pertinences of the matter (accidental to it) … For which reason, whereas I find some others to affect Lines and Figures; I choose rather to demonstrate universally from the nature of Proportions, and regular Progressions; because such Arithmetical Demonstrations are more Abstract, and therefore more universally applicable to particular occasion. Which is one main design that I aimed at in this Arithmetic of Infinites.

In his search for general methods of quadrature Wallis had one particular objective in mind: the quadrature of the circle. At Proposition 121 he began in earnest, by considering a quadrant of radius R, divided into a large number of parallel strips of width a (Wallis was content to write R/∞ = a) (Fig. 6.2).17 The height of a strip at distance ka from the centre of the circle is , so to calculate the ratio of the quadrant to its circumscribing square Wallis needed to evaluate the following ratio (in modern notation):

For Wallis the expression (R2 − (ka)2)1/2 posed a difficulty since he had no method of expanding it algebraically. Undeterred, he put his code-breaking skills to work and turned to numerical interpolation. For integer values of p and q he could evaluate

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Fig. 6.2 Wallis's diagram from Proposition 121 of the Arithmetica infinitorum showing a quadrant divided into an infinite number of infinitesimal strips, or indivisibles.

To avoid fractions Wallis worked with reciprocals of these ratios and set them out in the table shown below (Fig. 6.3). The names of the qth powers run along the top of the table and of the pth roots down the left-hand margin. The first row, of oth roots, or repeated is, is strictly meaningless but was inserted by Wallis to keep the symmetry of the table. For the circle Wallis needed and , in other words he needed to interpolate an additional column between the oth and 1st powers (aequalia and residua) and an additional row between the oth and 1st roots (nulla and subprimanorum). He therefore expanded his table and introduced the symbol □ to represent the ratio he sought, that of a square to an inscribed quadrant, in modern notation 4/π (Fig. 6.4). By this stage Wallis had changed his row and column headings, for by now he had recognized the sequences of figurate numbers (the triangular numbers: 1, 3, 6, 10, …, pyramidal numbers: 1, 4, 10, 20, …, etc.). From this point on he dropped any reference to geometry and devoted his entire attention to filling the gaps in the table, and thus finding a value for □, by numerical interpolation alone.18 First he derived a set of formulae: l(l + 1)/2

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Fig. 6.3 Table from the Arithmetica infinitorum Proposition 132.

for the triangular numbers, l(l + 1)(l + 2)/6 for the pyramidal numbers, and so on.19 Such numbers had long been known in European arithmetic, always, by definition, as integers: arrangements of pebbles, spheres or points.20 Wallis himself had derived his formula for triangular numbers by considering an array of points on a triangular lattice, but now he took the unprecedented step of allowing l to take non-integer values, separating mathematics from physical reality and following where the mathematics led him. In modern terms he was fitting a continuous polynomial to a set of discrete points, never questioning that there was an underlying continuity that allowed him to do so. Now Wallis's table looked like the one shown in Fig. 6.5 (note his use of ∞ again to represent an infinite quantity). Next Wallis noted that alternate entries in the completed rows could be generated by a simple pattern of multiplication. For example, in the laterales row, alternate entries are successive terms of while 1, 2, 3, 4, … are successive terms of . In each case the multipliers form a sequence of fractions, the numerators and denominators of which increase in arithmetic progression: Wallis called such sequences hypergeometric. Wallis's choice of multiplication as his interpolative tool set the course for all that followed: in the end it was to limit his room for manoeuvre, but initially he used it with great success. For example, in the

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Fig. 6.4 Table from the Arithmetica infinitorum Proposition 169.

third row of the table (between monadici and laterales) the known terms are and he could fill in the unknown terms by analogy as . A particular leap of faith is required in the first row where the first multiplication is . but Wallis easily persuaded himself that such a step was permissible. So far had he travelled from his starting point that he had now dropped all row and column headings and was working only with numerical entries (Fig. 6.6). The pages that describe the final stages of his search are some of Wallis's most engaging writing, and convey with a humility and openness rarely found in his work his sense of frustration as he approached the solution only

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Fig. 6.5 Table from the Arithmetica infinitorum Proposition 184.

to see it slip from his grasp:21 Although no small hope seemed to shine, this slippery Proteus whom we have in hand, here as so often above has escaped, and disappointed hope.

The Greek god Proteus was a deity who had the gift of knowing the future, but who often refused to give answers and if left unfettered escaped by assuming new shapes;22 it would be hard to find a better metaphor for the elusiveness, in Wallis's treatment, of the number we now know as π. Finally, however, studying the third row of the table (Fig. 6.6) Wallis had a brilliant flash of insight and saw how alternate entries, those generated by the two sequences

and

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Fig. 6.6 Table from the Arithmetica infinitorum Proposition 189.

must eventually approach each other.23 Hence he was able to find sequences of upper and lower bounds, for example:

Wallis argued (in the first algebraic proof of its kind) that for large enough (integer) values of z the difference between the fractions and could be made less than any pre-assigned quantity and that the upper and lower bounds could therefore ultimately be regarded as equal.24 There are hints here of a modern limit argument, but for Wallis, as he later explained, his reasoning was firmly grounded in his reading

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of Euclid:25 And when … [Euclid] had occasion to compare Quantities, wherein it was not easy by direct Demonstration, to prove their Equality; he takes this for a Foundation of his Process in such Cases: that those Magnitudes (or Quantities,) whose Difference may be proved to be Less than any Assignable are equal. For if unequal, their Difference, how small soever, may be so Multiplied, as to become Greater than either of them: And if not so, then it is nothing.

Hence Wallis arrived at a fraction infinite in both the numerator and the denominator, which he wrote as:26

Wallis's infinite fraction is now regarded as his finest achievement.27 The Arithmetica infinitorum, however, did not end there. Wallis showed his work to Brouncker, who came up with a remarkable alternative form:

Brouncker's work was the subject of a short but mathematically rich Scholium that deserves attention in its own right and will be described in detail in the next chapter.

Praise and criticism The Arithmetica infinitorum ended with diagrams that demonstrated Wallis's interpolations geometrically, with smooth curves constructed between the points representing the numbers in his tables. As the book began to go to press in the spring of 1655, the most important diagram was printed off on a separate sheet dedicated to Oughtred, with a brief indication of what the rest of the book would contain (Fig. 6.7).28 This and further sheets were sent to Oughtred as the book went through the press, and Oughtred, over eighty years old, greeted the work with warm praise. In it he saw the first realization of the hopes he had held out twenty years earlier, and wrote a glowing letter

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Fig. 6.7 The ‘printed paper inscribed to Mr Oughtred’ published at Easter 1655 describing the contents of the forthcoming Arithmetica infinitorum.

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to Wallis:29 I have with unspeakable delight, so far as my necessary businesses, the infirmness of my healthe, and the greatness of my age (approaching now to an end) would permit, perused your most learned papers, of several choice arguments, which you sent me: wherein I do first with thankfulness acknowledge to God, the Father of lights, the great light he hath given you; and next I gratulate you, even with admiration, the clearness and perspicacity of your understanding and genius, who have not only gone, but also opened a way into these profoundest mysteries of art, unknown and not thought out by the ancients. … I saw [in Cavalieri's theorems] a light breaking out for the discovery of wonders to be revealed to mankind, in this last age of the world:30 which light I did salute afar off, and now at a nearer distance embrace in your prosperous beginnings.

Not everyone was as enthusiastic as Oughtred. Others reacted more cautiously, and sometimes with fundamental criticisms of Wallis's methods. The earliest known comments are to be found on a scrap of paper amongst the mathematical notes of John Pell,31 and it is clear from his remarks that Pell was responding not to the full text but to the single sheet dedicated to Oughtred, printed at Easter 1655: May 26. 1655 Sir. I thanke you for yours of April 27 with that printed paper inscribed to Mr Oughtred. If his great age have not made him unwilling to looke upon things of that nature, perhaps he will make some reply. When it comes to your hand, I pray you to send it to mee. As also if the Author expresse himself more fully heere-after. Artists will not trouble themselves to make an enquiry concerning the truth of his new Theorems, till they be sure of the sense of it. They may soone find out the mysterie of continuing his numbers as farre as they desire and so may perceive that his Graver hath set 360 for 630. But out of that paper and those schemes, no man will be able to find what he means by aequabilis curva. He makes mention of a Probleme proposed by him, to many mathematicians, some years ago. Perhaps yt problem joined with this printed paper, might help toward the finding of his meaning. I never saw that Problem, nor heard of it till now. But I should be glad to see it, especially if it have an intelligible Definition of aequabilis curva, in such a sense as he would have his Readers understand in his new Theorems.

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Wallis's curva aequabilis was what would now be described as a smooth or continuous curve, passing through ordinates 1, 6, 30, 140 and 630 at equal intervals from the origin, but Wallis had as yet given no method for calculating intermediate points. The problem of finding the correct intermediate values between 1, 6, 30, 140, 630,… or, as Wallis wrote it, , or , was posed by Wallis as a numerical problem in 1652 to his Oxford colleagues Seth Ward, Lawrence Rooke, Richard Rawlinson, Robert Wood and Christopher Wren.32 The reason that Pell ‘never saw that Problem nor heard of it till now’ was that he was in Breda, not Oxford or London, until August 1652. Christian Huygens received a copy of the Arithmetica infinitorum in the summer of 1656. In his letter of thanks he expressed some uncertainty about the validity of Wallis's fraction for □, and also about the method of induction which, he said, was neither clear nor certain enough to resolve his doubt.33 A further query, about Wallis's upper and lower bounds for □, contains an elementary misunderstanding that suggests that he had not read Wallis's argument very carefully. He also complained, as Pell had done, that Wallis's curves on the final pages were not properly determined. Wallis was able to reply that his infinite fraction agreed to at least the ninth decimal place with values calculated by traditional methods,34 and soon put Huygens right on his misreading of the text. He also claimed that induction was not a new method but had been used by Briggs in his rules for interpolating tables, by Viète in his work on angular sections and by Euclid in the Elements. Wallis was in fact using ‘induction’ in two senses, which he did not distinguish: first, that an established pattern could be continued indefinitely (as assumed on occasion by both Euclid and Viète);35 second, that particular cases could be used to demonstrate a general principle, so that a single equilateral triangle, for instance, might stand for all such triangles (as assumed by Euclid). If Huygens had lingering doubts he did not say so, and the discussion ended with his brief acknowledgement to Wallis in September 1656.36 In England, however, Wallis faced an opponent who was not so easily dealt with. Thomas Hobbes had already drawn fire from Ward and Wallis for his attack on the state of the English universities in his Elementorum philosophiae in 1655. In the wide-ranging quarrel between Hobbes and Wallis, arguments about the nature and scope of mathematics were one of the issues at stake,37 and in 1656 the Arithmetica infinitorum was caught in the crossfire of the first round. Hobbes' opening attack on the book was in his Six lessons to the professors of the mathematicks of 1656; Wallis's immediate reply was his

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Due correction for Mr Hobbes, or school discipline for not saying his lessons right. The titles are sufficient indication of the quarrelsome and personal tone of the exchange. In Six lessons, Hobbes upheld geometry as the foundation of mathematical reasoning, and, in striking contrast to Oughtred's view that symbolism ‘plainly presenteth to the eye the whole course and process of every operation’,38 Hobbes complained that ‘Symboles serve only to make men go faster about, as greater Winde to a Winde-mill’.39 In lesson five, he explained at more length:40 I shall also add that symboles though they shorten the writing, yet they do not make the reader understand it sooner than if it were written in words. For the conception of the lines and figures (without which a man learneth nothing) must proceed from words either spoken or thought upon. So that there is a double labour of the mind, one to reduce your symbols to words, which are also symbols, another to attend to the ideas which they signifie.

Hobbes lacked Wallis's technical facility in mathematics: when he gave the sequence 0, 1, 3, 5, 7, … as a counterexample to Wallis's rule for the sum of an arithmetic progression, it was easy for Wallis to point out that this was no arithmetic progression at all. Hobbes was on firmer ground, however, in his attacks on the fundamental methods of the Arithmetica infinitorum: induction and the use of indivisibles. Hobbes saw more clearly than Wallis did the potential pitfalls of using induction to make claims about infinite processes, and railed against ‘Egregious logicians and geometricians, that think an Induction without a Numeration of all the particulars sufficient to infer a Conclusion universall, and fit to be received for a Geometricall Demonstration’.41 Wallis merely restated his case that induction was a perfectly valid method ‘if after enumeration of some particulars comes the general clause “and the like in other cases”’,42 and argued as he had with Huygens that without induction none of Euclid's propositions could be considered proved, for it was impossible to demonstrate every individual case. In the use of indivisibles, Hobbes saw another fundamental problem, and insisted that indivisibles must be either ‘somewhat or nothing’. He particularly objected to the way Wallis treated an arbitrarily narrow strip quasi linea (as if it were a line):43 ‘The triangle consists as it were’ (‘as it were’ is no phrase of a geometrician) ‘of an infinite number of straight lines.’ Does it so? Then by your own doctrine, which is, that ‘lines have no breadth’, the altitude of your triangle consisteth of an infinite number of ‘no altitudes’, that

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is of an infinite number of nothings, and consequently the area of your triangle has no quantity. If you say that by the parallels you mean infinitely little parallelograms, you are never the better; for if infinitely little, either they are nothing, or if somewhat, yet seeing that no two sides of a triangle are parallel, those parallels cannot be parallelograms.

These were difficulties that neither Wallis nor anyone else was to resolve satisfactorily for a long time. Wallis's response, ‘I do not mean precisely a line but a parallelogram whose breadth is very small, viz an aliquot part [divisor] of the whole figures altitude’,44 contradicted his own assertions elsewhere that they could be infinite in number. The fact was that Wallis was not really concerned with such niceties: for him the justification of his methods was that they worked. The arguments went on through two further pamphlets, Hobbes' ΣITΓMAI and Wallis's The undoing of Mr Hobs's points, but to no further avail.45 Wallis's attitude towards Hobbes was scathing: ‘And tis to be hoped you may, in time, learn the language;’ he wrote, ‘for you be come to great A already’.46 Criticisms from a mathematician who had progressed far beyond the letter A were soon to follow. Pierre de Fermat, like Huygens, received a copy of the Arithmetica infinitorum in the summer of 1656, and opened up a correspondence with Wallis the following year. The correspondence (described in greater detail in the next chapter) centred on problems in what is now known as number theory, but at the same time Fermat made a number of negative comments about the Arithmetica infinitorum. He claimed in April 1657 that he himself had found some of Wallis's quadratures many years since, though Wallis, he added patronizingly, was no doubt unaware of it.47 Fermat was correct on both points: he had corresponded with Torricelli on the quadrature of the hyperbola (and other curves) in 1646, but Wallis could not have known his results as Fermat had written nothing publicly on the subject since 1636.48 Fermat added that he was not fully persuaded of Wallis's result on the quadrature of the circle. More detailed criticisms were to follow: in August 1657 he wrote again and made it clear that he, like Hobbes, much preferred classical geometrical methods to Wallis's new ways:49 It is not that I do not approve it, but all his propositions could be proved in the usual, regular Archimedean way in many fewer words than his book contains. I do not know why he has preferred this method with Algebraic notation to the older way which is both more convincing and more elegant, as I hope to make him see at my first leisure.

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In an appendix entitled Remarques sur l'Arithmetique des Infinis du S. J. Wallis Fermat added four specific charges:50 1. That Wallis hoped to find the ratio of a sphere to the circumscribing cylinder. But, argued Fermat, this was possible only if one knew first the ratio of the circle to the square, precisely what Wallis was trying to find. 2. That it was impossible to interpolate new numbers into the sequence 1, 6, 30, 140,… for if the sequence was written in Wallis's way as , or , then the number between 1 and 6 must be found using a multiplier greater than , which was absurd. 3. That Wallis's inductions were not a satisfactory method of proof. 4. That the result for the sum of an arithmetic progression was not restricted to those progressions where the difference was equal to 1. The most fundamental of these criticisms was the third, already made by both Huygens and Hobbes, that Wallis's method of induction was not sufficiently convincing. We have already seen that Wallis himself considered induction to be both reasonable and well-founded, and he had no difficulty in answering this and Fermat's other points to his own satisfaction in a long letter completed in November 1657.51 Wallis did little more than restate what he had already written in the Arithmetica infinitorum itself, and the details of the argument are now less important than the underlying tensions that gave rise to it. Fermat, like Hobbes, was deeply uneasy with Wallis's methods and for similar reasons: both mistrusted Wallis's use of induction and both disliked the new analytic, or algebraic, approach to what they saw as geometric problems. When the correspondence between Wallis and Fermat came to an end the following year, Fermat was still holding out for classical geometric methods:52 We advise that you would lay aside (for some time at least) the Notes, Symbols, or Analytick Species (now since Vieta's time, in frequent use,) in the construction and demonstration of Geometrick Problems, and perform them in such method as Euclide and Apollonius were wont to do; that the neatness and elegance of Construction and Demonstration, by them so much affected, do not by degrees grow into disuse.

Wallis argued, however, that his intention was not to abandon the traditional methods but to show how they might be improved and

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extended:53 To the elegance and neatness of the Ancients way of Construction and Demonstration, I am no Enemy. And that these Propositions might be so demonstrated, I was so far from being ignorant, that I had again and again affirmed it; but had shewed also the reason why I chose to go a shorter way … [Fermat] doth wholly mistake the design of that Treatise; which was not so much to shew a Method of Demonstrating things already known; (which the Method that he commends, doth chiefly aim at,) as to shew a way of Investigation or finding out of things yet unknown: (Which the Ancients did studiously conceal.)… And that therefore I rather deserved thanks, than blame, when I did not only prove to be true what I had found out; but shewed also, how I found it, and how others might (by those Methods) find the like.

The dispute between Wallis and Fermat, like that between Wallis and Hobbes, appears to reflect an uneasy transition from old paradigms to new, from the classical, geometric and synthetic to the modern, symbolic and analytic. In the case of Fermat, however, the matter was not quite so simple: for all his avowed disdain for ‘Analytick Species’, Fermat's mathematics was deeply rooted in the analytic methods of Viète and he repeatedly went beyond the classical foundations he ostensibly espoused.54 Fermat had discovered some of Wallis's results many years before, for he too knew the formulae for the triangular polyhedral numbers,55 and the results that Wallis later discovered for sums of powers, that is

for large N, and for Fermat, as for Wallis, such results had opened the door to quadrature.56 To Fermat, the Arithmetica infinitorum must have come as something of a shock, not so much for the difference of Wallis's approach as for the similarities of his achievement. Fermat could not argue against Wallis's conclusions but instead challenged him on his foundations. He was even provoked into bestowing a rare favour on the mathematical community by writing up his own results on quadrature (some obtained up to twenty years earlier) in a treatise almost certainly written in 1658 or 1659.57 Unfortunately it remained unpublished until 1679, fourteen years after Fermat's death, and by then work that had been ground-breaking in the 1630s was no longer new and never gained the appreciation or influence it had deserved.

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The enlargement of the mathematical empire The criticisms of Hobbes and Fermat failed to deter Wallis, or others, from building on the ideas of the Arithmetica infinitorum. Over the next ten years or so, new results on quadrature, rectification and infinite series began to proliferate, many of them directly inspired by Wallis's work. One of the earliest successes of Wallis's methods was in finding the rectification, or length, of a curve. In Proposition 38 of the Arithmetica infinitorum Wallis had outlined a possible method for the rectification of a parabola: suppose, using modern conventions, that the parabola has equation a2y = x2, and take points a, 2a, 3a, 4a,…along the x-axis. The corresponding ordinates will be 1, 4, 9, 16,… with differences 1, 3, 5, 7,… and so (by Pythagoras' theorem) the lengths of curve cut by the ordinates will be, said Wallis, approximately . The length of any portion of the curve can therefore be approached by summing such a sequence for sufficiently small a. Wallis could not handle the expansion of (a2 + n2)1/2 and did not return to the problem, but he had shown in principle how the (geometric) problem of rectification might be handled arithmetically. The first successful rectification was demonstrated in 1657 by William Neile (1637–1670), then a student at Wadham College, Oxford, for the curve whose equation in modern notation is 9y2 = 4kx3. Neile's method was based on the summation of indivisibles but his argument was purely geometric. He constructed, on the same axes, two curves, which may be denoted A and B, whose equations in modern notation would be (A): 9y2 = 4kx3 and (B): y2 = 1 + kx. By dividing the areas enclosed by the curves into arbitrarily thin strips, Neile could show that the length of (A) from x = o to x = a was proportional to the area contained by curve (B) between the same ordinates. That area was easily found since (B) was a parabola. Neile did not actually calculate any lengths but showed that in principle the calculations were possible. Wallis later claimed that he had foreseen the possibility of rectifying such a curve but that Neile had taken up his hints before he had time to pursue them himself.58 This was perhaps a case of being wise after the event: Wallis had indeed suggested a method of rectification and had listed some of the higher parabolas that might be amenable to it, but Neile's curve was not one of them, and in any case Neile's geometric argument was very different in style from Wallis's. Nevertheless, Wallis immediately saw how to use the methods of the Arithmetica infinitorum to re-write Neile's argument algebraically, and it was

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Wallis who named the curve the ‘semicubical parabola’. Wallis published Neile's method with full attribution, alongside his own modification of it, as part of a long letter to Huygens appended to his Tractatus de cycloide in 1659.59 The rectification of curves was an idea whose time had come. William Brouncker, like Wallis, re-worked Neile's method algebraically, while in the Netherlands Hendrick van Heuraet arrived independently at the same result. Van Heuraet's method was published in the same year as Neile's in Frans van Schooten's new edition of Descartes' La géométrie (provoking the inevitable priority dispute between England and the Netherlands).60 In that same year, 1659, Fermat too turned his attention to rectification and solved the problem for both the semicubical parabola and the cycloid. Fermat's treatise on the subject, De linearum curvarum, was printed in 1660, the only work he published in his lifetime.61 Wallis later claimed that all these methods sprang directly or indirectly from his own:62 I will not disparage Mons. Fermat's Invention herein, nor his Demonstrations therof. But allow the Invention to be very Ingenious, and his Demonstrations to be good and full … Nor will I impute it as a fault in him that others had done the same thing before him: Or that he had (or might have had) the first hints of it from the Arithmetick of Infinites, (which I am sure he had read.)… And I do not at all doubt that this notion there hinted, gave the occasion (not to Mr Neil only, but) to all those others (mediately or immediately,) who have since attempted such Rectification of Curves (nothing in that way having been attempted before;) and even to that of Mons. Hugens (which he thinks did give the occasion to Mons. Heurats invention)…

Wallis's remarks about Fermat can be seen as an echo and a riposte to the snub he himself had received from Fermat in 1657: ‘I have read the Arithmetica infinitorum of Wallis and have great regard for its author … who no doubt did not know that I had pre-empted his work’.63 But he was overestimating his own importance: Van Heuraet had discovered his rectification independently, while Fermat in Toulouse had apparently never heard of either Van Heuraet's result or Neile's, for the first sentence of his De linearum curvarum reads: ‘Never, so far as I know, have mathematicians compared straight lines to purely geometrical curved lines.’ Fermat was prompted to write his treatise not by Neile or Wallis, but by a reference in Pascal's Lettres de A. Dettonville to Wren's 1658 rectification of the cycloid64

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(which Fermat would not have regarded as a geometrical curve).65 In any case, Fermat approached the problem in the ‘regular Archimedean way’, a refutation rather than an endorsement of Wallis's new methods. The initial successes of Neile, Van Heuraet and Fermat appear to have been independent of each other and to have owed little or nothing to Wallis, but the Arithmetica infinitorum showed immediately how the same results could be achieved and written algebraically, and for that Wallis could rightly claim credit. Rectification, though significant, never became as important as the search for general methods of quadrature. Wallis in the Arithmetica infinitorum had made enormous advances and had provided numerous examples, but still there were curves he had not been able to handle. One of them was the rectangular hyperbola,66 with equation in modern notation y = k/x, since Wallis's rule for summing nth powers involved the ratio 1/(n + 1) and therefore broke down for n = −1. Nicolaus Mercator overcame the problem by what amounted to a change of axes so that the equation became y = 1/(1 +x). By long division Mercator obtained:

and then used Wallis's methods to sum the powers of x. It was already known that areas bounded by the hyperbola and the x-axis had the properties of logarithms,67 so Mercator was able to deduce the relationship:

Mercator's work was published in his Logarithmotechnia in 1668 and Wallis, possibly at the instigation of Pell (see p. 152), wrote an account of it for the Philosophical transactions later the same year.68 The publication of Mercator's results immediately prompted two other mathematicians to reveal their work. First, Brouncker published his quadrature of the hyperbola,69 by a method he had devised at least twelve years earlier (described more fully in Chapter 7). Second, Newton hurriedly composed a treatise entitled De analysi per aequationes numero terminorum infinitas, to demonstrate that he too had arrived at the same result as Mercator.70 In De analysi Newton, like Mercator, obtained the infinite series for (1 + x)−1 by long division, but four years earlier, after reading the Arithmetica infinitorum, he had arrived at it by a different method, and in doing so had discovered, at the age of twentytwo, the general binomial theorem.

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Fig. 6.8 Newton's discovery of the general binomial theorem following his reading of the Arithmetica infinitorum (Cambridge University Library Add. 3958.3:72r).

Newton's story is perhaps best introduced in his own words:71 In the winter between the years 1664 and 1665 upon reading Dr Wallis's Arithmetica infinitorum and trying to interpole his progressions for squaring the circle. I found out first an infinite series for

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squaring the circle and then another infinite series for squaring the Hyperbola …

Newton's manuscripts confirm his account, for his 1664 notes on the Arithmetica infinitorum are followed without a break by his own investigations into the quadrature of the circle and hyperbola.72 Almost immediately, however, there were significant differences between Newton's work and Wallis's: where Wallis had worked only with numerical ratios, Newton introduced an algebraic variable, x, so that for Newton the coefficients of each term xn were clearly visible, whereas for Wallis they had been absorbed into a single sum. Furthermore, where Wallis had applied his method only to the quadrant taken as a whole, Newton set himself the more difficult task of finding partial areas.73 This advance has been described by later commentators as ‘freeing the upper bound’ of the integral to give the more general , 74 but neither Wallis nor Newton would have understood such terminology, and it is perhaps more helpful to consider the change from a seventeenth-century perspective. The crucial step was Newton's introduction of a variable that allowed him to calculate absolute areas, instead of expressing areas in terms of ratios as every previous author had done.75 The importance of this can hardly be overestimated: Newton in one stroke liberated mathematics from the concepts of ratio and proportion that had pervaded all Greek and early European mathematical thinking and opened the way to perceiving areas (and associated logarithmic and trigonometrical quantities) as functions of a free variable. Newton did not abandon Wallis's foundations altogether, for he relied on Wallis's calculations to make his own interpolations between curves in the sequence y = (1 − x2)0, y = (1 − x2)1, y = (1 − x2)2, …, and so quickly arrived at the infinite series for the partial area of a quadrant:76

This already took Newton well beyond anything Wallis had done. Nevertheless, Newton retained Wallis's idea of describing each of the coefficients as a multiple of the preceding one, and wrote the sequence just as Wallis might have done as: It has been remarked that Newton's use of was ‘a typically Wallisian flourish’,77 but it was more than a flourish: Newton was following Wallis's precepts exactly.

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For the area under a hyperbola, Newton took a different sequence of curves, y = (1 + x)0, y = (1 + x)1, y = (1 + x)2, …. Wallis had interpolated fractional powers between integers but had not considered doing what Newton now did, that is, extrapolating ‘backwards’ from positive integers to negative. Using the rule that each entry in the following table is the sum of the one above and one to the left of it, it is easy to extend it backwards as well as forwards:

Thus Newton was easily able to find the appropriate coefficients for the area under the hyperbola y = (1 + x)− 1, giving him the series that Mercator was also to find:

Newton returned to this work in the autumn of 1665 but now found a new and more powerful method of interpolating the above table.78 Wallis, as we have seen, used multiplication to obtain each entry from a previous one, but Newton saw a simpler method based on addition. He noted that (starting from any column) the rows take the pattern:79

Newton saw that the pattern could be used to find intermediate entries and, as Wallis had done, he spaced the known entries to allow for new ones. Each new entry was marked by the symbol * which would eventually assume different values in each place; thus for the third row of the table he had

From the algebraic pattern established for the third row Newton then needed to solve80

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Consistent values of d, e and f are easily found to be d = 0, intermediate values denoted by each successive * as

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and f = ¼ and from these Newton could calculate the , exactly the values found by Wallis.

