The Beginnings of Greek Mathematics (Synthese Historical Library)

SYNTHESE HISTORICAL LIBRARY

ARPAD SZABO Matl,ematicallnstilute, Hungari4R .Amdem!l

0/ Science4

TEXTS AND 8T11DIBS IN TUB m8TORY Oll' LOOIO AND PHILOSOPHY

Editor8: N. Kmrrzu.uiN, Oornell University. G. NUORBI,vns, UniverBily 0/ Leyden L. M. DB RIJ][, UniverBily 0/ Lsyden

THE BEGINNINGS OF

a1J

GREEK l\fATHEMATICS

Editorial Board: J. BEllO, Munich Institute 0/ Pechnology F. DEL PuNTA, Linacre Oollege, Ozford D. P. HBNRY, University 0/ Manchester J. HINTIXXA, ~cademy 0/ Finland and Btan/ord UniverBily B. MATES, University o/Oali/ornia, Berksley J. E. MURDOCH, Harvard University G. PATZIO, Univer8ity 0/ (}iJUingen

;!

..

:

D. REIDEL PUBLISHING .. COMPANY VOLUME 17

'J

DORDRECHT

I

HOLLAND'/BOSTON

I

U.S.A.

'------"

CONTENTS

(

I'

I

Pre/ace to the Engli8h edition

11

Note on re/erenus

12

Ohronological table

13

Introduction

15

Part 1.

The early history of the theory of irrationals

33

Current views of the theory's development The concept of dynamis The mathematical part of the TheaeletU8 The usage and chronology of dynamis

33 36 40 44 46 48 56 61 66

1.1 1.2 1.3 1.4 1.5 1.(; 1.7 1.8 1.9 1.10 1.11 1.12 1.13

Tetrago1i.i..mw8

The mean proportional The mathematicalecture delivered byTheodorus The mathematical discoveries of Plato's Theaetetus The 'independence' of Theaetetus A glance at some rival theories The so-called 'Theaetetus problem' The discovery of inoommensumbUity The problem of doubling the square 1.14 Doubling the square and the mea.n proportiona.l

pmf 2.1 2.2 2.S

The pre-Euolidean theorY of proportions .. Introduotion A survey oC tho most importa.nt terms Consona.nces and intervals I

.'

.;,

71

75 85

91 97

99 99 lOS 108

The diaBlema between two numbem A digression on the theory of musio End points and intervals piotured 88 'straight linea' Dipla8ion, hemiolion a.nd epitriton ~ The Euclidean algorithm ) 2.9 Theoa.non 2.10 Arithmetical operations on the oanon 2.11 The teohnical term for 'ratio' in geometry ~.A"qloyla as 'geometric proportion' 2.13 •A"cUoycw 2.14 The preposition el"d 2.15 The elliptio expression d"a A&yo" 2.16 The subsequent history of d"cUoyo1l as a mathemati-

2.4 2.6 2.6 2.7

cal term

2.17 Cuts of the canon and musical mea.ns 2.18 The"creation of the mathematical concept of Uyo, 2.19 A digression on th"e history of the word Myo, ~The application of the theory of proportions to arithmetic and geometry 2.21 The mean proportional in music, arithmetio and geometry 2.22 The construction of the mean proportional ~nclusion

Part 3.

3.1

114 119 124 128 134 137140 144 145 148 151 154 157 161 167 168 170 174

177 181

The construction of mathematics within a deduotive framework 185 '~roof' in Greek mathematics

3.2

The proof of incommensurability 3.3 The origin of anti-empirioism and indirect proof 3.4 Euolid's foundations 3.5 Aristotle and the foundations of mathematics 3.6 Hypothesei8 3.7 The 'assumptions' in dialeotio 3.8 How hypothesei8 were used 3.9 HypotheBei8 and the method ofindireot proof . 3.10 A question of priority S.l1 Zono, the inventor of dioJeotio

.- - _. -

250 Plato and the Elea.tica 253 Hypothesei8 and the foundations of mathematics 257 The defini'tion of 'unit' 261 Arithmetio a.nd the teaching of the Eleatics 265 The divisibility of numbers 268 The problem of the aitemata 271 uclid's wstulates 273 1e constructions of Oenopides 3.19 276 ~The first three postulates in the Elementa 3.21 The koinai ennoiai . ..• 280 282 3.22 The word ~lo>pa 287 3.23 Plato's dpoA~paTa o.nd Euolid's dE,wpaTa 291 3.24 "The whole is greater than the part" 299 3.25 A complex of axioms 302 3.26 The difference between postulates and axioms 304 Arithmetic o.nd geometry 3.27 307 3.28 The science of space 312 3.29 Tho foundations of geometry 3.30 A reconsideration of somo problems relating to early 317 Greek mathematics 3.12 3.13 3.14 3.15 3.16 3.17

185 199 216 220 226 232 236 239 243 244 248

Postscript Appendix. How the Pythagorea.ns discovered Proposition II.5 of the Elemen18 The prevoiling view 1 My own view 2 Elements of a Pythagorean theory about the areas of 3 parallelograms How to find 0. square with tho same area as a given 4 rectangle Conolusion 5 Index of names Subject index

330

332 332 334 335 347 352 355 357

--~---------------.-----------------"".jIIICI--"¥""--'"

CHRONOLOGICAL TABLE

NOTE ON REFERENCES

These investigations into the origins of Greek mathematics are concerned with the pre-Euclidean period. Individual scientific discoveries oannot be dated exactly within this period, but the following table should help the reader to start out with 0. rough idea of the ~hron?logy. I have tried to give the likely dates of those persons mentlOned In the text. (Only the dates referring to Plato and ~ circle are absolutel! certain.) In the right-hand column, I have mentioned some .mathem.atlcal discoveries whioh can be &88Ooiated with the person In question.

The following books are frequently referred to in the notes. Unless otherwise stated, the editions are those given below. Burkert, W. JVei8heit und JVissenschaft, Studien zu Pylhag0ra8, Philolaos und Platon, Nuremberg 1962. Diels, H. & Kranz, W. Fragmente der Vorsokratiker, 8th edn, Berlin 1956. Heath, T. L. Euclid'8 Elemenl8, 3 vols. Dover Publications 1956. lIUmmru 8cienlififJUt8, ed. J. L. Heiberg & H. G. Zeuthen, ToulouseParis: vol. I, 1912; vol. 2, 1913; vol. 3, 1915. Proclus, Diadochi in primum Euclidis ElementorumLibrumcommentarii. ed. G. Friedlein, Lipsioa 1873. Thaer, C. Die Elemente von Euclid, in Ostwald's Klassiker der Emden lVwenscha/len, Leipzig 1933-7. van der Waerden, B. L. Science Awakening (Noordhoft', Groningen, Holland, 1954; Science Editions, New' York, 1963). lDrwac1&ende JViasenscha/t, 4gyptische, babylonische unf griechiache Mathematik (Baeel-Stuttgart 1966) Zur Ge8chichte der griechischetl. Mathematik, ed. O. Becker, Wiesenschaftliche Buohgesellsohaft, Do.rmstadt 1065•.

6th Century Because of the tradition which attributes new mathematical discoveries to Thales, the concept of 'angle' must have been known at this time. The elaboration of this concept seems to have been a new achievement of the Greeks.

ThaleR (circa 639-54.6)

Ana.ximenes (circa 560-528) Pythagoras (circD. 510) Parmenides (0. little younger than Pythagoras) Epicharmus (flourished about 500)

I

II

I \I

The theory of odd and even was already generally known.

5th Century Zeno (0. pupil of Parmenides) IDppasus of Metapontum (0. pupil of Pythagoras)

His experiments with bronze disfurther verified the ratios

m

14

ClIRONOLOIJICAL TADLK

Oenopides (an older contemporary of Hippocrates of Chios) Hippocrates of Chios (active in Athens around 480)

8880ciated with oonsonances. I have therefore come to the conclusion that the original experiments with a monochord and measuring rod, to whioh the concept of dia8tema is due, must have taken place earlier. The first three postulates of Euolid. The construction of a mean proportioned to two straight lines (Elementa, Book VI. 13) WIl8 already known at this time.

4th Contury Archytll8 (about the same age as Plato) Plato (427-847) Eudoxus .(a younger contemporary of Plato) Aristotle (384-822) Eudemus (a pupil of Aristotle) Autolyous Euclid (around 800)

Doubling the (volume of a) cube. The fifth book of the Elements

Compiled the Elements

INTRODUCTION

This book is entitled The Beginnings 0/ Greek M atllematws. The reader should not, however, expect a systematic and comprehensive account of the oldest period of Greek science. The investigations presented here revise and extend the many works which. in the course of the lll8t few yeo.rs. have sought to throw light from different viewpoints on the early history of Greek mathemo.tics. However, I did not collect together these investigations supposing that a definitive new picture of the early history of Greek science could be presented on the basis of the historical conclusions reached here. On the contrary, I believe that today we are still at the beginning of the difficult work which one day will lead to a completely new picture of the historical development of Greek mathematics. To this task it is hoped that the present book may make a contribution which, although modest, nonetheleBB opens up some new avenues. First of all. I must give an account of what is new in this book.

• In the first place I would like to disCUBB the method used, for undoubtedly it is this which distinguishes the book from most of its predecessors. I want to illustrate the difference straight away with an example. and have chosen for this purpose van der Waerden's book1 which at present is the most widely read and. quite deservedly. the most highly thought of work on Greek mathematics. When the German edition of this book appeared more than ten years ago. I wrote in a review:' ·E~ WU6emcha#. agyptucM. babylonueM und griechucM Mathematik, translated from Dutch byH. Habioht with additions by the author. Boael-Btuttgart 1960. (The first Dutoh edition appeared in 1960 and the first English edition in 1964.) . ·,AclaBcienlianlmMalhemalica",m(8zeged. I1ungnry) 18 (1067) pp. 140-1.

INTRODUCfl0H

16

17

SZABO: TUB BKOINIUNOS Olr OBEKK AlATBIWATICS

.. I oxpect that all further historical researoh in this field will, for a long time to come, be based on van der Waerden's work." The time whioh has passed since then has only se"ed to bear out this prediotion. I should also emphasize that without van der Wa.erden's pioneering work, it would hardly have been possible to begin my own resea.rohes. (Furthermore, I do not bolieve that the undisputed merits of hiq work n.re made obsolete or in any way replaced by what is new in this book.) Dut however highly I esteem van der Waerden, I disagree in one essential respect with the method he employed. One oCthe greatest merits ofhis book lies in the fact that it uis based on actual study oCthe sources". This was an indispensable requirement. As the author himself emphasized: uSo many statements in books on the history of mathematics have been copied from other books unoritically and without studying the sources. Thus there are many tales in circulation which pasa for universally known truths." A glaring example of this is given, and then follows the passage against which I would liIee to argue from a methodological viewpoint. We are intel'08ted only in those parts of the following quotation which concern Greek mathematics, nonetheless the passage is given here in full for the sake of completenesa.1I To avoid such errors, I have checked aU the conolusions whioh I found in modern writel'8. This is not u.s diffioult u.s might appear, even if, as in my case, one cannot read either the Egyptian characters or cuneiform symbols and one is not a olassioal philologist. For reliable tra.nslations are obtainable of nearly all texts. For example, Neugebauer has tra.nalated and published all mathematioalouneiform texts. The Egyptian mathematioal texts have aU been translated into English or German. Plato, Euolid,' Archimedes, ••• of all these good translations exist in French, Gennan and English. Only in a few doubtful cases it beoame neoesaary to consult the Grock text. • B. L. van dar Waerden, Science AUKJl:ening (Noordhoff, Groningen, Holland, ID64, Solence Editions, Now York, 1003) p. O. 'Lot me take this opportunity to note something not directly relevant to tho contents oC this paaaage. The correct transoriptlon or this Greek name Is the ono UB8d throughout by van der Waerden, namely Eukleldes. I however, will always usa the English Corm Euclid, so 88 to dlstlngulah tho author or the RlemmU Crom the Hegu.roan phllOllOpher oC the B8m9 name (EuJdoldes), with whum hu WII8 often conrUlNld ovon In anolent times.

In Part 1 of this book a short passage from Plato (Theadelua, 14701(88) whioh is not particularly diffioult to understand from the linguistio point of view is discWJSed in detail. This is because. i~ seems to contain some interesting facts about two Greek mathematlclans-Theodorus and Thea.etetus. I am, it is true, a classical philologist, nonetheless I have read the passage in question not only in the original. but als; in 0. multitude of German, English, French, Ito.lian and even Hungarian translations. Before proceeding further, I want to recount my experiences with these. To my surprise, most translations I looked at (with some unimportant exceptions) were accurate on the whole. Even more astonishing was the fact that by and large they rendered the mathematical content of the passage faithfully. (Although they also contained some obvious mistakes' which load to an erroneous interpretation of the text.) It only became fully apparent to me that these translations were completely unreliable after I had studied in deta~ ~he interpretation of this passage, whioh regrettably has become a tradition among philologists and historians of the last hundred years. A tra.nalo.tion can be 'accurate on the whole' and still give rise to a completely wrong interpretation of the text. Let me give a small example of this. In the pasao.ge from Plato mentioned above, the technical ter~ dtWop', occurs more than once. If this word is translated as power (n:s It usually is), this is simply a mistake. It is true that the ~athematlcal content of the p888age will still be comprehensible, but the mcorreot use of this word inevitably suggests erroncous idcua about Greek mathematics. In clasaical times the Greeks did not as yet have the concept of power. Furthermore the Greek term dynamis has nothing to do with the Latin notion of potentia (ability or possibility). If, on the other hand, the word is translated as 'square', and this translation is justified by saying (as HeathO does) that the partioular 'power' found in Greek

• ApeltB' Bltmirable translation should be mentioned as an exception to this: PlGlonlJ Dialog Thdlle', translatod into Gonnan with annotations by O. ApoltB (Leipzig 1011). 'T L Hoath Euclid'/J EkmmU (Dovor PubUcations, 1966, Vol. 3, p. 11): 'Oom~ in equore Is in the Greek 6Mpa f1I1PJA"flO'. In earlier tranaJa· tlons (e.g. Williamson'e) 6twdJU' has boon translated 'in power', but as the partioular power represented_by 6Wap" In Greek geometry is equore, I have thought It rn'llt to UBlI the Io.ttor word throughout.'

2 SuW

18

SZABO: TIIB IIKOlNNlN09 Olr ORBKK. MATHEMATICS

geometry is the squa.re, then this is still mialeading for it implies the following two ideas, both of whioh o.re patently false: (a) The Greek word dynamis literally means 'power'. It is translated as 'squo.re', because in practice dynamis is always the second power. (6) If in mathematioa the word dynamiB is related to ·power'. then perhaps xtSpo, could at least in principle o.1so be construed as a kind of dynamis. Finally, if the word is tra.nslated as 'square' without a.ny further explanation, then the translation is undoubtedly correctj but not much is gained for the history of mathematics. The term Mval"t;, correctly translated as 'square' , reoJly becomes 0. key to understanding this whole pll.88ll.ge of Plato only if one also knows that this goometricoJ term did not originate in Plato's time. Even in the first half of the 4th century II.C. it had boon handed down from an earlier time. Furthermore one should remember that this scientific term could not have been coined without the prior discovery of some important mathematical facts. In other words one has to be clear about the etymology of this expression as a technical term in Greek mathematics. Only then does its correct translation become informative as far as understanding the text is concerned. If one knows this, the translation of this piece appears in a now and totally different light. Many questions whioh were previously discuBSed in great detail (and could still not be answered satisfa.ctorily), suddenly become insignifioant. Indeed they seem to be red herrings. In their pla.ce other problems emerge which could not even have been thought of before the discovery of the reo.1 meaning of the term. At this moment, however, I just want to sketch the method followed in this book, not give the results obtained by its use. So it should be omph8Bized that as far 8B the history of mathematics is concerned, translations of the source materio.1s are frequently unreliable, even when they are philologicoJly excellent. A good exo.mple of this is the pR888.Se from the TheaelelU8 mentioned above. This piece, 8B far as I have been able to oheck in the rather extensive teohnical literature, was never considered problemo.tio by philologists or historians. The whole pll8llage could be translated almost without objeotion. It didn't matter whether the expression MIIal'" was tro.ns1ated 8B 'power', 'liqua.re' or 'side of a square', since everyone knew that in the text it reforred to squares or to the sides of these squares. Furthermore this knowledge ensured that the mathema.tical content of the pll.88age W8B cor-

INTllODuarloN

19

rectly understood (at least in broa.d outline). One might have thought it just philological pedantry to take a lot of trouble OVer this single word. Yet the correct understanding ofthis word would have been important not for philology, but for the history of mathematics. It would have avoided a lot of mistakes in reconstructing the early history of Greek mathematics. It is perhaps worthwhile to recall here at least briefly the change which the interpretation of this expression in the history of Greek mathematics has undergone over the last hundred years. As far as I know, Tannery was the first to notice that the meaning of the term Mval"t; in this pll.88a.ge from Plato was in any way problematic. On the one hand it was clear that for some obscure reason this expression must have its usual meaning of 'square' here. On the other hand the same word in the very su.me pll.88a.ge BOOms to mean 'square root' as well. This idea is justified in as much as the p8B8&ge in question deals not only with squares, but o.1so with their sides. Nevertheless had the text boon ano.1ysed carefully enough, it would have been immediately apparent that there is no justification for supposing that the word M"al'" is Il.mbiguous. Tannery, however, instead of making 0. careful linguistic analysis of the text, chose the wrong way hi his paper of 18767 and conjectured that mathematical terminology had not yet been fixed in Plato's timej so MIIal'" could have meant 'square root' (racine rA.rrl.e.) as well as 'square' (rA.rre). This erroneous conjecture of Tannery survived for a. long time in the history of scienco, although he himself reoJised his mistake eight years later. It was simply impossible for one and the same word to have had two meanings at the same time. Unfortunately, however, he failed to find the correct interpretation of the pll.88a.ge this time o.s well. On the contro.ry, in this later work (of 1884) he propose dB that the difficulties of interpretation be done away with by substituting lnwaM for lJ6vap" throughout the text of the TheaelelU8. This proposoJ by Tannery is of course a heavy-handed and completely unnecessary a.dulteration of the 'P. Tnnnury, 'Lu nornhro nuptial dunB Platon', Revue Pllilollopllique I (1876) 170-88 (M4moiru .cienlifiquu, ed. J. L. Heiberg & H. O. Zeuthen, ToulouseParis 1912, Vol. I, pp. 28-38). Tho Incidental remarks cited in tho text are to bo found in M4,n. Scienl. 1,33, n. 2. • P. Tannery, 'Sur 10. Lunguo mathcSmatlque do Platon', Annalu de la Facult4 de. Leuru de Bordeauz, 1 (1884) 06-106 (M4m. Scient. 2, 91-104).