Newton, like Wallis, was essentially fitting a polynomial to the values in each row of the table, but now by a method of constant differences, a method he was to develop more explicitly in later years.81 Harriot had seen the constant difference properties of figurate numbers many years earlier, and following his lead both Warner and Pell had also taken up constant difference methods for interpolating tables. The pattern set out by Newton in 1665 is to be found in the unpublished manuscripts of all three,82 but it is unlikely that Newton at the age of twenty-two had seen them. The method was also well known to Briggs who wrote extensively on subtabulation in the introduction to his Arithmetica logarithmica in 1624.83 The evidence for Newton having read Briggs, however, is inconclusive, and the possibility must remain that Newton reinvented the method for himself.84 Newton's method enabled him to go far beyond Wallis for he could use it to find not just one but as many intermediate terms as he chose, and so to obtain the coefficients of x in the expansion of (1 + x)p/q for any (positive or negative) integers p and q. At this point Newton went back to Wallis for the last time. Once again he used Wallis's method of cumulative multiplication to write the coefficients as85

or, putting m = p/q, as

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Newton used the second form when he presented his result in the famous Epistola posterior to Oldenburg in October 1676.86 Harriot had written the identical formula more than half a century earlier for the case when m is an integer,87 but Newton's genius was to extend the formula to both negative and fractional indices. Wallis readily recognized that Newton had surpassed him and was the first to publish Newton's results, in some of the final chapters of A treatise of algebra.88 Newton fully acknowledged his debt to Wallis:89 like Wallis he saw that the quadrature of curves could be approached through numerical interpolation (and like Wallis he took for granted the underlying continuity on which such methods were based). Both Newton and Wallis searched the empty spaces in Wallis's table and emerged with rich treasures. Newton, however, went much further than Wallis in his use of algebra, and in doing so began to write mathematics in a new language. When Newton showed that an angle could be found from its sine from the series

or a number from its logarithm from the series

he was not just producing brilliant new results but changing the way mathematics was written and conceived. When Fermat in 1657 complained about ‘this method with Algebraic notation’ he sounded already as a voice from the past: only eight years later Newton read the Arithmetica infinitorum and took mathematics into a new and algebraic future. There is one further remarkable feature of the Arithmetica infinitorum that received little attention at the time, or since.90 Wallis knew that in his infinite fraction for 4/π he had found a number different in kind from any used or understood by mathematicians up to then. In an important Scholium sandwiched between Propositions 190 and 191 Wallis recognized that the numbers so far known—integers, rationals and surds—were inadequate for the quadrature of the circle, and that something other was required. Wallis argued that there were precedents for introducing new kinds of numbers to serve particular purposes: negative numbers to allow the (apparently impossible) subtraction of greater quantities from lesser; rationals to denote the (also apparently impossible) division of numbers by

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non-divisors, for example, ; surds to denote square roots of non-squares. Surds were of special interest to Wallis because they allowed the interpolation of a geometric mean between any two terms in a geometric progression. What Wallis needed in the Arithmetica infinitorum was a more difficult interpolation, between the terms of the hypergeometric progression where the numerators and denominators of the successive multipliers are themselves increasing. His unknown quantity, □, was the first such interpolation, between 1 and . Wallis recognized that such a number could not be found exactly, but neither, he pointed out, could a surd such as √18. In both cases, however, it was possible to approach the true value to any required degree of accuracy. Furthermore, it was clear that the new numbers could be handled by the usual operations of arithmetic. In short, Wallis was introducing not just a new kind of number, but a new concept of number which no longer depended on counting or measuring. According to Wallis, new numbers could, and indeed must, be introduced to allow the completion of any properly defined arithmetic process: that they could be evaluated as accurately as one chose and satisfied the usual laws of arithmetic was sufficient. In A treatise of algebra twenty years later, Wallis returned to this discussion and added some new thoughts on equations.91 He pointed out that the full solution of ‘ordinary’, or finite equations required negative, rational and surd numbers and those numbers ‘(commonly called) Imaginary’. The quadrature of the circle, however, forced mathematicians to numbers that could not be defined by any such equation:92 There must be some other way of Notation invented, (if we would express it in Numbers,) than either Negatives or Fractions; or (what are commonly called) Surd Roots, or the Roots of Ordinary Equations; or even the Imaginary Roots of such Impossible Equations in the ordinary forms;

Wallis's new numbers could only be defined by what he, following Newton, called ‘infinite equations’, or as we should now say, infinite series.93 Only forty years after Descartes described negative roots as ‘false’ Wallis was feeling his way towards the later distinction between algebraic numbers, those that satisfy ‘Ordinary Equations’, and those now called transcendental that cannot be so defined. For Wallis the necessary completeness of the laws of arithmetic not only allowed but demanded new kinds of number, and no modern mathematician could disagree. Wallis, the great interpolator, not

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only filled the spaces of his tables, but ventured further than anyone else in his time into the interstices of the number system itself. Wallis's methods, however controversial to begin with, were seen within twenty years to be justified by the discoveries of Newton and others. The doubts and criticisms of Fermat, Huygens and Hobbes were forgotten and the Arithmetica infinitorum came to be held in general approbation. In 1682 the astronomer Ismael Boulliau, a longstanding acquaintance of Wallis, published his Opus novum ad arithmeticam infinitorum in which he expanded at considerable length on the first part of the Arithmetica infinitorum (though he left untouched the more difficult later material) and Wallis noted that Boulliau found the work ‘sound and good’ (‘Only he thinks I have not done my invention so much honour as it doth deserve’).94 Five years after Wallis's death, David Gregory, Savilian Professor of Astronomy, wrote a summary of Wallis's life and work, and claimed that ‘the Arithmetica infinitorum has ever been acknowledged to be the foundation of all the Improvements that have been made in Geometry since that time’,95 an opinion that Wallis himself would certainly have endorsed. A fitting closing comment on the Arithmetica infinitorum is to be found almost two centuries later in Charles Babbage's unpublished essay ‘Of induction’ written in 1821.96 The concept of induction that had so troubled Wallis's contemporaries was by this time seen by Babbage as an essential feature of modern scientific thought: Few works afford so many examples of pure and unmixed induction as the Arithmetica infinitorum of Wallis and although more rigid methods of demonstration have been substituted by modern writers this most original production will never cease to be examined with attention by those who interest themselves in the history of analytical science or in examining those trains of thought which have contributed to its perfection.

7 Catching Proteus: the mathematics of William Brouncker William, Viscount Brouncker (c. 1620–1684) was once described by Sir Kenelm Digby as one of ‘the greatest mathematicians of the age’ (Fig. 7.1).1 He was chosen by Charles II as the inaugural President of the Royal Society, and held the post unopposed for fifteen years from 1662 to 1677. Today his name is not generally familiar, and it is not easy to understand why, among so many eminent and gifted colleagues, Brouncker was selected for such a prestigious position. His appointment may have been in part a reward for his political and personal loyalty to the King, but the fact that he was accepted and esteemed for so long by so many distinguished early Fellows for ‘his great abilities in all Natural and especially Mathematical knowledge’ suggests that history has underrated him as a mathematician.2 Brouncker's finest mathematical work was done in the 1650s, before the period of his Presidency, all of it in association with John Wallis. In many ways the partnership between Wallis and Brouncker was a curious one: Brouncker was a member of the aristocracy and a staunch Royalist, Wallis a puritan clergyman loyal to Parliament and the Commonwealth, and their collaboration is an example of the new determination during the 1650s to set mathematical and scientific progress beyond political and religious divisions. Wallis and Brouncker differed not only socially and politically but also in the way they handled mathematics. A study of their joint work provides fascinating insights into different forms of mathematical creativity and contrasting mathematical styles, and quickly puts paid to any notion that mathematics is a purely logical and impersonal subject. Brouncker was born about 1620 (the exact date is unknown) and is usually said to have entered Oxford at the age of sixteen but there is no evidence

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Fig. 7.1 William Brouncker (c. 1620–1684) by Sir Peter Lely. The diagram in the portrait appears to be Brouncker's rectification of the semicubical parabola. © The Royal Society.

that this was the case, and he told Aubrey that he was ‘of no university’.3 His father, Sir William Brouncker, was made viscount of Castle Lyons in Ireland in September 1645 but died only two months later, so that his son inherited the title at the age of twenty-five. Brouncker spent the civil war years in Oxford and in 1647 his intellectual prestige and loyalty to the King were rewarded with the degree of ‘Doctor of Physick’. Brouncker had neither completed the statutory fourteen years of study for medicine, nor ever practised it afterwards, but it was not uncommon at the time for such honours to be awarded for political service.4 According to Aubrey, ‘he addicted himselfe only to the study of Mathematicks, and was a very great artist in that learning’.5

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Wallis and Brouncker would first have met after the war, almost certainly through the scientific and mathematical meetings held at Gresham College. Wallis, never reluctant to promote his own interests, probably welcomed the social connection with Brouncker, and Aubrey, who disliked Wallis intensely, certainly ascribed such motives to him when he described him as: ‘a most illnatured man, an egregious lyer and backbiter, a flatterer and fawner on my Lord Brouncker and his Miss, that my Lord may keepe up his reputation’.6 It is possible that in addition to mathematics Wallis and Brouncker shared an interest in music: Brouncker translated and published Descartes' Musicae compendiae with his own commentary in 1653, and Wallis in his later years also wrote on music and edited and published Greek texts on harmony.7 As Wallis was resident in Oxford, their meetings can only have been occasional and in the later 1650s some of their discussions were carried on through letters that now give us a valuable insight into their working relationship.

Squaring the hyperbola and the circle The earliest known piece of mathematics by Brouncker is his quadrature of the hyperbola.8 Though not published until 1668, Wallis referred to it in 1657 in the dedication (to Brouncker) of his Adversus Meibomii,9 and Wallis's reference suggests that Brouncker's method was devised in connection with their joint work on the quadrature of the circle, and therefore no later than 1655. Wallis in 1657 hoped that Brouncker would soon present his discovery to the world, but Brouncker seems to have had no interest in doing so, and only when another method of quadrature (Mercator's) was published in 1668 was Brouncker finally persuaded, almost certainly by Wallis, to bring out his own results. Brouncker's brief paper in the Philosophical transactions gives us a glimpse of the skills he had already developed by the mid 1650s. His method of covering an area bounded by the hyperbola is best followed from his own diagram from his 1668 paper (Fig. 7.2). It is perhaps helpful to use modern conventions and think of A as the origin, AB as the x-axis with B at (1, 0) and AE as the y-axis with E at (0, 1), so that the equation of the curve EC is

The area sought is ABCE and Brouncker devised an elegant method of covering it with rectangles in what would now be recognized as a fractal pattern.

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Fig. 7.2 Brouncker's diagram for the quadrature of the hyperbola, Fig. II. and III. from his paper in the Philosophical transactions in 1668.

The area of rectangle AC is and Brouncker proceeded to form further rectangles by halving the intervals on the x-axis. Taking the point , he obtained rectangle dF with area , Halving again gave points and , and corresponding rectangles bn and fk, with areas and , respectively. Halving further gave points a, c, e, g and rectangles ap, cm, el, gh with areas , and so on.10 Furthermore, Brouncker attempted to prove, by appropriate grouping of terms, that the sum of this sequence was convergent, an attempt that has been described as ‘more soundly based than any later seventeenth-century convergence investigation’.11 Brouncker's quadrature of the hyperbola is a pleasing piece of mathematics, but Wallis's work on the quadrature of the circle was to move Brouncker

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to further and greater invention. We have seen that Wallis described his unknown quantity □ (or 4/π) as being as slippery as Proteus, but by persistent effort, he eventually succeeded in pinning it down as

At this point Wallis showed his work to Brouncker who came up with an entirely different kind of fraction:12

Wallis, in a section headed Idem aliter (‘The same another way’), described this new form as fractio, quae denominatorem habeat continue fractum (‘a fraction which has a denominator continually broken’), the first description of what has come to be known as a continued fraction. There was also an unexplained but correct remark that the sequence(1)

gave increasingly good approximations, alternately too large and too small. These were astonishing results. Wallis had led his readers through every twist and turn of his thinking on his way to finding his own infinite fraction for □, but Brouncker's new form came quite out of the blue. Nothing like it, so far as we know, had been seen in English mathematics up to that point, and certainly not in the Arithmetica infinitorum. Wallis realized that some explanation was needed but failed to persuade Brouncker to provide it and so attempted it himself.13 Unfortunately Wallis's efforts to reconstruct Brouncker's argument do not take us very far, but the beginning was clear: Brouncker started, said Wallis, from the identities

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Next, according to Wallis, Brouncker asked:14 by what fraction the factors should be increased to give products not equal to those squares reduced by 1, but to the squares themselves.

In other words Brouncker was looking for numbers that may be denoted A, B, C,…, a little larger than 1, 3, 5,…, with the property(2)

This was all the help Wallis could give, and from here he went straight to the final outcome without offering any further clue to the intermediate steps:15 [Brouncker] found that this is possible, if each factor is increased by a fraction with the denominator infinitely broken, in the form we have shown above.

That is to say:(3)

Wallis had explained what Brouncker had set out to do, but not why; what he had achieved but not how. There was not even any proof that Brouncker's first fraction, A, was indeed equivalent to □.16 All Wallis could add was a demonstration that successive partial values of Brouncker's fractions multiplied together did in fact approach the required squares. This demonstration seems to have been Wallis's attempt to provide some footing where he was uncomfortably out of his depth, and it is given here to show how he handled such things (note his use of Oughtred's notation Fq, Fc, etc.): Let the first number of any fraction be F, the next is F+2. The number between (to be squared) is F + 1. The product Fq + 2F is less than its square Fq + 2F + 1. Now one fraction is adjoined to each factor.

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Then two fractions are adjoined; the resulting fractions

It is perhaps not surprising that at this point Wallis felt he had gone far enough and he concluded with the comment Et sic quousque procedatur (‘And so it may be continued as far as you like’). Wallis never returned to the problem of how Brouncker arrived at his result,17 but succeeding generations of mathematicians have continued to be intrigued by Brouncker's fraction for □, and how it might have been derived from Wallis's. Euler saw that the reciprocal of Brouncker's fraction has the same partial convergents as

a series found by Leibniz in 1682,18 but the equivalence between Brouncker's fraction and Wallis's is much harder to prove and no formal demonstration was given until 1872, by the German mathematician Bauer, using the theory of determinants.19 Further proofs followed in the twentieth century, but all relied on modern notation and on concepts (especially of functions) that were eighty years or more into the future when Brouncker was at work.20 In 1655, even general algebraic expressions were still rare, and the examples in the Arithmetica infinitorum were among the first (Wallis's was one of them). Brouncker had no special techniques at his disposal, nor so far as is known, any previous work to serve as a model: like

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Wallis in the Arithmetica infinitorum, he had to rely only on his skills in arithmetic, some elementary algebra and his mathematical intuition. Any attempt to rediscover Brouncker's method inevitably involves a certain amount of guesswork, but it is perhaps instructive to make such an attempt within the confines of mid seventeenth-century notation and technique.21 What follows is my own suggestion as to how Brouncker might have proceeded. Starting, as Wallis said Brouncker did, from(4)

it is clear that each factor on the right must be increased, and it easy to find an upper bound for the first factor of each pair:(5)

Brouncker needed, however, to maintain the pattern set out in (2) where the second factor of 22 is the first factor of 42, and so on. A well-known approximation to √ (n2 + 1) is , so a reasonable starting point would have been(6)

This is already quite a good approximation, especially as the terms increase. Each product is a little too large,22 but it is possible to find an exact product

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(and a lower bound for the first factor) by increasing the denominator of the first fraction:(7)

These equations yield

and so on, that is:(8)

Brouncker still needed to maintain the diagonal symmetry of (2), and could do so by always keeping the final denominator twice the leading integer as in (6):(9)

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These pairs are a better approximation than (6) but are now slightly too small; once again an exact product (and an upper bound for the first factor) can be found by increasing the final denominator:(10)

yielding

and so on.

Manipulation beyond this level rapidly becomes unwieldy, but by now Brouncker would have had his pattern and could, like Wallis, assume an et sic quousque procedatur. This approach explains not only the natural emergence of continued fractions through the refinements made at (7), (10),…, but also the little noticed sequence of approximations alternately too large and too small at (1), since those approximations are simply the upper and lower bounds for the first factor from successive calculations (4), (5), (8), (10). …23 The simplicity of the procedure should not lead us to underestimate Brouncker's genius in devising this or something like it. Apart from anything else it is remarkable that he regarded infinite ‘continually broken’ fractions as acceptable mathematical entities. Such fractions had appeared only once before in print, in Cataldi's Trattato del modo brevissimo published in Bologna in 1613,24 but it seems very unlikely that Wallis or Brouncker in 1655 had ever seen the book,25 in which case Brouncker was inventing this strange new form for himself. Ironically, Brouncker's very success in re-writing Wallis's fraction has done much to obscure the full subtlety of his mathematics. Later commentators, including Wallis in A treatise of algebra, have concentrated only on the first of Brouncker's fractions, and in doing so have limited perceptions of his achievement to that alone.26 But as the above reconstruction or any similar attempt shows, it is impossible to arrive at Brouncker's fraction A without bringing the entire infinite sequence (3) of fractions B, C, D, E, … in its train.

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The sequence in fact immediately provides the solution to the problem that so perplexed later mathematicians, making it easy, as Wallis and Brouncker would have realized, to prove the equivalence of A and □:

Furthermore, it was precisely the sequence as a whole that Wallis needed, for it enabled him to write each row of his final table (Fig. 6.6) as a single sequence, thus:27

The use of Brouncker's fractions to complete Wallis's work fills the final pages of the Arithmetica infinitorum,28 but has passed almost unnoticed because readers of Wallis, like commentators on Brouncker, have arrived at the fraction for □ and failed to look any further.29 I believe, however, that the search for this completion of Wallis's interpolative process was Brouncker's true starting point. Certainly Brouncker had a very clear understanding of what Wallis was trying to do, as Wallis admitted when he ended the long Idem aliter with the words:30 ‘Up to here I have set out his Lordship's thinking with what brevity and clarity I could.’ Brouncker's fraction for □ was only the first of the sequence of fractions that emerged from his study of Wallis's progressions, and to credit him with that one alone is to miss the full significance of what he achieved. Wallis's and Brouncker's fractions for □ both converge too slowly to be of much use for calculating π. The later fractions in Brouncker's sequence,

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however, converge much faster, and all are rational multiples of π or its reciprocal:

Brouncker almost certainly made use of one of the later fractions to calculate bounds for π as31

in agreement with the values found by Van Ceulen using the traditional method of inscribed and circumscribed polygons.32 Once again the simplicity of Brouncker's result belies the imagination and effort that went into achieving it, for to calculate to this degree of accuracy, Brouncker must have discovered and used the method of evaluating continued fractions ‘from the top down’. If a continued fraction is denoted then the rth partial convergent is found from33

This pair of recurrence relations is given towards the end of the Idem aliter,34 using subscript notation: N1, N2, N3 and D1, D2, D3 for successive numerators and denominators, the first use of a notation that immediately rendered the algebraic handling of lengthy or infinite sequences more manageable.35 The next piece of work we have from Brouncker, though brief, again demonstrates his mathematical ease and competence. Brouncker and Wallis both paid close attention to William Neile's rectification of the semicubical parabola in 1659, and both reworked it for themselves in their preferred styles.

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Fig. 7.3 Brouncker's rectification of the semicubical parabola as published by Wallis in his Tractatus de cycloide et … de cissoide (Oxford 1659).

Wallis published all three methods, Neile's, Brouncker's and his own, in his De cycloide in 1659.36 Wallis's method retained the geometric and descriptive flavour of Neile's original but used some of the algebraic shorthand he had introduced in De sectionibus conicis, for example D for diameter, L for latus rectum. Brouncker went much further in converting Neile's argument entirely into algebra (Fig. 7.3).37 Retaining the concept of proportion he wrote the rule for the length of the curve 9y2 = 4kx3 from (o, o) to (a, c) as a to ‘length’ is as 27ac2 to (4a2 + 9c2) × √(4a2 + 9c2) minus 8a3

Modernizing the notation, this gives the absolute value of the length correctly as

Brouncker was indebted to Neile for the underlying method, but the transformation into algebra was very much his own.

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The challenges from Fermat The publication of Wallis's Arithmetica infinitorum in 1656 gave rise indirectly to Brouncker's last major piece of mathematical invention. In the summer of 1656 Fermat received a copy of the book from Kenelm Digby, then based in Paris, and was prompted to write to ‘Wallis and other English mathematicians’ with some of the number problems that had engaged him for many years. All the correspondence between England and France passed through Digby who used the services of his friend and collaborator Thomas White for the journeys between England and France, but long delays (as much as two months) meant that events often moved faster than the communication of them, and it was not always clear to the participants themselves (nor to the modern reader) what had or had not been done or understood elsewhere. Michael Mahoney has given an excellent account of Fermat's side of the story,38 and the following account attempts to untangle the sequence of events on the English side and to distinguish more carefully than has sometimes been done between the respective contributions of Wallis and Brouncker.39 Fermat had devoted much of his life to his researches into what is now known as number theory, but had so far failed to interest any of his fellow French mathematicians. Pascal, as recently as 1654, had rejected Fermat's offerings with the words: ‘I confess to you that they go right past me; I am capable only of admiring them and of begging you very humbly to take the first opportunity to complete them’.40 On reading the Arithmetica infinitorum in 1656 Fermat must have supposed that Wallis might be more interested in his ideas than Pascal was but, ever reluctant to reveal just how much he knew, he put his questions in the form of a public challenge from Narbonese (southern) France to Celtic (northern) France, England and the Netherlands. Fermat's target in northern France was Bernard Frenicle de Bessy, with whom he had corresponded for many years over such problems.41 The version sent to England in January 1657 was directed especially to Wallis:42 If it please them, let the following numerical problem be proposed to Wallis and the other English mathematicians. To find a cube which, added to all its aliquot parts [divisors] makes a square. For example the number 343 is a cube of side 7. All its aliquot parts are 1, 7, 49, which, added to 343, make the number 400, which is a square of side 20. Sought is another cube of the same kind.43

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Also sought is a square number which, added to all its aliquot parts, makes a cube number. We await the solutions which, if England or Belgian and Celtic Gaul cannot give them, Narbonian Gaul will give and will offer and speak as a pledge of growing friendship to Mr Digby.

Brouncker received the challenge in March 1657 and passed it on to Wallis who was dismissive:44 The question is almost of the same kind as those problems usually posed about numbers known as Perfect, Deficient or Abundant, or others of that kind, which with difficulty or not at all can be reduced to some general equation that takes account of all cases. Moreover, whatever the case, it finds me extremely busy with various affairs that do not allow me to attend to it immediately. But for the present he may at least have this response: one and the same number, 1, satisfies either question.45

To a mathematician of Fermat's standing, Wallis's response was an insult. The problems may have appeared to be little more than simple number puzzles, but Fermat's real interest was in the equations that arose from them,46 equations of the form:(11)

Perhaps Fermat realized that he needed to be more explicit, for in February 1657 he issued a second challenge:47 Given any non-square number, there are given infinitely many squares which, multiplied by the given number and added to unity, make a square. Example: 3 is given, a non-square number; 3 multiplied by the square 1 and added to unity makes 4, which is a square. Again, the same 3, multiplied by the square 16 and added to unity, makes 49, which is a square. And, in place of 1 and 16, one can find infinitely many squares with the same property; we seek, however, the general canon, given any non-square numbers. What, for example, is the square which, multiplied by 149, or 109, or 433, etc. and added to unity, makes a square?

The second challenge was not even passed on to Wallis, but Brouncker responded to both. His initial solution to the first challenge was hardly better

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than Wallis's. He simply divided Fermat's solution by 82 to give:48

There are infinitely many solutions of this type if one allows that are ‘aliquot parts’ of , but this was a questionable assumption, later contested bitterly by Frenicle. Fermat, it may be noted, had not actually specified that the solutions should be integers: to him it was probably too obvious to need stating, though in fact this was the first time that problems requiring only integer solutions had been posed.49 Brouncker's solution to the second challenge was also in rational numbers. The identity

immediately gave him(12)

Brouncker, treading unfamiliar ground and unaware of Fermat's own solutions, must have felt that he had dealt with the problem satisfactorily. When Fermat received Brouncker's response, he found an Englishman to translate it, but the young man could not understand mathematics and Fermat could not tell whether Brouncker had really solved the problem or not. The fact that Brouncker appeared to have found it easy made Fermat suspect that he had not.50 Wallis, despite his expressed distaste for such problems, was finally drawn in by Fermat's next move. In April 1657, and again in August that year, Fermat voiced his criticisms of the Arithmetica infinitorum.51 Wallis had no option but to reply, and thus the number challenges became part of a wider exchange. As the correspondence progressed, Fermat's tone changed from condescension to outright provocation:52 I venture to say to you [Digby], with respect and without diminishing in the slightest the high opinion I have of your nation, that the two letters of Milord Brouncker, however obscure and badly translated, contain no solution at all. It is not that I mean thereby to renew the jousts and ancient tilting of lances which the English once carried out against the French.…

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This was Fermat at his most disingenuous. For all his protestations to the contrary he was setting his challenges firmly into a context of national rivalry, and was raising the stakes in a way the English could not ignore. To keep the fires burning he went on to add another problem, clearly aimed at Wallis and Brouncker:53 It is proposed to split a cube number into two cubes. In the same way, to split a given number composed of two cubes into two other rational cubes. And, it may be asked, what England, and what Holland, think of this matter?

As late as September 1657, before this letter had yet arrived, Wallis had reiterated to Digby his opinion that Fermat's number problems had ‘more in them of labour than either Use or Difficulty’.54 He still had not even seen the second challenge, but was sure that Brouncker had answered it satisfactorily: ‘I know his Lordship so well, and his peculiar dexterity in things of that nature; that I have a very strong presumption of the accurateness of what he doth in such a way.’ A few days later Brouncker sent him a copy of the second challenge and of his own solution to it, and said that Wallis could, if he wished, send the solution to Digby in Latin so that Fermat could no longer complain about the use of English.55 Wallis obliged: having so far contributed nothing to the work he now took over the correspondence and sent a formal letter to Digby summarizing all the results so far.56 Fermat's letters rejecting Brouncker's efforts and criticizing the Arithmetica infinitorum finally reached London in early October and were immediately sent on to Wallis in Oxford.57 Brouncker realized that he had both misunderstood and trivialized Fermat's challenge and now worked intensively to find integer solutions to Na2 + 1 = l2. In less than three weeks he was able to send Wallis a long list of solutions.58 For N = 2 he found:(13)

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Brouncker wrote each of the solutions 2, 12, 70, 408,… as a multiple of the previous one:

and so, using ‘Q’ (quadratus) to indicate squaring, wrote the list of solutions (13) in concise form as

Wallis and Brouncker had used such multiplicative sequences extensively in the Arithmetica infinitorum so Wallis would have understood Brouncker's notation perfectly. There were further sequences of the same kind for N = 3, N = 5, and other values:

Just as with his sequence of continued fractions, Brouncker had come up with brilliant results but with no indication of how he had found them, and Wallis once again had to advise him that it would be expedient to provide an explanation of his method.59 This time Brouncker was more forthcoming and sent what is now the longest surviving example of his work, and at the same time evidently asked Wallis for suggestions as to how the method might be shortened.60 Interested for the first time, Wallis at last gave the second challenge serious attention. His and Brouncker's different approaches to the same problem demonstrate vividly their contrasting mathematical styles, and so the outlines of each are given here in some detail.61 Brouncker had already noted that putting R = r/s in his original solution (12) led to:62

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Thus, the integer solutions required by Fermat entailed finding pairs of numbers r and s such that |Nr2 − s2| divides 2rs. Wallis sought such pairs by working his way systematically and somewhat laboriously through the possible ways of writing Nr2 (or, as he put it, NQr) in the form s2 − 2u (or Qs − 2u) in the hope that he would see where 2u divided 2rs. Here is his first example, for N = 7:

Wallis noted the pattern in the final non-square term, 2u, and continued:

Once again the final terms increased regularly, and from this Wallis was able to predict where he would find the pairs he needed. After a while, realizing the need for generality, he translated his results into algebra, but his initial approach was entirely numerical, an exercise in pure pattern-spotting. By finding successive solutions for particular values of N, Wallis was eventually able to come up with a general result: given any non-square N and first solution integer r, let t = 2√(Nr2 + 1). Successive solutions are then given by(15)

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Setting N = 3 and r = 1, for example, gave the sequence 1, 4, 15, 56, 209,… exactly as obtained by Brouncker. Wallis's work took several pages to write out, and one cannot but admire his willingness to engage in repeated calculations without, at first, seeing where they would lead: here, as so often in Wallis's mathematics, we glimpse the mind of a code-breaker at work. Now compare this with Brouncker's brief explanation of his method, sent at Wallis's request. Brouncker illustrated it with N = 13, so he needed 13aa + 1 to be a square. Since 9aa < 13aa < 16aa he could assume that the required solution was of the form 3a + b (with b < a). Hence he had:

Putting a = 2b would make the left-hand side too large so now Brouncker had a double inequality, b < a < 2b, and he could put a = b + c (with c < b). Hence (after simplifying):

Now he needed c < b < 2c so he could put b = c + d (with d < c) and hence:

Continuing in this way he eventually reached

which has integer solutions h = 2j, j = 1. Substituting back up the chain Brouncker arrived at b = 109, a = 180, and the solution:

Brouncker's process was essentially the Euclidean algorithm for the greatest common divisor of 649 and 180, which guaranteed that it was finite.63 He would almost certainly have known Euclid's algorithm but made no mention of the similarity between Euclid's method and his own.64

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It is immediately clear that Brouncker's approach was very different from Wallis's. Where Wallis began with repeated numerical calculations, Brouncker's method was algebraic from the start. Wallis introduced algebraic notation only when he really needed to and retained much of Oughtred's outdated symbolism whereas Brouncker was fluent in the newer Cartesian notation. Wallis worked from the particular to the general, while Brouncker assumed the form of the solution at the outset. In Classical terms, Wallis's approach was a perfect example of synthesis, Brouncker's of analysis. The power of Brouncker's method was not appreciated, so far as we know, by any of his contemporaries beyond Wallis and Fermat.65 Wallis noted the important repeating patterns that emerge if the process is continued far enough, suggested some short cuts and refinements and was able to find the general form noted at (15).66 Otherwise the method was to remain undeveloped until taken up by Euler, seventy years later. Ironically it was Euler who at the same time largely wrote Brouncker out of history. He read the Latin translation of A treatise of algebra and wrote, as though Brouncker had never existed: ‘Such problems have been agitated between Wallis and Fermat… and the Englishman Pell devised for them a peculiar method described in Wallis's works’.67 Poor Brouncker! The equation Na2 ± 1 = l2 has been mistakenly but universally known ever since as ‘Pell's equation’.68 By the end of 1657 all the important results were in place, but the correspondence went on into the spring of 1658 due to the late intervention of Frenicle (who had long ago arrived at his own solutions to the first challenge).69 Frenicle knew nothing of Wallis's or Brouncker's work until he first saw their earliest responses in February 1658 and wrote a scathing reply.70 Wallis wrote back in early March with a long and pointless defence of the early solutions, but he knew by now that he had to show something better, and for the first time produced non-trivial solutions to the first challenge.71 His approach was once again numerical, not algebraic, but Frenicle also worked numerically and was suitably impressed. Finally, Frenicle challenged Wallis and Brouncker to prove the generality of their method by finding a solution for N = 313: this was to be the acid test.72 Brouncker passed it without difficulty and sent his solution back to Digby in a brief and modest note on 13 March:73 Within the space of an hour or two at most this morning I found that 313 × Q7170685; −1 = Q126862368 and therefore that 313 × Q(2 × 7170685 × 126862368 =) 1819380158564160, + 1 = Q32188120829134849. Which I thought fit to present you, because

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Mouns. Frenicle may thence perceive, that nothing is wanting in the perfect solution of that Probleme.

Neither brevity nor modesty were Wallis's style, and he followed up Brouncker's note with a much longer letter in Latin, confirming that Brouncker had indeed found a solution and that Frenicle should remain in no doubt that ‘we’ understood the method perfectly.74 These letters from Wallis and Brouncker in March 1658 containing Brouncker's numerical solutions set the seal upon their success. Frenicle conceded victory, and Digby wrote a long and effusive letter to Wallis:75 And I doubt not but that your last Letters of the 4 and 15 of March will make [Fermat and Frenicle] and all the world give as large and as full a deference to you. For although I had time, since receiving them, but to run them greedily over, yet I see enough of the redundant light in them to reverence, not a rising, but a noon day Sun in its very vertical point and highest Zenith.

Digby also wrote warmly to Brouncker but, exhausted by his praise of Wallis, very much more moderately and more briefly:76 I give you most humble and hearty thanks for yours, which I embrace with exceeding gladness, joy and respect … Now, neither [Frenicle] nor Mouns. Fermat, will have any more to cavil at, either your Lordship, or Dr Wallis; unto whom I have written at large (considering my inability of writing much at present) and do presume to beg your favour in conveying my Letter to him; which I leave open, that if you please you may cast your eye over it If I were not quite wearied out (I am yet so weak) with writing my Letter to Dr Wallis, your Lordship should not thus easily be delivered of my troubling you at this time: which for my mentioned reason I must not now further enlarge, but humbly kissing your hand, I rest, My Lord Your most humble and worthy servant Kenelm Digby

Fermat too acknowledged their success:77 I gladly recognise, and indeed rejoice, that the most illustrious gentlemen Viscount Brouncker and John Wallis have at last given legitimate

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solutions to the numerical problem I set forth. The most noble gentlemen did not wish even for a single moment to confess themselves unequal to the proposed problems.

This last remark was another of Fermat's barbed understatements, as Wallis had steadfastly ignored the problems for many months. Fermat went on to hope that the English mathematicians would maintain their new found skills:78 Truly, that both sides may from now on proceed in a fitting manner, the French will admit that the English satisfied the proposed problems. But let the English admit in turn that the problems were worthy of being proposed to them and let them not disdain in the future to examine and investigate more closely the nature of integers, and indeed to foster that subject, in which they have shown strength of mind and subtlety.

Fermat's plea fell on deaf ears. Neither Wallis nor Brouncker ever did any further work on number problems. As early as February 1658, Wallis was already planning to publish the entire correspondence, for at his request Van Schooten wrote a long letter in March outlining the Dutch responses for inclusion.79 Many of Wallis's own letters, formal, lengthy and in Latin, have the appearance of having been written with publication in mind from the start. By contrast, those of Brouncker, if they have survived at all, were little more than brief and informal notes, almost always in English. Wallis happily accepted the role of spokesman and chief correspondent, but without Brouncker he might have had nothing to report. Not only was Brouncker the first to take up Fermat's challenges, several months ahead of Wallis, but it was he who did the best and most important work. It was on Brouncker's foundations rather than Wallis's that future number theorists were to build. With such work to his credit, it might have been hoped that Brouncker would go on to even greater things, but it was not to be, for he left no mathematics of any great value from the period of his Presidency.80 The last significant reference to his mathematics is in a letter from Collins to James Gregory in February 1669:81 … the Square Roote here resembles somewhat of Division there supposing the Divisor equall to the quote and the Lord Brouncker asserts he can turne the square roote into an infinite Series …

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Collins' remark was based on the identity:

in which the divisor is the same as the quotient. If it is assumed that (1 − x2)1/2 can be expressed as an infinite series a+bx+cx2+dx3+…, then it is a simple matter to evaluate the coefficients as b = d = f = h = … = 0 and and so on. If this result had been available to Wallis in 1652, of course, he would not have needed to resort to interpolation, and we might not then have had either his infinite fraction or Brouncker's. For a number of reasons Brouncker's reputation has faded over time. He never had to earn his living from mathematics, so that although he responded with great skill and originality to the problems posed by Wallis and Fermat, he never needed as Wallis did, nor chose as Fermat did, to venture into wider fields. In later years he became occupied with other matters. After the Restoration in 1660 he became not only President of the Royal Society, but also a Member of Parliament, President of Gresham College and Commissioner for the Navy, so it was probably only during the relatively leisurely years of the 1650s that he had enough time to divert himself with mathematics. For Wallis, on the other hand, mathematics was a lifelong career that kept his name and his work in the public domain. Wallis also had a far keener sense than Brouncker did of the value of claiming and publishing results. Indeed, without Wallis's encouragement it is doubtful that we would have any of Brouncker's work at all. Unfortunately, Wallis's method of incorporating Brouncker's work, albeit acknowledged, into his own publications, meant that Brouncker's name was never as prominent as it might have been, and in the quarrel with Fermat, Wallis, as chief correspondent, came to be seen by contemporaries as the leading partner in the affair. This was certainly the view of Digby and Frenicle: the final adulatory letter from Frenicle mentioned Wallis by name no fewer than eleven times but Brouncker only once.82 Wallis's mathematics eventually filled three large volumes while Brouncker's entire published output amounts to no more than a dozen pages. In its own time Wallis's work was undoubtedly the more influential. The Arithmetica infinitorum played a major part in the development of English mathematics while the topics explored by Brouncker were to some extent always outside the mainstream. It was not until a century later that the deeper implications of Brouncker's work began to emerge. Euler, from 1759, recognized the close relationship between continued fractions and ‘Pell's

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83 equation’ by observing that Brouncker's algorithm for the latter produced the continued fraction expansion of Lagrange, following Euler, wrote three papers between 1768 and 1779 on continued fractions, the first of which included a definitive treatment of ‘Pell's equation’.84 If Brouncker's work lapsed into obscurity in the later seventeenth century it was perhaps because his successors, in England at least, were not yet ready for it.85

In his own day, Brouncker's genius if not fully understood was clearly recognized, most of all by Wallis, his closest mathematical colleague. The mathematical styles of the two men could hardly have been more different: Wallis persevering and systematic, and often, it has to be said, rather dull; Brouncker original, intuitive and sure-footed. Wallis's approach to the problems they worked on together was primarily numerical, while Brouncker slipped easily into algebraic notation, and was able to present an entire argument in generalized form, a striking achievement for its time. Wallis was a mathematician who worked ‘with much labour, and by many circuits and operations’,86 while Brouncker seemed to begin with a clear view of what he wanted and how to get there. Wallis was the synthesist; Brouncker the analyst. Wallis managed, in the end, to catch Proteus by circling in more and more closely, carefully covering every inch of ground; Brouncker simply took Proteus by the hand and kept his grip, unbemused by new or changing forms. For those who delight in original and unusual mathematics Brouncker's work still reveals a remarkable depth and richness; without it seventeenth-century mathematics would shine a little less brightly than it does.

8 ‘Many pretty things worth looking into’ This final chapter looks at some of the reactions to A treatise of algebra immediately after its publication and since, and draws together a few observations on Wallis's perspective on algebra and on history. We have little direct evidence of how Wallis's book was received by contemporary English readers, except in a letter written by Roger Cotes (1682–1716) to his uncle and tutor, John Smith, shortly before he went up to Cambridge in 1699. Cotes was especially interested in the material in the later chapters, relevant to the development of the calculus:1 I have Dr Wallis's Algebra I think I bought it very cheape I am very well pleased wth ye Book. The Dr's Buisness therein is to shew ye Original, Progress & Advancement of Algebra from time to time, and by what steps it hath attained to yt height at which it now is he give[s] a full Account of ye Methods used by Vieta Harriot Oughtred De-Chartes and Pell & others and of ye several methods of exhaustions, Indivisbles, Infinites, Approximations &c. amongst other things he speak's of squaring Curves and after other ways of approximations shewed he show's you this of Mr Newton he determin's it impossible to do ye business exactly. In my mind there are many pretty things in yt book worth looking into.

A notice of A treatise of algebra had appeared in the Philosophical transactions in July 1685, but it was not a review in the modern sense, simply a reprint of Wallis's Preface to the Reader outlining the contents of the book.2 (The Preface was in turn almost identical to Wallis's original Proposal of 1683 (Fig. 1.3).) The first detailed review was written by Leibniz and appeared in 1686 in the Acta eruditorum.3 Leibniz, like Wallis, summarized for his readers the contents of A treatise of algebra but also added some comments of his own. Not needing

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to proclaim English superiority at every turn, he was able to offer a slightly different slant on parts of Wallis's account. He had read the early Italian algebraists, for instance, more carefully than Wallis had, and so was able to add to Wallis's meagre list of sixteenth-century English mathematicians the name of Richard Wentworth who had once been a pupil of Tartaglia. On the subject of rectification he named Van Heuraet before either Wren or Neile, and considered Mercator (whom he regarded as German) to be prior to Newton in his discovery of the infinite series for log (1 + x), perhaps the first foreshadowing of the bitter disputes that were to arise later between Newton and Leibniz himself as a result of Newton's failure to publish. Leibniz was possibly the only continental mathematician to respect Wallis's opinion of Harriot, and he recounted Wallis's claims for Harriot at some length. Significantly, he stated that Wallis and Pell took such claims seriously, and may have heard as much from Pell himself when he met him in London in 1673, about the time that Wallis was beginning to write A treatise of algebra. Another continental mathematician who read the book in its original English edition was Christiaan Huygens, who made notes headed ‘Du livre de Wallis, Historia algebrae anglice’ as part of his writing on the three classical problems of antiquity.4 In connection with the quadrature of the circle he noted four chapters: Chapter 83, where Wallis ventured that the quadrature could not be carried out using numbers so far known; Chapter 79, where Wallis gave a detailed justification of the method of induction (about which Huygens had expressed serious doubts when it first appeared in the Arithmetica infinitorum in 1656); Chapter 95, where Huygens' own quadrature of the circle was mentioned in the context of Newton's new method by infinite series; and Chapter 10, with its fractional approximations for π. This last was of particular interest to Huygens since he himself had found similar approximations in 1680 by a rather shorter method (he called Wallis's method ‘bien longue’). He gave his notes on Chapter 10 the subheading ‘Développement du Numerus impossibilis en une fraction continue’, but this was to read into the chapter more than Wallis had put there, for although Wallis's method leads naturally to a continued fraction, Wallis himself had not taken it in that direction. The only continued fraction in A treatise of algebra was Brouncker's but Huygens made no mention of it. In general Wallis's history drew more attention than his mathematics. That there were a number of criticisms in this respect can be inferred from revisions he made when the book was translated into Latin in 1693. In the Latin edition Wallis justified his lengthy treatment of Oughtred,

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praised Kersey and one or two other English mathematicians whom he had previously neglected, and paid more attention to Bombelli. The most important changes, however, were in his discussion of Harriot and Descartes. Samuel Morland had written to Wallis in 1689 asking him to detail his accusations against Descartes,5 but others had been less inclined to give him the benefit of the doubt, and had taken their derision straight into print. The French mathematician Jean Prestet, in the second edition of his Elemens in 1689, wrote:6 It is only on vain conjectures or from envy that some have wanted to make believe even in his lifetime that [Descartes] took his method from others, and particularly from a certain English Harriot, whom he had never read, as he declared in one of his letters. And while Monsieur Wallis, a little too jealous of the glory with which France has acquitted herself in mathematics, has just renewed this ridiculous accusation, one is right not to believe it at all, for he speaks without proof.

The gauntlet was also taken up by Descartes' biographer, Adrien Baillet, who retorted with more enthusiasm than accuracy that the charges of plagiarism had been discounted long ago by Pell, Aylesbury and Warner.7 It was this remark of Baillet's that finally provoked Wallis into revealing Pell's influence in all he had written, in the piece entitled De Harrioto addenda added to the Latin translation of A treatise of algebra. There Wallis carefully refrained from making any charge of plagiarism, but continued to insist, correctly, that Harriot's work had preceded Descartes' by up to forty years. In the early eighteenth century Euler read A treatise of algebra in its Latin edition and, as we have seen, took inspiration from the mathematics of Brouncker that he found there, though like Huygens, he remained strangely oblivious of Brouncker himself. The other eighteenth-century responses that we know about were again to the history rather than the mathematics, and it is little surprise that English readers were better disposed towards the book in this respect than were their continental counterparts. Nicholas Saunderson (1682–1739), professor of mathematics at Cambridge, recommended it without criticism to those of his students who were interested in such things.8 Charles Hutton in 1796 tactfully remarked that one purpose of his own history was to counteract the ‘superficial and partial’ investigations of his predecessors, including ‘the celebrated Dr Wallis’,9 but Etienne Montucla writing at about the same time was scathing, and all too often correct, about Wallis's poor treatment of Cardano, Bombelli, Viète and Descartes.10

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Even in the nineteenth century English mathematicians refrained from too strident criticism: Augustus de Morgan in 1838 admitted that the book had some shortcomings but still considered it ‘full of interest’;11 Arthur Cayley, writing in the Encyclopaedia Britannica fifty years later, referred briefly to the views of Montucla and De Morgan but added no further comment,12 while Rouse Ball merely said that Wallis's account contained ‘a great deal of valuable information’.13 Moritz Cantor, on the other hand, dismissed Wallis's history (and Montucla's) as ‘inspired by excessive national pride’.14 Late nineteenth- and early twentieth-century historians generally disengaged themselves from the history in A treatise of algebra and concentrated on its mathematical content, but with some curious distortions of vision. Rouse Ball in 1888 thought the book ‘noteworthy as containing the first systematic use of formulae’ and cited as an example ‘v = st’, a formula nowhere to be found in it.15 The point that most interested Florian Cajori in 1894 was that ‘Wallis discusses the possibility of a fourth dimension’, but this too was a theme that the book barely touched on. Cajori also noted the attempts to give geometrical interpretations of complex numbers but said that Wallis ‘failed to discover a general and consistent representation’.16 This was not true, and Cajori can only have meant that Wallis had not come up with what later became the standard representation. Joseph Scott's long discussion of A treatise of algebra in his biography of Wallis in 1938 was almost entirely taken up with the account of Harriot, so much so that Scott claimed that Wallis's ‘best work consists of filling in the gaps which Harriot had left’.17 On the remaining seventy-five chapters of the book he was virtually silent. Perhaps the least helpful description of A treatise of algebra, however, is to be found in the 1974 Encyclopaedia Britannica article on Wallis (now on CD) which says: Wallis published, in 1685, his Treatise on Algebra [sic], an important study of equations that he applied to the properties of conoids, which are shaped almost like a cone. Moreover in this work he anticipated the concept of complex numbers.

The loose remark on conoids possibly refers to results in the Arithmetica infinitorum. The second sentence is simply untrue: complex numbers had been handled by Cardano, Bombelli and Harriot long before Wallis attempted his geometrical representations of such numbers. In recent years D. T. Whiteside and David Fowler have investigated some of the deeper mathematical content in A treatise of algebra. There is also a new awareness of its value, as of other early mathematical texts, in

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mathematics education. This book is therefore the latest, but almost certainly not the last, word on Wallis's ‘large discourse concerning algebra’.

Wallis's perspective on algebra One of the most remarkable features of A treatise of algebra is how little explicit discussion it contains about what algebra was, or how it changed in the course of its development. To the modern reader this is an unaccountable omission in a book that purports to be a history of the subject, but Wallis probably assumed that algebra was familiar enough to his readers to need no definition; furthermore, by retracing the steps by which algebra ‘hath attained the Heighth at which now it is’,18 his readers would come to understand for themselves how the subject had developed and improved. For a modern audience, however, it is perhaps helpful to draw out some of the implicit assumptions about what was meant by algebra at various times in the sixteenth and seventeenth centuries. In the opening pages of A treatise of algebra Wallis gave four names by which algebra was commonly known: analysis, aljabr, regula cosae and arsmagna. The only one he expanded upon at any length was analysis,19 but in doing so he shifted the meaning of the word in an interesting way. Viète had seen analysis as a conceptual tool for the rediscovery of Classical theorems; Wallis, however, discussed analysis as a property of arithmetic, in which subtraction, division and extraction of roots were the analytic counterparts (we should now say inverses) of the synthetic or compositive operations of addition, multiplication and composition of powers. This tells us much about Wallis whose frame of reference was entirely different from Viète's, and who saw algebra as primarily a generalization of arithmetic (recall his description of Viète's algebra as ‘specious arithmetick’). Wallis's description of analysis in fact brought his concept of algebra much closer to the traditional al-jabr, in which subtraction, division and extraction of roots were used to solve equations, by reversing the operations by which those equations had been composed in the first place. Wallis's discussion of analysis in his opening chapter also reveals another unspoken assumption: that algebra was essentially about finding unknown quantities through the solving of equations. Only once did Wallis come close to making such a definition explicit, at the beginning of his section on Harriot's algebra, where he said that in ‘the nature of equations … lyes the main Mystery of Algebra’.20 Recent historians have largely restricted

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themselves to a similar definition of early modern algebra as the study of equations,21 yet Wallis's treatment depicts a far richer landscape of algebraic activity. In fact the ‘nature of equations’ was discussed thoroughly in connection with the work of Oughtred, Harriot and Pell, but in the final third of A treatise of algebra was barely mentioned. Wallis's narrative therefore reflects the trend of actual historical development, in which the problem of the structure and solution of equations was initially a powerful motivating impulse (as it was to be again a century and a half later) but in the later seventeenth century was overshadowed by the richness of progress in other directions. From what has been described in this book it is clear that a comprehensive definition of algebra has at least to include: describing and solving quadratic equations (al-Khwārizmī c. 825); transforming equations by change of root (Cardano 1545); transcribing Euclid II into symbolic notation (Harriot c. 1600); finding general formulae for figurate numbers (Harriot c. 1600); defining a recurring pattern using subscript notation (Brouncker 1655); describing symbolically an infinite set of solutions to a problem in number theory (Brouncker and Wallis 1657); and handling infinite series for logarithmic and trigonometric quantities (Newton 1665). All of these topics were treated (though not always ascribed to their earliest authors) in A treatise of algebra and all may be considered algebraic activities. What all of them have in common is some movement from concrete to abstract and from the particular to the general: to do algebra, or to algebraicise, was then as now to work with essential mathematical properties or structures, independently of particular manifestations:22 And this Abstractio Mathematicae (as the Schools call it,) is of great use in all kind of Mathematical considerations, whereby we separate what is the proper subject of Inquiry, and upon which the Process proceeds, from the pertinences of the matter (accidental to it,) appertaining to the present case or particular construction.

The process of abstracting mathematical structure began long before the existence of mathematical notation as we now understand it; indeed, adequate or useful notation often lagged far behind the ideas it was needed to express. Good notational ideas were sometimes lost (Chuquet's index notation), others spread slowly and unevenly (Recorde's ‘equals’, or Harriot's inequality signs), and false starts (cossist notation and Viète's ‘Aquadratus’) hindered progress. By the beginning of the seventeenth century, however,

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it was finally becoming possible to represent mathematical ideas in concise and internally consistent symbolism, which in turn clarified and facilitated further thinking. There can be little doubt that the invention of appropriate notation hastened the spread of mathematical ideas, but it would be a mistake to think that early algebraic thinking could not develop without it, and in the case of Cardano it did so to an astonishing degree of sophistication. Wallis noted many developments and improvements in notation, but in his own work often fell back on what he had first learned, the notation of Oughtred. In this as in so many ways the impact of Wallis's first encounter with algebra was discernible for the rest of his life. Wallis's four names, analysis, al-jabr, regula cosae and ars magna, served well enough at the beginning of A treatise of algebra. Between them they adequately described the kind of algebra that had developed by about 1600 and with which Wallis began his studies in 1648. By the end of the seventeenth century, however, the last three names had fallen into disuse and analysis was taking on a new meaning, becoming associated almost entirely with the algebra of infinite series created by Newton and others. Wallis neither revised his descriptions nor introduced new ones, and perhaps never saw the need to do so: for him algebra remained what it had always been, generalized arithmetic. As far as Wallis was concerned, new algebraic entities went hand in hand with increasingly advanced arithmetic: ‘ordinary’ or finite equations with negative numbers, rationals and surds; ‘impossible’ equations with imaginary numbers; and infinite equations or, as we should now call them, infinite series with the new kind of number (transcendental) needed for the quadrature of the circle. For Wallis, applications of algebra to geometry were of secondary importance. It seems perverse to say this of the mathematician who produced the first systematic algebraic formulation of conics, but Wallis's motive was not primarily, as it might have been for Viète or Descartes, to investigate the geometric properties of curves, but to apply to them new methods based on arithmetic. Newton, who followed Wallis in approaching algebra through arithmetic, also appeared to share his view that algebra and geometry were best kept separate, but whereas Newton upheld a pure geometry unsullied by algebra,23 Wallis was interested in ‘pure Algebra, abstracted from Geometry’. Such concerns in either case came rather too late, for by the later years of the seventeenth century Wallis and Newton had both played major roles in ensuring that every aspect of mathematics, not just geometry, had come to be handled algebraically.

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Wallis's perspective on history The revitalization of English mathematics can conveniently be dated from 1631 when the Clavis and the Praxis first made available to English readers some of the ideas that had been coming to fruition in the sixteenth century in Italy and France. By the time Wallis wrote A treatise of algebra in the 1670s, English mathematics had emerged to take its place amongst the most advanced in Europe, and Wallis's own career mirrored this astonishingly rapid pace of change: over the same forty years he had progressed from his brother's arithmetic books to become one of the foremost mathematicians of his day. In writing a history of seventeenth-century mathematics he was in more than one way writing his own story. The structure of A treatise of algebra frequently reflects this. The opening chapters often lack immediacy, except where Wallis turned to the one subject that really caught his imagination, the origin and spread of the Hindu–Arabic numerals. His brief survey of algebra from al-Khwārizmī to Viète was necessary to set the scene, but lacked either substance or enthusiasm. The algebra in the book comes to life only at the point where Wallis and so many of his contemporaries started, with Oughtred's Clavis. After that Oughtred, and Wallis's friend Pell, and Pell's hero Harriot, occupy the substantial central portion of the book. The final quarter of the book is given over to Wallis's own achievements, to the Arithmetica infinitorum and to the new mathematics that arose from it, culminating with results revealed by Newton even as Wallis was writing. Wallis was clearly promoting not only his own mathematics but the work of his English contemporaries. He always had a keen sense of the benefits to both author and audience of publishing new ideas, and his efforts in this respect contributed significantly to the contemporary mathematical scene. Work by Neile, Wren and Brouncker had already appeared, fully acknowledged, in Wallis's publications during the 1650s, and A treatise of algebra gave him a new opportunity to ensure that the achievements of mathematicians both well known and obscure reached a wider audience. Unfortunately, he was censured more than once for too readily publishing other people's work; Aubrey once wrote bitterly of Wallis that:24 … he lies at watch, at Sir Christopher Wren's discourse, Mr Robert Hooke's, Dr William Holder, &c; putts downe their notions in his note booke, and then prints it, without owneing the authors … But though he does an injury to the inventors, he does good to

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learning, in publishing such curious notions, which the author (especially Sir Christopher Wren) might never have the leisure to write of himselfe.

Aubrey's irony obscures a kinder truth: that Wallis did much to ensure that the work of other mathematicians (many less eminent than Wren) was published and correctly attributed, and nothing but generosity can be claimed for his treatment of Harriot, Oughtred, Brouncker, Neile, Pell, Newton and many lesser figures. There can be little doubt that Wallis saw publication as a means of establishing priority, and in particular of proclaiming English successes abroad. An analysis of seventeenth-century nationalism is beyond the scope of this book,25 but Wallis was not alone in regarding English mathematical and scientific achievement as a matter for national pride. Among Pell's papers there are several copies of a 1678 pamphlet advertising the forthcoming English Atlas, which proclaim that ‘The Work is intended for the credit of our Nation’, just the kind of language that Wallis was using (and in which Pell probably encouraged him).26 Nor was aggressive nationalism a purely English phenomenon, for Fermat in 1658 had initially been far more provocative in this respect than Wallis. Such attitudes appear in striking contrast to the spirit that had prevailed in earlier centuries when western Europeans had often shared a common mathematical culture that moved easily across national boundaries. It was unfortunate for the later reception of A treatise of algebra that Wallis's often laudable efforts on behalf of his countrymen were so often marred by disregard or contempt for foreigners. His dislike of the French in particular must have sprung in part from his mathematical quarrels in the late 1650s, first with Fermat, and then with Pascal, who had behaved less than honourably towards both Wallis and Antoine Lalouvère over their work on the cycloid (Lalouvère was one of the few Frenchmen of whom Wallis afterward spoke with some regard). Two lesser but bitterly fought disputes with Dulaurens, and Leotaud can only have further inflamed his prejudices. Wallis's mistrust of foreigners and especially of Catholics, however, was rooted in more than personal quarrels. In common with many others at the time, he was particularly suspicious of the intentions of Catholic France, and his perceptions of the dangers of Catholic domination are illustrated in a letter (previously unpublished) to his friend Thomas Smith in 1698:27 I concur with you in considering the hardships of the Greek church under the Turkish oppression. And heartily wish them a more happy

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condition. But if they should change the Turkish slavery for that of the Romist I doubt they would change for the worse. For, certainly, the Protestants in Hungary are in much worse circumstances, under the Christian Emperor, than they were under the Turkish. And like oppressions there are in Poland, France and elsewhere, especially where the Jesuits rule.

Through his work as a code-breaker Wallis was more familiar than most with the potential for political treachery. (One of the very first letters he had deciphered was from the Catholic sympathizer Francis Windebank, whom Wallis would have regarded as a traitor and who had fled to France.)28 William Wallis, a great-great-grandson of the mathematician, later wrote with perception (if not punctuation) of Wallis's knowledge of foreign affairs:29 He must likewise have been well acquainted with what was passing in the several courts and countries in Europe for without this knowledge it must have been impossible to have explained many passages where only a hint was given and which was often enough to explain the writers meaning to the person wrote to who may be supposed to know something of the business but to a third person must without that knowledge have appeared unmeaning and unintelligible.

With his inside knowledge of political dealings against a background of suspicion and mistrust, it is hardly surprising that Wallis had few good things to say of France or its mathematicians. Wallis was at his best as a historian when he focused not on nationalities or personalities but on mathematics. This was so even in his much criticized treatment of Harriot once he forgot to harangue Descartes and concentrated instead on the implications of Harriot's algebra. But his historiographical skills were at their most refined in his study of Hindu–Arabic numerals where, with no English hobby-horse to ride, his evidence was thoroughly researched and carefully considered. There Wallis went beyond the introduction of new techniques to redefine the way the history of mathematics itself was understood. To the medieval mind mathematics was ancient and revealed knowledge handed down from one generation or civilization to the next, but by the second half of the seventeenth century the pace of change in mathematics (as in other subjects) had rendered such ideas untenable and history had to be seen and written differently. Wallis was the first mathematician to present mathematics as a system of living, changing ideas that arose and spread in complex and not always easily discernible ways,

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and his purpose was to trace the emergence and development of one particular branch of mathematics, algebra, and to show ‘the Original, Progress and Advancement thereof, from time to time’. He may not always have succeeded in his aim as well as he might have done but, for all its weaknesses, A treatise of algebra established the history of mathematics for the first time as a serious intellectual discipline. Wallis's strong sense of mathematics as a historical subject led him not only to investigate its past but to record its present for the benefit of posterity. In 1653 he deposited a collection of decoded letters in the Bodleian Library,30 and wrote that part of his purpose in doing so was that: ‘those who shall live hereafter, may perhaps be hence informed, what kind of ciphers they were which have been in use in this age’.31 His words may be applied equally to mathematics: those who shall live hereafter may be informed by Wallis, who was tireless in his efforts to publish and promote the best of English mathematics, what kind of mathematics it was that was in use in that age.

Notes Chapter 1 1. 2.

Wallis to Collins, April 1677, in Rigaud 1841, II, 606–607. The Italian Gino Loria wrote a history of British mathematics in 1913–15, but for the sixteenth and seventeenth centuries his account is rather less useful than Wallis's, see Loria 1913–15, 425–427. See also Pycior 1997, ‘Symbols, impossible numbers, and geometric entanglements: British algebra through the commentaries on Newton's Universal arithmetick’. Pycior's American usage of ‘through’ makes her title misleading for British readers: the commentaries are not the starting point of her discussion but her endpoint. For general histories of early modern algebra see Franci and Toti Rigatelli 1985; Van der Waerden 1985, 32–68; Parshall 1988; Pycior 1997, 10–39; Høyrup 1998; Bashmakova and Smirnova 2000, 49–90. 3. The most important source of biographical material on Wallis is the autobiography he wrote when he was eighty years old. The full version is in Bodleian Library MS Smith 31, ff. 38–50, and there is also an earlier, shorter draft in British Library Add MS 32499, ff. 375–376v, copied again in Bodleian Library MS Eng. misc. e. 475, ff. 256–274. Several eighteenth-century biographies of Wallis were based on his autobiography, and additional material was contributed by David Gregory in 1705, MS Smith 31, ff. 58–59, and William Wallis, great-greatgrandson of the mathematician, in 1786, MS Eng. misc. e. 475, ff. 275–349. For a complete list of eighteenthcentury biographies and publication details see Scriba 1970, 19–20. See also De Morgan 1838, XXVII, 41–43; Scott 1938 and 1960; Yule 1939; Scriba 1975. 4. Scriba 1970, 26–27; the relevant section is reprinted in Fauvel and Gray 1987, 316–317. 5. Scriba 1970, 40. 6. Scriba 1970, 27. 7. See Feingold 1984, 86–121; 86–90. 8. Bodleian Library MS e Musaeo 203, f. xxi. Wallis made two identical copies of letters deciphered up to 1653, now Bodleian Library MS e Musaeo 203 and MS Eng. misc. e. 475, ff. vii–243; letters deciphered between 1651 and 1701 can be found in British Library Add MS 32499, and letters from 1689 to 1703 in Bodleian Library MS Eng. misc. 382. See also Smith D. E. 1917; Kahn 1967, 166–169. 9. See Fauvel, Flood and Wilson 1999, 79–83, 97–99. 10. For a full list of Wallis's mathematical works see the bibliography of primary sources.