8ZABO: TUK BII:OINNU408 0., OIlIl:KK MATDlUIATICS

text, whioh is worthy of an amateur. But I still consider this second at.wmpt of his noteworthy for the following two reasons; firstly, becaUBO he clearly rejeoted his previous erroneous view about the ambiguity of the mathematico.l term ".wap", and secondly, because his proposal to al ter the text is clear proof that he bolieved the usuru interpretations of this piece to be unsatisfactory, even though its mathematical content WI18 correotly undel'8tood by and large. In 1889 Tannery made another vain attempt to deny his earlier view.' But his erroneous ideo. had been accepted by the beat soholars in the field, even though he himself had rurea.dy realis8d its inadequacy. Consequently it was just his attack on the authority of the text that was rejected,lo It is not surprising that Tannery's correct realization that 6tWap" could not mean 'square root' (or 'sido of 0. square' as well) did not provail, for he made no serious effort to elucidate the process by which concepts evolve, and whose result in this case was that a square could Le referred to as dynamis. Instead, in the long run he just oreated confusion with two startling and incompatible ideas. Unfortunately these n.re still ourrent, so I givo them hore in some detail. (1) In his paper of 1884, Tannery wanted to explain the fact that /J{wap', means the side of 0. square in Definition 4, Book X of the ElemenL9, as follows:" 8 P. Tannery, 'L'hypoth~e g4om4trique du Menon de Platon', Archiv I. Oesch. der Philo,ophis 2 (1880) 609-14 (Mlm. Scism. 2 4.00-6). It. is worth quoting the following from this work: 'Jo oltals mAme, oomme exemple typique, 10 passage de ThIWUte, au "Wap" est employOO dans Ie II6D8 de racinCI ~, tandia qua dans La lUpubUqua • • • Ie m&me mot signifie au contralru CtJJT4. Milia depuia, la poursuite de mes 'tudes aur lea variations qu'a pu subar la langue matMmatlque des Greea, m'G conduU d II,. oonclu.tiom 10"' d fGU oppa,l,. et je n 'h6aite plus dtblormala eta.' .0 Cf. T. L. Heath, Buclid', BIIlfMnl8, Vol. 2, p. 288; A History of Gred= MalM· matiC$, Oxford 1921, Vol. I, p. 209, n. 2: BIao B. L. van der Waerden, Scilmu Awakming, pp. 142, 166 ota. On page 142 of this last referenoe, one finds • .•• it Is not noo08ll&l'f, lUI Tannory does, to roplaoe tho word &VvO/'" by dtMJpm, Ilrout.ing'. 11 Sctt n. 8 above. Tannery of course Is also solely to blame for van der Waer· don's U96 (Scilmu AUllLbming, pp. 142, 166) oC suob expressions 88: ~p" (illlp"'.fI, IOre4), Lbe 'gmemliorl oC moan pmportlonala', 'the I1cmenJ1icm oC • •• goometrio, arithmetio and harmonio means' eta. by reason of his strange pam. phrase pouvoir una Gire (ala).

INTILOOUcrlON

21

"Soit un carre dont l'aire soit dcSterminoo, de trois pied8 par exam1,le, 10 c6t.C de ce carrO est, dans 10. langue mathematique clo.ssique, La 6tntapbr] (Ialigne qui peut) cotte aire de trois pieds. Pouvoir une aire (Mvaa{}al -r& xrue/ov) c'eat de meme, pour une ligne droite limitOO, 6tre telle que Ie carre construit sur elle ail. preoiscSment cette aire." I must confess that it remains a mystery to me how one could possibly 'explain' 6tntapbrJ and 6Vvaa{}a, by the equally obscure phro.ses 'la ligne qui peul' and 'pouvoirune aire'respeotively. Nonetheless this fine 'explanation' which the master-historian of soience gave did not lack adherents. It would be easy to show that this explanation of his was carried further, when the mathematical term 6tWap&t; was construed in German as 'generating force', and the term 6tntapbr] as 'that which generate.s'. Cloarly the line 80gment concerned is called 6tntapbrJ (that which generates), becau80 being tho side of a square it generatu a square. So muoh for the evolution of concepts I (Tho Greeks could never have used tho verb Mvaa{}a, to express this kind of 'goneration'.) So Tannery's one idea led to these kinds of mistakes. (2) His second attempted explanation in 1D02 had even more unfortunate consequences. It must be presumed that by now' he had forgotten what had been correctly established by him in 1884, namely that the technico.l term 6tWap', could not mean 'square-root' or 'side of a square'. Otherwise he could never have pursued the train of thought which I want to recall bere. In this later work Tannery slcotched tho method of approximation which the Greeks used to calculate the square-roots of non-square numbers (i. e. surds). He concluded his words on this topic with the following remark:l I " ••• en exprimant de plus en plus pres la vrueur de cotte moyenne, si 1'0n ne peut In. construiro que goom6triquement, si elle n'existe qu'en pui88ance, non en acle, pour employer Ie langa.ge dea Grecs." For the moment the only things that interest us in this quotation are the two words in italics. To appreciate their significa.nco, one should follow Tannery's own advice and translate them back into Greek. He helioved thn.t when he spol(o of puis8ance and acle, ho WII.8 using 0. 'quasi-Greek language of geometry', and his ideas ca.n be better under'1 P. 'rmlllury, 'n.. rtll .. 1111 111 InUHlqllo grocqulI tlunB lu cloSvuluJlIJUmunt. clu la math6motic)lIe puro',.BibliothucJ mGthematica, 8 (1002) 102,161-76. (Mlm. Smw., 8 pp. 68-9. Tho quotation la token from p. 82.)

22

8~ABO: TIlE BKOIHHIHOB 01' OlllWK 14A1.'UE.UATICS

tltood if one U8C8 the originuJ Greek words, namely d,wClp', for pui8,ance and biey£ta, IvreUx£,a for ade. In other words, Tannery is 8aying L1I1Lt c5tivallt' means the 'irrational 8quare-root of 0. number' in Greek, because such 0. number could be constructed (puiB8ance) geometrioally hut ~~ not exist 8.8 ~ (lvreUX£ta). In the first place, this conjectu~ of IllS 18 tot0.11y foreign to the way in which the early Greeks thought aLout mathematics. It is just o.n unsuccessful eonstruetion of his own. In the second place, the idca is wrong on two counts. (a) dUvall" never meant 'irrationa1square-root' in Greek; and (b) the origin of the rnathematica.l term lIVvall" has nothing to do with the Aristotelian contl'8Bt between dynamiB and entelechia. (Aristotle's dynamiB is not the same u.s dynamis in geometry.) Here I attempt to give an authentic account of the early history of Greek mathematics, not just in the sense that I have consulted the original texts, but uJso because I have endeavoured to elucidate the JlOW concepts of Greek mathematics in ,Iatu nascendi. Even in its earIicHt period 0. whole series of new concepts were introduced into Greek mathematics; concepts which do not correspond to anything in preGreek mathematics (or at le8.8t are not known to do so). Unfortunately JlO mathematico.1 texts have survived from the period during which mOlit of these evolved. . Yet since thcse same concepts also occur in the extant Greek mathematical texts which originated muoh later, I believe it i8 possible to olu~idate .at le~t in ~art the development of mathematical thought durlllg tillS earher penod by reconstructing the history of those concepts,

For example, there i8 no doubt that tbe axiomatic foundations of mathematics laid down by Euclid in the Elemenls are tbe oulmination of a lengthy pre-Euclidean development. Previous historians only drew IIpon Aristotle, who lived just before Euclid, or at best on Plato to ex plain this process. They never even contemplated the idea that Greek '''x~om~tics' could have originated in 0. 8till earlier time. Vague 'interILl:tIOIlH betweon the Pytlmgor6ans and Elcatica had altlo boon conjectured (by Tannery, for example, and by Rey - one ofhis followers), ILl though no one had determined preoisely what these 'interactions' cUlllprisod, nor which 8ide might have gained the most from them. Tho idea that the founding of this science on definitions and axioms is

IHTllODUCfJON

23

attributable to the Eleatic influence h8.8 not been considered previously. Yet this is what I hope to prove in the 8equel, mainly by analY8ing the history of concepts. A8 far 8.8 I am concerned, 0. 8peoial significance is attached to language in this o.nalysi8. I intend to investigate Euolid's mo.thematicallanguage from a historica.l point of view, for thi8 technical language provides living proof of the developmentuJ process whioh started long before the texts themselves came into being. For example, with the help of lingui8tio analy8is it can be 8hown that all the technical terms of the geometrical theory of proportions have their origins in music. Indeed I will demon8trate tho.t the concepts of 'ratio' (diastema or logos) and 'proportion' (a1'llzlo!]ia, i.e. 'ao.meness of ratio') , and even the expressions for operations on ratios, were all developed from considero.tions about and experiments in the theory of musio. From this I conclude that the musical theory of proportions must have developed up to a certain stage before the geometrical theory. For example, 'similarity of rectilinear figures' could only Le defined 8.8 'so.meness of ratios between the corrcsponding sides' after the theory of musio had made the concepts of 'ratio' (logos) and '8ILmenessofmtio' (analogia) available to geometry. In this case linguistio analysis helps us to reveal 0. historical connection between music and geometry whioh the literary sources only hint at. Although they report that music and geometry were twin disciplines of the Pythagoren.ns, only a historical investigation into terminology discloses that the theory of musio must have preceded the development of geometry, at le8.8t in the creation of its fundamental concopts. I should emphu.sizo, however, that in this work I am not pursuing the history of terminology for its own sake. Although for the most part philological methods are u8ed in this book, these are designed to contribute towards mathomatical and historical understanding, not to philology itself. I am convinced tho.t the signifioance of many important foots about science in antiquity 8imply cannot be appreciated either historically or mo.thematica.lly without using the philological preci8ion I have attempted here. Lot me illustrate this with an example. Reading Tha.er'sls outstanding translation of the Elemen18, one encounters in Book X the notion of 'straight lines commensurable in .. C. Thoor, 'Die Eltlmonto von Euklid', in Ost.wald's Klatail.:er def' E%Gdtm Wia8tm8cha/ten (Leipzig 1933-7).

24

•

SZABO: TBB BBOINNINOS 01' OIlEEI( KATBIWATICB

1H'l'RODUL'TION

26

•

than so years ago to mark out the boundaries within which our attempts at reconstructing pre-Euclidean mathematics have taken place. In his papcr, 'The Theory of Odd and Even', Becker showed that an ancient Pythagorean mathema can most likely be recovered in its original form from the Element8.l' This is the oldest example of deductive knowledge in Greek science, or at leOBt the oldest surviving example. It is important to us for two reasons. The first is that it can be dated fairly accurately. A fragment of Epicharmus clearly indioa.tes that he knew this theory. Iii Considering that he flourished around 500 B.C., the theory of odd and even must date from at least the beginning of the fifth century. However, the second reason is more important. It is that this theory cleo.rly culminated in the proof of the incommensurability of the dio.gonu.l and sidcs of a square. It scorns that linear incommensurability, at least in the speciu.) case of the sides and diagonoJ of 0. square, as well as the theory of odd and even must have already been known to Greek mathematicians at the beginning of the fifth century. (The opposing view which maintains that knowledge of irrational quantities dll,tos only "from the middle of the fifth century"lIl, is simply 0. tradition deriving from modern ofForts to fix tho construction of tho theory of irrationals around 400 B.a. Since one wants to place the 'further development' of the theory during Plato's younger years, the 'first case' of this discovery (namely the diagonal and sides of a square) has to be dated as late as possible, i.e. no earlier than the first half of the fifth century.) In connection with this discovery, Becker suggested four stages of historicoJ development. He thought that these could be traced in the changes which took place in arithmetic and the theory of proporMons

Having sketched at least in outline the methodological novelties to be found in this book, it remains for me to indioa.te brieJly the actuo.1 results of my investigations and the extent to whioh 0. different picture o.f the origins of Greek mathematics emerges from them. As I empha.slzcd above, a definitive new picture of early Greek scienco oannot as yet be pu.inted. Nonetheless, I believe that these investigations oan help to change significantly our conceptions of the development of preEUtllidol~n mathomatics. To ohu.raol.orize this new perspective in 0. few lines, I will start by describing the fundamentoJ work whioh tried more

.. 0, Becker, 'Quellen und Studien zur Goachiohw der Mathematik', A8lronomie und PhyBilc B, 3 (1934.) 633-53. Reprinted in the collection Zur Ge8chichU der griechiBcI,en Malhematik ed. O. Becker (Wisaenschllftliche BuehgesellsehaCt, Darmstadt 1065). U Cf. Also van der Woerden's Science Awaking (2nd English edition, Science Editionll, New York 1063) pp. 100-10. (The pllll8Ugein qUUllt.ion wUBunfurt.unal.u· Iy omitted Crom the Oennlln edition.) On the interpretation oCthe EplehannusCragmont, HIIO nlso K. Roinhardt Parmmulu und die Ge8c1iic/,te der griec/'iBcMn Philo8ophie, (2nd edition, ))'rnnkfort 1050) pp. 120 and 138. I. K. Gaiser, Plalons uRgeschriebme LeMe (Stuttgart 1963), p. 47., n.

squaro'. This translates the corresponding Greek phrase quite correotly and its mathematical meaning oan easily be understood. Yet if one studies only the modern translation and not the Greek text, the belief that it faithfully renders the original might mislead one into thinking that in antiquity there were two different theories of 'straight lines rationoJ in square' whioh led to the same result by different methods. The one seems to stem from questions about the existence of a geometric mean between straight lines (or numbers); the other dea.ls with the 'squaring of straight lines', not with the notion of geometrio mean. But this ideo. becomes 'untenable as soon as one reoJises that these two phrases, 'straight lines rational in IJquare' and 'squaring ofstraight lines' , both translate the Greek expression wOeiat. oovape& croPPBT(!O&. Also, it should be remembered that producing a dynamia by its very nature presupposes the uso of tho goometrie mean (of. pp. 07-8, 174-81). Therefore it oa.nnot be emphasized too strongly that research into the history of ancient mathematies is impossible without 0. very oa.reful and thorough investigation of its language. It should not be forgotten that mathematioal thought and languago were still very closely linked at that time. The mathematicoJ symbols which we oJl roly on nowadays, could oxpress practically nothing, thoy did not oven exist, so words had to be used. Furthermore these words, even the ones which later acquired speoioJ mathematicoJ meanings, were drawn m08tlyfrom everyday language or from the language of philosophy. So the oha.ra.cteristics of ancient mathematical thought, whioh sometimes provide 0. marked contrast to later ideas about mathematics, are only acoessible to us through language. "

II

~I

l !

26

UfJ:1l0UUI.."l'ION

8ZAUO: TWC UKOUUUN08 OM' OUJj;K. UATIllWATICS

during the fifth and the first lwJf of the fourth century. This dating is noteworthy because it has been retained with minor modifications to the present day. I will give the four stages here, together with those of their features which Beoker thought most important~ (1) The early Pythagorean stage: According to Becker the most significant achievement of this period was the theory of odd and even (although it did not as yet include the theorem that any number may be decomposed uniquely into prime faotors). Distinot from this was a theory of rational proportions (the proof of the incommensurability of the sides and diagonal of a unit square) which arose from musical and geometric questions. (2) The stage of Theodorus: This period sa.w the genera.1 definition of incommensurability by means of anlanaireai8 of magnitudes, also the proof of the irrationality of Vii (for n = 3, ••• , 17 and not 0. perfect square), and a general theory of proportions classified according to kind (straight lines, numbers, etc.). (3) The stage ofTheootetus: At tllia stage antanaireai8 was applied to number theory: the 'Euclidean algorithm' was discovered, and it was proved that any number can be decomposed uniquely into prime factors (Element8 VII. 30). (4) The stage of Eudoxus: This saw 0. general and abstract theory of proportions uniformly applicable to ratios of all kinds. Also sameness of ratio was defined in terms of the 'Eudoxus cut' (Elements V, definitions 5 and 7), and the axiom about the multiplication of magnitudes (otherwise known as the axiom of Archimedes) was formulated (Elements V, definition 4). As one can see, Beoker was simply interested in giving the chronology of arithmetic and the theory of proportions. Geometry is considered (lilly incidentally. l!'urthermore, the problem of incommcnsurability, which was a geometrio problem both in the minds of the ancients and in ita origin, is not just a matter of arithmetio. An immediate drawback of this attitude is that the most important geometer of the fifth century D.O., Hippocrates of Chioa, cannot be placed correctly in this ohronology. He must have lived before Theodorus and Theootetus, yet it is scarcely possible to believe that Hippocrates had no knowledge of the mathematical facts whioh Becker associates with their stages. It'urthermore, Becker's four stagea completely ignore the lliatorical problem connected with laying the foundations of mathematics and its

....

systematic construction. They leave open the question as to when Greek mathematicians were first able to lay the foundations of their subject on definitions and axioms. Becker himself dea.lt with this question in an earlier work which was to some extent superseded by his paper on the theory of odd and even.17 At that time (1927), "Q~ey'~r, Beclt_e!,_ s.t~~. had the view that 'Plato was the first to be cl~k~J!.are of the~g~rous ;;;th~di~l-p~;d~ for constructing the elements of mathemMj~1B' Boo:ring in mind these--- lierri eas of Beoker (which were formulated under the influence Zeuthen's onjecture on the matter), one is es with a. fifth - namely, the attempt inclined to supplemen . four to systematically construc mathematics first became possible under Plato's inOuence. Hence it 'took place direotly after the fourth stage, or perhaps during it. I believe that Becker's chronology has to he extensively reviscd in the light of my own investigations. First of u.ll, the most likely date of the earliest attempt to lay the foundatiolld of matheniatics and construct it in 0. deductive manner lies far back in the time of Becker's first stage. Following Becker, we might call this the Platonic stage. I believe I can show that the theory of odd and even presupposed the basic definitions of Euclidean arithmetic. Not only does the oldest Pythagorean mathema go back to the first half, if not to the beginning, of the fifth century D.O. but this is also the time at which the theoretical foundations of arithmetic were laid. Only the theoretical foundations of geometry are perhaps later, since Euclid's first three postulates are attributable to Oenopides (around the middle of the fifth century). Becker's second and third stages are omitted in my reconstruction. In view of the interpretation proposed in Part 1 of this book for the relevlLnt 1'IUI8age of Plato, it cannot be shown that Theodorus or Theaototus (as thoy appear in Plato) made any new contributions to matbematics. On the contrary, since the concept of dynamis is used in this passage, I have come to the conclusion that the discoveries which were once attributed to the oha.raoters Theodorus and Theaetetus in Plato's dialogue date in fact from pre-Platonlo times amI in all likelihood antedate Hippocrates of Chios. n Boo chapter 3.30, 8OcLion 11. O. Beoker, Mathema,iac1&S EmUnIl (Hullo 1927), p. 260, n. 2.

It

,

28

SZABO: rUB BBOINNINOS OJ!' OllBBK llATlIlWArlClS

1 have nothing to say about Becker's fourth stage, since my investigations are not directly concerned with the mathematical discoveries which modern research attributes to Eudoxus•. Instead of Becker's chronology, the following investigations enable us to divide up the development of early Greek mathematics into different stages. These will be briefly outlined here. . The oldest stage in the development of Greek mathematics which is still accessible to us is the musical theory of proportion. All the technical terms of the later generoJ theory originated in the musico.1 one. (This sto.ge is dealt with in chaptera 2.3-2.19.) I believe it would be 0. hopele88 undcrtaking to try and pin down this stage as having to.ken place in some particular century or half-century. Even if the fragments are included, our oldest texts touching on musico.l questions scarcely antedate the age of Plato. Yet thoso terms from the theory of music which had already been taken over into geometry at the time of Hippocrates of Chios must have been coined muoh earlier. It seems to me much more important that within this stage two periods can be clearly distinguished. In the earlier period experiments with the monochord took place (although these did not as yet involve 0. measuring rod). Also, some important terms in the theory of music originated at this time, namely diplruion dirutema (2:1), hemiolion dirutema (1Y2 = 3:2) and epitriton dirutema (1 1/3 = 4:3). These of course were later carried over into the theory of proportions. Finally the method of the 'Euolidean algorithm' (successive subtraction, antanairesia or anthyphairesia) was developed during this period. The later period was characterized by the introduction of the measuring rod whioh was divided into twelve intervals. This brought into being the new musical/mathematical concept of lO(Jo8 (the relation between two numbers). At t.ho next stage the mwdcal theory of proport.ion8 was applied and extended to arithmetic, particularly to the geometrized arithmetic of ,pane I ' "'1 I numb era, SImI ar pane numbers' eto. (See many of the propositions in Books VII, VIII and IX of the Elements.) There are two remarks to make in this connection. Euolidean arithmetio is predominn.ntly of musical origin not jU8t because, following a tradition developed in the theorf of musio, it uses straight lines (origin&lly 'sootions of II. string') to symbolize numbers, but also because it uses the method of suc~ve subtraction whioh was developed originally in the theory of mURIC. However, the theory of odd and oven oleo.rly derives from n.n I

INTRODUcrlON

29

'arithmetio of counting stones' (tpijrpo&), whioh did not originally contain the method of successive subtraction. It seems that only the arithmetio of the theory of propOrtions (just for numbers, of course) has developed from the musico.l theory of proportions. Actua.lly. this theory seems to have come after the theory of odd and even. But this is simply a conjecture on my part and not a proven foot. In my opinion it is extremely unlikely tho.t Book VII of the Elements goes back to Arohytas, for the problems with which it de&ls o.re very olosely linked to the theory ~f incommensuro.bility. They seem indeed to prepare the wo.y for this theory. Yet quadratic incommensurability (the general theory of dynameis, not just the particular case of the diagono.l of a square) mus~ have been known well before the time of Archytas, o.nd before the time of Hippocrates of Chios as well. . The third stage comprises the o.pplico.tion of the theory of proportions to geometry. This led to the precise definition of similarity 0/ rectilinear figures ('samene88 of the ratios of the corrosl'onding sides') and subsequently to the construotion by geometrio methods of the mean proportional. All this, of course, took place at the ~ime of the .early Pythagoreo.ns. The construction of the meo.n proportional led directly to the discovery of linear incommensurability. This construction (o.lreo.d~ well known to Hippocrates of Chios) implied the extension of the notIOn of 'ratio' to o.rbitrary quantities. whether the Greeks were o.ware of it or not. It is espeoio.lly noteworthy tho.t the discovery of incommensu~o.­ bility is due to 0. problem which &rOse origino.1ly in the theory of musIc. Moreover it is interesting that Tannery had &lreo.dy conjectured that "The problem of the irro.tionruity of V2 could just 88 well ho.ve origino.ted in the theory of music as in geometry."l" . The discovery of incommensurability prepared the way for 0. further stage in tho developmont of mathematics which is troated in Part 3 of this book under the heading, 'The construction of mathematics within 0. deduotive fro.mework'. To prove in an unobjectionable wll.y that incommensuro.bility reo.lly existed, it became necessary to develop 0. new teohnique of proof o.nd to lay the theoretical founda.tions of deductive mathematics. Aocording to my reconstruotion, these developmenta took place under the influence of the Eleo.tics. . II

Du r6le de 1& musique grocque dlUlllle d4veloppement. de la mathtSmat.iquo

pure. (MIm. 8ciml. a, 68-89 espeoially pp. 83-9). Tannery's conjooture was olso takun up by Bookor (Zur Guc1aidale der griechiscMn MalliemGcik. p •• (3).

:Ill

HZAIlc): Til" lIKlIlNNINUH Ott UJIKKK UATlIKUA1'ICtI

As one can S80, my 'chronology' fails to answer many qucstions about dates. I cannot pin-point any more accurately the time at which the theory of proportions waa first applied to arithmetic and then to geom(It.ry. Indeed tho idea. that it was applied to arithmetio first and only later to g~metry is just a conjeoture baaed on the fact that the theory of proportIons of numbers is closer to the original musical theory than the theory of proportions of arbitrary geometrioo.l quantities. Hence this 'chronology' serves only as an attempt to reconstruot in their likely order the succeasion of problem 8ituations whioh led from 'simpler' to 'more complicated' knowledge. It still remains fQr me to mention 0. problem which is not dealt with in this book, although 0. treatment of it could rightly be expected. One might expect that in a book entitled The Beginning8 of Greek M alMmatica the pre-history of this subject would at least be touched upon. Modern reseo.rch has shown that Greek mathematicians took over 0.101. of well developed mathematics from pre-Greek times. A well-known example of this is what is usually oo.lled Pythagoras' theorem. This could not have been discovered first by Pythagoras, because it had already been learned from much older, Babylonian texts. Similarly it en.n he shown that other portions of Greek mathematioo.llmowlodge are of oriental descent. Nevertheless I have deliberately omitted 0. discuBBion of what was horrowed from pre-Greek mat.hematics, mainly because1n my view we still do not know enough about Greek mathematics itself to be able to treat, this problem with any success. As an illustration of this, let me , ~:atlO~ here the problem concerning 'the geometrical algebra of the

;::;!~\l~noti~bat there are some interesting geometrio p. kSlI ~nd VI of the Elements which would normally be written out as algebr&lo formulae. From this he conoluded that they deal.t wit.h 'algebmiopropositions in geometric olothing', or as he put it - with gtDmdric algeb,,,. , However, he could not furni!iP satisfactory answers to suoh questions lUI wbether the earl! Greeks actu~lly bad an algebraic theory which Was

'0 cr. what. Neugebauur hllB to say about. the follOwing In 'St.udien zur OeIIchicht.e der antiken Algebra, m', Quellen tmd Bttulien zur O.cAicAU d4r Mathematik, Alltnmomie tmd Phyrik B, a (1936) 246.

IN'I'IWUU(Jl'IUN

later geometrized, how this theory might have been arrived at, and what the role of 'geometric algebra' might have been in early Greek science. On the other hand Neugebauer, the discoverer of 'Babylonian algebra', found Zcuthen's 'discovery' very opportune. According to him, the reason the Greeks 'geometrized algebra' is ..... on the one hand that after the discovery oC irrational quantities, they demanded that the validity of mathematics be secured by switching from the domain ofration&1 numbers to the domain of ratios of arbitrary quantitiesj and on the other hand that it then became necessary to reformulate 1neb • ,(/r;ullouMC A_A' . '_I.Kjf the ,esul18 of p,e-Greek ,olgeb" ralc"19ebra an T"_ "21 Evcr since then, Euclid's 'geometric algebra' has been regarded o.s 0. borrowing from Babylonian science or as 0. geometrical version of BabyIonian algebra. Only a thorough examination ofZeuthen's theory ab~ut 'algebraic propositions in geometric clothing'-w6uld allow one to de~lde the extent to which the above is 0. historico.l1y sound reconstruction. But until we have obtained a betterunderstandingoCGreek mathematics itself, and in particular have answered the question o.s to whether theso ProIJositions of 'geometrio algebra' really deal with problems originating in algebra and not in geometry, I think it best to put o.side 0.11 questions concerning the origins oC 'Greek geometric algebra'. (But seo tho ApJlondix to !.hill boole, pp. 332 ff.) Moreover, one must be careful not to misunderstand the phrWlo "8witching from the domain of rational number8 to the domain 0/ ratios 0/ "rbitr"ry quantities". If one undorstands it lUI saying that aftor the diKcovery of linear incommensurability it beoo.me necessary to extend the concept of , ratio' (which hitherto had been applied only to numbers) to incommensurable quantities as well, then there is no objection to Neugebauer's words. H on the other hand one understands by 'switching' some kind of 'additional geometrizo.tion', then I have some decided objections to raise. In the first place incommensurability was a geometrie problem to the Greeks right from the start. In the second place it is simply incorrect to suppose that the discovery of incommensurability shook the Greeks' 'faith in numbers' in any way. They knew very well that even, though 0. number could not be D.BBigned tothe length of the diagonal oC a square, for example, nonetheless this ao.me dio.gon.al could be expreBBCd numerically in terms oCthe square constructed on It. 11

Ibid., p. 260.

\

\

32

It

BZABO: TUlI: BKOINNINOB Olf OlU!:E1( lalATIUUlATlCB

1. THE EARLY HISTORY

not neoeaaa.ry to 'abWldon the domain of numbers' after this diljcov~ry•. The .expl~nation of the genesis of the mathematioaI concept dynamu given In tbJ8 book throws some new light on the above topio as woll. . WIl8

In conclusion I would like to list here those of my papers on which I have relied heavily in this new disoussion of early Greek mathematics. 'Wie ist die Mathematik zu einer deduktiven Wissenschaft gewor-. den t' Acta am. Acad. Sci. hung. (Budapest) 4 (1956) 109-51. 'Deiknymi, a.Is mathematisoher Terminus fUr beweisen.', Maia N. S. 10 (1958) 106-31. 'Die Grundlagen in der frUhgrieohisohen Mathematik' Studi Italiani di Filologia Olasaica (Florence) 30 (1958) 1-51. 'Der Alteate Versuoh einer definitorisch-axiomatisohen Grundlegung der Mathematik', Oliri8 (Bruge, Belgium) 14 (1962) 308-69. 'The Transformation of Mathematics into Deductive Science and the Beginnings of its Foundation of Definitions and Axioms' Scripta Afathemalica (New York) 37, 27-49 and 113-39. ' 'Anfii.nge des Euklidischen Axiomensystems'. Archive lor Bi8tory 0/ Exact Sciencea (Springer-Verlag) 1 (1960), 37-106. '~in Bolog fUr die voreudoxische Proportionen Lehre t '(AristoteJcs: Topu: 8 3, p. 158b 29-35). A,chiv /. BegriOsge&ehichte (Bonn) 9 (1964) 151-71. 'Der Ursprung des "Euklidisohen Verfahrens" , Math. Ann. ISO (1963) 203-17. 'Die frUhgrieohisohe Proportionenlehre', A,chive lor Hiatory 0/ Exact Sciencu (Springer-Verlag) 2 (1965) 197-270. , , 'Der mathematisohe Begriff dynamia', M aia N. S. 15 (1963) 219-56. Theaitetos· und daa Problem der Irra.tionalitAt in der grieohisohen Mathe~atikgeso~ohte. Acta. am. Acad. Sci. hung. (Budapest) 14 (1966) 303-58•. II I ~':,,' , . "j ":'

J'

~

I

I.,'

\

.

•

,~':.

OF THE THEORY OF IRRATIONALS

1.1

OUBBBNT VIBWS

OJ' THl!I TllBORY'S DEVELOPMENT ,-'-- .-.-- .. ---.-'~---:\

I In what follows I Will give an example of the method. I intend to use for

investigating the history of early Greek mathematics. I believe that amongst other things this method sheds new light on the development of the theory of irrationals. Let me begin by recalling what has been thought up to now about the history of this theory. Recognizing the existence of irrationals was an ~utstandi~hieve­ ment oC early Greek mathematics. Yet historioaI resea.roh has been unable to give a sa.tisfaotory a.ocount of how this discovery came about. The only thing that seems to be more or less established on the basis of previous investigations is "that Greak mathematioians knew of the existence of irrational quantities (or incommensurable ratios) from the middle of the fifth century".l If one surveys the relevant literature of the last fifty years, what stands out i8 that the circumstances surrounding this discovery remain obsoure. The prime example of irrationality in ancient texts is always the lengths of tho diagonal and sides of 0. square.' Hence it used to be thought that the discovery was 'undoubtedly' 8Uggested by the diagonal of a square.a Recently, however; historians have been more inclined to attribute the discovery to Hippa8u8 of Metapontum and claim that he was led to it (in the fifth century B.O.) by considering the doclec4hetl-

1

'.:

. .:t ",",

.

,: •• ,

,'"

1"'.1';

.:' ,

"." .! \. ..f., ; •• ':,,', •• _,"1: . ,/1.- t . l •• :,.,\!.,.~I:.;;C,}1::,!)L::r!r•. I;H.

··f

\,

,'I

,\. K. Gaiser, PlaloM ungNcAriebme LsAri, BtuLtgart 1063, p .. 4'll n. ..'. Aristat.1e, MeltJphy,iu. 983a19ff.' and .I063a14ff. ·or. also '1'.- L. Heath, MalhemGIiu in ArieIolle, Oxford 1949. p. 2: "Thelnoommensurable Is menLionod over and ovor again, but. the only case Is Loot. of lobe diagonal of 0. square In relation to ita side eta." . '. ,to • "..! I. • K. von FrItz in R~ie _ KlMmDAm AlIerlurnftDiu., od. WiaBowa, Kroll, u. ale A1813; SOO also W. Burken, WeUIM" unci W"'IMcho/', 81Udim IU p~. Philolaoe unci PkIlon, Nuremberg 1062, p: 436 D; , ,

3 BuW

34

./

BZAnO: TU!> lIKUINNINUS 01' OIUU.1t MATUIWATICS

l'AII1' I. TU,", KA1U.Y

ron.' This latter view represents 0. compromise between conflicting ILccounts handed down from antiquity.' On the one hand, it is known that the pentagram was an emblem of the Pythagorean&. On the other hand, 0. late BOurce credits Hipp&8US with working on the penta-dodecahedron. He was supposedly the first person to disclose the properties of this solid and is sa.id to have perished at sen. for this irreligious act.· Now it is argued that the incommensurnbility of the side and diag~l of 0. regular_pentagon (i.e. of the / faces of 0. pento.:aodeco.hedron) is easily seen. FurtliCrmore, in some anciont accounts the very mention of mo.themo.tico.1 irrntiono.1ity was held to be a 'terrible betrayal of the teaching of Pythagoras' amounting to a 'scandal'. This historico.1 reconstruotion (Hippuus' shocking discovery - his 'treason' and the divine retribution for it) supports the ancient tradition. In recent years, however, convincing arguments have been advanced against the credibility of this tradition. I mention only two of these here. Reidemeister hBS pointed out "that there is no mention pf any scan~ in the various texts of Plato and Aristotle which deo.1 with irrntiono.ls, even though its effects must still have been felt at )hat-t\me". 7 This leads one to think that the story of discovery andJ!~~!,!-y~is perhaps only 0. later legend arising from the ambiguity of the word 8e~TOt;. Then the explanation o.nd open mention of mathematical irrationo.1ity becomes rather a 'terrible breach of a holy tradition'.8 In the language , of mystical religious writings (partioularly those of the Neo-Pythagoreans) 8e~Ta meant the 'carefully guarded secret teachings which were dangerous to the uninitiated'. A non-mathematician might eMily

r.:----

• K. von Fritz, 'The Discovery or IncoJDrn8naurability by Hlppasua or Meta-

rpontum', Ann. Math. 48 (1946). See also S. Holler, 'Die Entdeclnmg der stetigen i Teilung durch die Pythagoreer', Ablaandlungm der Deuuchen Al:ddcmia der ( IVis6. %U Berlin, Ki4u. f. Malh., Phy6i1& urad Tec1anil&, 1968: reprinted In Zur . GuchiclJla der grieehisc1aen MalhcJmatil&, pp. 319-64. • Burkert, WeisMit und JVis6an6c1aafl, p. 436. • Iamblichua, On 1M Phil060phy of Pytlaagortu (tid. Deubner) 62.3-6, cr. Uellor, 'Die Entdeckung' (n. 4, above), p. 8. 'K. Reld8Dluiater, Dtu BzaJ:u DenJ:en der GtWc1um, Hamburg 1949, p. 30.

<:J ''r:~~~~~_,!eis~nd_ Wis~6Mc1aafl, ~.

I

1II~'I'1I1lY

01' TU,", 1'IIJo:Ull\' 0.' llUU'UUJIoAl.I>

have thought. that. such 0. secret. o.1so existed in t.he iIef!11TOV of mathe,. . . -'. ___~ matics. In view oftheso dubious parts of the tro.dition,~!:.t,Comes to the \ conclusion' that "The only certainty about the discovery of irrntion\ ality is that'rheodorusofCyrene proved that Vii' (forn = 3, ... , 17 \ and not a perfect square) is irrntional, Furthermore, the irrntionality of \ V2 WIl8 known earlier." Therefore, in this view, 0. "new epoch" in the history of the theory begins with Theodorus. Whether the square or the dodecahedron first led to the discovery of irrntiono.ls, subsequent development of the theory is linked to the name ofTheodorus. It had to be he who proved ;' further cases of irrationalit.y for _~~!!._~J.:!!.~_Y.L!Uld, yIJ. .../ - This remarkable dating of events is due to Yogt, who attempted more than fifty years ago to reconstruct the early history of the theory in the following three sto.ges. 10 (1) The younger Pytho.goreans discovered and proved (before 410 B.O.) that the lengths of the sides and diagonal of 0. square are incommensurable, and in so doing, they gave approximations to their ratio ( dUJPeT(!o, d'1n1 and du1IleT(!O' 8e(!rrro t;). (2) Theodorus of Cyreno (around 410-390) posed the inverse problem i of squaring, i.e. ho discovered the irrationality of square roots in general and proved this by generu.lizing tho methods of the Pythagoreans (cropPeT(!OV and otl m5pPeT(!OV; l1f1T&v and 8ef!TITov). (3) Thea.etetus of Athons (around 390-370) laid the foundations of a geneml theory of quadratic irrationals, and clB88ified them (drrro)f p~~tt Ilnd 1!'1TO)f dtJ)faptt or ~ov: plaT/, l~ Mo dJ'OpaT"')f, dnOTOp~). As one cUon soo, special significance attaches to the names ofTheodorus and Thea.etetus in Vogt's outline, Theodorus is supposed to have discovered 'the irrntionality of square roots in general,' and Thea.etetus to have 'laid the foundations of a general theory of qundmtic irrntionals', (No one would claim, however, that these words are sufficient to distinguish clearly between the contributions of Theodorus and

.