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11. Wallis's correspondence is being edited for publication by Christoph Scriba and Philip Beeley, see Wallis 2002. 12. Cajori 1929; Pycior 1987; Grant H. 1990, 1996; Probst 1993; Jesseph 1993, 1999. For the post-Restoration disputes between Hobbes and members of the Royal Society see also Shapin and Schaffer 1985. 13. Pascal 1658 and 1659; Wallis 1659; Lalouvère 1658 and 1660. See Tatton 1974. 336, 339. 14. Wallis's quarrel with Dulaurens arose over ‘Simon de Montfert's problem’, see Brouncker et al. 1658, letter 40; Dulaurens 1667, 249; Wallis 1668a, b and c. There are other letters relating to the affair throughout Hall and Hall 1965–86, vols IV and V. Solutions to the problem by Moore (1658) and Wren (undated) are preserved in MS Aubrey 10. 15. Wallis's quarrel with Leotaud was an extension of an earlier disagreement between Peletier and Clavius about the magnitude of the angle between a circle and its tangent, see Peletier 1557, Preface and 73–76; Vitellio 1572, 18; Clavius 1574, 132f; Wallis 1656a; Aynscom 1656; Leotaud 1662; Wallis to Leotaud, 17 February 1668, in Wallis 1685, Appendix III, 79–88. 16. Scriba 1970, 42. 17. The first history of mathematics was written by Eudemus (late fourth century BC) who, like Wallis in the seventeenth century, was aware of the many new discoveries made by his predecessors. See Fauvel and Gray 1987, 46–47. 18. The surviving mathematical papers of Charles Cavendish are preserved in the British Library as Harley MSS 6001–6002, 6083. 19. See Jacquot 1952a, Feingold 1999a. 20. British Library Add MS 4423, ff. 146–153v. 21. Collins to Pell, 9 April 1667; Collins to Brancker, June [1667], in Rigaud 1841, I, 126, 136. 22. Collins to Wallis, 17 June 1669, in Riguad 1841, II, 515–516. 23. See Scriba 1964, 51–53; Whiteside 1967–81, II, 277–294. 24. Collins to Wallis, 21 March 1671; Wallis to Collins, 14 November 1672; Collins to Wallis, 27 March 1673, in Riguad 1841, II, 526, 552, 556. For the Dutch and French authors mentioned by Collins (apart from Verstay whom I have not traced) see the bibliography of primary sources. 25. Wallis to Collins, 14 November 1672, in Riguad 1841, II, 552. 26. Collins to Baker, 10 February 1677 and 24 April 1677, in Riguad 1841, II, 14, 21. 27. Wallis to Collins, 11 January 1670; Collins to Vernon, [Jan/Feb 1671], in Riguad 1841, II, 519 and I, 161. The idea of publishing Wallis's works in the Netherlands came up again in 1689, see Morland to Wallis, 1689, MS Eng. lett. c. 291, ff. 37–38. 28. Collins to Baker, 10 February 1677, in Riguad 1841, II, 15. 29. Collins to Baker, 24 April 1677, in Riguad 1841, II, 21. 30. Wallis to Collins, 3 November 1668, in Rigaud 1841, II, 507. 31. Wallis to Collins, 29 March, 8 April, 12 April, 6 May, 27 September 1673. 11 September 1676, in Rigaud 1841, II, 557–586, 591–600. 32. Collins to Baker, 24 April 1677, in Rigaud 1841, II, 21.

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33. The Proposal was reprinted at the beginning of A treatise of algebra following the (unpaginated) preface and contents. 34. Wallis 1685, Preface. 35. Such long production times were not at all unusual. Kersey's Elements of algebra (1673–74) took some twelve years to write and a further six or seven to publish. Boulliau's Opus novum (1682), Clark's Oughtredus explicatus (1682), and Baker's Geometrical key (1684), were all begun twenty to thirty years before they were eventually published. 36. For a full account of the Bodleian Library at this period see Philip 1983. 37. For many years the Savile Library was housed in the Savilian Professors' study between the Schools of Geometry and Astronomy (where the Lower Reading Room reserve desk now stands), but was incorporated into the main Bodleian Library in the nineteenth century. The first catalogue of the Savile Library was Bernard 1697. 38. Of the 238 volumes of the Digby collection at least 40 contain medieval mathematical texts, making it by far the richest single collection of such material in England. The contents are not included in the Bodleian Library's Summary catalogue of western manuscripts but are catalogued separately in Macray 1883. 39. Greaves travelled to Constantinople on Laud's behalf during the 1630s. The Laud manuscripts are catalogued in Coxe 1853. 40. The manuscripts in the Selden collection are mainly of Greek and oriental origin, but there are also a few important medieval Latin texts. The majority of the Arabic and Persian texts were acquired from the estate of John Greaves after his death in 1652. 41. Ashmole's manuscripts were at first held with his collection of ‘curiosities’ (acquired from John Tradescant in 1659) in the original Ashmolean museum, which opened in 1683 next door to the Bodleian Library (in the building that now houses the Museum for the History of Science). The manuscripts were transferred to the Bodleian Library in 1860, and are catalogued in Black 1845. 42. See Greaves 1648; 1650a,b; Pococke 1650; 1656; 1661; 1663. 43. Isidor, Migne LXXXII, cols 153–184; 155–169. 44. Bede (ascribed), Migne XC, cols 647–653; 650. 45. Molland 1995, 214, 221–223. 46. Regiomontanus 1537; Rose 1975, 95–98. 47. Baldi 1998; Rose 1975, 253–269; Moyer 1999. 48. Baldi 1707. 49. MS Savile 29; Goulding 1999. 123–125. Savile did present sound historical arguments about the identity of Euclid, see Goulding 1999, 96–103. 50. Zilsel 1945; Lilley 1958, 3–37; Crombie 1975; Molland 1978; Molland 1983, 141–148. 51. Vossius 1650. De scientiis mathematicis is the third book of Vossius's trilogy on the history of contemporary arts and sciences, De quatuor artibus popularibus de philologia et de scientiis mathematicis. The main title page describes the third book as De … scientiis mathematicis, cui operi subjungitur, chronologia mathematicorum, but at the beginning of the book itself the title is given differently as De universae matheseos natura

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52. 53.

54. 55. 56. 57.

NOTES

ac constitutione liber; cui subjungitur chronologia mathematicorum. For convenience I have used, as Wallis did, the abbreviated title (and running head) De scientiis mathematicis. Wallis owned a copy of the 1660 edition, now Bodleian Library Savile G.21, and all references in the present book are to that edition. In Wallis's copy of De scientiis mathematicis, page 177 in particular, from alFarghānī to ibn Ezra, is heavily annotated at every paragraph. For further details of the Arab writers mentioned by Wallis see Stedall 2001, 76–77. Wallis could find little evidence of algebra in Classical authors, and the debate about the existence of Greek ‘geometrical algebra’ will not be taken up in this book. The subject was discussed at some length in the twentieth century, see Unguru 1975, 1979; Van der Waerden 1976; Freudenthal 1977; Weil 1978; Mueller 1981, 43–44, 50–52; Berggren 1984, 394–410. Wallis 1685, 15. Wallis 1685, 272. Wallis 1685,128. Oughtred to Wallis, 17 August 1655, in Rigaud 1841, I, 87–88.

Chapter 2 1.

2. 3. 4. 5. 6. 7. 8.

9.

Wallis's chief biographer, J. F. Scott, disposed of the opening chapters of A treatise of algebra in one sentence: ‘[Wallis's] account of the history of mathematics in antiquity is very comprehensive and gives evidence of a close study of the Classical literature of the sciences’; Scott made no mention at all of Wallis's researches on the medieval period. See Scott 1936, 335, reprinted as Scott 1938, 133. For commentary on selected paragraphs from Wallis's Chapters 1, 2 and 6 see Molland 1994, 215–218, and for detailed discussion of Chapters 2 to 4 see Stedall 2001. The title of the present chapter is taken from the opening sentence of Wallis's Chapter 14, Wallis 1685, 64. ‘Neque enim ipsam tradimus scientiam; sed de ea scribimus’; Vossius 1650, 37. For the life and work of Vossius see Rademaker 1981. See Leland 1549 and Leland 1975. Leland's Itinerary, MS 5107–5112 and Collecteana, MS 5102–5106. The notes were first edited and published by Bodleian Librarian Thomas Hearne as Itinerary of John Leland the antiquary in 1710–12 and Collecteana in 1715. Summarium, Bale 1548; Catalogus, Bale 1557–59. Bale's notebook, MS Selden Supra 64. The contents were eventually published as Index Britannia scriptorum in Bale 1902. ‘Before these times the Arabic Language, and Greek itself, being but little known in these Parts, Mathematical Learning was but very rare, and slenderly improved in Europe. We had indeed in England, Althelmus or Adelmus, whom Vossius placeth about the year 680; and Walfridus Ripponensis, placed by him at 690; and Bede (the most eminent of that Age) at 730; and Albinus or Alcuinus, (a Scholar of Bede) at 760’; Wallis 1685, 6. See Vossius 1650, 171, 312, 395; Migne LXXXIX.

NOTES

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22.

23.

223

See Vossius 1650, 395; Migne XCV. See Vossius 1650, 171, 312; Migne XC. For an assessment of Bede's mathematics see Jones C. 1970. Migne XC, cols. 277f, 293f. See Vossius 1650, 171; Migne C, CI. Migne CI, cols 1143–1160; Folkerts 1978; Hadley 1992. MS Bodl. 309, ff. 3v–62, 68–80, from the Abbey of the Holy Trinity, Vendome, France c. 1075. Wallis 1685, 5. ‘As Adelardus, (a monk of Bath) whom Vossius placeth about the year 1130 who for that purpose travelled into Spain, Egypt, and Arabia; and (as Vossius tells us) translated Euclid (and some other Arabic authors) out of Arabic into Latin, Anno hoc MCXXX.’; Wallis 1685, 5. For modern scholarship on Adelard see Burnett 1987. Adelard I is a close translation of the entire work; Adelard II was the most popular version but omits many of the proofs; Adelard III is a commentary rather than a translation. Recent scholarship has suggested that Adelard II should in fact be ascribed to Robert of Chester, see Busard and Folkerts 1992. MS Savile 19 (Adelard II); MS Trinity College 47 (Adelard I). 'About the same time (or somewhat sooner) Guilielmus de Conchis (William Shelley) is said to have travelled into Spain to furnish himself with Arabic and Mathematical Learning; and brought from thence divers Arabic Books. And, soon after, Daniel Merlacus (Morley), about the year 1180 made several Journeys into Spain on the like account, where (at Toledo) Arabic and Mathematical Learning were in great request (brought thither by the Moors) which in other parts of Europe were scarce known. And these brought with them that kind of Learning into England very early, with store of Arabic Books … About the same time were Johannes Sarisburiensis, Rogerus Infans, and divers others of the English.'; Wallis 1685, 5–6. See Sudhoff 1918; Birkenmajer 1970, 45–51. The only manuscript copy of the preface now surviving is in the British Library, MS Arundel 377. ‘A particular account of these Travels of Shelley and Morley was a while since to be seen in two Prefaces, to two Manuscript Books of theirs in the Library of Corpus-Christi College in Oxford, but hath lately (by some unknown hand) been cut out, and carried away;… Who ever hath them, would do a kindness (by some way or other) to restore them, or at least a Copy of them.’; Wallis 1685, 6. The ‘Explicit Will de Conchys’ led to a mistaken identification in Coxe 1852, where the manuscript is catalogued as ‘three books of the Norman philosopher William de Conches, alias Shelley’. The error was pointed out by H. Nash in a letter to Corpus Christi librarian, Charles Plummer, written 25 March 1889. The letter (preserved with MS CCC 95) begins: ‘I have been to see the BM MS (Arundel 377) of Daniel de Morley. It is the same book as the one in your library and it is then also followed by a dialogue between the Duke of Normandy (D) and the Philosopher (P) of Gul. De Conchis. Coxe confounded the two. A passage which I copied from your MS fol 15b occurs on the

224

24. 25. 26.

27. 28. 29. 30.

31. 32.

33. 34. 35.

36. 37.

NOTES

last folio of the Arundel 377, where the division between the two is quite distinct. You will see the ‘incipit’ of Gul de Conchis in the Arundel catalogue. I mention this as you may like to make a note of the fact in your copy of Coxe's catalogue. The beginning (missing in the CCC MS) contains a delightful little piece of autobiography [of Morley].’ Migne CIC, cols 407–409. Russell J.C. 1932; French 1996. MS Digby 40. Because of this manuscript, Roger has acquired two separate entries in the Dictionary of national biography: ‘Roger Infans (fl. 1124)’ and ‘Roger of Hereford (fl. 1178)’. The mismatching dates stem from the figure 1124, which appears in the margin of the Tractatus de computo in MS Digby 40, but which was meant as part of the calculation, not as a date of writing. Roger's name was anglicized by John Leland to ‘Yonge’; Wallis later went further and gave his name as ‘Roger Child’, see Wallis 1693–99, II, 6. MS Selden Supra 31, ff. 32–204; see also Migne CLXXXIX, col. 649f; col. 659f. Prefacio Roberti translatoris, MS Selden Supra 31, ff. 32–33; Migne CLXXXIX, col. 657 f. See Thorndike 1923, II, 14–98, 155–187; Haskins 1924, 3–66, 113, 129; Rose 1975, 76–89; Lindberg 1978, 52–90. ‘And Robertus Retinensis (Robert of Reading) who travelling into Spain on the account of the Mathematics, did there translate the Alcoran out of Arabic into Latin, in the year 1143. (as appears by his Epilogue to that Translation, and the Preface of Petrus Cluniacensis thereunto.)’; Wallis 1685, 5. The Dictionary of national biography still carries two articles on Robert, headed ‘Chester, Robert (fl. 1182)’ and ‘Robert the Englishman, (de Ketenes, de Retines) (fl. 1143)’. The date 1182 in the former arises from the dating system then in use in Spain; it was in fact the year we would now denote as 1144, which at least brings the two Roberts into the same time frame. See note 18. The Canons of Arzachel (al-Zarqālī of Cordoba, d. 1100) explained the use of the Toledan tables, compiled between 1062 and 1078 from the earlier tables of al-Khwārizmī (c. 830), al-Battānī (c. 888) and Thābit ibn Qurra (c. 870). The Canons are to be found twice in MS Savile 21 (at ff. 27–41v and ff. 63–103); the second version is the translation by Robert of Chester, there described as Robertus Cestrensis. See Karpinski 1915, 49–63; Hughes 1982. Wallis 1657b, Chapters 6–9. Wallis 1657b, Chapters 7–8. Conrad Dasypodius, writing at the end of the sixteenth century, put forward the now discounted idea that modern numerals were derived from Greek alphabetic numerals, Dasypodius 1593–96. The relevant section is quoted by Smith D. E. and Karpinski 1911, 33, note 2, and Dasypodius's table is also reproduced in Ifrah 1998, 358. Archimedes 1676. Wallis drew on the earlier editions of Geschauff [Ventorius], Commandino and Rivaltus. Heath 1931, 305–309.

NOTES

225

38. Wallis 1688a. Book I and the first thirteen propositions of Book II of Pappus have long been considered lost, though Jones A. 1986, 46–47, suggests that Book I is extant in Arabic. Wallis edited the second part of Book II from a manuscript in the Savile Library, now MS Savile 9, ff. 41–48. Books III–VIII were first translated and published by Commandino in 1588. See also Wallis 1688b, edited from MS Savile 10, ff. 132–140. 39. The Greek word ψηφοζ means ‘pebble’, the equivalent of calculus in Latin. Planudes' text has not been translated into English, but for the Greek text and a German translation see Planudes 1865 and for a French translation see Planudes 1981. 40. Folkerts 1997, 8–25. The treatise is thought to have been called Kitāb fi 'l-jam ‘wa’l-tafrīq (‘Treatise on gathering [addition] and dispersion [subtraction]’). 41. See Allard 1987; Folkerts 1997, 6–7. The three surviving twelfth-century redactions of al-Khwārizmī's text are the Liber ysagogarum alchorismi, Liber alchorismi and Liber pulveris. 42. See Halliwell 1839, 73–83; Steele 1922, 72–80. 43. There are eleven copies of the Carmen in the Bodleian Library, seven in the Digby collection alone, and another in the Savile manuscripts. 44. ‘Solstitium quinis horam praecedit in annis, Cumque diemfaciant viginti quatuor horae, Annis viginti centumque dies datur una. Solstitium legimus Christo nascente fuisse. Centum viginti decies jam praeteriere Anni. Sic denis praecedit meta diebus.’

45.

46. 47. 48. 49.

As a calendrical work the Massa computi could be written in either Hindu–Arabic or Roman numerals, and copies survive in both forms, see note 56. The year of Sacrobosco's death comes from his tombstone in the Convent of St Mathurin in Paris, but the last three words of the Latin inscription, ‘M christi bis C. quarto deno quater’, are ambiguous and may be read as ‘four tens plus four’ (giving 1244) or ‘four fourteens’ (giving 1256). Vossius chose the second interpretation, in which Wallis followed him. Modern scholars remain uncertain and have suggested other possibilities, see Pedersen O. 1985, 186–192. In Wallis 1693, 6, Wallis argued on etymological grounds that Sacrobosco came from Halifax in Yorkshire, but there is no historical evidence to support this suggestion. For the little that is known of Sacrobosco see Pedersen O. 1985. ‘Omnia quae a primitiva origine rerum’, in MS Savile 17, ff. 94v–104. There is an early English translation in Steele 1922, 33–51. For a modern critical edition see Pedersen F. S. 1983. An early English translator of Sacrobosco's Algorismus struggled to explain the word algorism as deriving either from algos (art) and rithmus (number) (hence the Latin ars numerandi), or from gogos (introduction) and rithmus, or from a mythical Indian king Algus, the supposed inventor of the art; see Steele 1922, 33. In an early English commentary on Ville Dieu's Carmen de algorismo, an anonymous writer expounded similar ideas about the origin of the word algorism: ‘Ther was a kyng of Inde, the quich heyth Algor, and he made this craft. And after his name he called

226

50. 51. 52. 53.

54. 55. 56. 57. 58. 59. 60.

NOTES

hit algorym; or els another cause is quy it is called Algorym, for the latyn word of hit Algorismus comes of Algos, grece, quid est ars, latine, craft on Englis, and rides, quid est numerus, latine, or nombur on Englys, … quasi ars numerandi.’; see Steele 1922, 3–32. Kircher 1665, 44–47. Kircher mistakenly dated the Alphonsine tables at 1252 because they referred to the ‘Alphonsine era’ which began in 1252 with the coronation of Alphonso X of Léon and Castile, but they were actually compiled 1263–72 and did not reach Paris and Oxford until about 1320; see North 1989a, 327–359. ‘Non nisi anni sunt CCCL, saltem infra Quadringentes, quod eas Sifras accepimus.’; Vossius 1650, 34. See Netz 1999. MS Savile 20 and MS Selden Supra 25 both contain copies of Boethius's Arithmetica which use Roman figures but in both copies Arabic numerals have been added alongside or in the margins. Wallis would certainly have known the first of these and probably the second also. ‘To this way of Arithmetic, by these Numeral Figures, they give the peculiar name of Algorism, (a word which, I believe, is not to be found any where used more anciently, nor for any other, than this way of Practical Arithmetic,) being an Arabic name, compounded by them of their Arabic article Al, with the Greek’ Aριθμóζ. (in like manner as Ptolemy's Almagist, is by them so called from Al and μεγíστη) The Arabic name of Algorithm, or Algorism, being of the same age with us, as is the Arabic way of Calculation, or Practical Arithmetic.'; Wallis 1685, 9. Wallis's derivation of algorism has been described in Molland 1994, 217, as ‘eccentric’, but it is also instructive, for it shows that although Wallis recognized al-Khwārizmī as the inventor of algebra, he had lost sight of him as a writer on arithmetic. ‘Compotus est sciencia numeracionis et divisionis’, in MS Savile 21, ff. 127–142v. Grosseteste wrote the original version of this treatise c. 1210, corrected it 1215–1219 and revised it again in 1244; the version in MS Savile 21 is from 1215–1219. ‘Compotus est sciencia considerans tempora’, in MS Savile 17, ff. 141–174v. Wallis copied the date of Sacrobosco's text on the flyleaf at the front of the volume: ‘Ab incarnatione domini elapsi sunt 1235’. Wallis knew copies of the Massa computi in MS Savile 17, ff. 175–184v and MS Savile 21, ff. 161–175. In the former it has been copied twice, once with Roman numerals, once with Arabic. ‘Algorismus Jordani, tam in integris quam in fractionibus, demonstratus’, in MS Savile 21, ff. 143–150. MS Savile 21, ff. 143–160v. Thomson 1940, 22–36, includes a facsimile of a fragment from this manuscript. The handwriting has been identified with near certainty as that of Robert Grosseteste, see Hunt 1955, 133–134; Clanchy 1979, 128; Southern 1986, 107. See note 32 and North 1986, 114–117. Wallis knew Robert of Chester's translation of the Toledan tables and associated Canons in MS Savile 21, ff. 63–103. He found references to Robert, Bishop of Chester in the Historia de gestis regum anglorum (ending at 1129) of Simeon of Durham and

NOTES

61. 62. 63. 64. 65. 66. 67.

227

in the Abbreviationes chronicorum (ending at 1201) of Ralph de Diceto (both published for the first time in the seventeenth century in Twysden 1652) but noted that Bishop Robert lived too early (c. 1085) to have travelled to Spain in 1140. He also discounted identification with Robert Cestria who lived too late (c. 1380). See Wallis 1685, 12; Stedall 2001, 102–103. Wallis 1683; Wallis 1685, 12–13. Churton 1800, 1232. Baker 1822–41, I, 631; Gough 1865. It is possible that the initials read by both Wallis and Baker as ‘W.R.’ could be ‘W.K.’: it would not be the first time that Wallis had mistaken a medieval looped ‘K’ for an ‘R’ (note his earlier reading of Retenensis for Ketenensis). Charles Tracy FSA, personal communication to the Rector of Helmdon, received 13 March 2000. Ron Baxter, personal communication to the author, 29 June 2000. '… these Studies were strangely advanced, and especially in England, where (beside those above mentioned) we had Clement Langthon, whom Vossius placeth about 1170; Gervasius Tilburiensis, about 1210; Johannes de Sacro Bosco, about 1232; Robertus Lincolniensis (Robert Grosthead) about the same time; Roger Bacon, about 1255; Johannes Peccam (or Johannes Cantauriensis) about 1276; Odingtonus, about 1280; Johannes Bacondorpius, about 1330; Robert Holcot (or de Northamptona) about 1340; Johannes Estwood (de Ashenden), about 1347; Climitonus Langley, about 1350; Nicolaus Linnensis, about 1355; John Killingworth, about 1360; Richard Lavingham, about 1370; Simon Bredon, about 1386; John Sommer, about 1390; John Walter, about 1400; William Batecombe, about 1410; William Buttoner, about 1460; who were, many of them, very eminent, as in other kinds of Learning, so particularly in the Mathematics; and divers of their Works are extant in our Libraries, which have not yet been printed. Besides others whom Vossius mentions not: As Adamus de Marisco (Adam Marsh), contemporary with Grosthead Bishop of Lincoln, intimate with him, and commended by him; Bradwardine and Read, and divers others about that Age.'; Wallis 1685, 6. In their modern forms the names in Wallis's list are: Clemens Langthorn, Gervase of Tilbury, Johannes Sacrobosco, Robert Grosseteste, Roger Bacon, John Pecham, Walter Odington, John Baconthorpe, Robert Holcot, John Ashenden, Richard of Kilvington, Nicholas of Lynn, John Killingworth, Richard Lavenham, Simon Bredon, John Somer, John Walter, William Batecombe, William of Worcester or Botoner, Adam Marsh, Thomas Bradwardine, William Rede. Those not discussed in the main text (because there is little or no evidence of any mathematical writing) are Clemens Langthorn (c. 1170), presbyter of Llanthony Abbey; Gervase of Tilbury (c. 1210) whose Otia imperialia included geography, natural history and folklore; Walter Odington (fl. 1301–1330), author of De speculatione musice; Robert Holcot (fl. 1343–1349), author of biblical commentaries; and William of Worcester or Botoner (1415–1482?), antiquary. For dates, biographies and bibliographies see Emden 1957; Kretzmann, Kenny and Pinborg 1982, 853–892; Sharpe 1997. Other references that have been found useful

228

68. 69. 70. 71. 72.

73. 74.

75. 76. 77. 78. 79.

NOTES

include: Pedersen O. 1985 (Sacrobosco); Thomson 1940, Hunt 1955, Clanchy 1979, Southern 1986 (Grosseteste); North 1976, III, 238–270 (Odington); Xiberta 1927; North 1992b, 105–106 (Baconthorpe); Smalley 1956, Thorndike 1957, Tachau 1995 (Holcot); Snedegar 1988 (Ashenden); Kretzmann and Kretzmann 1990 (Kilvington); North 1988, 87–133 (Lynn and Somer); North 1989a, 343–346; North 1992b, 124–127 (Killingworth); Talbot 1962 (Bredon); North 1986, 126–130 (Walter); North 1989a, 337–342 (Batecombe); North 1986, 186–195 (Botoner); Clagett 1959, 220–222, 230–234; North 1992a, 79–82 (Bradwardine); North 1989a, 332–336 (Rede). North 1989b, 44–46; North 1992b, 131–132. Pecham is best known for his Perspectiva communis (on optics), Tractatus de numeris (on the mystical properties of numbers) and Tractatus de sphera. Bacon 1928, 117–127; Grant E. 1974, 90–94. North 1989b, 46–48; North 1992b, 132. John Ashenden is also sometimes referred to as John Eastwood. Wallis noted five spellings of Eastwood (Estwood, Estwyde, Eshwood, Eshuid, Eschuyde) and five of Ashenden (Ashendon, Eshendon, Ashenton, Aysden, Estemdene); Emden's Biographical register of the University of Oxford lists five additional variants of Eastwood and no fewer than twenty-four of Ashenden, and I have found further spellings not mentioned by either. I have attempted to correlate Wallis's spellings with those in the manuscripts in order to trace his sources, but without any great success: the best identifications are the unique forms [Estomdene] in the colophon to the second book of Ashenden's Summa judicialis in MS Savile 25 at f. 163, and [Aysden] in a later hand at f. 164v. See also Snedegar 1988. Rede, bursar and later subwarden of Merton, endowed the library with 100 books and £100, together with astronomical items that were still there in 1615 and are assumed to be among those there today. Some of Bradwardine's papers were bought from his Executors by Rede (see note 73), who bound them with papers of his own and bequeathed them to Merton. The volume was later acquired by Thomas Allen and was eventually given to the Bodleian Library as part of the Digby collection, MS Digby 176. Most of the Merton medieval mathematical manuscripts that have survived have at some time made a similar journey across the road to the Bodleian Library. Bredon's astrolabe is now owned by Oriel College, and may be seen in the Museum for the History of Science in Oxford. Ashenden 1489. Wallis probably knew the printed copy in MS Ashmole 576. Both tracts are to be found in MS Digby 176 alongside others concerned with predicted conjunctions of planets and the disasters that might arise from them, a poignant reminder of the terrible consequences of the Black Death of 1348. Somer's calender, his Opusculum tertium calendarii, was made for Joan, mother of Richard II, and survives in many copies. John Killingworth the astronomer was confused by Vossius and others with an earlier John Killingworth who was at Merton from 1381 to 1384.

NOTES

229

80. Richard Swineshead was variously known as ‘Suuinsete’, ‘Suiseth’ or ‘Suiset’. He was not always distinguished from his contemporaries Roger and John Swineshead, so that his first name sometimes appears as Ioannes, Rudiger, Reyner or Raimundus. In Swineshead 1520, his name is given as Ricardus in the title but Raimundus in the colophon. 81. MS Savile 29, f. 3v. 82. North 1999, 33; North 1976. 83. Padua c. 1477, Pavia 1498 and Venice 1520. See Clagett 1959, 290–304; North 1992a, 89–92. 84. The Arithmetica of Boethius was based on the Introductio arithmeticae of Nichomachus (c. AD 100), and was essentially a treatise on Pythagorean number relationships. See Boethius 1983 and Boethius 1999. 85. Wallis 1685, 2. 86. Al-Khwārizmī's exact sources are not known, but Babylonian, Greek and Indian influences are evident in his work, see Toomer 1973, 360; Høyrup 1986; Høyrup 1993. See also Chemla 1994 for evidence that Islamic mathematicians drew on Indian and Chinese sources. 87. See Saliba 1973; Van der Waerden 1985, 4–5; Bashmakova and Smirnova 2000, 50. 88. Høyrup 1986, 475–477; Høyrup 1993, 10–14. 89. For a translation of al-Khwārizmī's text see Karpinski 1915, 66–157; Chapters 1–6 of Karpinski's translation are reprinted in Grant E. 1974, 106–111. 90. For a translation and commentary on Abū-Kāmil's Al-jabr see Levey 1966. 91. Wallis 1685, 3. 92. There was no printed edition of the Liber abbaci until the nineteenth century, see Boncompagni 1862, I. 93. For a detailed comparison of Leonard and Abū Kāmil see Levey 1966, 217–220. 94. Robert of Chester had used the term substantia for a square. 95. Franci and Toti Rigatelli, 1985; Van Egmond 1994. 96. Karpinski 1929; Høyrup 1999. 97. Van Egmond 1978; Franci and Toti Rigatelli 1985, 30–32. 98. Franci and Toti Rigatelli 1985, 57–61; Smith F. K. C. 1994, 114–115. 99. Franci and Toti Rigatelli 1985, 36–39. 100. Smith F. K. C. 1994, 119–124. 101. Franci and Toti Rigatelli 1985, 51–52. 102. Parshall 1988, 139–140. 103. See Flegg, Hay and Moss 1985. 104. 'Primo. A numero radicum incipe, eumque dimidiatum, loco eius pone dimidium illius, quod in loco suo stet, donec consumata fit tota operatio. Secundo. Multiplica, dimidium illud positum, quadrate. Tertio. Adde vel Subtrahe iuxta signi additorum, aut signi subtractorum, erigentiam. Quarto. Invenienda est radix quadrata, ex summa additionis tuae, vel ex subtractionis tuae relico. Quinto. Adde aut Subtrahe iuxta signi aut exempli tui exigentiam.

230

105. 106.

107. 108. 109. 110. 111.

112. 113. 114. 115. 116. 117. 118. 119. 120.