--~

I

• Ibid., p. 439. H. Vost in Bibliothaca MalhcJmatica, Z6chr. f. Ouch. d. math. Ww., a Series, 10 (1909/10) 97-166 and 14 (1914/16) 9-29. See 8180 C. Thaer, 'AnUko Matho. matik, 1908-1930' in C. Bureian, JaAreabericlJla Uber die FortschriUo der 1:laui. BCMn AlterIum6wWan6c1aall, Jahrgang 1943, Vol 283, Leipzig. 10

36

IIZABO: TUlI: IU'U1NIUNOIi 0., Ollll:lI:K MATUKMATIVS

I'AUT I. TllB KAJU.... 1I111TOILY 0' Tll~ 'l'UJ::OltY 01' IIUlATI0NALS

Thenotetus.) This reconstruotion of the history of irrationals is still found ~ay in textbooks on Greok mathematics. Tanneryll argued for it even earlier, and it waa doubtless suggested by Plato's dialogue PheadelU8 or rather by the mathematical part of this dialogue (147~~. (( 14SB). As Vogt wrote,JJ "The mathomatica.1 part ofthePheaetelU8 is the ) birth certificate of irrationals, drawn up by a contemporary." .~ ' / This interpretation of the text led to even more far-reaching conclu-' sions in later researoh, especially where Theaetetus himself waa concerned, for this same passage of Plato is the most important source of information about the mathematica.1 achievements of the young Theaetetus. However, the so-called 'Theaetetus problem' is not our main concern here.1 S11 The paasage haa to be examined from an entirely different viewpoint to see first whether it really supports the idea that '''l'hcodorus waa the fll'Ht to poso the inverso problem of squaring", or that "he discovered the irrntionality of square roots ,~n.~ral", and then whether it allows 0. different reconstruction of th, carlyjhistory of the theory. I

r

c..-

1.2

TIIB CONOEPT OF 'UYNAmS'

In this next section we will inspect more closely the text itself, together with a translation. But since my translation, BB well BB my interpretation of the text, is baaed primarily on the definition of a single word, I want to refaoe the translation with some remarks about this dofinition in the hope t 10. these will make it eBBier to understand. The mathematical term 6wap" oeours more than once in thEge, an? .the co:2ponding verb 6.waa{}a, ooours there BB well. 1 It is my opinIOn that .nless one understands this term correotly, on cannot give a sati o.otory interpretation of this paasage from Plato, or recon'Sur la languo matMmatlquo do Platon', Mllm. Scient. 2, 91-104. RibliotlldCtJ Afal1lemalica, 10 (111011/10) 131. , .... Jfor t.ho lIu·uallud "J'huuut."LUI' prublom' uf. my l'I1JltJr, ..1'huul.... 1.oa uml dUll J roblum der Irratlonalltlt. in dor grluobisohon Mat.bematlkgoachichte" Acta tllII. Acad. Soi.l,un(1. (Budapost.) 14 (11166) 303-68. ' 1I'1'hu vorb dtlraaOa, OOOUrII uguln In t.ho Orook ....xt. t.hill tlmu In Ito non. technical, ewryrlaY IJenu: d de"P~ lJ",dpno; ra~ ladx" yl""aIJa" 'tho number whloh can bo obtainod lIS t.ho produot of t.wo oqual faowrII'. II It

37

struot tho development of the theory oC irrationals. It scems romarkable that so little attention WBB paid to it in earlier researoh. I WBB finally able to establish the following fa.cts about this term." 66vap" is frequently t~rt!l~ BB 'power,' but this translation is misleading - in fact it is ~ownrightlla18e, for Greek mathemo.tioio.ns did not BB yet have 0. generalnotion-6f 'powor' .15 The Platonio term alJE'IICI f\comes close to our concept of 'power' but docs not correspond to it, since there was only a second o.nd third alJEf/ in Greek (see HaTa T(!tTTf' alJEf/v in Plato). Furthermore, such later terms BB OOvapo6.wap" (a'), 6tWapoxvpo, (ai ) and xvpoxvpo, (a8 ) which occur in Diopho.ntus,l7 show that 66vap', "cannot mean 'power'. Since HVPO, undoubtedly means 'cubic', the L word ".wap', can only mean 'square' in mathomatical contexts. (This applies ~ Diophantus' ~sage of th~ ~ord aa w~II.!:.>. . . i More Important than Just estabhshmg that 6tWap', (10 Its techmca.1 ~ sense) meana squlU'O is to cxplu,in how this usage comes about, i.e. to \show how 0. word which meant something entirely different (force, \capa.city or power) in everydo.y language, came to have this unusual 'meaning in the specialized language of mathematics. It is not just the noun 6.wap", but also the verb 66vaa{}a, which has 0. special mooning in mathematics. The latter means 'to have the value' 'to be worth' or 'to amount to', and is used only in connection with

I

I.

~ A. Szab6, 'Ocr mathomatilloho Begrlft 6.n.apl, und dll8 sog. goomot.rlschu :r.lit.tel' Maio, N. B. 16 (1963) pp. 219-66. I am indobted to Prof688or Fran90is Laasorre for bringing to my at.tention tbo papor by Baortblein In Rllein. Mw. 108 (11166) 36ft. Thore III mucb uBOful pbilologlcal material in this work, nonetheless, ito But.bor fails to recognlzu that dtWap" 118 a mathomati0l11 tonn ('valuo of t.bo squaru' or 'square') and Arlstotlo's IJ.waPI, (os opposod to enB1'(Jeia or mlB' lulUJiaj havo absolutely notbing to do wlt.h oacb oth~r. 16 For oxamplo Holberg In his Latin t.ranslation of tho ElemBn14 (Lipslao 1888·-1 P I II) nlllClul'lt nnOnlLIClII X. 2 .IH 'UU(ltuu IK.t.lln!.i" oommnnRurultihlll 111.0: (etWeia, 6vvdpe, cnipl'"oo,); alBoThnranaro.Cardlni (Pitagorici, Vol. 2, Floronco, 11162, p. 77) trlUlBlatas a part of thu Platonlo taxt. In quost.ion os 'Toodoro qui, oi uvuvu ulIIIHtnlit.o clllno flgllftl n •• utlvlI unll pohln~(I IIUO.' II Republic, Book IX, 687d. 17 Cf. T. L. Hootb. A Hutory 0/ (he/:l: Mathematics, Oxford 1921, Vol. 2, flp. 4"7-H.

38

SZAUO: TU~ IIKIJINNINUII Olr UJtKKK MATJlJWATICB

( squaring. UI It should be not.iced that in mathematics the word 66.aafJa, was used originally about trana/ormalions 0/ 81lr/acea in the plane. A rectangle was transformed into a square of the same area and the side of the resulting square was described as follows - this straight line (w{)eia) 10hen squared [i.o. if ono were to construct a square on tho straight line in question] has the same t1t1lue (rat»' 6VvQTa&) as the previO'lUl reclangle (Tip 1Uf!,elopbqJ Wro •• •).1' It should also be pointed out that this vorb Was only used about tho tro.nsformation of a rectangle into n. square of the same area and in this context it indicated the value of t.hat area. It would have been impo88ible for example, to express tho value of the square of a triangle using this verb. There is no evidence of any such usage in the language of ancient mathematics. (Moreover, a triangle was always transformed first into a rectangle of the same arelL, and then the rectangle was transformed into a square. This can be scen from propositions 1.42 and 11.14 of the Elemenl8. So it scoma thtLt i - !5v"aaDa, was only used in tho latter step, whon it expl'e88ed tl,e value 01 tile square of the rectangle.) Therefore, this particular transformation I must have been considered especially important for some reason. Thero is no other way to account for the fact that 66.aafJa, and 66.al'" were . used to express 'samoness of value' only in the special case 01 the ret:lanule and square. //"'- l~ my opinion the mathematical oxprCH8ion66.aafJm/nltlst have been , dOrlved from tho lanb'llago of finMcc. J uat. as when converting one kind (If 1Il0noy into another, t.he value was expressed by this word,1O 80 in geometry, whon transforming ono figure into another, the same word l'xpl'08lied the value 0/ a reclangie when squared. Furthormore, just as

I

L

(-'-7

/, l

,s cr. F. Rudio, Der lleric/tl des Simpliciu. fiber die Quadraturen des Anliphon uwl Ilippol:rales, r.uipzlg 1907, p. 139 (800 tho indol[ undor 6uJ'WJ80,), also T. 1.. Hunth, (od,), ArchimedtlIJ, J)ovur ruprint, p. olx!. .. Cf. T. L.lIonth, (od.) ArchimedcIJ, Dovurroprint, p.olxi: "Tho vorbdalroa80, (with or without raov) hWi thu 88JU18 of hoing and, when HraaIia, is used alonu it iH fi,lIC1wOII hy dl...a"ta r"" Utn noollllnt.ivlI, thll8 'he ItqIUInJ (on II Ij'raiuhl li,UI j ,.. "'I"al rectangle contained by • .• is: (",Dela) rao.. dalroTa, 1'41 1Uf"%o/,b(~

tbr6 ... •10

'0 ".e

l'Alll' I. l'UK

!tho term """aI'"

~AIt.LY

39

lUlITOllY Ol!' 'l'UK TIIHOllY Olr llUtATIONALB

meant 'value' in general!l in the language of finance,

! 80 in geomet.ry it came to mean 'the value of the square of a rectangle',

,It

'tho valuo of the square' and finally just 'lIqua.re'. ---------1 should be emphasized, however, th'ftt: these moaning1 apply only to mat.hematicn.lla.yguage. Neither ~a~ n~r 6Vval'" could o~er take on t.hese mcanin~ everyday GreeyIn th181nstanco the speCial usage of mat.hematicians did not lead to a corresponding everyday usage. (A square was novor cn.l1ed &Wap', in everyday Greok.) It is important to point this out because we will see in Part 2 of this book t.hat the oxpression dva lO"t»' (or dvaAoyo,,) is 0.180 of mathematical origin. It was coined for 0. particular mathematical purpose and meant 'equal (when taken) in logoi' or 'in the same ratio'. Unlike 66.al'" it had originally no meaning outsido mathematics, so it could subsequently be takon over into everyday language with tho extended meaning of 'similar'. In geometry 'similarity of rectilinear figures' was defined U.8 d"aAo"la (i.e. 'having sides in tho same ratio to each other'), and when 'similarity' camo to be expressed by dvaAo"la outside mat.hematics as well,4!lo technicu.1 term was boing used to oxpress a very common notion, woll known in everyday Ii~ 66.al"" on tho other hand, never acquired an overyday meaning of~his kind, bocauso 'tho transformation of a rect.anglo into 0. square of tho samo area' nover figured as prominently in everyday experience (~it occurred there at all) IL8 it did in t.ho pure science of geometr@~t mo emphasizo oneo mor~hap. tho mathematical conccpt 6Vval'" and its derivation from the language of finance laave notl,ing to, do 10ilh everyday lile. "6.al'" and 6VvaaDa, had lite meanings explained above, only in tJleorelic:al geometnJ. Defore turning to the text itself (Theaetetua, 147c-148b), I would like to mako ono furt.her remark. In the part of tho Theaelelua which we are going to discU88, 66.al'''; is sometimes translated as 'the sido of a square' or U8 'squaro root'. I ~a.ve shown in a previous work22 that this translation is com 1'Iotely unfoundcd-:Tlloexj'-rcssion-avvap,,-ii'i'miilliOmBtrcs ~nover meant any~iOOp£ 'value 0/ a square' or 'square'. That this word IU18 heon translated 1\8 'Ride of a squaro' or 'Rquare-root', only goes t~n

C',UllIrlUrn this with XUliophull (.lhaabatJie 1.6. 6): "tho alyA~ [Iln Asilltin

"UIII, tltu JluiJl'UW ahukul] amuu,,'.., 10, ill worth or hall 1M value of (du.al'a,) 7Y:

Attio obols", or Dom08tonea (34,23) "tho stBtur of CyziOUB had a valu. (l1"""al'o) of 28 draohmos".

.. It'ol'llxt&UlI'IIl III I'luturuh (lAJI:urUIIII 0 Ulld Suluu Ill): "1111 "nvu olily val,", to monoy": 6tivapav dALY'lJl or,;; "oplapan u 800 n. 14 nbovo.

' ' ' Kell.

(j

..,,,,all

40

8ZAD~: TUB JJXU1NN1N08 Olr OllWlJ( IoIATWWATICS

PAlLT 1. TUB BAl1LY HISTORY 01" TUX TUKOllY 01" IIULATJONALS

to show how superfioially and arbit.mrily the mathematical part of the Theaddua has been treated in the literature.II

there are infinitely many squares (6twdpe,,), we should try to group these under one name by which we could refer to all squared reotangles (6vv&pe,,).

,/

This is enougb about the word 6.wap" for no"!:...I will have to ret.urn to it whe~ come to give 0. detailed interpretation of the text.

1.S

41

Socrates And did you manage to do this t

TUB IU.TBBMATIO PART OJ!' THB 'TBBABTETUS'

The.aeIelua

'P46,0l', W X~eaTe" vtW )IS Wr(J) rpa{VeTa,' tUde ~'v6twB11e&' leonal' orOJl ~al ath'ol, ~pil' ba)lXo, el(fifiDs 6,a).8)lopivo", 11'01 TS ~al Tq; aq; 0PWvvprp TOVTrp EWNeaTe'.

TheaetdU8

It seems easy when you put it like that, Soorates. For you are asking something like what your namesake, tho other So orates, and mysolf were disoUBSing here recently.

Yes, I think BO, but see for yourself.

Socrates J.l:ye.

Go on. Theaeldua

Socrates

To" de,Dpo" nana 61xo 6,e}.,&pop£JI· We divided all numbers into t.wo TOV ph 6V11ap£JlOV raov ladN" )lty- groups. Those which can be tleaDa" TCP TeT(!aywvrp TO axijpa ! obtai nod us the product. of two dnetNdO'avTe~ TeTed)lCIJV&v Te Nal i equal factors, we compared in lO'&nJ.ev(!ov n(!oaebrop£JI. sbape to a square and called them (equilateral) square numberB."

What was that, Theaetetus t Theaeldua

Heel 6vvapewv T, ~piv geo6C1Jeo, 86e lYea'l's, Tij, Te Tebr060, ni(!, Nal nBJITin060, [lurorpa/Jl(J)v] 8n pqNe, 00 aVppeT(!O' TfI nOO,a/q., ~al oVTCIJ NaTO pial' ~ n(!0atemSpEVO' pix(!, Tij, imaNa,6eNM060,' b 6A TOVTn nCIJ, biaXeTo' ~piv om, elaij).{}1 T' TO,oih'ov, Ine,d~ Ibre&(!o, TO n).iJ{}o, al 6vJIdpe" IrpalWlno, ne.eoOijJlm avllafJeiJl el, lv, 8Tll' ndao, TOVTa, n(!Oflayo(!eVaopn TO, 6vJItipe".

"I---.

Theodorus was drawing some squares (mel6twdpeC/JJI) to demonstrate to us that the lengtAs 0/ the sides of those having an area of three or five square feet are not commen8Urable ( p~1U' 00 aVppeTeO,) with the length of the sides of a unit square. He disou88Cd the coso of each square (haa-n", scil. 6.wap,l') individually until he reached the oo.se of the square with area seventeen square foot. Then for BOrne reBSOn he stopped. Now it ooourred to us that since

I

-'

~

ye.

Socrates Very good.

Theaelelus Since" numbers wich can be decomposed into 'two equal f'actors' (lllwn, la",) are simply callud nT~ detD~ In t.ho Elemenl8 (Definition VII. 0), I used ( to tJtink tJmt Plato's tenn 'squilaterolsquare numbtlr' (TeTeUY~ TIl KQllad~) 'wus pleonastio. Prof. H. Chomiss, howevor, bas made me realise tJtat tJtis conoIusion is aomewhat host-yo Tho literal meaning of TUeUYQWOV is 'quadrangular'. Tho l'08triotod moaning 'square'. which becumu accopted after Euclid, was in reality not as common, ovon in post-Platonic t.imes, as I had tacit.ly 888umcd. Chomisa bas correotly polnLod out that evidonce for this can be found in Timaeu4 Iilib7; AristotJe, MelGp/&ym. I064b2; Horon, De/iniliona 100; Proclus; Com· fMnIary on Eudid (od. Friodloln), 166. 10-1. ;" U

I am pleased to report tJaat van dor W8CIlden bas accepLod tJds BUggostlon of mine in Zur GucAWIu der griec1aUclam MalMmaIu; p. 264.

"--

Nal

I

t"

'1I J

42

suuO: TUB

1'ov TO{VVV peTa~v TOVTOV, wv "al Ta T(!la "alTa nlvTB "al nii~ 6~ deMvol'o," rao~ lad"e, rev/aDa" cUl' iJ n).elccw IAanov&x" IJ l.tdnrov nAeova,,,, "{yveTa,, l'e{Crov 68 "al IldTTWV del nkvea a"TOv ne(!tAal,pdv£" T~ neopJi"£' aJ ampaT' cm£,,,daavTe, neOI'~"'1 Oe,{}pov beaUaapw.

:---~

PeTa

ToiirOj'

!

-~

TOV

I

l

p.

w

•AeeO'Td y' dvlJewnrov, nalc5e,· (iiC1Te po, 60"el 8eo6roeo, 00" boto, 1'01, tpeovc5opO(!Tvelot' laealJat.

Socrates

-J

Socrates Well done, lads I I think that Theodorus was right (to praise you ...) etc.

Excellent, but what comes next t We will begin our interpretu.tion of tho text by showing that the mathematical portion of tho Theaetelu8 and the rcst ofthedio.1oguo form 0. coherent whole. The ahove text hcgins with tho young Thclwtctus 88Serting thl"t after the example just given, Socrates' question is easy,' he is asking "something like what your namesake •.. and myself wcre discussing here recently". These words are the tenuous conncction between the mathematical pu.rt and tho convorsation which prccedes it. Howcvcr, the mathematical part cannot be properly undcrstood without talting into account whu.t precedes it. Socmtcs began by I18king tho young man 'What is knowlcdg6~' (146c), and Theo.etetus responded firat by enumerating different kinds of Imowl(l(lgo. Rut Socrates WIl.8 not RIl.tisficd with this answer. J nHteu.c:l of lion enumeration, ho domlmdcd II. comprehensivo and genoml definition

TheaetetU8 {Jaot PEv reappal To" laonAweov "a2 In{nE60v Oe,{}pov TeTeayrov{Coval, l'ii"O' we,adpeiJa, Haa, 68

43

ITeeop~"71' 6IJ11dpE'" W, 'P~"E\ Those which yield 0. rectangular ptV ou ctvpplTeovr; beeivae" TO-~ number wl,en 8quaredZ' we denoted l 6'lncnl60t, a cSVvaVTae. Kal nee &8 6IJ11dp£er;, for although the tu.tter \ Ta C1Teeea a.uo Totoiirov. are not commensurable with the \ former in length, nonetheless the areas (lnml60tr;) w},ic}, 1I,eyenclose When 8quared· (a 6tSvaVTat),· are commensurable. We o.1so tried to I ~o something similar for solids.