NOTES

Modum extrahendi hunc tibi, mi bone Lector, formavi, ita ut memoriae tenaciter haerere possit adminiculo dictionis huius AMASIAS.'; Stifel 1544, 240v. See Stifel 1544, 249v for negative exponents, and 53–54v for his exploration of fractional exponents. Scheubel 1551; Peletier 1554; Borrell [Buteo] 1559; Ramus 1560; Aurel 1552; Perez de Moya 1562; Mennher 1556; Nuez 1564; Peucer 1556; Recorde 1557. The only one of these texts I have been unable to inspect is Perez de Moya's Arithmetica practica et speculativa of 1562, though I have seen his Tratado de mathematicas of 1573. For texts published in France by Scheubel, Peletier, Buteo and Ramus see Van Egmond 1988; Cifoletti 1996, 128–140. Mennher wrote in French, while Nuez wrote originally in Portuguese and then translated into Spanish, but both published at Antwerp. Recorde's text is unpaginated. This extract is from the section entitled ‘The arte of cossicke numbers’. Wallis 1685, 63. Rigaud 1841, II, 607. Norman 1584. Perhaps Collins gave Wallis some guidance here, for in the Additions and emendations written while A treatise of algebra was in press Wallis was able to add the correct title and date of The whetstone. In the Additions and emendations he also alluded to Recorde's The grounde of artes of 1543 and The pathway to knowledge 1551, but seemed to think they were the same book under different titles. He later acquired a copy of The whetstone, now in the Savile Library with IOHN WALLIS on the spine and Wallis's annotations inside (Savile H.12). For instance, Clavius 1608 (Germany), Cataldi 1618 (Italy), Henrion 1623 (France) and others. Cossist notation still occasionally appeared up to thirty or forty years later, for example in Renaldini 1655; Brasser 1663. Wallis 1685, 3. Pacioli 1494, I, dist. VIII, tractate 5. Smith F. K. C. 1996. Cardano 1993, xviii–xxii. Cardano 1545, Chapter XXXVII; Cardano 1993, 217–221. ‘duc. 5p:Rm:15 in 5m:Rm:15, dimissis incrutiationibus, fit 25 m:m:15, quod est p:15, igitur hoc productum est 40.’; Cardano 1545, 66; Cardano 1993, 219–220. ‘Cum vero diligenter considerassem in his, visum est mihi, ut etiam ultra trangredi liceret.’; Cardano 1545, 14v; Cardano 1993, 48. ‘Est etiam transmutationis via, qua ante demonstrationem universalia capitula multa inveni, atque inter reliqua, cubi aequalis quadratis et numero, et cubi cum quadratis, aequalis numero, velut cum conamur hanc solvere quaestionem, duos invenios numeros, quorum aggregatum aequale fit alterius quadrato, et ex uno in alterum ducto, producatur 8, una enim via pervenes ad 1 cubum p:8 rebus, aequalem 64, hac igitur inventa aestimatione, si diviseris 8 per eam, prodibit reliqua aequatio, ex qua in capituli illius cogitationem perveni.’; Cardano 1545, 15v–16; Cardano 1993, 51–52.

NOTES

231

121. ‘Quaestiones igitur alio ingenio cognitas ad ignotas transfere positiones, nec capitulorum inventio finem est habitura, non tamen extra haec, ex una quaestione, generalia poteris assequi.’; Cardano 1545, 16; Cardano 1993, 52. 122. Fully worked examples of the method may be seen in the Ars magna at Chapter XXXIX amongst problems V to XIII, Cardano 1993, 239–253. 123. ‘et ideo complementum in his operationibus, est quasi extremum, ad quod pervenit perfectio humani intellectus, vel potius imaginationis, in hoc enim cognosces illorum differentiam.’; Cardano 1545, 75v; Cardano 1993, 246. 124. By handling cube roots of what would now be called complex conjugates Bombelli was able to show that an apparently ‘impossible’ quantity might in fact be real, for example: . Thus it was possible in certain cases to find a real root of an ‘irreducible’ cubic. Bombelli's examples were carefully chosen, however, for finding the cube root of a complex number is a non-trivial matter and in the case of ‘irreducible’ cubics leads back to the original equation. 125. Reich 1968; Jayawardene 1973. 126. The most comprehensive assessment of the Ars magna, as of so many early algebra texts, is to be found in Charles Hutton's Mathematical and philosophical dictionary, see Hutton 1796, 68–73; reprinted in Hutton 1812, 206–224. See also Gliozzi 1971, 65; Smith F. K. C. 1999. 127. For example, Parshall 1988, 143, 149; Pycior 1997, 10, 12. Parshall discussed Cardano's work in the context of ‘natural selection of ideas’ in a mathematical environment, an approach that possibly begs more questions than it answers about how such an environment is created. Pycior, searching for the background to nineteenth-century symbolic algebra, dwelt almost exclusively on the acceptance or otherwise of negative and imaginary numbers. In both cases the need to support a particular theory or point of view has perhaps prevented an appreciation of Cardano's innovations from a sixteenth-century perspective. 128. Pell translated the crucial sentence quoted in note 121 as: ‘Therefore transfer to unknown positions, questions otherwise known, ye invention will be endlesse…’, British Library Add MS 4409, f. 261. 129. British Library Add MS 4409, ff. 261–261v. 130. Wallis 1685, 1, opening sentence. 131. For Viète's life and work see De Morgan 1838, XXVI, 311–318; Busard 1975. 132. See Bombelli 1572; Diophantus 1575; Apollonius 1566; Pappus 1588. For Viète's use of Diophantus see Reich 1968; Jayawardene 1973. 133. For an excellent description of Viète's aims and methods see Mahoney 1973, 26–48. 134. For modern translations into English of the Isagoge see Viète 1968, 313–353 and Viète 1983, 11–32. 135. For the oriental origins of Viète's method see Rashed 1974; Chemla 1994. 136. ‘Nullum non problema solvere’; Viète 1646, 12; Viète 1983, 32. 137. Ad logisticem speciosam notae priores, 1631; Zeteticorum libri quinque, 1593; De aequationum recognitione et emendatione tractatus duo, 1615; De numerosa potestatum

232

138. 139. 140. 141.

142. 143. 144. 145. 146. 147. 148. 149. 150.

NOTES

ad exegesin resolutione 1600; Effectionum geometricarum canonica recensio 1592; Supplementum geometriae, 1593; Ad angularium sectionum analyticen theoremata, 1615. For English translations see Viète 1983. Van Egmond 1988 begins: ‘Algebra as we know it began in France around the beginning of the seventeenth century … it was the conceptual transformations wrought by François Viète and his immediate followers that established algebra in the form we know today.’ Vossius 1650, 314. The Bodleian Library owns a 1523 edition of the Summa printed at Toscolano. Wallis also mentioned Francesco di Ghaligai (whom he called Caligarius) whose Summa di arithmetica containing chapters on Arcibra was published in Italian in Florence in 1521. Wallis himself appeared not to have seen the book but quoted Vossius who gave a date of 1515. The 1521 edition of Ghaligai's Summa now in the Bodleian Library [Don.e.283] was a later acquisition that he could not have seen, but there is a 1552 edition in the Savile Library which he seems to have missed [Savile M.17, bound with Peletier 1554 but not listed in Bernard 1697]. The 1552 edition is entitled Practica d'arithmetica, but apart from some changes in notation the text is essentially the same as Ghaligai 1521. Ries wrote his Coss in 1523 and revised the manuscript in 1539 but it was never published, though other works by Ries did appear in print. For Rudolff, see Rudolff 1525. Wallis 1685, 62. Wallis to Collins 12 April 1673, Riguad 1841, II, 573. Scheubel 1551; Peletier 1554 [Savile M.17]; Borrell 1559 [Savile Cc.11]; Nuñez 1567 [Savile Aa.7]. Bombelli's L'algebra, surprisingly, seems never to have been in the Savile collection. It was not listed in Bernard 1697 and has not been added to the Savile collection since, though there is a copy in the main Bodleian Library. Ramus 1560 [Savile Y.31, edition of 1586]; Salignac 1580 [Savile V.22]; Stevin 1585 [Savile Q.10]; Clavius 1608 [Savile Bb.7]; Henisch 1605. Viète's activities as a cryptanalyst as well as his legal training may have influenced his algebra, see Pesic 1997. Wallis 1685, 66. Wallis 1685, 66.

Chapter 3 1.

For biographical information on Oughtred see Aubrey 1898, II, 106–114; Cajori 1916; Willmoth 1993, 43–61. The year of his birth, 1573, is inferred from a contemporary portrait that gives his age as 73 in 1646; see Fig. 3.1. Aubrey gave Oughtred's date of birth as 5 March 1574 (or 1575 in the Julian calendar), but also said that Oughtred was 88 when he died in June 1660, implying tha he was born in 1572 (English) or 1573 (Julian). Aubrey's inconsistency has been reflected in dates

NOTES

2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

233

given by later commentators: Cajori 1916 gave 1573–1575, while Scott 1974 gave 1575. Aubrey 1898, II, 106. The registrar at Eton was appointed to assist the two Bursars who were elected from among the Fellows. Oughtred 1596. Hales to Oughtred, in Riguad 1841, 1, 3. Oughtred's margin notes to Viète's De aequationum recognitione (1615), were copied out by John Pell and are in British Library Add MS 4423, ff. 146–153v. Wallis 1685a, 157. Tapp published The arte of navigation in 1596 and The seaman's kalender in 1602. The arte of navigation was based on a translation made by Richard Eden from Spanish, just as The pathway to knowledge in 1613 was based on William Philip's translation from Dutch. For the period 1600–1630 the Savile Library contains a number of algebra texts written in Latin and a few in French, but none in Italian, German, Spanish or Dutch. ‘Cum tibi, illustrissimi, tui patris jussu, in disciplinis Mathematicis exponendis deservierim, nihil magis in votis mihi fuit, quam ut optime id fide, hoc est, via Analytica (quae quidem doctrinae est) praestarem. Atque hanc ob causam, tum Euclidis demonstrationes, inter legendum, ad formam Analyticum revocavi, cujus in XIX capite hujus tractatus nonnulla habentur exempla;’; Oughtred 1631, Preface, first page. ‘Postremo ut matheseos studiosis quasi Ariadnes filum porrigerem, quo ad intima harum Scientiarum adiuta deducantur, et ad optimos antiquissimosque authores, Euclidem, Archimedem, Apollonium Pergaeum magnum ilium Geometram, Ptolomaeum, ac reliquos, facilius penitiusque intelligendos, dirigantur:’; Oughtred 1631, Preface, second page. Mahoney 1973, 27, note 3, claims that Chapter XVI of the Clavis was taken directly from Chapter V of Viète's Isagoge but although Oughtred adopted Viete's notation he never took up his vocabulary, and in fact both chapters were based on the contents of many earlier texts. Oughtred 1631, 46. Harriot had written the propositions of Euclid II algebraically many years before, see British Library Add MS 6785, ff. 153–156. Harriot's version was never published but may have been communicated to Oughtred by Cavendish. Oughtred 1631, 49–51. Oughtred 1631, 50–51. The symbol ± was used frequently by Harriot, but had not previously appeared in print. At the end of the Isagoge Viète announced his intention of using algebra to solve the ancient problem of trisecting an angle, see Viète 1983, 32, but his Ad angularium sectionum was not published until 1615, with demonstrations supplied by Alexander Anderson; see Viète 1983, 418–450. Hutton 1796, 91–92 or Hutton 1812, 286–288. Hutton's account of Viète's algebra omits any mention of the application of algebra to Classical geometry; see Hutton 1796, 83–88 or Hutton 1812, 260–273. Robinson to Oughtred, 11 June [probably 1633], in Riguad 1841, 1, 16.

234 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

NOTES

Derand to Cavendish, 11 February 1635, in Riguad 1841, I, 23. Mersenne's opinion was later recounted by John Twysden, see Mersenne 1932-, IV, 367. Robinson to Oughtred, 2 July 1636, in Riguad 1841, 1, 26. Aubrey 1898, 11, 108, 284. Price to Oughtred, 2 June 1642, in Riguad 1841, 1, 59. Oughtred to Price, 6 June 1642, in Riguad 1841, 1, 60–61. Aubrey 1898, II, 111. Aubrey 1898, II, 110. Oughtred 1647, Preface. For a full account of the sequestration proceedings against Oughtred, see Willmoth 1993. 56–60. Ward was deprived of his Cambridge Fellowship for writing against the Covenant in 1644. He later swore the Oath of Allegiance to the Commonwealth when appointed Savilian Professor of Astronomy at Oxford in 1649, but never committed himself entirely to the new regime and was eventually rewarded with a bishopric after the Restoration. The same metaphor was later taken up by Newton: see Newton 1967–81, II, 393. MS Aubrey 6, f. 40; Aubrey 1898, II, 113–114. In 1652 there were two University printers, Leonard Lichfield and Henry Hall, see Madan 1908. British Library Add MS 4431, f. 109. Boyle 1744, I, 24. King 1830, I, 227. Newton to Hawes, 25 May 1694, no. 452 in Turnbull 1959–77, III, 364. A copy of the 1652 Clavis owned by Newton was in the Turner Collection secretly sold by Keele University in 1998. A copy of the 1653 edition arrived at the Bodleian Library from an anonymous source in 1950. Collins to unknown recipient, [1674], in Riguad 1841, II, 477. The date of this letter is inferred from its mention of the recent reprinting of Collins' Merchants accounts, see Collins 1653. (See also note 53.) Collins to Gregory, 25 March 1671, in Turnbull 1939, 180. In Riguad 1841, II, 219, incorrect punctuation makes it appear that Gibson died in 1650. The sentence should read ‘… were lent to Gibson, deceased, in anno 1650…’. Collins to Wallis, February 1667, in Riguad 1841, II, 484. Collins to Wallis, February 1667, in Riguad 1841, II, 483. Collins to Wallis, [probably January 1667], in Riguad 1841, II, 468. Collins to Wallis, 2 February 1667, in Riguad 1841, II, 470. Among ‘sundry tracts of Algebra expected from beyond the sea’, were books by Chauveau, Hudde, Tacquet, Renaldini, Fermat and Descartes. Little is known about Chauveau (see Tannery 1895), and the expected publication never appeared. However, there is a manuscript copy of Chauveau's Traicte d'algebre among the surviving papers of Charles Cavendish, British Library MS Harley 6083, ff. 350–379, and another among

NOTES

45. 46. 47. 48. 49. 50. 51.

52. 53.

54. 55. 56. 57. 58. 59.

235

Pell's papers, British Library Add MS 4407, ff. 31–37v. Nothing by Hudde or Descartes was published after 1659. For the other writers mentioned by Collins see Tacquet 1669; Renaldini 1669; Fermat 1679. Collins also mentioned the forthcoming work of the English mathematicians Pell and Kersey; see Pell and Rahn 1668; Kersey 1673–74. Collins to Wallis, 2 February 1667, in Riguad 1841, II, 471. Robert Anderson, although a weaver by trade, was also the author of a book on gunnery, Anderson and Street 1674. Robert Anderson appears to have been of a somewhat critical disposition, for he also greatly disliked the work of Collins' friend Michael Dary, see Anderson 1670. Pitts and Collins were discussing a print run of 1000 copies. This may be deduced from Wallis to Collins, 5 February 1667, in Riguad 1841, II, 476, in which Wallis mentioned that the copyright should be sold to Pitts for £40, or 9½d per book. Wallis to Collins, 5 February 1667, in Riguad 1841, II, 474. Collins to Wallis, February 1667, in Riguad 1841, II, 482. It seems that Bunning (mentioned by Collins) and Clark (mentioned by Thompson) were not the only authors of commentaries on the Clavis. In 1671 Collins noted that the late Dr (Richard) Rawlinson had also written on the Clavis, but Collins held him in little regard: ‘One Isles, a bookseller, bought some of his books: and Anderson, a weaver, in company of Mr Streete, bought more of them; and they have seen some of his writings, for which a great rate was demanded; and if I meet Streete accidentally, I shall with no great appetite inquire where they are.’ Collins concluded that he was not keen to recommend any of the three commentaries, for he knew that Kersey's forthcoming book was better than any of them; Collins to Vernon, [Jan/Feb 1671], in Riguad 1841, I, 151–154. The date of this letter is inferred from the mention of 67 issues of the Philosphical transactions: Issue 67 appeared on 16 January 1671, and Issue 68 on 20 February 1671. Collins to Wallis, 2 February 1667 and February 1667, in Riguad 1841, II, 471, 483. Collins to unknown recipient, [1674], in Riguad 1841, II, 477–481. Rigaud assumed that the letter was written to Wallis, and placed it with the 1667 correspondence between Collins and Wallis on the reprinting of the Clavis. The style of address, however, is not that customarily used by Collins to Wallis, who is also mentioned by name in the text. (See also note 39.) Collins mentioned Cataldi, Geysius, Glorioso and Alsted. I have been unable to identify Geysius, but for the others see Cataldi 1602, 1610, 1618; Glorioso 1627–39; Alsted 1620. We have no evidence as to whether Oughtred read any of these texts, but we do now know that he read Viète, see note 5. Grisio 1641. A comma is missing between ‘Maghet’ and ‘Grisio’ in Riguad 1841, II, 480. Girard 1629. Viète 1615a. See note 5. Robert Hooke's copy of the Clavis is now in the British Library, BL.529.b.19 (4,5).

236

NOTES

60. Oldenburg to Glanvill, 3 October 1668, no. 970 in Hall and Hall 1965–86, IV, 75. 61. Wallis 1685, 175. 62. Savile Z.16, bound in leather, has just a few neatly written notes; Savile Z.19 has Wallis's writing on almost every page and also contains a small insert in Oughtred's hand; Savile Z.24 contains many of the annotations from Savile Z.19, transcribed and supplemented. 63. Wallis 1685, 121, 177; Wallis to Collins, 29 March 1673 and 8 April 1673, in Riguad 1841, II, 558, 559, 561. 64. Wallis 1657c. 65. Wallis to Collins, 12 April 1673, in Riguad 1841, II, 564–566. 66. Wallis's 1648 discourse on angular sections later formed Chapters 1–5 of his Treatise of angular sections. Chapters 6–9 were written about 1665, and the whole was published as an appendix to Wallis 1685. For a full discussion of the treatise see Scriba 1966. 67. Wallis 1685, 121. 68. Wallis also claimed that he had already seen how to factorize biquadratics into quadratics, a claim discussed further in Chapter 5. 69. Aubrey 1898, II, 282. 70. Wallis 1685, 67–125. 71. Wallis 1685, 67. 72. MS Savile 101, f. 14. 73. Wallis to Gregory, 4 April 1692, MS Savile 101, f. 15. 74. Whiteside 1967–81, I, 17 and note 7. Whiteside suggested that Newton was endorsing the 1694 English translation of the Clavis, but Wallis's letter to Gregory shows that he was rallying support for a corrected Latin edition. For further evidence of Wallis's propensity to suggest the wording of letters that went out over other people's signatures, see Fig. 3.7 and Chapter 6, note 60. 75. Whiteside, personal communication, 14 November 1998, suggests that the translation would have been done by a minor mathematician in need of the income, perhaps Joseph Raphson or John Colson.

Chapter 4 1. 2. 3. 4. 5.

Montucla 1799–1802, II, 111; Cantor 1894–1908, III, 4. Tanner 1967b, 238, 270–273; Lohne 1966, 186; Scott 1938, 133–145. For biographies of Harriot see Stevens 1900; Shirley 1983. For an account of this and related voyages see Milton 2000. Most of Harriot's surviving manuscripts are now held in the British Library, as Add MSS 6782–6789; copies are held in the University Libraries of Cambridge, Durham and Oxford. A smaller selection of papers remains at Petworth House, as MSS HMC 240–241; copies are held in the West Sussex Record Office, Chichester. For a survey of the scientific papers and their contents see Lohne 1979; for the history of the papers and attempts at publication see Shirley 1983, 1–38.

NOTES

6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22.

237

Harvey 1593, 190; Hues 1594, 166. For other contemporary references to Harriot see Quinn and Shirley 1969. For biographies of Harriot's companions see Shirley 1983, 358–379, 388–424. Add MS 6788, f. 117–117v, reproduced in Pepper 1967a, 290. Pepper dates the letter to 1586, but Viète was dismissed from the court in Paris in 1584 and lived in or near Tours for the next ten years. He returned to Paris only in 1594, and only from that year were his books printed there. It seems more plausible, therefore, that Torporley's letter was written some time after 1594. John Pell to an unknown recipient, 12 October 1642, Halliwell 1841, xv; Aubrey 1898, II, 263. Add MS 6782, ff. 482–483. Harriot's notes on Viète's Zetetica (1593) can be found, for example, in British Library Add MS 6782, ff. 438–481 and on Apollonius Gallus (1600) in Add MS 6785, ff. 50–72. Recorde 1557; see also Cajori 1928–29, 297–298. The symbols < and > were simplified into their modern forms within a few years of Harriot's death. The simplified forms can already be seen in Warner's hand in British Library Add MS 4394, f. 392, and in Torporley's in Sion College MS Arc L.40.2/L.40. The modern division sign was not introduced until some years later, in Pell and Rahn 1668. See Cajori 1928–29, 229–250. The first appearance of the ± sign in print was in Oughtred 1631; see Cajori 1928–29, 245. Viète 1646, 7–8; Viète 1983, 23. ‘Oporteat A plano/B addere Z quadratum/G. Summa erit G in A planum + Z quadratum/B in G’, and ‘Vel, B in G oporteat adplicere ad A planum/D. Ducta utraque magnitudine in D, ortiva erit B in G in D/A plano’; Viète 1646, 8. Harriot's versions of these statements are in British Library Add MS 6784, f. 324. Instructions on (i) changing terms from one side to another and (ii) reducing the leading coefficient to one were given by al-Khwārizmī in Al-jabr w'al-muqābala and were repeated in many sixteenth-century texts, for example Peletier 1554; Mennher 1556; Recorde 1557; Nũnez 1564. Advice on (iii) division by excess powers of the unknown can be found in, for example, Recorde 1557; the earliest instance I have found of it given as an explicit rule is in Perez de Moya 1573. Rules (i) to (iii) were the first of the ten rules for reducing equations given by Simon Stevin in Stevin 1585, 63–65. Viète 1646, 9; Viète 1983, 25–27. ‘Proponatur A quadratum minus D plano aequari G quadrato minus B in A. Dico A quadratum plus B in A aequari G quadrato plus D plano, neque per istam transpositionem sub contraria adfectionis nota aequalitatem immutari’; Viète 1646, 9. Harriot's version is in British Library Add MS 6784, f. 325. Harriot did not attempt to prove that all polynomials can be written as products of factors, nor did he pose the question, for he was interested only in the restricted class of polynomial equations with one or more positive real roots.

238

NOTES

23. Cajori 1928, 317–320; Pycior 1997, 54–64; 57, 64. 24. The discriminant of the equation x3 − 3p2x = 2q2 is q6 − p6, which can be used to discover the number of distinct real roots. In Harriot's equations, b and c are assumed positive, so the conditions c > b, c = b and c < b correspond to q6 − p6 > 0, q6 − p6 = 0 and q6 − p6 < 0. 25. Cardano and later writers sometimes used the transformation x′ = k/x to eliminate the cube term from a quartic already lacking a linear term (see Cardano 1993, 246), but Harriot used x′ = x − a to eliminate the cube term from a general quartic x4 + 4ax3 + bx2 + cx + d = 0. 26. The original of this letter is now lost but it is reproduced in Stevens 1900, 121–122; Shirley 1983, 1–2. 27. The original Will is lost but the Probate Copy was found in the late nineteenth century by the American researcher Henry Stevens, see Stevens 1900, 165–178; Tanner 1967a, 1–16. For full transcripts of the Will see Stevens 1900, 193–203; Tanner 1967b, 244–247. 28. Shirley 1983, 413–414. 29. British Library Add MS 6789, ff. 448–450; reprinted in Tanner 1969, 346–349. 30. The Congestor survives in two manuscript copies. The first was held for many years at Sion College in London, where Torporley spent his final years, but was transferred along with all other Sion College manuscripts to Lambeth Palace Library in 1996. The Sion College MS catalogue listed it under the title Congestor analiticus; Anthony Wood in his account of Torporley referred to it as Congestor opus mathematicum; Wood 1691–92, II, 525. Its modern shelfmark is Sion College MS Arc L.40.2/L.40, ff. 1–34v. The second copy is in the Macclesfield collection, acquired by Cambridge University Library in 2001. For a detailed account of the history and contents of the Congestor see Tanner 1977. 31. Congestor, f. 5. 32. Congestor, ff. 5–25. 33. Congestor, ff. 26–34v; see Tanner 1977, 419–428. 34. ‘ … me licet hostis inter alia convitia et hoc criminaretur domino Petworthiae quod essem dialecticus ignarus’, Sion College MS Arc L.40.2/L.40, f. 11; Halliwell 1841, 114. 35. ‘… hominis per eos in coelum sublati…’, Sion College MS Arc L.40.2/L.40, f. 8; Halliwell 1841, 110. 36. ‘Ad mathematices studiosos’, Harriot 1631, 180 and British Library Add MS 4395, f. 92, reproduced in Prins 1992, Fig. 5. In manuscript the paragraph is entitled (verso) ‘Praefatio ad Opus Harrioti’ and so was originally intended as a preface rather than an endnote. 37. Bodleian Library MS Rigaud 35, f. 183. Rigaud's identification was accepted without question by Tanner in Tanner 1967a, 9; Tanner 1969, 342–345. 38. Torporley formed the letter ‘c’ like an ‘r’ so that his algebra appears, oddly, to use the letters a, b, r, d. His ‘r’, on the other hand, resembled the Greek ‘k’. His ‘e’ was also written in the Greek style as ‘Q ’. None of these features is present in the endorsement of Warner's paragraph, and the writing is altogether more looped and flowing than Torporley's.

NOTES

239

39. Lohne 1966, 204, also tentatively suggested that the handwriting of the imprimatur was Aylesbury's. I have compared the handwriting of the endorsement in British Library Add MS 4395, f. 92, with that in the list of papers handed over to Torporley, in Add MS 6789, ff. 448–450. The two scripts are very similar, but the formation of certain letters, in particular the letter p, is different, so the identification cannot be made with certainty. 40. Aylesbury to Percy, 5 April 1632, Add MS 4396, f. 90; Halliwell 1841, 71. 41. For a modern assessment of Warner and his work see Prins 1992. 42. Harriot 1631, ‘Praefatio ad analystas’, opening paragraph, translated in Stevens 1900, 151–152. 43. Harriot 1631, 10. The Praxis helped to standardize both the inequality signs and Robert Recorde's = sign. 44. British Library Add MS 6783, f. 183. Harriot marked his sheets in the top right or top left corner as d.1), d.2), d.3), etc. 45. The first appearance of a negative root in Harriot's Treatise on equations is a = −f in a sheet marked d.7.2°), British Library Add MS 6783, f. 204. Harriot used 2° (secundo) to indicate that this sheet was a later addition to d.7). 46. Harriot 1631, 12–15. 47. Harriot 1631, 16–26. 48. Harriot 1631, 29–46. 49. Harriot 1631, 52–77. 50. ‘reductiones tamen earum cum in autographis obscurius traditae sint, ad meliorem inquisitionem referendae sunt.’; Harriot 1631, 46. 51. British Library Add MS 6783, f. 174. 52. For a similar example from Harriot's previous folio, see Seltman 2000, 166; see also Lohne 1966, 200. 53. Sion College MS Arc L.40.2/E.10, ff. 7–12. The Latin text, not always accurately transcribed, can be found in Halliwell 1841, 109–116. 54. Sion College MS Arc L.40.2/L.40, ff. 35–54v. 55. Torporley's Summary has received even less attention than his Corrector. Stevens 1900, 170, briefly described it as comprising ‘examples of Algebraic processes’. Seaton 1956 mentioned it in a non-mathematical context. Tanner 1974, 100–101, was the first to recognize its importance: ‘Torporley has meticulously crammed symbol for symbol well over 150 large sized pages of Harriot's mathematical writings into the twenty folios (35 to 55)’; see also further references in Tanner 1980, 137, 148. 56. Sion College MS Arc L.40.2/L.40, f. 42v. Problems 16, 17 and 18 appear in Section 3 of the Praxis, 43–45. 57. Tanner 1974, 100, wrote: ‘It is, however, astonishing and gratifying for us who are thus enabled to gather together in correct sequence correlated pages with relatively little further toil.’ In note 48, Tanner made a preliminary list of the relevant manuscript sheets. 58. ‘Primo accurata tractatio irrationalius surdorum sive, ut ille vocat eos, radicalium numerorum’; Sion College MS Arc L.40.2/ E.10, f. 8; Halliwell 1841, 110. 59. Sion College MS Arc L.40.2/L.40, ff. 35v–41.

240

NOTES

60. 'In ipso analyticae artificio contentus trimembri divisione inscribit primam ejus partem ita. De generatione aequationum canonicarum sub paragrapho d) compaginatis ad illud argumentum chartis 21 cum appendiculis duobus de multiplicatione radicum. Secunda pars autem sub titulo De resolutione per reductionem, habet paragraphum e) chartas 29. Item f α) chartas 7: f β) chartas quoque 7: et succedens illis in chartarum numeratione, f γ) ad chartam f 18 γ) cum appendicula sub lemmata duplici non illa contemnenda licet a suis omissa: Deinde f δ) chartae 8. f Q)chartae 4. f ζ) item 4: Postremo chartae novem continentes reductiones veterum ad Harrioti methodum revocatas. Sed tertiam partem (non ita studio dissentiendi) cum Vieta inscribit. De numerosa potestatum resolutione, et recte merito. Non totus fere est Vietaeus per exempla singula. Et supposito paragrapho a) et in chartis 13 sunt exempla tria quadratica quorum primum est suum, duo reliqua sunt Vietae, quinque cubicae omnia Vietae praeter primum. Et quinque quadrato quadratica quorum quartum est suum, reliqua Vietae. Et sunt ista secundum Vietae methodum aequationum omnino affirmantium. Altera ejus pars sub paragrapho b) in chartis 12 habet cum Vieta habet analyticam potestatum affectarum negate quadratica b 1) b 2) b 3) cubica b 4) ad b 10) quadrato-quadratica b 10) b 11) b 12). Tertia ejus pars sub paragrapho c) habet 18 chartas, tractat analysin potestatum avulsurum cum Vieta, ubi radices sunt multiplices et singularum limites demonstrantur. Exempla hujus sunt quadratica duo, cubico-cubica 4, quadrato-quadratica duo.' Sion College MS Arc L.40.2/E.10, ff. 9–9v; Halliwell 1841, 111. 61. Halliwell has wrongly transcribed this phrase as ‘cum Vieta in suo libro’ instead of ‘cum Vieta inscribit’; see Halliwell 1841, 111. 62. In Halliwell's transcript, the letter a is missing, but it is clear in the original. Harriot himself did not give a letter to his first section, but Torporley naturally adopted the letter a in keeping with Harriot's usage in subsequent sections. 63. The second subdivision of Part three. 64. The third subdivision of Part three. 65. ‘Avulsed powers’ is the term used by Viète for equations of the form bxm − xn = 0 when m < n. 66. See, for example, British Library Add MS 6002, ff. 4. 28v, 43–44. Photocopies of Cavendish's mathematical papers are kept alongside the Harriot manuscripts in the University Libraries of Cambridge, Durham and Oxford. 67. See Gaukroger 1995, 331. Beaugrand wrote three pamphlets against Descartes, see Descartes 1897–1913, II, 460, 508. 68. ‘Et ainsi i'ay commencé où [Viète] avoit achevé; ce qeu i'ay fait toutesfois sans y penser, car i'ay plus feűilleté Viete depuis que i'ay receu vostre derniere, que ie n'avois iamais fait auparavant, l'ayant trouvé icy par hazard entre les mains d'un de mes amis’; Descartes to Mersenne, [? December 1637] in Descartes 1897–1913, I, 477–480; 479–480. ‘… c'est Henriotti, que ie pensois que Gillot eust emporté avec luy,…. I'avois eu desir de voir ce livre, a cause qu'on m'avoit dit qu'il contenoit un calcul pour la geometrie, qui estoit fort semblable au mien’; Descartes to [Huygens], [December 1638], in Descartes 1897–1913, II, 455–457; 456. 69. The convention of using xx for x squared was retained on the continent for much of the seventeenth century and also by Newton, the first English writer fully to adopt

NOTES

70. 71. 72.

73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84.

85.

241

Descartes' notation. English writers both before and after Newton used Harriot's a and e, rather than Descartes' x, y and z; see, for example, Moore 1650; Gibson 1655; Brouncker 1658, 1668; Dary 1664; Pell and Rahn 1668; Kersey 1673–74; Leybourn 1690; Alexander 1693. Harriot, British Library Add MS 6783, f. 130; Descartes 1897–1913, VI, 449. Both Harriot and Descartes went on to solve the final quartic for both positive and negative roots. See also the same method in the Praxis, 102–116. Wallis 1685, 198. ‘I purpose, God willing, to set forth other peeces of Mr Harriot, wherein, by reson of my owne incumbrances I must of necessity desire the help of Mr W.’: Aylesbury to Percy, 5 April 1632, Add MS 4396, f. 90; reprinted in Halliwell 1841, 71, but wrongly supposed by Halliwell to be from Torporley to Percy. In 1635, Hartlib recorded in his Ephemerides Henry Gellibrand's remark that ‘Mr Warner hase all Hariots MS. and is setting some of them forth’; Hartlib papers, Ephemerides 1635, 29/3/41A. Hartlib papers, Ephemerides 1639, 30/4/9B. British Library Add MS 4278, f. 321, printed in Hervey 1952, 89 and Lohne 1966, 203. In Add MS 6083, ff. 403–455 there is a complete copy in Cavendish's hand of Harriot's De numeris triangularibus, with references to ‘Mr. Hariot's loose papers’ at ff. 403v, 404, 429v. An anonymous writer later remarked that ‘coming to London, [Aylesbury] found his Library, wherein were many rare and curious books, plundered’; MS Rawlinson B.158, f. 153. See Shirley 1983, 7–9. Pell to Cavendish, 7/17 September 1644, British Library Add MS 4280, 107–109, printed in Halliwell 1841, 80. Hartlib papers, Ephemerides 1650, 28/1/62A. Ward's main interest appears to have been in Warner's optics, which he later accused Hobbes of plagiarizing; see Ward 1654, Introduction and 247. Wallis to Aubrey, 20 July 1683, MS Aubrey 13, f. 242. Collins to Gregory, 25 March 1671, in Rigaud 1841, II, 219. See Chapter 3, note 40. Thorndike to Pell, 23 December 1652, and a conversation between Pell and Thorndike, recorded by Pell, 17 January 1653, both in British Library Add MS 4279, ff. 275–276v. Thorndike's letter of 23 December is printed in Halliwell 1841, 94. Hartlib papers, Ephemerides 1653, 28/2/49A. Northamptonshire Record Office, IL 3422, bundle VI, ff. 1–23. The papers were first noticed in 1997 by Dr Timothy Raylor who happened to recognize Warner's handwriting; see Clucas 1997, 6–7. Two letters in Isham's Correspondence imply that something was delivered from Thynne to Isham in 1651, but are secretive as to what it might have been; see Isham Correspondence, Northamptonshire Record Office, numbers 305 and 306. All the sheets except f. 5 are in the hand of Warner. Sheets 4, 12 and 18–23, on geometry or optics, are marked ‘T. H.’. Sheets 2, 6, 9 and 14–17 are also concerned with mathematical or optical problems known to have been studied by Harriot.

242

NOTES

86. IL 3422, bundle VI, f. 11. 87. Wallis described Harriot's method as follows: 'To each part of his Quadratick Equation, aa ± 2ba = ± cc; [Harriot] adds, the Square of half the Coefficient, bb, thereby making the Unknown part, a Compleat Square in Species equal to a Known Quantity.

And consequently, the Square Root of that, equal to the Square Root of this.

88. 89.

90. 91. 92. 93. 94. 95. 96.

97. 98. 99.

which being known; the value of a is known also.'Wallis 1685, 134. Collins to Gregory, early 1668, in Gregory J. 1939, 45. ‘An inventorie of the papers of Mr Warner’, reprinted in Halliwell 1841, 95. The inventory lists 23 items, most of them on coinage or logarithmic tables. Item 18 is entitled De resectione spatii, a topic treated more than once by Harriot and found several times among Warner's papers. Item 22 is ‘A bundle intituled “Mr Protheroe” ’. For a detailed analysis of the inventory see Prins 1992, 25–27. Collins to Vernon, [Jan/Feb 1671], in Rigaud 1841, I, 153. For the dating of this letter see Chapter 3, note 51. Neither Monsieur Garibal nor the piece that he owned has so far been identified. Hartlib papers, Ephemerides 1653, 28/2/80A. For other contemporary speculations about Harriot's Will and papers see Collins to Vernon, [Jan/Feb 1671], in Rigaud 1841, 1, 153 and Aubrey 1898, I, 285. For a full account of the Royal Society searches see Shirley 1983, 7–9. Wallis to Aubrey, 8 March 1684, MS Aubrey 13, f. 245; ‘Thomas Harriot’ in Wood 1691–92, I, unpaginated. Wallis to Aubrey, MS Aubrey 13, f. 245. The mathematical papers of Warner and Pell are now classified together in the British Library catalogue as ‘The Pell collection’. Pell's mathematical papers are British Library Add MSS 4397–4404, 4407–4431. Warner's papers are British Library Add MSS 4394–4396, and are strangely described in the catalogue as ‘Pell collection, second series: mathematical collection of John Pell chiefly in the hand of Warner’. British Library Add MS 4394, f. 392. See, for instance, British Library Add MS 4413, f. 224, Add MS 4415, f. 83. ‘Quibus usque adeo exacte consentiunt ea, quae habet Cartesius in magna parte libri tertii Geometriae suae, ut ex Harrioto descripta non levis suspicio sit. Certe Pellius et Wallisius id pro certe habere videntur’; ‘to [Harriot's principles], everything that Descartes has in the greater part of the third book of his Geometria, so far agrees exactly, so that the suspicion that he wrote it from Harriot cannot be taken lightly. Certainly Pell and Wallis seem to hold that view for certain.’; Leibniz 1686, 285. Leibniz became acquainted with Pell when he visited Boyle on 2 February 1673; see Leibniz 1899,

NOTES

100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112.

113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123.

243

74–78; also Journal Book of the Royal Society for 24 April 1712, cited in Manuel 1980, 322. Wallis 1656b, ‘Dedicatio’; Wallis 1693–99, I, 294, translation JS. Wallis 1685, 125–126. Wallis 1685, 128. Wallis 1685, 134. Wallis 1685, 129. Lohne 1966, 186, was the first to note that Wallis ‘gives a description which conforms much better with Harriot's algebraical manuscripts (notably Ms. Add 6783) than with the printed praxis’; Lohne went on to question Wallis's moral character and trustworthiness. Wallis to Morland, 12 March 1689, in Wallis 1693–99, II, 209–210. For sheet d.13.2°), see Lohne 1966, 195–196. The full text is reprinted in Shirley 1983, 10–11, but the shelfmark is incorrectly quoted. The copy used and annotated by Wallis was originally owned by Charles Cavendish, and given by him to Robert Payne. After Payne's death the book was acquired for the Savile Library where it is now Savile O.9. Wallis to Aubrey, 20 July 1683, MS Aubrey 13, f. 242. Wallis 1685, 198. Wallis's detailed reference to Harriot's treatise in his 1689 letter to Morland implies that the material was still in his possession four years after Pell's death. This raises questions as yet unanswered about the subsequent fate of that material. ‘Certe nemo omnium judicaverit, haec ante ab Harrioto non fuisse tradita. Quid Pellius ipse senserit, ego aliquatenus intelligo; ut qui me hac de re saepius compellavit; & ex cujus ore descripsi quod hac de re dixi; eique postquam erat descriptum, ostendi, (examinandum, immutandum, emendandum pro arbitrio suo, siquid alias dictum malit) antequam prelo subjiceretur, totumque illud quod inde prodiit, assentiente & approbante Pellio dictum est.’; Wallis 1693–99, II, opening paragraph of ‘De Harrioto addenda’, following the preface (unpaginated). Pell to Cavendish, 2/12 March 1646, British Library Add MS 4280, ff. 117–118, printed in Hervey 1952, 77–79. Pell 1638. Collins to Oldenburg for Leibniz, [1676], in Rigaud 1841, I, 247. Pell 1682 and Descartes 1682. For translations of both pieces see Fauvel and Gray 1987, 310–314. Wallis 1685, Chapters 35–38. Wallis 1685, 135, 141, 142, 144, 200, 208–209, respectively. Wallis 1685, Chapters 32–34. Wallis wrote the necessary conditions by hand into his own copy of the Praxis, Savile O.9. Wallis 1685, 199–200. Sion College MS Arc L.40.2/L.40, f. 43, 47v, 54v. Pycior 1997, 54–64; 57, 64, for example, dismissed the Praxis as ‘little more than a basic introduction to an equation theory that recognized only positive real roots’.

244

NOTES

124. Hutton 1796, 91 and Hutton 1812, 286. 125. Collins to Vernon, [Jan/Feb 1671], in Rigaud 1841, I, 153.

Chapter 5 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

There is as yet no accurate modern biography of Pell. For Aubrey's information on Pell, much of it provided or checked by Pell himself, see Aubrey 1898, II, 121–131. The original nomination, signed by Brouncker, is preserved in British Library Add MS 4423, f. 237. Pell made notes on the work of almost every sixteenth- or seventeenth-century English or continental mathematician of note, including Cardano, Nuñez, Ramus, Bombelli, Stevin, Viète, Harriot, Oughtred, Briggs, Gellibrand, Hérigone, Bachet, Descartes, Fermat, Girard, Gibson, Barrow, Mercator, Wallis, Huygens, Billy, Frenicle, Stampioen, Dary, Baker and others. Collins to Beale, 20 August 1672, in Rigaud 1841, I, 196–197. For the seventeenth-century history of ‘Pell’s equation' see Chapter 7. For a comprehensive account of Pell's publications see Malcolm 2000. Collins to Oldenburg for Leibniz, [1676], in Rigaud 1841, I, 247. Busby already held the papers of Walter Warner, see Chapter 4, note 98. British Library Add MS 4397–4404, 4407–4431. British Library Add MS 4278–4280. Pell's early booklets are simply made from one or two sheets of paper folded to make four or eight pages. There are several of them in, for example, British Library Add MS 4431. British Library Add MS 4397, f. 1. We do not have Pell's letter to Briggs, but its contents can be deduced from Briggs' reply, see Briggs to Pell, 25 October 1628, British Library Add MS 4398, f. 137; the letter is printed in Halliwell 1841, 55–57. British Library Add MS 4431, f. 70. British Library Add MS 4431, f. 73. Aubrey 1898, II, 129–130. John Williams (1582–1650) became Bishop of Lincoln in 1621, but was suspended and imprisoned in 1637. For Hartlib and his aims see Dury 1651, Preface; Turnbull 1947; Clucas 1991. Pell 1638. Pell admitted his authorship of the Idea in a letter to Mersenne in 1639, and publicly in his inaugural lecture at Breda in 1646, see Pell to Mersenne, 2 November 1639, in Mersenne 1932–, VIII, 580–581 and Breda 1647, 181. The only known surviving copy of the Idea in English is in the British Library, pressmark 528 n. 20 (5*). A single copy of the Latin version has also recently been discovered, see the article on Pell by Scriba in the new DNB. In 1650 the Idea was reprinted as an appendix to The reformed librarie keeper, by Hartlib's closest associate, John Dury, see Dury 1650, 33–46. In 1682 it was reprinted in Latin, with the responses of Mersenne and Descartes, in Philosophical collections5, 127–145. See also Wallis P. J. 1967; Fauvel and Gray 1987, 310–313; Malcolm 2000, 280–282.

NOTES

245

20. According to William Brereton, Pell wrote ‘a quire of papers’ on the subject; Collins to Wallis, 2 February 1667, in Rigaud 1841, II, 474. Amongst Pell's papers there are several lists of mathematical texts; see, for instance, British Library Add MS 4394, f. 108; Add MS 4407, ff. 44–53v; Add MS 4415, ff. 316–317. 21. Hartlib papers, Ephemerides, 1639, 30/4/9B. 22. Aubrey 1898. II, 291–293. 23. See Chapter 4, note 84. 24. William Boswell (d. 1650) had once been a Secretary to John Williams, see note 17; he was appointed Ambassador to The Hague in 1632. 25. See Van Maanen 1986. 26. Pell to Cavendish, 7/17 September 1644, British Library Add MS 4280, f. 109. 27. See Toomer 1996, 183–187. Pell worked on Books V–VII of Apollonius using an Arabic text borrowed from Christian Ravius, lecturer in oriental languages at Utrecht. It is not clear how much Arabic Pell knew, and it was not impossible for a mathematically astute reader to edit Apollonius with little knowledge of the language, as Halley later also did; see Halley 1706; Halley 1710. 28. Pell to Cavendish, 2/12 March 1646, British Library Add MS 4280, ff. 117–118; Cavendish to Pell, 21/31 March 1646, Add MS 4278, f. 246. Both letters are printed in Hervey 1952, 77–80. 29. Sassen 1966; Lindeboom 1971. 30. Breda 1647, 168–183. The poster in Fig. 5.3 is now in the archives of Lincoln Cathedral. It was deposited there in 1660 by Michael Honeywood, Dean of Lincoln, who like many other Anglican clergymen lived in exile in the Netherlands during the interregnum. 31. Silas Titus (1623–1704) fought for Parliament in the early battles of the Civil War but became a devoted Royalist after attending Charles I in captivity at Holdenby. In 1649 he was sent as one of four representatives of the English Presbyterians to the negotiations between Charles II and the Scottish Commissioners in Breda, see Gardiner 1894, I, 203–228; 205. 32. ‘Quant à moy je ne fais que venir icy, ne l'ayant jamais veu autrement que quelques fois chez Monsieur Pell à Breda qui ne me l'a jamais voulu prester, ny m'en dire une sentence definitive encor qu'il l'ayt eu assez long temps.’; Huygens to Mersenne, 20 April 1648, in Huygens 1888–1950, II, 566; see also Van Maanen 1986, 345 and note 58. 33. Vaughan 1839. 34. Thompson to Pell, 27 November 1658, British Library Add MS 4279, f. 273; printed in Vaughan 1839, II, 478–479 and in Halliwell 1841, 95–96. 35. Pell was ordained as Deacon on 31 March 1661 and as Priest on the 10 June following, on both occasions by Robert Sanderson, Bishop of Lincoln; see Binall 1961, 68, 70. 36. Rumours have persisted that some of Pell's papers may still be at Brereton Hall, but there are none there, nor are there any relating to the house at that period in the Cheshire Record Office. For the history of the Brereton family see Moir 1949. 37. See Scriba 1974. 38. See Malcolm 2000, 286–287.

246 39. 40. 41. 42.

43. 44. 45. 46.

47. 48. 49. 50. 51. 52. 53. 54.

NOTES

Pell to Brancker, 11 April 1666, British Library Add MS 4278, f. 80, cited in Scriba 1974, 268. British Library Add MS 4414, f. 2. Collins to Wallis, 2 February 1667, in Rigaud 1841, II, 472. The Principia matheseos universalis of the Danish mathematician Erasmus Bartholin was an introductory text on algebra first published in Leiden in 1651 and reprinted by Van Schooten ten years later, see Bartholin 1651 and Van Schooten 1659–61, II, 1–48. A ‘Cambridge scholar’ discussed Bartholin's Principia matheseos universalis with Collins in a London bookshop early in 1667, and declared it superior to the opening of the Pell and Rahn Introduction to algebra, see Collins to Pell, 9 April 1667, Rigaud 1841, I, 125–126. Whiteside 1967–81, II, 279, suggested that the scholar may have been Barrow. Many years later Newton endorsed Bartholin's text as one of the books to be read as a prelude to his own Principia, see Newton to Bentley, c. July 1691. in Turnbull 1959–77, III, 155–156 (Turnbull, in note 5, suggested that the Bartholin text was the Selecta geometrica of 1674 (actually 1664) but Newton specifically referred to the 1661 Van Schooten edition). Bartholin visited England and was personally known to some of the English mathematicians: he sent both Wallis and Oldenburg copies of his treatise on Iceland spar (Bartholin 1669) and Wallis's copy, signed by Bartholin, is in the Bodleian Library as Savile G.25. Collins to Brancker, June [1667], in Rigaud 1841, I, 134–135. Pell and Rahn 1668, 80. Pell and Rahn 1668, 100; Diophantus, Arithmetica, II, problem 25. British Library Add MS 4418, f. 210. The scrap is 10 cm × 12 cm in size, in Pell's handwriting on both sides. Printing of the Arithmetica infinitorum was not completed until the summer of 1655 (the [Dedicatio] is dated 19 July 1655), so the Quadratum circuli referred to in Pell's note was not the full text but the printed sheet dedicated to Oughtred that eventually appeared at the beginning of the Arithmetica infinitorum after the Title page and ‘Dedicatio’; see also Wallis 1693–99, I, 362. Wallis noted that he received this sheet from the printers at Easter 1655. which is consistent with it being sent to Pell or Hobbes at the end of April 1655. Wallis MS Savile 101, ff. 82–83v; Wallis's notes are on Pascal's Lettres a Dettonville published in 1659. See British Library Add MS 4411, ff. 359–367v; Add MS 4412 ff. 197v–202; Add MS 4413, ff. 38–52; Add MS 4425, ff. 161, 196–206v, 367–368, and elsewhere. British Library Add MS 4413, f. 52. British Library Add MS 4425, f. 367. British Library Add MS 4425, f. 368, my italics. ‘Ex Iohannis Wallisii autographo exscripsi, Aprilis 14. 1663. quod reddidi Cap. Tito. Novemb. 14. 1663’; British Library Add MS 4425, f. 161. See also Wallis 1685, 225–227. ‘Ergo conveniunt DIW et MIP in omnibus coefficientibus aequtionis’; British Library Add MS 4425, f. 206v, see also ff. 202v, 203v. British Library Add MS 4411, ff. 359–367v. The eventual solutions were a = 2.525 513 986 744 158, b = 2.969 152 768 619 848, c = 3.240 580 681 617 174. See Wallis 1685, 225–252.

NOTES

247

55. Gregory to Collins, 17 January 1672 and 6 August 1672, in Rigaud 1841, II, 231, 242. 56. Hooke 1935, 322. Mr Baker was Thomas Baker, a friend of Collins and author of The geometrical key (1684). It may have been this diary entry that led Halliwell 1841. xii, wrongly to attribute the solution to Collins himself; Halliwell also suggested, but without offering supporting evidence, that the problem originated with Harriot. 57. Wallis 1685, Chapters 60–63. Chapter 60 gives the general solution exactly as in British Library Add MS 4425, f. 161 (see note 52); Chapter 61 is then a line by line commentary on Chapter 60; Chapters 62 and 63 contain the lengthy working out of numerical solutions from British Library Add MS 4411, ff. 359–367v (see note 54). 58. Wallis 1685, 225. 59. The mathematical friendship between Pell and Titus continued over many years, for Pell noted that he borrowed sheets of Dechales' Cursus seu mundus mathematicus (1674) from Titus in July 1675, and returned them the following year, British Library Add MS 4416, ff. 54–56. In 1679, Titus became the implacable political opponent of Lord Stafford, once the young William Howard for whom Oughtred had written his Clavis mathematicae, and helped to bring about Stafford's execution on 31 December 1680, fifty years almost to the day after Oughtred had dedicated the Clavis to him on 1 January 1631. 60. This work was first published in the second edition of Horrocks 1673. For a modern assessment of Chapters 10 and 11 of A treatise of algebra see Fowler 1990. 61. British Library Add MS 4416, ff. 31–31v. 62. Metius 1611, 69. This is an earlier publication of the ratio than that in Metius 1625, 88–89, cited in the DSB article by Struik. See also Metius 1626, 88–89. 63. Wallis 1685, 36. 64. Wallis 1685, 36. 65. Aubrey 1898, 1,198–203. 66. Collins to Baker, 19 August 1676, in Rigaud 1841, II, 8–9. This is the only mention in the Rigaud letters to the problem of finding fractional equivalents for π. Other references to ‘Dr Davenant's problem’ are to an entirely different problem, to do with polynomial equations. 67. Descartes 1637, 383–386; Wallis 1685, 208–212. 68. Wallis 1685, 209. The same claim was made by Wallis in letters to Collins, 8 April 1673 and 12 April 1673, in Rigaud 1841, II, 561, 576. 69. Cavendish to Pell, 21/31 March 1646 and 1 June 1646, British Library Add MS 4278, ff. 246, 255; Pell to Cavendish, 4/14 May 1646, Add MS 4280, f. 118. 70. British Library Add MS 6083, ff. 100v–101. 71. British Library Add MS 4425, f. 57, printed in Halliwell 1841, 105, but citing a pagination no longer in use. The item printed by Halliwell immediately afterwards is headed ‘Note on the solving of equations by John Pell’ but the original in Add MS 4432, f. 26, is not in Pell's handwriting and there is no reason to ascribe it to him.

248

NOTES

72. Wallis to Collins, 29 March 1673, in Rigaud 1841, II, 559. 73. Collins knew Kinckhuysen's work better than Wallis did, and told Wallis about one of Kinckhuysen's results on cubic equations, see Wallis 1685, 181. 74. For what little biographical information we have on Kinckhuysen see Kempenaars 1990. 75. Kinckhuysen 1661, 96; Whiteside 1967–81, II, 354. See also Scriba 1964, 48. 76. Kempenaars 1990, 246. 77. Collins to Pell, 28 August 1666 and 9 April 1667; Collins to Brancker, June [1667], in Rigaud 1841, I, 118, 126, 135–136. See also Scriba 1964, 49–50; Whiteside 1967–81, II, 279. 78. Mercator's Latin translation with Newton's annotations is interleaved into a copy of the Stel-Konst now held in the Bodleian Library in Oxford, shelfmark Savile G.20; both the translation and the annotations are reproduced in Whiteside 1967–81, II, 295–447. See also Scriba 1964, 50–53; Whiteside 1967–81, II, 280–291. 79. Newton to Collins, 6 February 1670, in Turnbull 1959–77, I, 24. Whiteside 1967–81, II, 281 and note II, assumed that the German gunner was translating Kinckhuysen's ‘analytic conics’ or De grondt der meet-konst but Pell's description (see note 80) makes it plain that it was the Geometria. 80. Pell for Oldenburg, 9 April 1672, British Library Add MS 4407, f. 118. 81. Oldenburg to Huygens, 6 May 1672, in Hall and Hall 1965–86, IX, 54–55. 82. Collins to Pell, 17 December 1667, British Library Add MS 4278, f. 326. 83. Collins to Pell, 6 February 1668, British Library Add MS 4278, f. 331. 84. Wallis 1685, 264–273; see also Wallis to Collins, 6 May 1673, in Rigaud 1841, II, 578–579. 85. Wallis 1685, 266; see also Schubring 1998, 140–142. If the construction of √bc is regarded as the construction of the sine of an angle, then the analagous construction of √−bc gives rise to the tangent of the same angle. 86. The modern parametric representation of a circle is in terms of sines (and cosines) while that of a hyperbola is in terms of tangents (and secants). 87. Wallis 1685, 272. 88. Collins to an unknown recipient, [1674], in Rigaud 1841, II, 481. 89. Collins for Leibniz, [1676], in Rigaud 1841, I, 248. 90. See Collins to Wallis, 2 February 1667, in Rigaud 1841, II, 472–473, and Collins for Leibniz, [1676], in Rigaud 1841, I, 243, 247–248. See also Collins to Gregory, [1670] and 25 March 1671; Collins to Wallis, 21 March 1671, in Rigaud 1841, II, 197, 219–220, 526–527. 91. Collins for Leibniz, [1676], in Rigaud 1841, I, 243–247; Collins to Gregory, 25 March 1671, in Rigaud 1841, II, 219. 92. Collins to Gregory, [1670] and 25 March 1671, in Rigaud 1841, II, 195, 220. See also Collins for Tschirnhaus, 30 September 1675, in Rigaud 1841, I, 216. 93. Wallis to Collins, 16 September 1676, in Rigaud 1841, II, 601–603. 94. Collins' correspondence is part of the Macclesfield Collection of letters and papers, which was in private hands from the early eighteenth century until it was acquired by Cambridge University Library in 2001. The letters have been inaccessible to

NOTES

95. 96. 97. 98. 99. 100. 101. 102.

249

researchers since they were transcribed and edited (not always correctly) by Rigaud in the 1830s, but will shortly be opened up for the first time to modern scholarship. Collins to Wallis, 20 September 1677: ‘You lye under a censure from diverse for printing discourses that come to you in private Letters without permission or consent as is said of the parties concerned’, Turnbull 1959–77, II, 242. See also Aubrey 1898, II, 281–282. Gregory to Collins, 23 July 1675, in Rigaud 1841, II, 268. British Library Add MS 4415, f. 2. It may have been Pell who encouraged Wallis to write a review of Mercator's Logarithmotechnia for the Philosophical transactions, see Wallis 1668d. Collins to Pell, 9 April 1667, in Rigaud 1841, I, 126. Collins to Beale, 20 August 1672, in Rigaud 1841, I, 196–197. See Jesseph 1999, 4. See Ross 1984, 7, 61–62; Aiton 1985, 91–94 Collins to Beale, 20 August 1672, in Rigaud 1841, I, 197.

Chapter 6 1. 2. 3. 4. 5. 6. 7.

8. 9.

The Arithmetica infinitorum was printed in 1655 but first appeared in 1656 in Wallis's first set of collected works, the Operum mathematicorum, Wallis 1656–57, II, 1–199. It was reprinted in his second set of collected works, the Opera mathematica, Wallis 1693–99, I, 355–478. Wallis 1685, 280–363. There are detailed accounts of the content of the Arithmetica infinitorum, but not of its contemporary significance and influence, in Nunn 1910–11 and Scott 1938, 26–64. Wallis 1656c, [Dedicatio]. See Giusti 1980; Andersen 1985. Andersen 1985, 355–358. Oughtred to Keylway, 26 October 1645, in Rigaud 1841, I, 65–66. Oughtred's correspondent in Paris was almost certainly Charles Cavendish, for in 1655 he told Wallis that Cavendish had shown him ‘a written paper sent out of France, in which were some very few excellent new theorems, wrought by the way, as I suppose, of Cavalieri’; Oughtred to Wallis, 17 August 1655, in Rigaud 1841, I, 87–88. Among Cavendish's papers in the British Library is a handwritten treatise entitled Elemens des indivisibles, Harley MS 6083, ff. 279–302. The text, in French, includes definitions, propositions and diagrams based on Cavalieri's, and it is tempting to conclude that this was the same ‘written paper’ that Cavendish showed to Oughtred. ‘Si sumatur series quantitatum Arithmetice-proportionalium (sive juxta naturalem numerorum consecutionem) continue crescentium, a puncto vel o inchoatarum, & numero quidem vel finitarum vel infnitarum (nulla enim discrimnis causa erit,) erit illa ad seriem totidem maximae aequalium, ut 1 ad 2.’; Wallis 1656c, Proposition 2. ‘Ergo, Triangulum ad Parallelogrammum (super aequali base, aeque altum,) erit ut 1 ad 2.’; ‘Item, Pyramidoeides vel Conoeides Parabolicum (sive erectum sit sive inclinatum,) ad

250

10. 11.

12. 13. 14.

15. 16. 17. 18. 19.

20. 21. 22.

23.