On the other hu.nd, those numbers which lie between the previous ones (e.g. throe, five and in general any number which cannot be obtained &8 the product of two equal fn,clora but only ns the product of a greater and a smaller or a smaller and a greater, and BO is always enclosed by 0. greater and a lesser side)2I wBf compared to rectangles and called I them rectangular numbers. -.J ',~

I /(dlltC1Ta. dlla t:l TO

PAltT 1. THB BAilLY UlIlTOll.Y 01" TUB TlU:OllY 01" JUll.AT10NJ.LS

BKOUUUNOS 01" OllEKK MATUKMATJC8

Now those straight lines whose 8lJfUJre ia an (equilateral) 8quare

number,u we denoted by pipeo,.

U In this context 'sido' (nh;ved) mllanB 'factor' as well, just 08 it does in Defini· tioll VII. 16 of the Elemenl8. From this wo can 88e that a product of two factors WUB in gonoml viowod as a recl4ngls. Moreover tho term 'rectangular numoor' (lreeOI'91n1' or neoI'91n1' in our text) does not occur in Euclid. He only uses tho concept of 'plane number' (hrlnedo, ,Je,{}p6,) which can refer to &q1UITS '"""OOT808 woll 08 rer-langt,'ar onM. (J.ot. mo ad.1 anothor important point horo. i IIIlVu conoluded that thll concepts 'Illano Jlumoor' and 'silllilar plano numbor' IIIUllt bo old and in any event pro.Platonlo, bocauso Plato frequently usos nkved to mean both 'the 8ida of a parallelogram' and 'fuctor'. Now these concepts ('l'lllno 1m moor' and 'similar plano numoor') are usod particularly in Book VIII of tho Eletnenu and this book is attributed to Archytos by van der Waerden ('Die Arithmetik der Pythagoreur', MaCh. Ann. 120 (19(7-49) 127-53). Apart from tho fact that I disagroo with him on this for other reasOnB, it 800mB to me that the concepts 'similar plane and solid numoors' could not possibly have buen first. introduced by Archytas.) It The words in italics, although unambiguous, aro somowhat brief and caro· lussly written. Their moaning is 08 follows: 'Thoso straight lines which when squnrt'd (Te'tf!O)JUWICOualll) form a flgul'U corresponding to a square numbor

dunowd by l'iJx~ •• .'. 1.'urthurrnoro thu tenn rm' 'lIquuro numbor' is rodun· dant. horo. For la6nleveo, leal hrln~ ,JeIO,16, moans Iiwrally 'oquilawral and plane numoor'. 11 Again tho monning of thu Orook toxt roquiros that tho vurb n't(1ayw"iCoualll bo added. So the sontonoo is boat translated 08 'those straight lines on the other hand, which yield·o. rectangu1ar number whon squared (TeTeayw/lICoUfJIII), wo donotod by 6vvd/18', • .... a 18 We should not forgot that in mathematics tho procisa meaning of 6.wa Oal W08 'to IIlWo thu valuo whun squared' or 'to amount to whon squllrod'.

WI!

\

45

l'ART 1. TUB KARLY DlSTOllY 01" TUB TUKOIlY 01" llULATlONAJ.8

44

SZABO: TUB BKUIHHUIUti 01' OJlKJo:J[ KATBJWATICS

of knowledge and immediately gave an illustration of what. he wanted by means of an example. If one were asked. 'What. is clay", t.hen one could not answer by enumerating the kinds of clay used. by each particullLl" craftsman. One should rather t.ry to give a general and comprehensive definition of clay (147aff). The young man replies to this example by saying t.hat. he now understands; Socrates' question is quite easy because it resembles something which he was discussing recently with his friend. '-So we can see t.hat. t.he central point of the preceding conversation is ~~ contrast. 'enumeration of particular instances' with a 'comprehensive dcscril)tion (definition) of the same phenomenon'. (For this reason one should reject the view found among some historians that Plato is giving an example of a mathematical investigation by the young Theaetetus in the dialogue. They would say tilat the example is rather farfetched; it is supposed to serve 88 an introduction to a philosophical discussion, hut is not. well suited to the purpose etc., etc.) According to the text, "Thcodorus was drawing some squares ... Illwing an area of three of five square foot ... He discussed ••• each square ••. until he reached ... a square with area seventeen square foot."'So we can soo tho.t in this seotion 'the enumero.tion of individual instances' consists of squares (6vJlaIJB&,), just. 88 previously Theaetetus had enumerated different. kinds of knowledge and Socrates had'enullI11raLted the Idnds of olo.y usod by difforent oraftsmcn. • Before investigating more fully what Theodorus might. have intended hy enumerating squares, I must. make a further remark about the term MvaIJ&,. Thus far I have given its preoise definition (as a mathematical term), but have not indicated how this booJR on the historY of mathe1lI1~t.i~ will proceed to do this now'fin~ my opinion~ helps us~ understand Plato's text b~ \ . / \...-.:-J

7

t-I

/

1.4 'J'UE USAOE AND onRONOLOOY OJ!' 'DYNAMIS'

The term 6thoaIJ" is used in the text. without. any explanation, as though it. were}iiiilpletely famili~r&~d could be understood unambiguously by the h,ad~r. From t.his wJ can infer that the concept 6thoaIJ" ~ ('the value of the square of a rectangle' or 'square'})oannot have origilIU,tcd at. the time of Plato. Indeed one can cite other passages from

Plato which at.test to the same fact.. Of these I shall ment.ion hcre only Politicus 266 a.5-b7, a passage which will be discussed in detail later from anoth~int-O:2.· w. In it the concept 6thoaIJ" is used in a ~ath~r complica~l~y on wo , which would ho.rdly ho.ve been p088lble If Plato had not4Jeen- 8 to assume that. t.he word (and its jechnical meo.ning) was well known, and familiar to the educated ~8f_~~ I~Y Exactly when in re-Platonio times the wo~ dynamt8 was. c:omed , we do not Imow 8.t present. here is no text earher than t.he wrltm~s of Plo.to in whioh-it·OcCuralnl -teohniealsense. The most. we can do IS to appeal to Foudemus'. account (as transmitted by Simplioius) of the 28 squaring of lines by Hippocru.tes of Chios. It begi.ns as follows: : '!Ie prepared 0. foundation for himself and erected on It first a proposition which served his purpose, namely that. similar segments of a circle have t.he same ru.tio to each other as the squares 0/ th.eir lHues" (xol al POdE&l;

aUTwP OOvaJJB')'

j---

•

This question may be considered. as evidence for t~e p.re-Pla~mc use of tho mathematical term dynamis. I should mentIon In p8.88mg that whon I 08tahlished the preci86 moonings of 6VvolJ" and 6vJloaDa, in mathematics, I bad in mind (apart from Arohimedes) mainly the usage of Simplicius-Eudemus (or ru.ther, of Hippocrates). '1'hero is, of courso, o.bundant evidence of tho same sort for thcRe two expressions being used in the senses exp~aincd above in t!le. whole of later mo.thematicalliterature. An interostmg example of thIS IS the for' mulo.tion given by Athenu.ous of Pytho.gorll.H' 1 t lcorum. 30 T(!,YWJlOV oVaat d(!IJO)'owlou " ~ d~ yow/av WrOTB/rovda fdOJl 6thoaTW Tai, 1lE(!ux l;. "The square of the hypotenuse of a right-anglcd triangle is equal to (rdO'il 6thoaTw) the sum of the (squares of) the otber two sides." It might be concluded from this that 6thoaaDa& and 6thoaIJ&, w~re ~m­ mon toohnical terms, used constantly in Groek mo.thematlcs smco before Plato. But this assortion becomes questionable when the Elements are considered. 6thoaaDa, in the sense of 'to 'have tI,e value wl,en squared', does not. occur in Euolid. He only uses the derivc~ ter":,s 61woIJ£& croIJIJt:r(!Ot or dcroIJIJt:r(!O& (cf. Book X, definition 2) winch Will I. cr. O. Becker, Grundlagen der Mathematik in Uuchichllicher Erllwieldung, Frolburg-MUnchen 1054, p. 20: also his article In QueUen und Sttulien wr Oeechic/lle der Malh. ota. n, 8 (1036) 417. '0 AUlllnaoU8 X 418r.

,HI

II:tAllt): Ttl.,; IIJ.:tJlNN1NUIi

()If

hu.ve to be colUlidercd later. turiilermoru II. square is never denoted by 6~p,~ in his writing~. ,rhis fact is noticeab e because one can easily find a proposition of Euclid in which the te MJ1Iap,~ would at least have suggested itself. lnm referring to Proposition D, Dook X. Tho second part of this proposition deals with "squares on straight lines incommensurable in length" (Tn dno Teall P1f~, davpphewlI eV/Je,{jj" TBTea)'wpa) which "do not havo the mtio to each other, of a square numoor to a square number". In tho TheaeJel1l8 this kind of TBTeaywpa is called 6wap,~. According to Theactetus, Theodorus sketched 6v"apel~ of area three, five, etc., square fcot, for his pupils, i.e. he constructed TBTea)'WPa which did not ho.vo the nLtio to each other of 0. square number to 0. square number, and then IIhowcd that the sides of theso squares were incommensurable in length with the side of the unit square. So undoubtedly ICuclid could luwe ( IUlcd 6u"apel~ to denote these squares, but he never does. In Euclid, II. Mllarejs.a.lwa.y8-ealJed-ure.~lIn'O!j.l believe that this is because EuclidIl.IWayS strove for the greatest possible genemlity. Tho terms heeOP7J)(E~, e&pPo~ and eoppoe,61~ (cf. Book I, definition 22) also do not [ occur in his propositions; instead ho always spealts of parallelograms. So just as lTEe&p7J)(e~, edf'Po~ and eoppoe,61~ are special cases of the pll.rlLllelogrom, 6vJ'apl~ (according to tho previous definition) is aspecin.1 CIUle of TBTeayw"OJ', Hence all sqtlc&res arc TBTeaywpa, but those which "ILVO the samo area as certain rectangles, are called 6tJ11apel,.

1.5

'TETRAOONISMOS'

These speCUlations about TBTea)'WIIOII and 6Vvap,~ also suggest 0. plausible conjecture about the relalive chronology of the two concepts, namely that 6Vvapl~ (the value of the square of a rectangle) must be of \Iater origin than TBTea"CI)POV Clripa, the other usual term for square in Greek geometrtV1o coin a phrase like TeTeQyWPOP Clripo, one need only to have distingUished between the most important kinds of po.rallelogroms (rectangles, squal'C8, ete.). On the other hand, the concept of MJ1Iop&~ presupposes that one already knows how to tra.nsform a reotangle into a square of the same area. In other words, this piece of knowledge must have originated at the same time as the mathematical ooncept dynamis.

1\

l'A I"r I. 1'11" l>:AIlI.Y lIUrI'''U Y Ulf 1'U" 1'UI1:IJII\

UIII"'!>. MA1'II~AlA1'IClt

lit· IIII'A •• "" ... ~...

..,

This conolusion follows immediately from tho simI)le facts established about the mooning of dynamis, and it leads to still further observations which in my opinio"'are of4a.rnmount importan91 not only for understanding our text )rom Plato, but also for the whole history of early Greok mathematics. ' The question we are interested in is how the 'tra.nsforming of 0. rectangle into a square of the same a.rea.' was expressed in Greek. The correct technical name for it was TBT(!aywpl(e,l', or the noun TeTeayCl)IIlt1p&~. It is no accident that this other important term (TETea • ywpICe,,,) occur8 right by 6tJ1Iapel~ in the text which we are discussing, since the dynamis was obtained by means of telragonismos. The two concepts are very closely linked. Furthermore, there is an important statement about telragoni.mzos by Aristotle which enables us to be more precise about tho relative chronology suggested above. Defore di8cussing it, however, I wo.nt to make a brief remark about English usage. The Greek words TBTeayCl)J'ICe,,, o.nd TeTeayCl)"'t1p&~ are usually trans:--; g Jated by 'squaring'. However, this translation can easily be mislcad'{ng. 'Squaring' can sometimes mean 'raising a number to the second power' or ·constructing a square on 0. given line segment', whereas the Greeks, even when they were not just talking about TBTeOYOJlIlctpo, TOO 1ttS)(,wu o.lways understeod by TETeayCl)"&ctpd~ the 'tmnsforming of a rectangle into a squure of the same area'. (Of course tetragoni8f1lO8 could IIso leiiCffrom othcrrootilincar figures to a square of the same area, by way of a reclangle.) lIenee I have consistently tried to avoid tho U80 of the word 'squaring' in this work. In the M elapl,ysiC8 Aristotle says the following about telragoni.mzos :31 "What is TeTeayOPICe,,,t It is the finding of a mean proportional" (Tl lCln TBTeayCl)"{Ce,,, ••• plt17J~ dJeect'~). The meaning of this assertion and its context are explo.ined as follows in the accompanying commenta.ry y Ross.32 "The definition, the 8quaring of a rectangle is tM finding of ~ I geometrical mean between the sidu, is an abbreviated form of the syl/ I logism: a rectangle can be 8quared beca1l8e a mean can be found belween its J

t

r

j

rh

l.aide8." II

II

Metaphyaiu 996b 18-21. AmtotUJ" Melaphy8ic.t, Oxford 1924, Vol. I, p, 229.

--

48

BZABO: TUB 81£0lNtlIN08 Oil' OllKKK ldATHBIUTIC8

l'AII" 1. TUB XAltLY IIltl'l'OllY Olr TUX TUBOUY 011 l1L1LAT10NALII

Aristotle himself also explains the same idea in more detail, in anothor work." "Definitions are usually like conclusions. For example, what is telragonism08' Phe conalrudion 0/ a 6guarB equal in area to a redanflle (hee6I"1xs,). This kind of definition is a conclusion. But he who maintains that telragonism08 is the finding of a mean proportional, o.lBO specifies the rationale behind it." (d 6A UyQW 8T' lad" d Tereayru",(fpor; plt1f1r; eiJeea", TOO nea"paror; Uys, TO aIT&OtI). These statements by Aristotle are important because they ahow us . that his contemporaries (if they were well versed in the mathematics of the time) must have been aware of tbe impossibility of transforming a \ rectangle into a square of the same area. without ~nstructing a mean \ proportional between two given line segments~ 80 the dyrUJmis (the value of the square of a rectangle) is obtained by constructing a mean proportional between two sides of the rectangle, for this ia just what tetragonism08 (the transformation of a. rectangle into a. square of the same area) comprises. Thus my previous conjecture to the effect that dynamis a.nd telragoni8mos originated at the slLme time inevitably loads to the conclusion that the creation of the concept of dynamiB must have coincided with the discovery of how to conatrud a mean proporlional between any two line segments. To understand why this entirely new concept evolved at a particular time in the development of Greek mathematics, ono hILI to look more closely at the problem of the mean proportional.

I

1.6

THE lIBAN PROPORTIONAL

Euclid discusses the construction of 0. mean proportional between any two lino segments in Proposition IS, Book VI (of the Elements). It is nowhere indicated in this book that tho construction finds its most important application in lelraUonUm06. This in itself is not particularly surprising. More interesting ia the fact that the same construction is di.......d BOmewhero e1.. in the BkmmaU. namely in T~4 of nook II. Tho problem with which this latter propositio '/ deals the Ill)" Anima II. 2. 41311 13-20. II On this BOO ohapter 1.6, 'Tho Moan Proportional', os woll to t.hie book.

B8

tho appendix

49

most general form of telragonismoB. It is 'to obtain from a. given rectilinear figure, a square having tho same area'. Sinee this is accomplished by first construoting a. rectangle of tho same a.rea and then finding 0. mc&n proportional between two of its sides, one would expeot the construction of a mean proportional to be fully discussed in this proposition. In fact. the construction of the desired line segment (the side of the square in question) in Proposit.ion n.14 is blLlico.lly the same as t.he construction of a. mean proportional between any two given line segments in Proposition VI.IS.* Even more noteworthy is the fact. that no use is mado of proportions in Proposition 11.14. It is never stated that the line segment which provides a solution to the problem, is 0. mean proportional. Instead Euclid gives an entirely different account of why it can be used to construct the required square. Indeed the proof of Proposition 11.14 gives the impression that knowledge of the mean proportional is perhalls" unnecessa.ry for oarrying out tetragonism08. Heiberg hl13 disou88ed both t.hese propositions together with the plLBBo.ges from Aristotle quoted in tho previous scction, and has como to what is clearly the correct conclusion.sa He argues that the J»lL88l.Lges from Aristotle unequivocally attest to he fact that the transformation of a reotanglo into 0. square of tho same area WILl originally accomplished by constructing 0. mean proporional. Thisoriginal method is given in Proposition VI.13. The o.lternative account offered in Proposition 11.14 must be the original in a later disguise. It scoma that Euclid, or whichever one of his predeC0880rs gave the propositions in Book II their final form, wanted to a.void using the theory of proportions. Henao 0. new proof that a rectangle can be transformed into 0. square of tho same area. WILl supplied, which did not mention that. the aide of the square obtained was 0. mean proportional between the two sides of tho original rectangle.

G

... Correction: compare this OIIIIOrtion with tho appendix to this book. Tho above diacIJ8IIIon wos wrlt.ten bofonl I oe.mo into po!III08IIion of tho information oontainud thurtlln. N MathematiBcl&U hi Ari6tolele8, Abhandlungen zur Geschichte dor math. WillllOll8ohal\on, No. ]8, fAlipzig ]1)04, p. 20. 800 also T. L. Hoath, Mathematic6 in Arilltutla, Oxford 1040, flp. 101-:1. 000 should not. fnrgot. that. t.ho const-ructlon oC tho moan proportional (i.o. Proposition VI. 13) must already havo boon known to Hippocrates oC Chi08, of. n. 37 below.