NOTES

Prisma vel Cylindrum (super aequali base, aeque-altum,) est ut 1 ad 2,’; Wallis 1656c, Propositions 3 and 4. Wallis 1656c, ‘Dedicatio’: ‘Ut enim ille suam, Geometria indivisibilium, ita Ego, methodum nostram, Arithmetica Infinitorum, nominandam duxi.’ Superscript notation for negative powers was first introduced by Nicolas Chuquet in his unpublished Triparty of 1484; see Cajori 1928–29, I, 102. Modern superscript notation for fractional and negative powers, a natural extension of the notation introduced by Descartes for integer powers, was first used by Newton, in Newton 1676a,b. Wallis 1656c, Propositions 53, 87. See Chapter 2, note 105. Torricelli had published his result for the solid of revolution of the rectangular hyperbola in Torricelli 1644, 115–116. Between 1644 and 1647 he obtained further results but never published them and they would not have been known to Wallis. Aristotle had claimed that the length of an arc could never be expressed in a finite ratio to the length of a straight line, that is, that rectification (straightening) was impossible. Descartes repeated the assertion: ‘Car, encore qu'on n'y puisse recevoir aucunes lignes qui semblent a des chordes, c'est a dire qui devienent tantost droites & tantost courbes, a cause que, la proportion qui est entre les droites & les courbes n'estant pas connuë & mesme, ie croy, ne le pouvant ester par les hommes, on ne pourrait rien conclure de là qui fust exact & assuré’; Descartes 1897–1913, VI, 412. Wallis 1656b, Proposition 48; Wallis 1656c, Proposition 45. Wallis 1685, 291–292. Wallis had introduced the symbol ∞ in exactly the same context on the opening page of De sectionibus conicis. See also Whiteside 1961a, 236–241. Formulae for the figurate numbers were first written down by Harriot in the form (nn + n)/2, (nnn + 3nn + 2n)/ 6, etc.; see, for example, his De numeris triangularibus et inde de progressionibus arithmeticis magisteria magna, undated, British Library Add MS 6782, ff. 107–146; 108–109. Folio 108 is reproduced in Lohne 1979, 294. It is not impossible that Wallis knew of Harriot's formulae but his own method of deriving them was considerably more laborious than Harriot's. The same formulae were known and described verbally by Fermat in 1636, see Mahoney 1973, 229–231, but Wallis was almost certainly unaware of Fermat's results. See Edwards A. 1987, 1–19. ‘Quamquam enim hanc spes non exigua visa est affulsisse, lubricus tamen quem prae manibus habemus Proteus tam hic quam superius non raro elapsus, spem fefellit.’ Wallis 1656c, Proposition 189, Scholium. ‘But we, shouting, fell upon Proteus, and threw our hands around him; nor had the old man forgotten his wily art. First he became a lion with noble mane, and then a serpent, a panther, a wild boar, he became rushing water and a soaring-leaved tree. But we held him firmly with patient mind until at last the cunning old man wearied and asked: “Which god conspired with you to seize me against my will?”’ Homer, Odyssey, Book IV. Wallis 1656c, Proposition 191.

NOTES

251

24. Although Wallis's conclusion was correct, his reasoning was not watertight: it is true that the fractions and approach each other as z becomes larger, but Wallis failed to take into account the increasing quantity that multiplies both of them. 25. Wallis 1685, 282, his italics. Wallis referred to Euclid ‘Book X onwards’. The key proposition is X.1, which is the foundation of several subsequent propositions. Mahoney 1973, 226 note 23, suggested that Newton in the Principia was the first to present this argument as a general lemma, but both Euclid and Wallis stated it explicitly. 26. A fraction with infinitely large numerator and denominator is strictly meaningless but Wallis had carefully described the process by which he arrived at it, and in that sense it was more soundly based than his earlier nonrigorous use of infinite quantities. The same fraction for 4/π was independently discovered by Mengoli in about 1659 and published in Mengoli 1672. 27. Nunn 1910–11 and Scott 1938, 26–64 both ended their detailed expositions of the Arithmetica infinitorum with the fraction for 4/π. 28. Wallis 1656c, following the Title page and ‘Dedicatio’, 195; Wallis 1693–99, I, 362, 477. 29. Oughtred to Wallis, 17 August 1655, in Rigaud 1841, I, 87–88. 30. Belief in the imminence of Doomsday was commonplace throughout Oughtred's lifetime but millenarianism was particularly strong during the Interregnum, 1649–60. The year 1656 was thought by some to be an especially likely date as it represented the number of years supposed to have elapsed between the Creation and the Flood, see Thomas 1971, 140–144. 31. British Library Add MS 4418, f. 210. 32. In 1652 Rooke was at Wadham College, Oxford, and was appointed Gresham Professor of Astronomy that same year. Rawlinson was at The Queen's College, Wood at Lincoln College and Wren at All Souls. 33. Huygens to Wallis, 21 July 1656, no. 316 in Huygens 1888–1950, I, 458–460. 34. Wallis to Huygens, 22 August 1656, no. 325, ibid., 476–480. 35. In his investigation of angular sections, Viète set up a table of ratios of double, triple, quadruple and quintuple angles and claimed that it could be continued in the same way indefinitely, exactly the sense in which Wallis used the term ‘induction’; see Viète 1983, 424. As a similar example of induction in Euclid, Wallis cited Elements V. 34. 36. Huygens to Wallis, no. 337, in Huygens 1888–1950, I, 494–495. 37. See Cajori 1929; Pycior 1987; Grant H. 1990, 1996; Probst 1993; Jesseph 1993, 1999. 38. Oughtred 1647, ‘Preface’. 39. Hobbes 1656, Introduction. 40. Hobbes 1656, 54. 41. Hobbes 1656, 46. 42. Wallis 1656d, 42. 43. Hobbes, 1656, 46. 44. Wallis 1656d, 43.

252 45. 46. 47. 48. 49.

50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

61. 62. 63. 64.

NOTES

Hobbes 1657; Wallis 1657d. Wallis 1656d, 49. Fermat to Digby, 20 April 1657, in Brouncker et al. 1658, letter 4. Mahoney 1973, 217, note 5. ‘Ce n'est pas que je ne l'approuve, mais toutes ses propositions pouvant estre demonstrées via ordinaria, legitima et Archimedea en beacoup moins de parolles, que n'en contient son livre. Je ne sçay pas, pourquoy il à preferé cette maniere par notes Algebriques à l'ancienne, qui est & plus convainquante, & plus elegante, ansi que j'espere luy faire voir à mon premier loisir.’; Fermat to Digby, 15 August 1657, in Brouncker et al. 1658, letter 12. For Fermat's application (and modification) of Archimedean methods to the quadrature of the spiral r = aθ and other curves in 1636, see Mahoney 1973, 218–228 and 233–239. Fermat to Wallis, August 1657, in Brouncker et al. 1658, letter 13. Wallis to Fermat, 21 November 1657, ibid., letter 16. Fermat to Digby, June 1658, ibid., letter 46, translated in Wallis 1685, 305. Wallis 1685, 305–306. See Mahoney 1973, 26–71. Fermat to Roberval, 4 November 1636, quoted in Mahoney 1973, 230–231. Fermat used the results only for n an integer or the reciprocal of an integer, whereas Wallis extended them to general fractional powers. For Fermat's derivation and use of his results, see Mahoney 1973, 231–239. De aequationum localium transmutatione et emendatione ad multimodam curvilineorum inter se vel cum rectilineis comparationem, Fermat 1679, 44–57. For discussion of Fermat's work on quadrature see Mahoney 1973, 244–267. Wallis 1685, 292. Wallis to Huygens, 24 November 1659, in Wallis 1659, 75–123; 91–96 and Wallis 1693–99, I, 542–569; 550–554. Aubrey in his biographical notes on Neile wrote: ‘Enquire of Dr Wallis of [Neile's] rare invention printed in one of his bookes: never before found out by man’, Aubrey 1898, II, 94. Van Heuraet 1659; see Van Maanen 1984. Wallis's publication of Neile's result was intended to settle the question of priority between Neile and Van Heuraet. The dispute flared again in 1673, however, at which point Wallis again took up the argument on Neile's behalf and persuaded both Brouncker and Wren to back him; see Wallis 1673. The supporting letter from Brouncker, preserved by the Royal Society, is actually in Wallis's hand, see Hall and Hall 1965–86. X, 291–292. I have recently discovered a draft of the letter from Wren amongst the papers of Pell, and that too is in Wallis's hand, British Library Add MS 4428, f. 314. It seems, therefore, that Wallis drafted both letters, and Wren and Brouncker needed to do no more than give a nod of consent. Fermat 1660. Fermat's De linearum curvarum was heavily annotated by Wallis in his copy of Fermat 1679 (Bodleian Library. Savile B.7). For discussion of Fermat's methods of rectification see Mahoney 1973, 267–281. Wallis 1685, 297, 298. Fermat to Digby, 20 April 1657, in Brouncker et al. 1658, letter 4. Pascal 1659, 90f.

NOTES

253

65. Descartes defined ‘geometric’ curves as those whose relationship to the axis could be expressed by a single equation, and believed the rectification of such curves to be impossible, see note 14, but excluded mechanical curves such as the spiral or quadratrix, which required more complex rules for their generation, see Fauvel and Gray, 1987. 344–345: Bos 1993, 37–57. Fermat, using a similar distinction, would not have considered the cycloid a ‘geometrical curve’, thus conveniently invalidating Wren's rectification. For further discussion on the construction of curves in the seventeenth century see Bos 1993, 23–36. 66. Wallis 1656c, Proposition 103. 67. The logarithmic properties of areas under a hyperbola were first noticed by the Belgian Jesuit A. A. de Sarasa from his reading of Gregory St Vincent's Opus geometricum of 1647; see Burn 2001. 68. Wallis 1668d. 69. Brouncker 1668. 70. Newton 1669. Though composed and sent to Barrow and Collins in 1669, Newton's De analysi remained unpublished until it appeared in Newton 1711. 71. Whiteside 1967–81, I, 8, note 22; see also Newton 1676b, 111, 130. 72. Newton 1664. 73. Wallis never attempted to find partial areas of a quadrant. Leibniz pointed out this limitation of Wallis's method after the Arithmetica infinitorum was reprinted in 1695; see Leibniz 1696, 252 and Whiteside 1961a, 322–324. 74. Whiteside 1967–81, I, 106, note 52; Cohen 1974, 46. 75. Before Newton, a method of quadrature was literally a method of finding an equivalent square, that is, of comparing one area with another. 76. Newton 1664, 108. 77. Newton 1664, 108, note 58. 78. Newton 1665; see also Whiteside 1961c and Dennis and Confrey 1996. 79. Newton 1665, 130. 80. Newton 1665, 131. 81. Newton 1675–76, 52–69. 82. See, for example: British Library Add MS 6782, ff. 112, 116 (Harriot); Add MS 4396, f. 77 (Warner); Add MS 4415, f. 113 (Pell). The latter, a long folded sheet, contains an exceptionally long constant difference table in 60 rows and 4 columns. Numerous smaller examples are to be found throughout Pell's papers. 83. Briggs 1624; see Whiteside 1961b. 84. Fraser 1927, 58, suggested that Newton drew his ideas from Briggs, but Whiteside 1967–81, I, 13, note 32, regards his evidence as ‘flimsy and circumstantial’. 85. Newton actually used x and y as his integers. 86. Newton to Oldenburg, 24 October 1676, letters 188, 189 in Turnbull 1959–77, II, 110–163. 87. British Library Add MS 6782, f. 110. Harriot's formula, in keeping with the contemporary understanding of figurate numbers, was intended only for positive values of m. Like Newton later, however, he also extended his tables to include negative values of m, see Add MS 6782, f. 330, reproduced in Lohne 1979, 294.

254

NOTES

88. Wallis 1685, 330–346. Wallis published the Epistola prior almost in its entirety together with some supporting material from the Epistola posterior (Turnbull 1959–77, III, 220, note 4, is inaccurate on this point). 89. Newton 1676b, 111, 130; Newton to Wallis, July 1695, letter 519 in Turnbull 1959–77, IV, 140. 90. There is no reference to this discussion in Pycior 1997, for example, even though the development of concepts of number in early algebra is a central theme of the book. Pycior (p. 125) briefly mentions the later, related, discussion in Wallis 1685 but misses the main purpose of it, which was to go beyond numbers already known. 91. Wallis 1685, 315–317. 92. Wallis 1685, 317. 93. Whiteside 1967–81, II, 240–241, note 127. 94. Wallis 1685, 310–311. 95. MS Smith 31, f. 58. 96. British Library Add MS 37202; Dubbey 1978, 109–114.

Chapter 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Digby to White, 8 May 1658, in Brouncker et al. 1658, letter 41. See Scott and Hartley 1960,150–151. Aubrey 1898, I, 128–129. The misconception that Brouncker studied at Oxford seems to have arisen from his later honorary award of ‘Doctor of Physick’. Frank 1997, 508–509. Aubrey 1898, I, 128–129. Aubrey 1949, 160 (but for some reason not included in Aubrey 1898, I, 403–405). Brouncker 1653; Wallis 1677; Wallis 1698a, b,c; Wallis 1693–99, III, 1–508. Brouncker 1668. Wallis 1693–99, I, 229–290; 231–232. For a detailed account of Brouncker's method see Coolidge 1949, 136–146; 141–146. Whiteside 1961a, 264. Wallis 1656c, 182; Wallis 1693–99, I, 469–470. Wallis 1656c, 182–184; Wallis 1693–99, I, 470–471. ‘Quaerebat igitur qua ratione augendi erant factores, ut prodirent rectangula, non quadratis illis unitate minutis, sed ipsis quadratis aequalia.’; Wallis 1656c, 183; Wallis 1693–99, I, 470. ‘Invenit autem id fieri posse, si utrique factores fractione augeantur, quae denominatorem haberet continue fractum in infinitum, ad eam formam quam superius exhibuimus:’; Wallis 1656c, 183; Wallis 1693–99, I, 470. Wallis always denoted Brouncker's first fraction by a small square, but used B, C, D,… for the subsequent fractions. He used A for an altogether different purpose later on, see note 27. In A treatise of algebra Wallis merely quoted Brouncker's result without any attempt at explanation; Wallis 1685, 317–318.

NOTES

18. 19. 20. 21.

22. 23.

24. 25.

26. 27. 28. 29. 30.

255

Leibniz 1682a, b; Euler 1748, paragraph 369. Bauer 1872. Brun 1951; Hofmann 1960; Whiteside 1961a, 210–213; Dutka 1981. The remarks in Scott and Hartley 1960, 149, that Brouncker ‘had merely to find a particular function’ and that the details are ‘readily reconstructed’ not only evade the issue, but belie the true nature and originality of Brouncker's achievement. Brouncker, like so many other seventeenth-century mathematicians, has been best understood by Professor D. T. Whiteside, who has created (but never published) his own reconstruction of Brouncker's work in the Latin, and even the typeface, of the original. I am grateful to him for generously sharing his insights into this rich piece of mathematics. The excess for each (2n)2 is Further support for the plausibility of this reconstruction can be found in Wallis's description of its essential features, though in another context, later in the same Scholium: ‘The first fraction increases the quantity, and indeed as far as this, that what was just less becomes just greater. And keeping the same numerator of this fraction, if the denominator is increased (since there is adjoined a second fraction) the first fraction, and therefore the whole quantity, is decreased by the addition of the second. And this decrease will be less (and therefore the total quantity greater) as the denominator of this second fraction (keeping the numerator) is increased, which is done by addition of a third fraction. Therefore the third fraction decreases the second, therefore increases the first and thus also, therefore, the whole quantity. Therefore adjoining fractions in odd places increases, in even places decreases the quantity.’ Wallis 1693–99, I, 475. Cataldi 1613, 70. A facsimile of the latter can be seen in Fowler 1994, 734–735, a good mathematical and historical introduction to continued fractions. Of some two dozen books and treatises that Cataldi published between 1572 and 1622 (all in Italian), none is to be found even now in the Bodleian Library, suggesting that they never came into the hands of the sixteenth- and seventeenth-century Oxford collectors of mathematical texts. There are copies of most of them in the British Library, but their relatively modern and uniform bindings suggest that they were later acquisitions. None of the papers cited in note 20 makes any direct reference to the existence of Brouncker's fractions other than the first. Dutka 1981, for example, refers in the title and throughout his paper to Brouncker's fraction in the singular only. Wallis 1656c, 187–188; Wallis 1693–99, I, 473. Wallis used the letter A indiscriminately to denote the first term of each sequence, see note 16. Wallis 1656c, 184–191; Wallis 1693–99, 1, 471–474. The completion of Wallis's Proposition 190 is described only in Whiteside 1961a, 241. Nunn 1910–11 and Scott 1938, 26–64, in otherwise detailed expositions of the Arithmetica infinitorum make no mention either of the Proposition or its completion in the final pages. ‘Atque hactenus Nobilissimi viri mentem, quanta potui brevitate simul atque perspicuitate exposui’; Wallis 1656c, 193; Wallis 1693–99, I, 476.

256 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

43. 44.

45.

46.

NOTES

Wallis to Digby, 6 June 1657, Brouncker et al. 1658, letter 5. Van Ceulen 1619. The required starting values are p0 = q−1 = 0; p−1 = q0 = 1. Wallis 1656c, 191–192; Wallis 1693–99, I, 474–475. Earlier in the Arithmetica infinitorum Wallis had denoted the terms of an infinite sequence as a, b, c, etc. Note also the labelling of Brouncker's fractions as B, C, D,… Cajori 1928–29 does not mention the introduction of subscript notation. Wallis 1659, 91–96 and Wallis 1693–99, I, 550–554. For further details of Brouncker's method see Coolidge 1949, 139–141. Mahoney 1973, 332–347. See for instance Weil 1983, 81, 92–97, 100, where Wallis and Brouncker are routinely mentioned in tandem. For a more careful distinction between the work of Wallis and Brouncker see Dickson 1919–23, II, 351–353. Pascal to Fermat, 27 October 1654, quoted in Mahoney 1973, 334. Mahoney 1973, 293–295, 337; Weil 1983, 51–91; Goldstein C. 1995, 21–33. ‘Proponatur (si placet) Wallisio et reliquis Angliae Mathematicis, sequens quaesitio numerica. Invenire Cubum, qui additus omnibus suis partibus aliquotis conficiat Quadratum. Exempli gratia. Numerus 343 est Cubus, a latere 7. Omnes ipsius partes aliquotae sunt 1, 7, 49; quae adjunctae ipsi 343, conficiunt numerum 400, qui est quadratus a latere 20. Quaeritur alius cubus numerus ejusdem naturae. Quaeritur etiam numerus Quadratus qui additus omnibus suis partibus aliquotis conficiet numerum Cubum. Has solutiones expectamus; quas si Anglia aut Gallia Belgica et Celtica non dederint, dabit Galli Narbonensis, easque in pignus nascentis amicitiae D. Digby offeret et dicabit.’; Brouncker to Wallis, 5 March 1657, in Brouncker et al. 1658, letter 1. From Fermat's example, it may be assumed that he meant the cube of a prime (which would give rise to an equation of the form given at (11) in the text). The ‘aliquot parts’, or divisors, are then simply the lower powers of the same prime. ‘Est autem ea quaestio eiusdem fere generis cum iis quae de numeris Perfectis (ut loquuntur) et Deficientibus aut Redundantibus exponi solent Problematis, aliisque id genus, quae ad universalem aliquam aequationem, quae ad omnes casus respiciat, vix aut ne vix reducuntur. Quicquid autem id sit, cum jam me multiplici negotio occupatissimum deprehendat, non vacat ei statim attendere. Id saltem impraesentiarum habeat responsi loco; Unum eundemque numerum 1, utrique quaesito satisfacere.’; Wallis to Brouncker, 7 March 1657, in Brouncker et al. 1658, letter 2. It is not quite clear what Wallis actually meant by his throwaway remark that the number 1 satisfied both of Fermat's requirements. Fermat wanted primes p such that 1 +p + p2 + p3 was a square or 1 + p + p2 was a cube. 1 is not a prime and certainly does not satisfy the second equation, so Wallis must have had in mind the trivial solution p = o, or 1 = 12 and 1 = 13. There is in fact only one solution to 1 + p + p2 + p3 = q2 for prime p, the one given by Fermat (p = 7). For proof see Mahoney 1973, 337–338 or Weil 1983, 87–91. If 1 + p + p2 + p3 = q2, then (1 + p)(1 + p2) = q2. Each of the factors on the left-hand side is even but they are otherwise mutually prime and may be replaced (since their product is a square) by (2r2)(2s2). The problem therefore reduces to finding integer

NOTES

47.

48. 49. 50. 51. 52.

53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

257

solutions to 2r2 − 1 = p and 2s2 − 1 = p2, the latter a special cases of Na2 ± 1 = l2 with a and l integers. ‘Dato quovis numero non-quadrato, dantur infiniti quadrati qui in datum numerum ducti, adscita unitate, conficiant quadratum. Exemplum. Datur 3, numerus non-quadratus; ille ductus in quadratum 1, adscita unitate, conficit 4, qui est quadratus. Item idem 3 ductus in quadratum 16, adscita unitate, facit 49, qui est quadratus. Et loco 1 et 16, possunt alii infiniti quadrati idem praestantes inveniri. Sed Canonem Generalem, Dato quovis numero non-quadrato, inquirimus. Quaeratur, verbi gratia, quadratus, qui ductus in 149, aut 109, aut 433, etc. adscita unitate conficiat quadratum.’; Brouncker to Wallis, 11 September 1657, in Brouncker et al. 1658, letter 8. Wallis to Digby, 27 September 1657, ibid., letter 9. Although the term ‘Diophantine’ is now often used to refer to problems with integer solutions, Diophantus himself observed no such restriction and was satisfied with rational solutions. Fermat to Digby, 6 June 1657, in Brouncker et al. 1658, letter 11. Fermat to Digby, 20 April 1657, ibid., letter 4. ‘J’ose Vous dire avec respect et sans rien abbattre de la haute opinion, que j'ay de votre Nation, que les deux lettres de My Lord Brouncker, quoy qu'obscures a mon egard et mal traduites, n'en contiennent point aucune solution. Ce n'est pas que je pretende par là renouveller les joustes et les anciens coups de lances, que les Anglois ont autrefois fait contre les Francois.'; Fermat to Digby, 15 August 1657, ibid., letter 12. ‘Proponatur itaque, datum numerum cubum in duos cubos rationales dividere. Item, Datum numerum ex duobus cubis compositum in duos alios cubos rationales dividere. Et inquiratur, quid ea de re Anglia, quid Hollandia censeat?’; Fermat to Digby, 15 August 1657, ibid., letter 12, postscript. Wallis to Digby, 3 September 1657, ibid., letter 7. ‘Poteris, si tibi videbitur, Latine transmittere, ut non cur de Idiomate Anglicano deinceps conquerantur, vel ob illud haereant.’; Brouncker to Wallis, 11 September 1657, ibid., letter 8. Wallis to Digby, 27 September 1657, ibid., letter 9. Fermat to Digby, 6 June 1657 and 15 August 1657; Remarques sur I'Arithmetique des Infinis, undated but apparently written in August 1657, in Brouncker et al. 1658, letters 11, 12, 13. Brouncker to Wallis, 22 October 1657, ibid., letter 14. Wallis to Brouncker, 21 November 1657, in Brouncker et al. 1658, letter 15. We have no record of this request, which may have been made verbally, but it is clear from letter 17 that Wallis was responding to it. Both are to be found in Wallis to Brouncker, 17 December 1657, ibid., letter 17. Wallis's method fills the main letter, Brouncker's method is added as a postscript. Brouncker to Wallis, 22 October 1657, ibid., letter 14. Wallis wrote r and s where Brouncker had originally used a and e. Brouncker also wrote a2 [ne2 for what is now denoted |a2 − ne2|. Note Brouncker's use of superscript notation: Wallis in his September letter to Digby had retained the older form Aq; Fermat used no notation at all but wrote his challenges verbally.

258

NOTES

63. See Euclid VII. 1−3. The algorithm leads easily to the continued fraction for In fact if a and l are solutions of Na2 ± 1 = l2, then and Brouncker's process defines the first period of the infinite continued fraction for . 64. Bachet had devised a similar technique for the solution of the Diophantine equation Ax − By = 1, and published it in the second edition of his Problemes plaisans et delectables, 1624, 18−33. Although Bachet used consecutive capital letters for each new number calculated, his method was described verbally not algebraically. For details see Dickson 1919–23, II, 44−45. Weil 1983. 93, has suggested that Bachet's text served as a model for Brouncker, and this may have been so but Brouncker's work was very much more sophisticated than Bachet's who, like Euclid, was dealing only with linear forms. 65. Fermat never disclosed his own method. Weil 1983, 93, supposed that it was similar to Brouncker's, see Weil 1983, 93. For further discussion of possible methods used by Fermat see Mahoney 1973, 328−332. 66. Wallis to Brouncker, 20 January 1658 and 6 April 1658, in Brouncker et al. 1658, letters 19 and 29; Wallis 1685, Chapter 98. The repetitions noted by Wallis correspond to the (palindromic) periods in the continued fraction for . 67. Euler to Christian Goldbach, 10 August 1730, here quoted from the translation in Weil 1983, 174. Weil suggested that Euler read Brouncker's method in the Commercium epistolicum, but if so it should have been more obvious that the method was Brouncker's. There are many references to Pell, however, in Chapters 57−60 of A treatise of algebra, and in particular to his treatment of indeterminate Diophantine equations (though not Na2 + 1 = l2). The Latin translation of A treatise of algebra precedes the Commercium epistolicum in the second volume of Wallis's Opera mathematica (1693−99), almost certainly the edition read by Euler. 68. For its full history see ‘Pell equation’, Chapter XII in Dickson 1919−23, II, 341−400. For a modern treatment in terms of quadratic forms see Weil 1983, 92−99. 69. Frenicle 1657. 70. Frenicle to Digby, 3 February 1658: Digby to Wallis, 6 February 1658, in Brouncker et al. 1658, letters 22 and 21. 71. Wallis to Digby, 4 March 1658, ibid., letter 23. 72. Frenicle to Digby, between 6 and 10 February 1658, ibid., letter 26. 73. Brouncker to Digby, 13 March 1658, ibid., letter 27. 74. Wallis to Digby, 15 March 1658, ibid., letter 28. 75. Digby to Wallis, 4 May 1658, ibid., letter 36. 76. Digby to Brouncker, 4 May 1658, ibid., letter 35. 77. ‘Illustrissimos Viros Vicecomitem Brouncker et Johannem Wallisium quaestionum numericarum a me propositarum solutiones tandem dedisse legitimas libens agnosco. imo et gaudeo. Noluerunt Viri Clarissimi vel unico momento impares sese aut quaestionibus propositis consiteri;’; Fermat to Digby, 19 June 1658, ibid., letter 46. 78. ‘Verum ut deinceps ingenue utrimque agamus. fatentur Galli propositis quaestionibus satisfecisse Anglos: Sed fateantur vicissim Angli quaestiones ipsas dignas fuisse quae ipsis proponerentur, nec dedignentur in posterum numerorum integrorum naturam accuratius

NOTES

79. 80.

81. 82. 83. 84. 85.

86.

259

examinare et introspicere, imo et doctrinam istam qua pollent ingenii vi et subtilitate, propagare.’; Fermat to Digby, 19 June 1658, ibid., letter 46. Van Schooten to Wallis, 18 March 1658, ibid., letter 33. Brouncker did some work on the cycloidal pendulum from 1661 onwards in connection with ideas put forward by Huygens. It is described in Scott and Hartley 1960, 150, as ‘largely uninspired’. The only other surviving work by Brouncker is a brief refutation of a paper by Thomas Hobbes, also written in 1661, and also sent to Huygens. Details may be found in the bibliography compiled by Whiteside in Scott and Hartley 1960, 157. Turnbull 1939, 66. Frenicle to Digby, 8 May 1658, in Brouncker et al. 1658, letter 43. Euler 1765. For accounts of the work of both Euler and Lagrange and full references see Smith H. J. S. 1894, I, 193–195: Dickson 1919–23. II, xii, 354–364; Weil 1983, 229–232, 314–316. Huygens in ‘Descriptio automati planetarii’ (1691) used continued fractions calculated by the Euclidean algorithm to find gear ratios for his planetary models. He had first written down such fractions in 1680, see Huygens 1888–1950, XX, 389–394 and XXI, 628–643. Huygens commented on the ordinary fraction approximations for π in Chapters 10 and 11 of A treatise of algebra, but made no mention of the continued fraction devised by Brouncker. Actually Digby's description of Fermat, Roberval and Descartes, in Digby to Wallis, 10 February 1658, in Brouncker et al. 1658, letter 24.

Chapter 8 1. 2. 3. 4. 5. 6.

7.

Cotes to Smith, 31 December 1698, in Edleston 1850, 190–202, 191. John Smith's son Robert (1689–1768) became Master of Trinity College, Cambridge, and endowed the Smith's Prize competition in mathematics; see Barrow-Green 1999, 273. Wallis 1685b. Leibniz 1686. Huygens 1888–1950, XX, 367–403, 389–394. Wallis to Morland, 8 January 1689; Morland to Wallis, 12 March 1689, in Wallis 1693–99, II, 206–213. ‘Ce n'est que sur de vaines conjectures ou par un mouvement d'envie que des gens ont voulu faire croire de son vivant meme qu'il avoit tiré sa méthode des autres, & particuliérement d'un certain Harriot Anglois, qu'il n'avoit jamais lu, comme il le déclare dane une de ses Lettres. Et lorsque Monsieur Wallis, un peu trop jaloux de la gloire que la France s'est acquise dans les Mathématiques, vient renouveller cette accusation ridicule, on est en droit de ne le point croire, puis qu'il parle sans preuve.’; Prestet 1689, II, Preface (unpaginated). ‘… la conformité de ses sentimens avec ceux de Harriot touchant la nature des Equations a paru un préjugé raisonnable, pour faire croire qu'il avoit quelque obligation à cet Auteur, quoy qu'il ne l'eut point fait connoitre en public. Celuy qui découvrit le prémier cette conformité fut Mylord Candische, qui se trouva pour lors à Paris, et qui la montra à M. de Roberval. La chose

260

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

NOTES

devint ensuite toute publique par le zèle que M. de Roberval faisoit paroitre à diminuer par tout la gloire de M. Descartes. Mais M. Pell, Mathématicien Anglois, le Chevalier Ailesbury, qui avoit été l'exécuteur testamentaire de Harriot et le dépositaire de ses papiers, et meme Guill Warner, qui a fair imprimer son livre, jugeoient plus favourablement de M. Descartes, rejetant tout l'avantage de la conformité sur la personne de Harriot, à qui il étoit assez glorieux que M. Descartes se fut rencontré avec luy.’; Baillet 1691, Livre VIII, 541. The story is not in the abridged English translation, Baillet 1693. Saunderson 1740, 49. Hutton 1796, vi. Montucla 1799–1802, I, III.3 and II, IV.6. De Morgan 1838, 42. Cayley 1888, 331–332. Ball 1888, 292. Cantor 1894–1908, III, 4. Ball 1888, 292–293. Ball may have been confusing A treatise of algebra with Wallis's De motu of 1669. Cajori 1894, 184. Scott 1938, 133–165; 156. Wallis 1685, title page. Wallis 1685, 1–2. Wallis 1685, 128. Van der Waerden 1985; Pycior 1997; Bashmakova and Smirnova 2000. Wallis 1685, 292. Whiteside 1967–81, V, 429. Aubrey 1898, II, 281–282. See also Collins to Wallis 20 September 1677: ‘You lye under a censure from diverse for printing discourses that come to you in private Letters without permission or consent as is said of the parties concerned’, in Turnbull 1959–77, II, 242. Pycior 1997, 111, speaks of the rise of English nationality, but a sense of nationality and nationalism are not the same thing; English nationality was established long before the seventeenth century. The English Atlas was compiled with the assistance of Wren, Pell and Hooke, and published by Moses Pitt, see Pitt 1680–83. Several of the advertising pamphlets from 1678 are to be found amongst Pell's papers, in British Library Add MS 4394, f. 405 and elsewhere; Pell often used the blank spaces in them for his calculations. Wallis to Smith, 21 December 1698, MS Smith 54, f. 55. Scriba 1970, 38. William Wallis, MS Eng. misc. e. 475, ff. 275–276. The letters decoded by Wallis up to 1653 are to be found in MS e. musaeo 203 and in Eng. misc. e. 475, i–243. Depositing material in a public library was in itself a form of publication, see Bennett 2001, 217. MS Eng. misc. e. 475, f. xxxi.