4 SlaW

50

'-J'

SZAUO: TUB lIXUIHHINUS o~ OlUlKK MATUBllATICS

IT 1. TUJ.: BAilLY IUn'OILY Olf '1'Ull: TUXOIlY 0 .. lllIlA'rIOHAUI

(brbr£IJOl; , ,(},.,or;), and the other two are said to be its Bides ~factors, nJ.£v{!al)" • Any number which can be written as tho product of two factors and hence can be represented geometrically by 0. parallelogram (usually a rectangle), fo.1ls under this definition. For example 6 is 0. plano number which can be represented gcometrico.1ly by a rectangle wit.h sides (pleurai) oflength 1 and 6 oroflength 2 and 3, since 2 . 3 = 6 = 1 ·6. (Of courso square numbers can also be represented in rectangular form. For 4 = 1 . 4 as well u.s 2 . 2. Similarly 9 = 1 • 9 and 9 = 3 . 3. i Hence the squares 2· 2 and ~ . 3 ~an serve as dynam~is, i.e. as tho 1 values which tho rectangles With Sides 1 and 4, and Sides 1 and 9, Lrespectivcly, tal<e when squo.rod.) One need only think how common the term nleveal (meaning factor) is in mathematics, to roruise at once that the concept of £!!IDILJlumber must be vcr)! j)1~ Euclid's othor difinition is still more interesting. I am refelTing to Definitio1& Vll.22 "Simil(tr plane number8 (8pOIO, bcbre60t Ue&{}pol) are those whoso sides (nlev(!a/) are in the same ratio."311 AU square numbcrs (I, 4, 9 .•. , since they co.n uJways be written u.s tho product of two equal flLctors a.nd so have sides in the same ratio, i. e. 1:1 = 2:2 = 3:3 .•. ) are similar plane numbers in this definition, as are such pairs of numbers as 2 and 8 and 3 and 12. Ea.ch member of the pair can be written as a product of two factors (1 . 2, 2 . 4, and 1 • S, 2 . 6 respectivcly) ancl the rat.ios of these factors ILre the samo ( (i. o. 1:2 = 2:4 and 1:3 = 2:6).38a Clca...y thoy arc called similar plane I numbers because numbers of this kind can be pictured as 8imilar rec\,_ tangles. Carrying this out for the example above, we get Fig. 1.

f

Let me add that in my opinion Heiberg was quite right in defending t.he view that Proposition VI.lS antedates Proposition 11.14. Certo.inly no one as yet has refuted his arguments for this view, or even cast doubt on them. Furthermore there are other arguments whioh can be ':'--advanced in support of his position. For as has been rightly observed: "The geometrio construction of 0. mean proportiono.l [i.e. Proposition VI.13] was completely familiar to Archytas and the Pythagoreans, and it must already have been lmown to Hippoorates ••. " (on this topio, see the references oited in n. 37 below). Consequently the pil17Jr; W{!ectlr; of whioh Aristotle speaks (in MeJaphysic8 996b 18-21 and De Anima 112, 41Sa IS-20), must o.Iso have been known at the time of Hippocrates. I do not see why these passages from Aristotle oannot 'be conneoted' with Greek mathematics of around 400 B.O. (perhaps only becauso of an obsolete viow held by ,rannery,s sohool, according to which the theory of irrationals first became current sbortly before 400

I

B.C.).

[

At the moment our interest in the problem of tho mean proportional is confined to investigating the extent to which the discovery of 0. method for constructing a mean proportional between any two line segments must of necessity havo led to the new mathematica.1 concept of dynami~y conjeeture is that the creation of the concept dynamis must be ~ated to tllis discovery':Y First let us remember that the mean proportional was a much disCU&!Cd I)roblem in Groek arithmetic as well. For example, in Book VIII of tho Elements, Proposition 11 states that there is exactly one mean proportional number between any two square numbers. The next proposition (VIII.12) establishes that there are two mean proportiono.l numbers between two cube numbers. Of greater interest to us are the two propositions concerning the oxistence of mean proportional numbers. Thcse are: Proposition V111.18 "There exists 0. mean proportionaJ numbcrbotweenany two similar plane numbers" (~vo opo/oo." In,m~wv de,fprop) and Proposition VII1.20 "If there exists a mean proportional numbor betwoen two numbers, then these two are 8imilar plane numbers". 'I~o lUulorstlLJUl thoso t.wo propositions, ono must hll,ve grasllotl tho interesting concepts of plane number and similar plane number which occur in early Grock mathematics. Tho cloftnition given in the Elemenl8 is UII follows: Definition V1l.16 "If two numbors 0.1'0 multiplied togot.her to form a third, then the resulting number is co.1led 0. plane number ..--------.

---------_

---_._----_.

-......

51

llEl~ z ,

and

,r"'1--=@"""'Iz 1L-__@_,_..... 6

3

1:2.3:6

1:2 -2:4

Fig. 1

•• I havo profurrod to usu 'in tho SQmo ratio' 88 a translation of dvdAoyov, ur thu mom UHIIRI 'proportion"l'. For un nxplllllld.lon of thiH IJIIlt.hnmati· cal tonn, 868 pp. 148-67 If. • On tho notion of. 'simllar plano number' cf. ulso T. L. UClUth, Euclid's JlJlfltllflflllf, V"l. 2, p. ID3: '''1'11111111 IIf Smyrm& rtlln"rl'H (p. :111. 12) t.Il1&t., "mong planu numbers, all squlll'tlS uro similar, while of iTEeol"IHe" thosu uru similur whoBe sides, "uU is, ,he numbers conlaining ,hem, are propor'ional." hIHIA"tAl

-1

-

52

8ZAUO: TUK DKOINNINOS Off UllmUt MATUEUATICB

Proposition VIII.IS, whioh was quoted above, states that betwoen any two similar plane numbers there always exists a mean proportional number. For example, there exists 0. mean proportional number betwoon three and twelve. In this oa.se the required number is 8iz. since 3:6 = 6:12. It is worthwhile using this simple example to il~trate the t difference betwoon ancient and modern ways of thinking oday the I, mean I)roportional would be obtained by mUltiplying the tw numbers togethor and taking the square root of their product. (6 = ys:-i2). In antiquity. however, the numbers were first. divided into the faotors whioh had led them to be classified as 8imilar plane number8 in the first pln,ce (here 3 = 1 • 3 and 12 = 2 • 6; of course if 12 were divided into factors other than 2 . 6. it would not be similar to 3). Then the mean proportional was obtained by multiplying two different 'sides' (factors) which were not similar together (3 • 2 = 6 = 1 • 6). '1'he above example could also bo understood in the following way. Wo are given 0. number-redangle of sides 3 and 12 which is to be transfunned into 0. square of the SILme arelL. The fon.sibility of this do})Onds on whether the two sides of the rectangle are similar plane numbers. Jf the two sides of the rectangle can be decomposed into factors in such a way that the ratios of the factors are tho slLme (i. e. if 0. smlLller rectlLnglo can be constructed from each side of the original one, and these Hrnnllor ones are similar to eaoh othcr), then the problem can be solved. I'I'01'uHition VIII .20 ILhovo ill the llonverllO of PropnHitinn VIII. I R. 1 t Htntes that "if 0. mean pro})ortional number exists botwoon two num( I,ors, then these two are similar plane numbers". So 0. mean proportionlLl number can only be found between two simillLr plane numbers. It is nowellSY to see that at the time when Propositions VIII.IS and !!II W(lrO known, hut Proposition VI.13 (the construotion of 0. mean prol'III,tional betwoon any two line segments) was not, an arbitmry rectangle could not be transfonned into 0. square of the same areo.. 37 This t.mnHformlLtion could only be curried out in the ollSe of reotangles WhORO n JII my opinion Propotdt.ions VIIr. 18 ond 20 Bre olc1ur t.hBn ProposiUon \' I. 1:1. Concorning t.ho lut.tor, ( ugruu wiLt, vun dur WLWrdon who' BBYS till'" "The geometrical construct.ion of the moan proportional (I'rop. VI. 13) WQ8 quito familiar to Archytaa and the Pythagoreans. Furthennore, it must already IlIwtl bUlln known to I1ippocrutull of 01,108," (Zur Geachiolde dar griflchillollell JlfalhflnlDli1.:, p. 226, n. 28.) On t.ho other hond, I cannot aceupt his dating of Buok VIII. Hia viow Is thBt "Tho problems of Book VIII are very olosoly

l'AlL'" I, '1'UK ~AllLY Ull>"1'OILY 01' TUX '1'1110:014\' 01' umA1'lONA1.li

53

sides were simil r plane n ~rs. As we would BlLy today, the transformation could be carried out y if the area. of the rectangle was a perfcct square. If, owover.a tangle with sides of. say, length one and three say given, it could not be transfonned into a square of the same area unless one knew how to construct a mean proportional by geometric methods. The sides can only be decomposed into two flLOtors in one way. namely 1 = 1 . 1 and 3 = 1 • 3 and since 1: 1 oF 1 :3, the lengths of the sides are not similar plane numbers. Whon subsequently 0. geometrio method of constructing 0. melLn proportional between any two line segments WIlS found. it became possible to transfonn any rectangle into a square of the same area. Thenceforth there was no longer any need to worry about whether the rectangle concerned had sides which were similar plane numbers or not. Instead a melLn proportional WIlS construoted by geometric methods which used similar right-angle triangles (as in the proof of Proposition VI.13). Of COUl'8O, as soon as the pOB8ibility of this construction became known, tho question IlS to wltat exac.lly the sides o/such a 8quare (i. e. 0/ a 8quare equal in area 10 a redano1e wl,ose Bides were not similar plane numbera) were, mllHt also have been raiscd. By Proposition VIII.20, a mean proportional number only existed betwoon simillLr plane numbers. The newly discovered fact that a mean proportional also existed 00tween two numhers which were not similar could only be reconciled with this ProI)osition by introduoing the notion of linear incommensurability. As the Greeks interpreted it. the mean proportional between two numbers which were not similar plano numbers was not a number. The question is still left open as to whether (and if so, how) the linear incommonsurability of 0. mean proportional between two such line segments could be rigorously proved. But tho abovo considerations are important boco.use they show that the problem of transfonning 0 . ) 1 J\ reotanglo into 0. square of the same ILrea, which in its most general form IV'! Unked to tI,e t.hoory of irrat.ionals which BCCordlng to a well. founded opinIon (ltlu), (lid no' erill' "n'il II/lOrtly btl/ore 100." 'rlao nut-hor of ..laiR 'wIIIIfounded opinion' Is nono othur thun Vost who uncrit.ically udoptud 'J'onnery'8 views and misinterpreted t.he mathemat.ical section of the TheaeUtU8 lUI "t.ho birt.h cortificato of irrat.ionBls". Furt.honnore, I remain uneon· vincod by tllO argumun .... which Purl)OIi. to show "hut. Archyt.ulI WOB t.ho But.hor of Book VIII.

54

l'A1I.T 1. 'rUK KAILLY 1IIS1'OItY Olf TIlK TUKOILY 01' IIU1.ATlONAU

8ZAUO; TU" UKUINNINOS OJl' OIlKKK &IATU":lIATU:8

(~8 oquivalent to the problem of finding

0.

1. 7

meo.n proportiono.1 betwcen

\~ ~ny two line segments,led to the problem oflinearincommensurability. .

My conjeoture is that the discovery of how to construot a mea.n proportiono.1 betweon any two line segments o.1so prompted the introduction 0/ the new malhematical conupt 01 'dyrwmis'. Of course I ca.nnot produce any dooumenta.ry evidence to support this conjecture, bemuse no Grcok mathcmatical texts have survived from the pre-Platonic times during ( which the concept originated. NonethelC88, I believe that it can be I established by 0. study oC tho word dyrwmis itself. I 'l'he mathematical term dyrwmis denotes by definition an area - 'the va.lue of the square of a reotangle'. The problem remains lIS to why it was Cound noC6B8ll.r)' to use this curious new term for ,~ square whiob bod the same area as some reota.ngle, and also what plU'ticular proporUes of the square led to its receiving this extraonlinary nu.me. I ca.n only think that u.s long u.s only those rectangleH whose sidcs were similar plane numbcrs wore transformcd into squareB oC the samo area, there wu.s no relUlOn to introduce" new namo for tho resulting squares. These were porfoctly.....QJ:dinu.r.y...r~yCOJla axtil'aTa, since their sidcs were whole numbers. Matters changed, however, when 0. new general method (the geometri~ig' oC 0. mean proportion ILl) was discovered. Any reota.ngle could now be 'squa.red', Cor 0. mean proportional always existed betweon any two line segments. Hence, (l.8 Aristotle emphu.sized, tetragonismos is equivalent to finding the In addition the Greeks must have been aware that the squo.res obtained in this way were frequently such that their sides were not numbers. , (A mean proportional number only existed between two similar plu.no numbers. The mean proportiona.l bctween two numbers which were not of this kind W(l.8 itself not 0. numbor.) Thus squares having the sume area as reota.ngles, whoso sides were not similar plane numbers, must havo aroused considerable interest. The lengths of their sides could not be lI.88igned 0. number, even though their areu.s could be caloulated exactly. This may have occasioned the surprising use of dyrwmis;j denote their area.. 'I'his amounts to saying that the concept of dyrwmis was introdu ILt tho sarno time it wu.s discovered thILt linearly inoommonHurable line ( Begme,nts exist, whose squares (dyrwmeis) ca.n nevertheless be meas. ured. ~

f

I'itn}'l

55

'!'HB MATlfBMATlCS LECTURB DBLlVBRBD BY THEODORUS

The above considerations have led us to conjecture that the oreation of the mathematical ooncept dynamis islinlced to the discovery oflinear incommensurability. But this concept is, by all appearences, rather old. In any case, it must a.ntedate Plato, since he usea it naturally and without feeling any need to explain it further. In this case, however, it ca.nnot be maintained that "the mathematical part of the TI,eaelelus is the birth certificate oC irrationo.ls" or that "Thcodorus discovered the irrationality of Bquare roots in general" (Vogt). It would perhaps be better to emphasize that Plato never a.otuu.lly says that Theodorus showed his pupils something new. 38 The impression he gives is rather that Thcodorus was telling the Athenian youths something which must ho.ve been common knowledge amongst the mo.thematicians oC tbe time. In fact, we shoJI soon see that hardly any new mathematical discoveries can be attributed to Theodorus, at lco.st on the basis oC this passage from Plato. To prove this wo have to return to 0. deta.iled interpretation of the text quoted above. According to Plo.to, "Theodorus was drawing some squares (dynameis) ••• having an area of three of five squu.re Coot ... . He disoussed each square (btdtTTfJ" scil. 6u..QI"v) individua.lly until he rea.ohed 0. squo.re with area seventeen square feet, when for some reason he stopped." Previous commentators have invariably asked how Theodorus constructed those squares; more partioularly, how he constructed their sides, which he wanted his pupils to see here linearly incommensurable. The answer given was either to say that it did not muoh matter how those sides were constructed," or to adopt Anderhub's interesting attempt at reconstructing the method used.'o Anderhub's ideo. was to

"I{. von Fritz, Ann. Mall" 40 (1946) 244. 800 also tho papor by WOSII8rstoln in OlCJUictJl QlUJlUrly, N. B. 8 (1968) 166-79: "n is at loaat. concoivablo that. Plato moans no moro than t.hut. Thoodorus WlI8 domonst.rat.lng not a now discovery or his own, but. somothing which though known to pror_Iona) mathomatlcians might bo now anti int.orostlng to his young hearors." It n. I •• van dur Wuurdon, lGrllltichentle lVU88fllC/,a/I, I" 236. H. J. Andorhub, "Joco·Seria, Aus dm Papierm rinu reuenden Ka'll/mannu," lCallo·Worku oditlon, Wicsbadon 1941.

'0

J

-'r'-

'

I

56

8~AUc):

TUK

UKUJ~H1HUIi

l'AI~T 1. 'l'JUI KA1U.Y UlIiTOILY 011 TU" 'l'IlKOIlY Olf IlIltA'l'IOHAI.lI

Olr UR.KKK. UATUJWATICII

construct first 0. right-o.nglo, iS08celes trio.ngle wh08e hypotenuse hOO the longth V2, then to dro.w o.nother right-o.ngle trio.ngle with hypotenuse V3o.nd whose other two sides ho.d the lengths 1 and He continued this os in Fig. 2, untilsixteon suoh trio.ngles ho.d been constructed; and the hypotenuse of these provided 0. geometrio representation of V2, yw.

1'2:'

va: ... ,

Ii'ig.2

~\'his

interesting reconstruction gains some additiono.l plausibility from the fo.ct that it seems to explo.in wby Theodorus stopped at V17. J Nonetheless I olo.im tho.t historically it is simply misleading. My reosons aro u.s follows. (1) This o.ttempt to oscerto.in how Theodorus might have constructed the squares in question does not proceed from the concept of dynami8 which is tbo key to understanding the text. It simply ignores the fact that here is o.n interesting mathemo.tico.1 concept whose origin as well as mCll,ning hos to be understood before one co.n begin to interpret this pW!sage from Plo.to. (2) Plato's words undoubtedly imply tho.t Theodorus must first have produced these dynameis somehow o.nd thon direoted his pupils' o.ttention to their sidea. On tho other ho.nd, Anderhub's construotion yields tho sides straight o.way. The whole construction is bosed on the to.cit·~·. (and orroneous) assumption that dynami8 is o.otuo.lly tho 8ide of 0. ~/ squll,re. (3) Plato does not indicate how Thoodorus construoted the dynamei8, 80 this construction must olearly ho.vo been both simple and well IwnwlI by tho stlLndarUs ofth()so times. Andorhub, howevor, pla008 tho construction itself in the foreground, o.lthough by o.ll appearences it WILM only of 8Ocondary importance to Thcodorus.

57

(4) Instead of to.king the trouble to understo.nd tho ancient mathemo.tioo.l concept of dynami8, Alluorbub without a socond thought identifies it with our modem notion of V6; ... yw. (5) It is oven morc unfortunate tho.t Anderhub o.lso tried to explain why Thoodorus stopped at the dynamis of o.rea 17 squo.re foot, or rathor at VI7 (although this is surely not of centro.l importance in the context of Plo.to's text). This fact also contributes towo.rUs obsouring tho distinotion betwoon 0. 'dynamis of areo. 17 square feet' and the modern notion of •Vrr'. ".-For these roo.sons I believe that Anderhub's reconstruction is historico.1ly inadequate.') Of course I do not dispute that there are 0. number of different ways in which one might obto.in squares ho.ving as areas whole numbers betwoon 3 and 17• Nevertheless I believe that the best way of explaining the text is to start with a definite and (so it sooms to me) very simple conjecture, namely that Theodorus oblained these 8IJU4rea (dynamei8) by traM/arming certain rectangles into 8quarea 0/ tl,e 8ame area, and that he wed Propositio1l8 VI.13 and 17 0/ the Elements to do thi8. (This possibility, of COUI'8O, hos frequently boon considered by other rcseo.rchers.)U Apo.rt from the fo.ot that, o.a we sho.ll soon see, this conjecture greatly facilitates further interpretation, there are at leost three clues in the text itself which point to ita being correct. These are os follows: (1) Theaetetus speaks of dynamei8 and, os we o.lroady know, tho word dynami8 means 'the value of the square of n. roctanglo'. Furthermore dynafllei8 wereorigino.lly obtained by transforming rectangles into squares of the same area. (2) The correct geometrical term for 'transforming 0. rectangle into 0. square of the same o.roo.' is 'tetragonizein' in Greek. As noted above, it cannot be accidontal that this term also occurs in Theo.otetus' short spooeh. Furthermore in my opinion the use of this word by Theaetetus suggests that the discu88ion between the two youths (Theaetetus and the 'young Socrates') was preceded by 0. tetragonismos which wos ca.rried out by Tbeodorus. (3) The dynameis investigated by Theodorus are wholo numbers between 3 and 17. Tbeodorus sooms to have started out with these

vs.

II B. L. van der Waerdun. ErtDCJCl&ende IViII6m6cJUJ/t. p, 236. '}'ho proJlOlliLion referred to thero Is II. 14 which i8 equivalent. to VI. 13.

i

I

-J

5S

SZABO: TIJJ~ 1I1!:UIIHIIHUlt Olf Oll.b:KK AlA'rUII:61ATIC8

whulo numbers and then to have construoted dynamei8 whose arena ~\correaponded to them. If one reoaJls Euolid's definition (VIII.16) of plane number which was discussed above, it seems olear that within tho context of the old Pythagorean arithmetio, these whole numbers were } represented geometrically by rectangles. t - Thus I believe that tho further interpretation of tho text can be based on this conjecture. Of course Theodorus would not have had to I transform all numberreotangles betwoon 3 and 17 into squares. A singlo I oxample would suffice to demonstrate the genero.l method. Perhaps he ( took the rectangle of area 3 and showea-liow-lt~oo·uid ~t;;~formed \ into 0. square of the same area. by construoting 0. mean proportiono.1 IJCtween its two sides. This same method works for all the remaining \ cU.'IC8, The construction of all dynameis botween 3 and 17 could havo been imagined simply by ano.1ogy with the construotion of 0. single one. So, when the text states that each of these squares was considered individu0.11y ("ard. plav l"dl1T7]V neOCU(!otSp811O,), it should not be taken to mean that Theodorus aotu0.11y constructed oaoh one of them. I shall have more to say about this matter later. For tho moment I want to recall t.wo points whioh modern researchers have made about the 0.11eged 'mathematical discoveries' of Theodorus. ....,;;-.. (1) Theodorus is supposed to havo beon the first to discover and prove the irrationality of square roots between Va- and V17 •4I (2) Nevertheless he is only supposed to have recognized the irrationILlity of the particular cases 3, 17. Subsequently "Thenetctus, who was still quite young at the time, grasped the genero.1 concept of irrationality and so laid the groundwork for a general theory of irrationals."" Up to now Theodorus has never been given credit for knowing 0. comprehensive dofinition ofirrationality. As Heath wrote u - - "It does not appear, however, that he reached any definition of a su:U in genero.1 or proved any genero.1 proposition about 0.11 surda." The impartial reader, knowing what WI18 said above about the concept of dynamis, will from the outset receive the second point at least with a certain sceptioism. It sooms unlikely that Theodorus discovored Vlr. Vir. .., of irrationo.1it.y, since he only tho particular CtJ8e8

V

vs: vcr. .. " V

VB,

"ur.

Vl't

5U

used freely not only tho concopt of dynamia but also the technical term pq"B' dmSl'pBTeo, whioh is found in Book X of tho Elemenls. Perhaps 'fheo.etetus' worda just moan that for some rel180n he stopped his individual demonstrations when he roached 17, rather than that ho was only able to construot dynamei8 up to 17. Once a singlo rectangle with sides which are not similar plane numbers has been transformed into asquo.re of the samo area. by constructing 0. mean proportional between its two sides, there scems to be no reo.son why 0.11 suoh rectangles cannot be transformed into squares. Furthermore, if it is known that the side of the square obtained is incommensurable in length, there seems to be no reason why this result cannot be genero.1ized to all such squares. The first point, howevor, is just na questionable when one thinks of tho senKO in which it is maintained that Theodorus proved the irrationality of V3, ~ , , . , V17. It is U88umed, (Iuite without justifieut.ion, that only the irrationality of V2(tho linear incommensurability of tho diagonul of a square with its sido) could be proved before Theodoru8. He is supposed to havo distinguished himself not just by recognizing that VB, V5. ... , VT7 wore irrational, but also by being ablo to provo tlus fact. If this WlHurnption is accepted, tho 'proofs of Theodorus' become very important. Honco a lot of trouble hl18 been taken to discovor theso l)roofs, i.o. "to find a mothod of proving the irrationality of V5. ...• VI7. wluoh is ano.1ogous to other methods used in Greek mathematics" ." I do not wish to deny that Theodorus could prove tho mathematical assertions which ho mado in front of his pupils. Indeed, shortly I shall try to show what kind of proofs he might have given. Nonetheless, I must emphasize that no such proofs are mentioned in Plato's account. Vogt already observed quite correctly that "Plato nowhere tells us Theodorus' mothod of proof. Ho does not oven hint at it."'· Furt.hermore the proper teohnical term for mathematical proof, the verb lJBi1nIVpl, does not occur in Theo.otetus' speech. (That Theodorus novertheless at lonat indicated somo kind of proof, can be inferred from the vorh dnntpa/Wll. This word often took on the meaning of 'prove' in overyday speech, C8peciu.lly in mathomat.icu.l contexts.) Hut an interpret.u.LioJl which places the proofs of Theodorus in the foreground distorts the

vs:

H. VogL, op. ciL. (n. 10 I100VU), p. 01.

E. Frank, PlalQ wid die 609. PY'1iagOf'Uf', Halle 1923, p. 228-9. •• T. L. Heath, A History 0/ GruID M",hemaliu, Oxrord 1921, Vol. I, p. 203. II

l'AU'l' 1.1'1111: U"ILI,V IIIK'rOIlY ltV 'flllt 1'IIKOII\' tl~' IIUlA'1'IuNAJ,K

U

Boo vogt (n, 10 above) •

.1 Ibid.

00

II~ADc):

l'Anl' 1. TuK HAIlLY 1I11l'l'OItY 011' l'UK TlIKOILY oar llIltA1:10NAUI

TUK DUU1NN1NUli 011' UltKKK AlATUJoaIAT1CII

meaning of Plato's text, for the point of the text is something other thlLl} th08O. Although I will try to recoll8truct the steps ofTheodorus' proof, it is not my intention by doing so to attach as much importance to them as is usually done in the history of mathematics. (In other wOl'ds, I do not believe that Theodorus had to invent the proof which I shall try to reconstruct here. Such 0. proof might already have existed ; in the mathematico.l tradition, just a.s the concept of dynamis WtLB j handed down to Theodorus a.nd not invented by him.) So let us consider ~what Theodorus' method of proof might have boon, even though this WILlI not of primary importa.nce to Plato. According to Theaetetus, "Theodorus WILlI dmwing some dynameiB to demonstrate to us that the lengths of the sides of those having an area of three of five square feet, are not commensurable with the length of the sides of 1.1. unit square. He discussed the CtLBO of each dynamiB individually until he reached the co.s6 of 0. square with area sevonteen square feet •.• ". The question now is how this was most simply demonstrated or •proved' , in the schools of the time. I believe that the following two stells suffice: (1) The recta.ngle numbers 3 (= 1 ·3), 6 (= 1 . 5), ... , 17 ~ (= 1 . 17), or at least olle illustrative example of them, were transformed intodynameis by use ofProposition VI.IS. Of course this proposi: Hon (and perhaps Proposition VI.17 which states that a:b = b:o implies 0.0 = bl , as weH) would have to be proved corredly. In the la.st analysis it is only the proof which shows that the dynamei8 produced ! \ have the same areas &8 the rectangles concerned. Then the sides of I \ those squares will be mean proportionala betwoon the two sides of the j \ corresponding reotangles. \ (2) The proof that the sides of the individual dynameis were linearly , ~ incommensurable might have used Propositions VIll.18 and 20 of tho • Elements. According to these propositions a mean proportional number exists only betwoon two similar pla.ne numbers. Since the sides of the recta.ngle concerned were not similar pla.ne numbers, the sides of the cOlTCSponding dynameis could not be numbers, i.e. magnitudes comJIIontlUftLble with the unit longth. This 'proof' needed to have been thorough and detailed in only ono ''«'JIII""t. EILCh dynnmis hotwoon 3 and 17 must Imve boon considorod individually (JeaTQ plav btdCfn]V neoal(!oVpE1'o, 88 the text says) to show that the sides of ea.ch rectangle, having the same area &8 the indi-

i

1

I t

61

vidual dynameiB, could not be similar pla.ne numbers. In other words, Thcodorus took 0.11 whole numbers between 3 o.nd 17 (excluding of course the perfect squares 4, 9, and 16) and showed that none of these could be decomposed into two factors which wcre similar plane numbers. In this way he mado the linear incommensurability of the sides of the dynameiB being disou88cd seem sufficiently plausible to his pupils. The sides of these dynameis are mean proportionals between plano numbers which are not similar, and moan proportional numbers only exist between similar plane numbers. The above, however, is not a rigorous proof by any moana. Notice that in the text not only are the concepts of dynamis a.nd p~Jee, oV mSppeTeO& used freely (furthermore Theo.etetus does no~ explain, what ~ dynamiB is, or how Theodorus obtained these dynamet8), b.ut In a.d
1.8

TJlB )(A'l'JlEMA'l'IOAL DlSCOVBIUES 011' I'l,ATO'S TIIRAl~1'E'l'US ;--

----------

In the light of the ahove, there seems to bO no sound historico.l justificl~tion for the idea. that 0. new chapter in the history of the thcory of irrationals began with Theodorus. In any event tillS ideo. is not justified by that part of Plato's TI,eaeletus whioh is the sole reference given for it. It is clear that neither the creation of the mathematical concept of dynamis nor the method of constructing individual dynameis can be attributed to Theodorus; so the question remains as to what his epochmaking new discoveries were. We concluded from the text that 'fheodorus was able to prove the linear incommensurability of the sides of dyna,neis Ltltwoon 3 o.nd 17. But it is mhdoll.ding to attach el'0oh·ffio.killg importance to these proofs (which are bardly even mentioned by l'IILto, and anywlI,y were prohably quite simple a.nd closely connected with the coll8truction of the moon proportional) o.nd even more so to suspect that Theodorus could only give prools lor the particUlar casu

62

SZAUO: 'I'UK IIKU1NN1NU8 01' OllKKK. MA'rUJWAT1C8

between Va and VI7. The following considerations make one relLlise immediately how unfounded this suspicion is. The theory that Theodorus could only prove particular cases of linear incommensurability has to be supplemented by another one, namely that Theu.etetus could handle all p088ible eases. This, however, docs not follow from Plato's text. Thou.etetus docs not report any proofs of his own. It is only fanciful to discuss any 'proofs ofTheo.etetus' in connection with this pB88&ge from Plato, when the text iiBelf offers no evidence fUi' them. ' - This brings me to the second point in my revision of the interpretation ofFlato. It is my contention that, on the basis of Plato's text, new mlLthematica) discoveries can no more be attributed to Theu.etetus than thoy can to Theodorus of Cyrene. This is the view for which I want to lL..gue in the following interpretation. Jf Theodorus' purpose was not to arrive at the proofs, then the question rumnins o.s to why he enumeruted all dynameis between 3 and 17 and proved linear incommensunLhility for twelve of them. In my opinion, this question can only be answered if one does not lose sight of the rulntionship between the mnthematioal section and the rest of tho TJ,eaetetU8 (as suggested in chapter 1.3 above). Ono should not forget tho way in which Theu.otetus W68 reminded of his mlLthematica) experience. Socrates had just told him that enumernting ono by one the kinds of clay used by different craftsmen did not provide a correct answer to the question 'What is clay "Bearing in mind those things which are common to all kinds of clay, one should rathcr give 68 a definition, 'Clay is earth mixed with water'. Thus Soorates' example resembles the mathematical part of the dialogue, in that the enumeration of individual cases is followed by the colleoting together of theso same CIl8C8. Having sketched Theodorus' enumeration, Tbeactetus says "Now it occurred to us that since thore are infinitely many dynameis, we slwllifl try 10 group these under one heading by wl,ich we could refer to all 'dyna.meis· ... 'rhus t.ho enumerution uf individUaLl tly,wmeia anel the 8OmowhlLt detailed proofs given by 1~heodorus seem to be 0. preparation for the c:omprehonsive classification which W68 then given by Theu.etetus and his friend. 'l'he quest.ion is whether Theodorus W68 delioorutely prel'lLring the way for this from the outset, perhaps evon asking them for it,

t

l'AIlT I. TUK KAltLY UIII'I'OllY Olf 'J'U" TlIKOItY 01.' IIUlA'l'lONA1.1I

63

or whether the two youths, on their own initiative, spontaneously thought of it. as the text ("Now it occurred to us •.. ") seems to imply. This matter will 00 taken up later. For the moment I want to point out 0. small diffemnoo ootween Socrates' example on the one hand, and the mat.hcmatical·cffort& ofTheodorus and his pupils on the other. In Socrates' example 0. correot definition of elll.y was sought only after the enumeration of the different kinds of clay, whereas Theo.etet.us and his friend were just seeking a vantage point from which they could cl088ify 0.11 dynamei8. Before going on to investigate in more det.u.il whether. or to what extent, it can be said that Theodorus' pupils were acting independently of their teacher. we shall take 0. closer look at wha.t they aotually did. ThelLCtetus and his friend sta.rted with numbers. As the text says, "We divided all numbers into two groups ... ". Tlus is easy to understand, since Theodorus started out from numbers as well. He first transformed tho numoor rectlLllgles between 3 and 17 into squares, and then directed his pupils' attention to the sides of these squares. Furt.hcrmore, in drawing 0. distinction ootween square and reclallyuiar numbers. the pupils were following 0. distinction which must at IOlLSt have been hinted at by their teacher. Theodorus could only provo that the sides of the d!lnamei8 3, 5, 6 . •• were linearly incommommnLLle. He must either have p088Cd over 4. 9, and 16 in silence, or Imve explicitly emllh68ized that thcso co.sesdiffered from the remaining ones in that tho lIides of the rectangular numoors concerned were similar plane numbers. After dividing all numbors into two groups (square and rectangular), the youths turned their attention to the dynameis which served to illustrato these numbers. First of all they called the sides of those dynameis which corresponded to square numbers 'lengths' (pi/"o,), The reason for this is obvious. Theodorus bad previously shown that the sides of dynameis with area 3, 5. etc. square-foot wcre 'not measurable in lerl!Jt1,' (p~"e, en) aVPP£Tf!O' linearly incommensurublc). So the sidcs of tho other squares (with area 4, 9 or 16 square foot) were called 'Iungt.ha' (pipeo,), hOCU.U80 unliku the dYJULlI1uis :1, 5, Ct.ll. thoy wore 'me68uruble in length' (i.e. linearly commensurable). At most, one could object that this nomenclature issomowhat inexact and imprecise. Thu sides of t.11080 dynarneill col'I'C8ponding to K'iUlLfC numbers should not really be co.lled Pit'"", but rather p~x£' aVPpnfl0t;.

S~AUc); 'l'UK UKUINN1NUS O¥ UlLKKK lIA1'IIKUA1'1~

Euclid &Iso uses the precise term (p~"e, aVPJ.&eTf!O' or auvpperf!O')' for example in Definition 3 of Book X. Moreover, the correct form of the tenn occurs in Theo.etetus' speech as well (Jn1"e, od aVPJ.&eTf!O" 147d 4-5). Anyway, <hough Th0&6tetus' inexact use of this mathematical term is annoying. it should not surprise us too much. for there is another striking terminologioal inconsistency in Theaetetus' speech. It does not affect his meaning. however. any more than the preceding example. In 147e7 he uses TeredyowO' TB "al UtonAevf!O' to mean square number (which Euclid calls TeredycoJlo, de'{}po,). By itself. this way of describing square numbers could be interpreted as simply an antiquated form of the usu&l term. but in 1480.6 Th0&6tetus uses 1aonAevf!O' Hal br~o, to mean the same thing. The word brbre60, is not only completely superfluous in this context. but o.imost troublesome as well. Anbre6o, a(!'{}pd, meo.ns not only squu.re number. but altiO reotu.nguJar number (hEf!O~'"I')' Furthermore. it occurs in a plL88D.ge which deals only with plane numbers (i. e. numbers which can be decomposed into two flLctors). Theo.etetus does not mention solid numbers (UTeeeol dedlpol) until later, and then only in passing. It seems that we have to put up with these inexactitudes in mo.themo.tic&l terminology on the part of '1~heo.etetus.

I l

'I'here is another interesting and importu.nt fact about Th0&6tetus' unusual use of words. which should be established here. It was rightly ('mphnsized in earlier reseo.rch that there is 0. close conneotion between tho word used by TheaQtetus ( pij"o,) and the correot technioal term for the same notion (p~He' aVpJ.&eTeo,) used in Book X of the "Elements"n. This observation has to be supplemented by a relative chronology of the two terms. It hardly seems possible that the preoise technical term "~"E' avppeTeO, was coined after Th0&6tetus (in Plato's dialogue) had just used the word pijHO, for the same concept. A moment's thought makes one realize that this order of events is extremely unlikely. H the word pij"o, by itself was used earlier for the concept oflineo.r commensurability. this vague and by no mea.ns80lf-expl&natory description could never have evolved into the completely clear and preoise term ~Hec aVpJleTe O , . Clearly those terms developed in the following three steps: 1) When linear incommensurability was discovered, the term ~"e, ciaVJlJlETf!O' was coined for it.