Bibliographies 1.Primary sources: manuscripts Manuscripts are held in the Bodleian Library, Oxford, unless otherwise stated. The relevant catalogues are listed at the end of the section. Adelard of Bath (translator), Euclid's Elements. MS Savile 19 and Trinity College Oxford MS Trinity College 47. Aristarchus, De magnitudinibus et distantiis solis et lunae, MS Savile 10, ff. 132–140. Ashenden, John, Summa astrologiae judicialis de accidentibus mundi, in MS Savile 25A. MS Ashmole 576 contains the edition printed at Venice 1489. Ashenden, John and Rede, William, De significatione coniunctionis que erit anno Christi 1365, MS Digby 176, ff. 34–40. Ashenden, John and Rede, William, Prognosticationes eclipseos lunae 1345, MS Digby 176, ff. 9–15. Aubrey, John, on Harriot's papers, MS Aubrey 13, f. 432. Babbage, Charles, 1821, The philosophy of analysis, British Library Add MS 37202. Bale, John, notebook, MS Selden Supra 64. Bede, De temporibus, MS Bodl. 309, ff. 68–80. Bede, De ratione temporum, MS Bodl. 309, ff. 3v–62. Boethius, Severinus, De arithmetica, in MS Savile 20 and MS Selden Supra 25. Cavendish, Charles, mathematical papers, British Library Harley MSS 6001–6002, 6083, 6796. Chauveau, Traicte d'algebre, British Library Harley MS 6083, ff. 350–379; British Library Add MS 4407 ff. 31–37v. Conches, Guillaume de, Dragmaticon, Corpus Christi College Oxford MS CCC 95, ff. 15v–56v. Gregory, David, brief biography of Wallis, MS Smith 31, ff. 58–59. Grosseteste, Robert, Compotus est sciencia numeracionis et divisionis, MS Savile 21, ff. 127–142v. Harriot, Thomas, mathematical papers, British Library Add MSS 6782–6789 and Petworth Leconfield 240–241. Harriot, Thomas, Treatise on equations, dispersed through British Library Add MSS 6782–6783. Harriot, Thomas, Examinatio Stifelius de numeris diagonalibus, British Library Add MS 6782, ff. 84–94.

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Harriot, Thomas, De numeris triangularibus et de inde progressionibus arithmeticis, British Library Add MS 6782, ff. 107–146. Hartlib, Samuel, The Hartlib papers, Sheffield University Library, and CD. Jordanus, Algorismus Jordani tam in integris quam in fractionibus, MS Savile 21, ff. 143–150. Leland, John, Collecteana, MS 5102–5106. Leland, John, Itinerary, MS 5107–5112. Morland, Samuel, letter to Wallis, MS Eng. lett. c. 291, ff. 37–38. Morley, Daniel, Liber de naturis inferiorum et superiorum, Corpus Christi College Oxford MS CCC 95 and British Library MS Arundel 377. Oughtred, William, letter to Seth Ward, MS Aubrey 6, f. 40. Pappus of Alexandria, Collectiones mathematicae, MS Savile 9, ff. 41–227. Pell, John, correspondence 1631–1683, British Library Add MSS 4278–4280. Pell, John, mathematical papers, British Library Add MSS 4397–4404, 4407–4431. Peter of Cluny, Tractatus adversus sectam Saracenorum, MS Selden Supra 31, ff. 5–15. Planudes, Maximus, ψηφoφoρiα, MS Gr. Laud 51 and MS Cromw. 12, ff. 1–52. Richard of Wallingford, De sinibus demonstratis, in MS Digby 168, 178, 190. Rigaud, Stephen Peter, Thomas Harriot, MS Rigaud 9. Rigaud, Stephen Peter, Hadley and Harriot papers, MS Rigaud 35. Rigaud, Stephen Peter, Rigaud letters, II, F – M. MS Rigaud 61. Robert of Chester, Prefacio Roberti translatoris, MS Selden Supra 31, ff. 32–33. Robert of Chester, translation of the Toledan tables and Canones tabularum of al-Zarquālī, MS Savile 21, ff. 63–103. Robert of Chester, translation of the Qur'an, MS Selden Supra 31, ff. 32–204. Roger of Hereford, Tractatus de computo, MS Digby 40, ff. 25–51. Sacrobosco, Johannes, Omnia quae a primitiva origine rerum, MS Savile 17, ff. 94v–104. Sacrobosco, Johannes, Compotus est sciencia considerans tempora, MS Savile 17, ff. 141–174v. Savile, Henry, Auctores mathematici, MS Savile 28, f. 28vf. Savile, Henry, History of mathematics, MS Savile 29, ff. 29–65v. Savile, Henry, Proemium, MS Savile 29, f. 2f. Smith, Thomas, letters to Wallis 1690–99, MS Smith 66. Torporley, Nathaniel. Congestor analyticus, Lambeth Palace Library, Sion College MS Arc L.40.2/L.40, ff. 1–34v. Torporley, Nathaniel, Corrector analyticus artis posthumae Thomae Harrioti, Lambeth Palace Library, Sion College MS Arc L40.2/E10, ff. 7–12. Torporley, Nathaniel, Summary, Lambeth Palace Library, Sion College MS Arc L.40.2/L.40, f. 35–54v.

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Ville Dieu, Alexandre, Carmen de algorismo, MS Savile 17, ff. 104–109v, plus several copies in the Digby collection, see Macray 1883. Ville Dieu, Alexandre, De computo ecclesiastico (or Massa compoti), MS Savile 17, ff. 175–184v, MS Savile 21 ff. 161–175. Wallis, John, autobiography (incomplete), British Library Add MS 32499. Wallis, John, ‘Memorials of my life’, MS Smith 31, ff. 38–50. Wallis, John, ‘Dr. Wallis's life written by himself ’, transcribed from the original by William Wallis, MS Eng. misc. e. 475, ff. 256–274. Wallis, John, ‘A collection of letters and other papers which were at severall times intercepted, written in cipher’, MS e Musaeo 203. Wallis, John, ‘A collection of letters and other papers intercepted in cipher during the late warres in England’, transcribed by William Wallis, MS Eng. misc. e. 475, ff. i–243. Wallis, John, ‘Letter book of Dr John Wallis, 1651–1701’, British Library Add MS 32499. Wallis, John, ‘A collection of deciphered letters[1689–1703]’, MS Eng. misc. c. 382. Wallis, John, notes on Lettres a Dettonville (c. 1659), MS Savile 101, ff. 71–94. Wallis, John, letter on the printing of Oughtred's Clavis (1693), MS Savile 101, ff. 13–16. Wallis, William, biographical material on John Wallis, MS Eng. Misc. e. 475, ff. 274–349. Warner, Walter, mathematical papers, British Library Add MSS 4394–4396 and Northamptonshire Record Office IL 3422, VI, ff. 1–23. al-Zarqālī, Canones tabularum, MS Savile 21, ff. 27–41v, ff. 63–103. Manuscript catalogues Bernard, Edward, 1697, ‘Librorum impressorum quos museum savilianum itidem servat catalogus’, Catalogi manuscriptorum Angliae et Hiberniae, Oxford: Bodleian Library. Black, William Henry, 1845, A descriptive, analytical and critical catalogue of the manuscripts bequeathed unto the university of Oxford by Elias Ashmole, Oxford: University Press. Bodleian Library, 1980, A summary catalogue of western manuscripts in the Bodleian Library at Oxford, 7 vols (Index in vol VII), Oxford: Bodleian Library. British Library, 1977, The British Library catalogue of additions to the manuscripts 1756–1782, Additional manuscripts 4101–5017, London: British Museum. Coxe, Henry O., 1852, Catalogus codicum manuscriptorum qui in collegiis aulisque oxoniensibus hodie adservantur, Oxford: Bodleian Library. Coxe, Henry O., 1853, Catalogi codicum manuscriptorum bibliothecae Bodleianae, pars secunda, codices latinos et miscellaneos Laudinos complectens, Oxford: Bodleian Library.

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Index Abu Kamil 36 Adelard of Bath 21 Albury, Surrey 55 Alcuin 21 Aldhelm 21 algebra 212–214; cossist 6, 36–45, 48, 51, 54, 62, 212; and geometry 16, 46, 50, 52, 54, 61, 62, 67, 72, 137, 170, 214; rule of 40, 41, 43, 50, 60–61 algebraic variable 177 Al-jabr 35–36, 52, 212, 214 Allen, Thomas 12, 228 n. 74 analysis 50–52, 54, 59, 157, 171, 212, 214 analytic art Seeanalysis Anderson, Robert 74, 76, 235 n. 51 angular sections 51, 61, 62, 78, 82, 146, 233 n. 16, 251 n. 35 Apollonius 24, 50, 121, 132, 147, 171 Arab mathematics seeIslamic mathematics Archimedes 14, 24, 34, 143–144 Arithmetica infinitorum seeWallis, John Ashenden, John 33 Ashmole, Elias 12 Aubrey, John 63, 64, 68–69, 79, 114, 116, 119–120, 127, 131, 145, 184, 185, 215–216 Aurel, Marco 41 Aylesbury, Thomas 90, 97–100, 113, 116, 131, 210 Babbage, Charles 182 Bachet, Claude Gaspar 258 n. 64 Bacon, Roger 13, 33, 34 Baconthorpe, John 34 Baillet, Adrien 210 Bainbridge, John 7 Baker, Thomas 142, 221 n. 35 Balam, Richard 71 Baldi, Bernadino 13 Bale, John 20, 34 Ball, W. W. Rouse 211 Barrow, Isaac 9, 253 n. 70 Bartholin, Erasmus 9, 137 Batecombe, William 33 Beaugrand. Jean 111 Beaune, Florimond de 9 Bede 13, 21 Benedetto of Florence 37 Bernard, Edward 149 binomial theorem 175–180 Birch, Thomas 116 Bodleian Library 12, 20, 21, 23, 26, 28, 53, 69, 78, 81, 83, 218

Boethius 34 Bombelli, Rafael 48, 50, 53, 54, 123, 210, 211 Borrell, Jean 41, 53 Boswell, William 132 Botoner, William 227 n. 67 Boulliau, Ismael 182, 221 n. 35 Boyle, Robert 70, 87 Bradwardine, Thomas 33, 34 Brancker, Thomas 135–136, 148–150 Brasser, Franciscus 9 Breda 113, 132–134, 141, 168, 244 n. 19, 245 n. 31 Bredon, Simon 33 Brereton Hall, Cheshire 134, 135, 245 n. 36 Brereton, William 134, 135, 141, 147 Briggs, Henry 7, 126, 127, 131, 143, 168, 179 British Library 91, 93, 102, 106, 127, 129, 135, 136 Brouncker, William 4, 9, 16, 18, 115, 126, 134, 148, 174, 175, 183–207, 209, 210, 213, 215, 216, 252 n. 60; Commercium epistolicum 4, 205 Bunning, Mr 74 Busby, Richard 116, 127 Buteo seeBorrell, Jean Cajori, Florian 211 Cambridge 55, 63, 127, 128, 146, 176, 210, 238 n. 30, 248 n. 94; Emmanuel College 2, 77 Cantor, Moritz 88, 211 Cardano Girolamo 13, 34, 51, 77, 78, 101, 123, 124, 146, 210, 211, 213, 214; Ars magna 45–50, 53

290

INDEX

Carmelites 34 Cataldi, Antonio 192, 235 n. 54 Cavalieri, Bonaventura 111, 155–157, 249 n. 7 Cavendish, Charles 7, 57, 111–113, 116, 121, 127, 131, 132, 146, 243 n. 108, 249 n. 7 Cayley, Arthur 211 Cestria, Robert 227 n. 60 Ceulen, Ludolph van 143–144, 194 Charles II 70, 132, 134, 183, 184 Chauveau, François 234 n. 44 Chuquet, Nicolas 38, 214 civil war 2, 5, 63, 64, 134, 184 Clark, Gilbert 75, 221 n. 35 Clavius, Christoph 54, 220 n. 3 Collins, John 8–10, 44, 72, 73–77, 78, 80, 114–115, 121, 126, 127, 135–137, 142, 145, 146–147, 148, 150, 151, 152, 154, 205 Colson, John 236, n. 75 completing the square 35. 115, 119, 241 n. 87 Conches, Guillaume de 22 constant difference methods 178–179 Cotes, Roger 208 Cromwell, Oliver 134 Dardi of Pisa 37 Dary, Michael 9–10, 73, 76, 146–147 Dasypodius, Conrad 224 n. 35 Davenant, Edward 145, 247 n. 66 Davis, Richard 10 Dechales, Milliet 247 n. 59 Dee, John 89 Descartes, René 6, 47, 77, 79, 88, 111, 114, 121, 123, 132, 146–147, 153, 154, 185, 210, 214, 217; La géométrie 8, 111–112, 124, 174 Diceto, Ralph de 226–227 n. 60 Digby, Kenelm 12, 183, 196, 199, 203, 206 Digges, Leonard and Thomas 43, 89 Diophantus 6, 49, 50, 63, 101, 121, 129, 132, 139, 257 n. 49 Dulaurens, Francis 5, 9, 117, 216 Dury, John 130 Eastwood, John seeAshenden, John equations 94, 100, 123, 181, 212; canonical 101–104, 110; cubic 37–38, 45, 46, 51, 53, 77, 95–96, 107, 122; impossible 39, 118, 151; indeterminate 138–139, 140, 147; numerical solution of 51, 67, 72, 80, 94, 100, 107, 110, 151; ‘Pell's’ 126, 197–204, 206–207, 257 n. 63; quadratic 35–37, 40, 45, 61, 71, 80, 95, 115, 213, 241 n. 87; quartic 37, 45, 51, 95, 96, 107; quartics as products of quadratics 53, 146–147 Euclid 34, 61, 67, 74, 165, 168, 171, 213, 233 n. 13 Euclidean algorithm 144, 202 Euler, Leonhard 189, 203. 206–207, 210 Ferguson, John 9

Fermat, Pierre de 4, 6. 117, 170–175, 182, 196–205, 216 Ferrari, Ludovico 45, 53, 146 Ferro, Scipione del 45, 53 Fibonacci seeLeonard of Pisa Fiore, Mario 45 Fobbing, Essex 134 Foster, Samuel 72, 73 fractions, continued 187–194, 207, 209, 258 n. 63, 257 n. 66; for π 143–146, 209, 247 n. 66; for 4/π 155, 160–165, 180, 187–194, 251 n. 26 France 6, 21, 40, 111, 215, 216–217, 249 n. 7 Francesca, Piero della 37 Franciscans 33 Frenicle de Bessy, Bernard 5, 196–205 Gerard of Cremona 36 Gerardi, Paolo 37, 45 Germany 6, 40, 59 Gervase of Tilbury 227 n. 67 Ghaligai, Francesco di 38, 232 n. 141 Gibson, Thomas 72, 73, 86, 114 Girard, Albert 63, 76 Goddard, Jonathan 134 Gosselin, Guillaume 59 Greaves, John 3, 12 Gregory, David 82–84, 182 Gregory, James 72, 115, 142, 152, 205 Gresham College 7, 73, 126, 134, 185, 205

INDEX

Grosseteste, Robert 27, 28, 33 Gunter, Edmund 144 Haak, Theodore 121, 128, 132, 134 Halley, Edmund 85–86, 245 n. 27 Harriot, Thomas 16, 18, 48, 72, 79, 88–125, 152, 153, 179–180, 209, 210, 211, 213, 215, 217, 247 n. 56; Operationes logisticae 92–94, 101, 108, 109; papers 17, 98–99, 102, 106, 111–116, 119–120, 236 n. 5; Praxis 71, 88, 93, 97–107, 108, 109, 118–119, 122, 124, 215; Treatise on equations 94–97, 102, 106, 110–111, 113, 116, 118–120; Will 97, 116 Hartlib, Samuel 70, 113, 114, 116, 127, 128, 131, 134 Helmdon mantelpiece 29–32 Henisch, Georg 23 Hermann of Carinthia 23 Heuraet, Hendrick van 174–175, 209 Hindu–Arabic numerals 14, 23–32, 36, 215, 217 historiography 17, 215–218 Hobbes, Thomas 5, 114, 141, 154, 168–173, 182, 259 n. 80 Holcot, Robert 227 n. 67 Holder, William 215 Hooke, Robert 76, 87, 142, 215 Howard, Thomas 7, 57 Howard, William 57, 59, 70, 247 n. 59 Hudde, Johann 9, 234 n. 44 Hues, Robert 89, 97 Hutton, Charles 62, 124, 210 Huygens, Christiaan 4, 134, 149, 168, 170, 171, 174, 182, 209, 210 Indian mathematics 12, 24–25 indivisibles 155–157, 160, 169 induction 158, 168, 169, 171, 182, 209 Infans, Roger seeRoger of Hereford infinite sequences and series 158–165, 172, 175–180, 186–194, 199–202, 209, 213, 214 interpolation 160–165, 177–179, 181, 193 Isham, Justinian 114–115, 116 Isidor of Seville 13 Islamic mathematics 12, 13, 14, 16, 51, 52, 53, 101 Italy 6, 22, 34, 40, 52, 59, 215 Jacob of Florence 37 John of Salisbury 22 Jordanus 27, 53 al-Karajī 36 Kersey, John 9, 74, 210, 221 n. 35, 235 n. 51 al-Khwārizmī 26, 53, 213, 215, 237 n. 19; Al-jabr 23, 35–36, 38, 44; treatise on Indian numerals 24–25, 27 Killingworth, John (c. 1415–1445) 34 Killingworth, John (fl. 1381–1384) 228 n. 79 Kinckhuysen, Gerard 9, 147–150, 153; Algebra ofte stel–konst 9, 147–149 Kircher, Athanasius 27–28

291

Kneller, Godfrey 3 Lagrange, Joseph Louis 207 Lalouvère, Antoine 216 Lambeth Palace Library 238 n. 30 Langthorn, Clemens 227 n. 67 Lansberg, Philip 143 Laud, William 12 Lavenham, Richard 34 Leibniz, Gottfried 4, 116, 121, 154, 189, 208–209, 253 n. 73 Leland, John 20 Lely, Peter 184 Leonardo of Pisa 16, 36, 53; Liber abbaci 36–37, 38, 52 Leotaud, Vincent 5, 216 Liber abbaci seeLeonardo of Pisa Lichfield, Leonard 70, 73, 82–84, 156 Locke, John 70 Longomontanus, Christian Severin 132 Lower, William 90, 96, 116 Lynne, Nicholas 33–34 Macclesfield papers 238 n. 30, 248 n. 94 Marsh, Adam 33 Mengoli, Pietro 251 n. 26 Mennher, Valentin 41, 59, 237 n. 19 Mercator, Nicolaus 148, 152, 175–178, 186, 209, 249 n. 97 Mersenne, Marin 63, 111, 127, 244 n. 19 Metius, Adriaen Anthonisz 144 Metius, Adriaen 144–145 Montucla, Etienne 88, 210, 211 Moore, Jonas 70, 71, 87 Morgan, Augustus de 211

292

INDEX

Morland, Samuel 119, 210 Morley, Daniel 21–22 national rivalry 5, 84, 199, 216–217 Naunton, Robert 116 Nave, Annibale della 45 Neile, Paul 134 Neile, William 173–175, 194–195, 209, 215–216 Netherlands 6, 8, 9, 18, 19, 111, 132, 174 Newton, Isaac 9, 16, 70, 84, 87, 148–149, 155, 175–180, 209, 213, 214, 215, 216; Epistolae 10, 80 Nichomachus 229 n. 84 Norman, Robert 44 Northumberland, ninth earl of seePercy, Henry notation 38, 44, 48–49, 60, 71, 72, 80, 169, 170, 213–214; = 41, 214, 239 n. 43; × 60; ÷ 137–138; ± 61, 92; < > 72, 90–91, 101, 138, 214, 237 n. 13, 239 n. 43; ∞ 159, 160–164; Brouncker's 186, 200, 220, 257 n. 62; cossist 38, 44, 214; Descartes' 72, 111, 117, 203; Fermat's 257 n. 62; Harriot's 90–94, 117, 123–124; Oughtred's 60–61, 62, 67, 76, 80, 81, 87, 117, 203, 214; Pell's 137–138, 140; subscript 194, 213; superscript 72, 76, 123, 158. 250 n. 11, 257 n. 62; Viète's 51–52, 60, 80, 214 number theory 170, 196–205, 213 numbers, concepts of 181–182, 214; figurate 160–164, 172; imaginary 150, 211 Nuñez, Pedro 41, 53, 237 n. 19 Odington, Walter 227 n. 67 Oldenburg, Henry 77, 149, 180 Onslow, Richard 64–65, 66, 67 Oughtred, Benjamin 55 Oughtred, William 16, 54, 139, 144, 152, 157, 165–167, 210, 213, 215, 216; Clavis mathematicae 3, 7, 17, 55–87, 215; treatise on sundials 55, 67, 70 Oxford, University of 3, 6, 12, 32, 52, 70, 88, 168, 183; All Soul's College 251 n. 32; Corpus Christi College 22; Lincoln College 251 n. 32; manuscripts 12, 21, 22, 23, 27, 34, 53, 82; Merton College 12, 33, 65, 228 n. 74; New College 84; Oriel College 88, 227 n. 75; The Queen's College 251 n. 32; Trinity College 21; Wadham College 70, 251 n. 32 Pacioli, Luca 38, 39, 45, 53 Pappus 6, 24, 50 Pascal, Blaise 5, 117, 174, 196, 216 Pecham, John 33 Peletier, Jacques 41, 53, 220 n. 15, 237 n. 19 Pell, John 2, 16, 18, 19, 49, 76, 77, 112–113, 121–124, 126–154, 167, 179, 203, 209, 210, 213, 215, 216; Idea 121, 128–131, 154; Introduction to algebra 73, 126, 135–140, 148; papers 70, 127, 153, 216, 242 n. 96, 252 n. 60; tables 127–129, 131–132; three–column method 137–140, 141, 150, 154 Pepys, Samuel 3

Percy, Henry 7, 89, 98–100, 111, 115 Perez de Moya, Juan 41, 237 n. 19 Petri, Nicolaus 57 Petworth 98, 99, 111 Peucer, Kaspar 4 Phillip, William 57 Philosophical transactions 175, 208, 249 n. 97 Pits, John 20 Pitts, Moses 73–75, 136, 260 n. 26 Planudes, Maximus 24, 25 Playford, John 10, 11 Pococke, Edward 12 polynomials, factorization of 72, 95, 101–104, 112, 118, 122–124 powers, fractional and negative 41, 158, 250 n. 11 Praxis seeHarriot, Thomas Prestet, Jean 210 Prosdocimus of Padua 53

INDEX

Proteus 163, 183, 187 Protheroe, John 90, 97–99, 115–116 Ptolemy 34 publishing of mathematical texts 8–10, 11, 73–75, 148–150, 206, 215–216, 235 n. 48, 260 n. 30 quadrature 158, 170, 173, 175; of the circle 156–164, 177, 181, 186, 209, 214; of the hyperbola 175, 177–178, 185–186 Rahn, Johann 9, 135–137, 150, 153 Ralegh, Walter 89 Ramus, Peter 41, 54 Raphson, Joseph 236, n. 75 Ravius, Christian 245 n. 27 Rawlinson, Richard 168, 235 n. 51 Recorde, Robert 6, 41, 53, 237 n. 19; Whetstone of witte 6, 41–43, 44 rectification 158, 173, 184, 194, 195 Rede, William 33 Regiomontanus 13 regula cosae 36–45, 52 Renaldini, Carlo 234 n. 44 Ries, Adam 53 Rigaud, Stephen Peter 99 Robert of Chester 6, 22–23, 27, 36, 224 n. 30, 226 n. 60 Roberval, Gilles Personne de 112, 120 Robinson, William 62–64 Roger of Hereford 22, 224 n. 26 Rook, Lawrence 134, 168 roots, imaginary 46, 49, 95, 96, 105–106, 118–119, 122, 151, 181; negative 72, 79, 80, 95, 96, 103, 117–119, 122–124, 151, 181 Royal Society 8, 9, 10, 85, 113, 116, 126, 134, 135, 148, 149, 183, 184, 205, 252 n. 60 Rudolff, Christoph 38, 53 rule of signs 47, 72 Sacrobosco, Johannes 26, 53; Algorismus 26 St Vincent, Gregory 134, 253 n. 67 Salignac, Bernard 54 Sarasa, Alphonso Antonio de 253, n. 67 Saunderson, Nicholas 210 Savile Library 12, 21, 26–28, 53, 54, 77–78, 81, 82, 119, 221 n. 37, 232 n. 146, 243 n. 108 Savile, Henry 7, 12, 13 Savilian Professors 1, 3, 7, 12, 68, 70, 79, 82, 86, 149 Scarborough, Charles 2, 63, 68, 70, 126, 134 Scheubel, Johann 41, 53 Schooten, Francis van 8, 174, 205 Scott, Joseph 212 Selden, John 12 Sheldon, Gilbert 134–135 Shelley, William seeConches, Guillaume de Sicily 21, 36 Sidney, Robert 115

293

Simeon of Durham 226 n. 60 Simon de Montfert's problem 220 n. 14 Sion College 108, 238 n. 30 Smith, John (1618–1652) 77–79, 146 Smith, John (1659–1715) 208 Smith, Thomas 216 Snel, Willebrord 144 Somer, John 33–34 Spain 6, 12, 21, 24, 27, 28, 40 species 51, 60, 71, 101, 171–172 specious arithmetic 54, 71, 212 Stevin, Simon 49, 54, 101, 237 n. 19 Steyning Grammar School, Sussex 127 Stifel, Michael 40, 53, 99 Streete, Thomas 235 n. 51 Swineshead, Richard 34 Swineshead, Roger and John 228 n. 80 symbolism seenotation Syon House 89 tables, Alphonsine 27, 226 n. 50; of antilogarithms 127, 131–132, 157; constant difference 176, 178–179, 253 n. 82; of primes 99, 136; of square numbers 26, 129; Toledan 27, 224 n. 32; trigonometrical 127, 151 Tacquet, André 234–235 n.44 Tapp, John 58 Tartaglia, Niccolo 45, 59, 101, 209 Thorndike, Herbert 114–116, 132 Thurloe, John 134 Thynne, Charles 114 Titus, Silas 133, 134–135, 141–143, 145

294

INDEX

Torporley, Nathaniel 90, 97–100, 107–111, 124; Congestor 98–99, 101; Corrector 107–111; Summary 107–111 Torricelli, Evangelista 8, 157, 170 Tovey, Nathaniel 114 trisection seeangular sections Turner, Peter 3 Twysden, John 73, 227 n. 60 Vaughan, John 116 Viète, François 6, 7, 8, 48, 50–52, 54, 57, 60, 62, 79, 90, 100–101, 110, 111, 123, 124, 131, 151, 153, 157, 168, 172, 210, 212, 214, 215; De aequationum recognitione 7, 51, 57, 76; Isagoge 51, 92–94; De potestatum resolutione 51, 94–95; Zeteticorum 81, 90 Ville Dieu, Alexandre de 25, 26; Carmen 25–27 Vossius, Gerard John 13, 19, 34, 52, 152; De scientiis mathematicis 14, 15, 19, 22, 27, 221 n. 51 Waller, William 2 Wallingford, Richard 34 Wallis, John 2–5, 68–87, 116, 141–153, 155–182, 185–207, 212–218; Arithmetica infinitorum 4, 16, 18, 139–141, 155–182, 196, 198, 200, 206, 211, 215; Commercium epistolicum 4, 205; code-breaking skills 2–3, 12, 79, 159, 202, 217, 218; De Harrioto addenda 121, 210; inaugural lecture 5; Mathesis universalis 4, 5, 8, 12, 24; Opera mathematica 4; Operum mathematicorum 4; portrait by Kneller 3; proposal for printing A treatise of algebra 10, 208; and publishing 215–216; De sectionibus conicis 4, 117, 159, 195; Tractatus de cycloide 4, 195; A treatise of algebra 1–2, 8–18, 52–54, 80–82, 116–123, 142147, 150–153, 155, 192, 208–218 Wallis, William 217, 219 n. 3 Walter, John 34 Ward, Seth 2, 3, 63, 64, 67–70, 73, 87, 114, 120 Warner, Walter 89, 97, 99–108, 113–116, 120, 123, 131–132, 179, 210, 242 n. 96 Wentworth, Richard 209 Westminster School 116, 127 Wilfrid of Ripon 21 William of Worcester seeBotoner, William Williams, John 128, 244 n. 17, 245 n. 24 Wood, Anthony à 116 Wood, Robert 65, 67, 68, 70, 168 Wren, Christopher 70, 87, 126, 134, 145, 168, 174, 209, 215–216, 252 n. 60 Xylander, Wilhelm 251 n. 26 Zurich 134, 135, 139