(

" B. L. van dar Woorden, Enmtlhmde WialJ~cAa/', p. 23(.

i'AWl: 1. 'l'U" ll:A1U.Y

IIU~l'OILV

ott TU" l'Illl:Oll\: Ott lIL1lAl'IONAI.I;

lit>

(2) Only then did it become necessary to give 0. name to the usual ca.sa of linear commensurability. To contrast it with incommensurability. the preoise and exo.ct term p~He, aVppeTeO, was originally used. (Symmdr08 of course is the opposite of asymmdr08. The dative mekei remains the so.me in both cases and serves to qualify the adjective.) (3) Theaetetus' curious description of linear commensurability (mekoa) must be 0. later abbreviation of the precise term p~"e, aVppeTeot;·

We are fo.ccd with the interesting fo.ct that an abbreviation (pij"o,) which cclo.rly postdates the precise term (p~He, croPPeT(!O,) occurs in Plato, whereas the earlier term is preserved in Euclid's later text. Furthermore. it should be noted that as, far as I know, Thooctetus' way of describing linear commensurability iB unknown in ancient ( mo.themILticn.llitcro.ture. Usuo.lly it iB called mekei 81Jmmetr08 (except in the passage immediately following our part of the Theaetetus. 148b6-7 which deals once more with neel ToV ~"ov, Hal Tijt; cSvvapEco,). Obviously this is 0. case in whioh Theo.otetu8 hOB somewhat unnecessarily o.~d arbitrarily abbreviated 0. cleo.r. unambiguous and precise term. Theaetetus' co.reless and inexo.ct use of words in the last part of his speeoh iB even more troublesomc. however (although I Bhould stress that once again the meaning of the text remains completely unambiguous). Having described the sides of those dynameU corresponding to BqUo.re numbers as 'commensurable in length'. the two youthB turned their attention to those other dynamei8 which corresponded to rectangular numbers. The sides of these they called simply 6v"ap", because these taller straight linea, all/lOugh incommensurable with the other8 in length. are nonet/,ele.ss commensurable with them in the area wl,ic1, they enclose when squared. This explanation clearly tells us how to understand tho peculiar namo dv"ap". The correct term of COU1'8O is not simply 6v"ap", but n~ther cSvvape, aVpper(!o, (i.e. commensurable with respect to the BqUo.re constructed on them), just as previously the term mekoa aymmetros would have had to be used instead of mekoa if one were speaking preoisely. Furthermore, in this case 88 well. the correet and preciso form of tho term is to be found in Book X of tho Elemenl8. . Yet again the question has to be raised as to the relative chronology of these two ways (dynamis and dynamri aymmetros) of describing 0. straight line incommensurable in length but measurable in square. 6Saab6

ijij

SZAUO: TUb: UKtJIHNINtJlI OIr tJ11XKK. lIlATUlWA'tlC8

In my opinion, the posaibility tbat such aline segment was originally described only by the word dynamis, and that this vague term later evolved into the precise and unambiguous term dY7lllmei, aymmetrOB, bas to be decisively rejected, for we aJready know the clear and precise meaning of the word dynamis. In geometry it describes ILn area and means 'the vILlue of the squo.ro of a rectangle' (more generally 'the vwue of 0. squo.re' 01' 'square'), never anything else. To use this so.me word on its own, as Thea.etetua does, in the sense of dynamei symmetroa (i.e. to apply it to a straight line, not an area) is such an inexact use of mo.themo.tico.l terminology that it aJmost leo.ds to elTOr and compels one to gucss the true meaning of the term from its context. But let us not be too hard on the young man for what seems to be characteriatic impreoision in his use of mo.thematico.l terminology. At the moment it is more important for us to decide on the relative chronology 'of the two terms. Of course in this case, as in the pl'6vious one, the exo.ct o.nd precise term dvval'£' ClVI'I'£T(!Ot; (found in Book X of the Elementa) mUlJt be the older of the two. The vague term 66vapct; must have como later and could never have been a correct technico.l term~nce again Thea.etetus has superfluously and lI,rbitrarily o.bbrevio.ted \. coml)lotely oloar and unambiguous soientific term~ 1.9 THE 'INDEPENDENOE' Oil' THEAETETUS

-

We now want to look more closely at the question of how far Theo.otetus and his friend co.n be said to have o.cted independently of their teacherThcodorus. In partioular we want to start out by investigating the extent to whioh the classification of numbers proposed by them was new. Although it has been maintained on the basis of our passage from Plato that Theo.otetua o.lso stated o.nd proved. somo mo.themo.tico.l theorems, we cannot invcstigo.te this claim at the moment. According to Plato, Theodorus' pupils wanted to give all 'dynameis' a name which would pick out the feature they had in common. So our first task will be to look at their 'd08oriptions' or 'definitions'. The youths commenced their 'independent work' by dividing all numbers into two groups, aquare and rectangular numbers. Apart from

l'All'l' I, TUJo: lllAIU.Y Hllfl'OIl\: 0,,"

TllJ~

l'UIWltY 0"" ll111A'l'lOHALIi

07

Frank,4a no one has been willing to credit Theactetus with inventing this cllLBSifico.tion. Wo nced not cite the o.rguments against tbis view here. Anyone who is just 0. little better versed in Greek mo.themo.tics will prefer Vogt's opinion. He wrote on this question,411 "The first steP'] the classification of numbers a.s being either squo.re or rectangulo.r u.s \7 well u.s the o.ssocio.tion of numbcrs with figures, wu.s not a per80nal ~ acl~ievement 0/ TheaetetUB. It, wu.s deviscd by the Pytbo.goreo.ns and wu.s common knowledge amongst number theorists of tho.t time." But if Thca.etetua (lid not invent this clo.ssifico.tion, he could only have bee" rC3ponsible for the two concepts 'commensuro.ble in length' o.nd 'commensurable in square'. I hoove already indicated that the abbrevio.tions (mek08 o.nd dynamis) which he used for these concepts must be lOoter forms of the correct and precise terms, i. e. the expressions used by Theo.etetus presuppose the eo.rlier existence of the precise terms. But for the so.l(e of completeness, we should consider the concepts themselves, not just their namcs. So let us ask whether these two concepts could ho.ve originated with Plato's Tlteaelelus. 'fheILCtetus' own words imply tho.t tho concept 'commenaurable in length' (p~"E' C1VPPET(!ot;) was not created independently by him. According to him, Theodorus had shown tlllLt dY7lllnteis of o.reo. 3 or 5 square feet were not commensuro.ble in length (I'~"E' OV ClVPP£T(!Ot). From this, of course, it follows that Theodorus must have known that the sides of the other kind of square (those with areo. 4. 9, 16, etc. square feet) were commensurable in length. Henee neither the concept nor the name p~xe, ClVI'I'ETeOt; could have origino.ted with Theo.otetus. He ml~y have used this name on his own, but the concept wa.s already known at least to his teacher. Furthermore, u.s I have shown above, the young man abbreviated the precise o.nd correct name. " E, FrlUlk, op. cit. (n. 42 abovo), pp. 258-9: ''Tho IUltithesis of 8qtUJT8 and reclangulor numbers, according to Speusippus, derives from tho Pythagorean tablu of oppoaiws, Tho fuot that it doos not roally duLo bac), to thu early Pytha· Goreans but only to Plato's timo, is proved by this picco of mathematical wOl'k which, 88 Plato himsulC says in tho Tlleaelaus, was the discovery 0/ t/,i8 mathe· matician • ..... [Translation) On tho other hand, Frajese (PeriOOico di MatJle· matiche Sorio IV, 44 (1966) 4) hoa very properly observed that "i1 giovinetto attribuiscc 0. 80 al.o88o (0 0.1 collega) nientllmeno cho 10. ripartiziono dui numllri in quadrati e rettangolari, con III. nomenclatura relativa: C080 inconupibile". n Or. cit, (n, 10 abovo), p. 113.

68

IIUlI(J: TUb: UKUlflfllNUIt Olf IlJlKb:K .IATU.h:MA'l'ICd

'this leaves the question of whether the concopt 6twaJl£' aVJll'eTf!O' (commensurable in square) originated with Theaetetus. On the fa.oe of it, this seems rather unlikely. Theodorus showed his pupils that the sides of dynamei8 with area 3, 5, etc. square feet were not commensurable in length; yet he also knew that the dynameia he was considering hnd areas of 3, 5, etc. square fcot. It would be astonishing if this did nut prompt him into thinking that tholJO straight lin08 whoso linear incommensurability he had just demonstrated to his pupils could most clUoIily bo measured by tho squares construoted on them. Moreover I have also made a caso for tho conjeoture that Theaetetus' inexact description (dynamis) of a length measurable only in square mUllt be a later form of the preci80 term (dynamei By1nmelr08). All this spclLI(s for the fact that the notion of 'quadratio commensurability' could hardly have been a now oreation of Theaetetus. In fact, I believe there are oIher texts of Plato from which it can be proved conclusively that the notion of qundratic commensurability (/SvVUIl£& aVl'll£-reo,), like tho concept of dynamis itself, originated in pre-Platonio times, Let me digreas here and give this proof.

• 'I:ho first piece of evidonce for tho llre-l)hLtonic existenco of this concopt is a ourious and rather complicated pun in Plato's Polilicru (2G6 1I..')-b7), whioh is hardly intelligible without a knowledge of contemporary Greek mathematics, Some l,hilologists who have studied this Hection of Platol o have correctly recognized that the pun depends on tho ambiguity of tho word dynamis. In everyday speech dynamis meant 'lLbility' , whereas in geometry it meant 'square.' I want to go one step further tban this and maintain that knowledge of quadratio commensurahility was also an indispensable prerequisite for understanding tho play on words. If knowledge ofthis conccpt had not been common and widesprco.d, Plato could never have thought of making suoh a pun. Let us now look at the pun itself. Tho prohlom which tho 'Elootin Strnngor' ILOd his intorlocutor would like to solve in Plato's Politicru, is one of classification, namely how to

•• cr, L. Cwnpooll, 2'/,e Sop/,ia'" arid l'oli,icua 0/ Plmo, oxronl 1867, l'P' 30-3j II. Loisegang, Die Plalondeulung tkr Oegmwcui. lCarlsruho 1929; 08 woll nil '\1',,11.'11 nuLtIit to his "nUlIIl"tiCln or this (lllll/tllg.. ,

I'AU'I' I, 'l'U" "AIU.Y ll11t1'OII~ 011' 'j'UI( 'j'IU.OIlY 011' JlUlA'j'WflAl.1t

60

distinguish between men and pigs. The simplest way would be to sa.y that men walk on two feet, whereUB pigs walk on lour feet. Dut this is too simple, Tho 'Eleatio Stranger' jokingly suggests a somewhat more lcamed method of classification, His interlocutor is the samo 'young Socrates' who appeam in the Tlu:aelelus as a pupil of the mathematician Theodorus. (Incidentally it is also mentioned in this dialogue that Theootetuy ,~nd tho 'young SocrlLtes' shared a kecn interest in geomotry , 266a6-7.) For this reason the Stranger l'roposes that the distinction betwoon men n,nd pigs indicntcd above should he mw:lo according to lI,e diagonal, mul tl,e diagofUJI 01 (111e square construe/ell on) tI,e previous diagonal. (rii lJ&oJlh(!tp lJq,zov xo; nd.4v Tii nj, lJ&ol'h(!ov lJ&ol'i-r(!tp.) Naturally the young mM is at fimt bewildered by this curious suggestion and no more understamls what the Stronger is really driving at than we do oumelvcs, However, wo can make it ou.sicr for ourselves to understand the pun which follows, if we keep in mind the sentenco "Man is (distinguished) by his ability to walk (X4Ta lJ6voJl'v, ., el, TfpI no(!elov) on two feet," 'fho phrase 'ability (to walk on) two feet' would mostlil(ely be tmnslated back into tho overday Greek of l~lato's time by /Svvaf'£' lJmov,. Now 0. geometer would immediately think of the diagonal of the unit square on hearing this expression, fur this linen,rly incommensurablo straight line was also denoted by lJvvaJlE' lJmov" i. e. 'two feet when melLtiured by the area of tho squaro constructed on it'. The pun is in fact mude possible by tho ambiguity of tho oxprcasions lJ6vUIl&t; ('ability' ami 'square') and lJmov, ('1100 legged' and 'mea8Uring two feet'). This is why the Stranger jokingly suggests that man should be classified by the diagonal (or more precisely, by the diagonal of the unit square). We have succeeded in making sense of the first part of the pun, but the joke goes further. Wo still havo to lool( more closely at the second part oCthe pun, to KCO how the Eleatic Stranger wanted to cha.racterize tho four-Ieggedness (four-footedness) of pigs 'by means of 0. diagonal'. His suggestion is that ono should construct a. square on tbe diagonal of tho unit sqUl\ro (HOO l~ig, 3), This diagonal, it WM jokingly suggested. yielded tho numerical value of our human iJwaJltt; (ability) (to walk on) two feet, Then one should n.sk how many feet the diagonal of this second s'lulLro hu.tt, Now this quclHticm (!I~nnot lIu ILIIHwerml in tho wll.y t1mt I~ geometer of Plato's time would have correctly answered it, for a geomotor would I1l\VO 8l\id thn.t, sineo this second diagonal is a linearly com-

70

8:tAUc); l'U~ UKUIIUUNUti 011 UIlKKK UATUJWATIC8

Fig. 3 1 'l'htl diagonal c1ulldpe, c1lnou, whioh is t.wo Coot. whon squared. 2 Tho diagO=]1l1 I'r)xe, 6{1rO~ (t.wo Coot. in longth) which moaauros twice two Coot when squared (J«Ird c1u.op," 4toOl" ,,~ In' nocSol" c11, mtplMIIla).

mell.Yurable quantity, it should be described as having 0. length 0/ two leel (11~~e, 6/nov,). On tho other hand, the Eleatic Strunger (just for tho sake of the joke, of course) insistB that this second diagonal is to bu mell.
PAll'!' I, l'U": .a.:AIlLY UJIITOIlY 01.<' TUbll'mWIlV Olr 11l1lA'l'lONALS

71

The second piece of evidence for the pre-Platonic existence of the concept of qUadratio commensurability (applied to certain linearly incommensurable straight lines) is to be found in Plato's Republic, Book VIII 546c4.-5. Since this posso.ge will be treo.ted in fulllawr (cf. pp. 89 ff.), I do not want to go into muoh detail here. Let me just say the following. The number 50 is desoribed in thi15 passage as the 'square on the (une;tpressable) diagonal 0/ five'. By this Plato means that if 0. length of five units is ossigned to the sides of 0. square, then the diagonal of that square is expressed by the number which we would call V50. In auch 0. case, however, the Greeks did not wish to o.ssign 0. number to the length of the diagonal (for, according to them, the length of a linearl y incommensuruble straight line was not 0. number). They preferred to BlLy that the square on the diagonal of 0. square whose sides had length five, is 50. It is therefor" clear that 50 could only be described in this way, becaulitl it was krlOwn that linearly incommensurable straiuht lines uisted which were nonetheless measurable in square. I believe that I have conclusively proved the existence of the concelJt of quadratic commensurability of linearly incommensurable straight lines in pre-Platonic timos. Although the correct technical term for this concept (6t1J1ape, (/VPPBT(!O,) does not occur in either of the references I have cited, the ClLBUal way in which a pUll is made about it in the Politicus and the curious expression 'the squlLre on the diagonal of five' (instead of 50) which is used in the Jlepublic shows tho.t it must already have been current in Plato's time. Hence this concept cannot have beon introduced by Theo.etetus,

1.10

A GLANOE AT SOMB RIVAl. 'l'UKOllIES

Thus far my interpretation of Plato hILS led to the following conclusions, It appears that Theodorus did not intend to teach his pupils about new scientific msult.a of his own when he gave the lecture on mo.themlLtics sketched by I'lato. 'rhu leoture 80011115 rather to havo just Leen an exercise. Theodorus exhibited various dynameis which had areas hutWtlll1l :1 "lUI 17 "CIIIlLl'U-rtlll!.. rn HO cluing, hn IOlulo hiH HtuclontH ILWILro of the fact that most of thoso had sides which were incommensurable in Inng!.h. Although no oxplioit mention is mude of what Theodorus

72

I'AUT I. TUE I'AIlI.Y uurrOILY 01' TUK TllHOILY

IIUUe): 1'UI' UKUlHHINGIi Olf OUJ.:Js:K UA1'UJatA'!'IOS

hoped to accomplish by tIlls enumeration (indeed tlus question cannot be definitively answered on the basis of the interpretation given up to now),51 one can at 16&8t say that Ius demonstration gave the youths food for thought. They im~cdiately came to the conclusion that there were infinitely many dynameis of which Theodorus had only enumerated 0. few, and they wanted to elll88ify 0.11 of them. One's first reaction is naturally to suppose that the youths extended in some way the teachings of Thcodorus which may well have consisted of scientific achievements of an earlier period. It might be that the clll88ification of dynameis was 0. piece of independent work carried out by Theodorus' pupils. This idea has to be abandoned, however, ns soon 88 it is realised that three very important concepts ('square and redan{lular numbers' , '1IU't,surable in length' and 'measurable in square') which not only wore nccC88ary for this clll.88ification, but o.1so oo.n be viewed in part ns resulting from it, existed long before Plato's time. 'Commensurable in length' and 'commensurable in dynamis' oo.n be said to result from the olo.ssifico.tion, because both concepts presuppose that one hns alrel\dy learnt to distinguish (cI088ify) the sides ofindividuo.1 dynameis (tho values of the squares of rectangles). If all numbers are represented ns reotlLngles (plane numbers) and these number rectangles are transformed into squares of the same area (dynameis), then the sides of the squares obtained are of two Idnds, those wluch are 'metuurable in lengt1,' and those wWoh are only 'metlBurable in 8quare'. IfTheaetetus had invented or introduced these two concepts, he would indeed have had to be credited with the elll88ification of dynameis. However, since it has been shown above that both concepts must have been known long before Plato's time, Theootetus could not have discovered this c1ll88ifieation; at most he could only havo rediscovered it, in the way that to.1ented students of mathematics oec88iono.1ly rediscover the oarlicr results of their predecessors. These conclusions by no means exhaust my interpretation of the mathematical part of the Theadelus. Indeed I will return later to the question of how this pll88o.ge was understood in antiquity. But first I want to raise three points which are bnsed on the conclusions reached up to now. Theso contradict somo viows which are found &I

I shall try to answtlr this quustion in tho noxt. ohllptor.

()Io'

JltllA'l'JONAI,S

73

today in the best and most authoritative works on Greekmathematics.&2 (1) First of 0.11 I cannot agreo with the interpretation which o.sserts that oxact definitions of tho concepts 'meMurablo', 'measurable in square'. 'rationu.l', and 'irrational' are due to Theo.etetus. For in tho dialogue he only tu.lks about the first two ofthcso, namely 'mensurable' (or more precisely 'linearly commonsurablo') and 'mensurablo in square' (or 'commensurable by the square constructed on it'). Furthermore, both of these concepts undoubtedly antedate Plato and Theo.etetus only refers to thom in an inexo.ct, imprecise and wmost misleading way (i.e. ho uses mekos and dynamis instead of the correct terms mekei symmelr08 and dynamei symmetr08). (2) In addition I cannot agree with the view that the mathomatical pnsso.ge concludes with Theaetetus stating 'very briefly but nonethe1088 clearly' tho fulluwing thconlfn: "I:)tmight lines which form a sflUlLro whose area is I" whole but not 0. square number, have no common meMure with tho unit of length." This theorem Clm indeed be reconstructed from Theaetetus' words, but thero is nothing in tho dialogue to suggest that he intended to o.ssert 0. theorom. It hus heon extracted from tho following part of tho text. "'1'h080 straight lines which yiold u. rectangular number when squared, we donoted by dynameis. For although these are not commensurablo in length, nonethclC88 tho areas which thoy enclose when squared. aro commensurablo." (Cf. p. 43 above.) The arbitrary way in which this theorem hns beon extracted from the words just quoted should also be mentioned. The same words could bo used with more justico to reconstruct the following, more complote, theorem: "Straight lines whose squares are whole but not square numbers, havo no common measure with the unit of Icngth. but thoy can be mCIL'iured by their sCluares." It should be omphlL'ii~ed

J

IS On 1.110 following cf. VIlR dor Wucrdon'lI Erwaclu:nde Wia.msc1.a/l, pp. 233ft. Wld pp. 271ft. 88 well 88 his papor 'Arit.hmot.ik dor Pythagorour' in tho ooll(.c· tion Zur GuchidIU der grieclai6clu:n MalhemtUil:. pp. 203-04. I should also munt.ion huro t.I1Ut. ufl.or I hud ftnhlhUlI writing thUllO IiIlUII, l'rof,!II>IOr n. ChtlmillS sent me the following communlcut.ion in connect.ion wit.h tho papor of mino mtlntionod in n. 1211 ubovo: "I hud myself urguod in Plato IJ8 MalhemaliciJan tbut. t.IlU JlUIIIIUSU ('l'lulllltut.UIt 1470-14&1) dOtIll 1I11t. iOllicutu llRytlaing ,,!tout. I' dilcovery or a new !Umorlltralicm." I hopo t.bat tho above discussion will servo to conftnn tbill thosis of Profllllllor Cbomlss, of which I W88 previously unUWl1ru.

-

SZABO: TIIK IJJ~UINHIN(l8 OM Oll11:KK .U1'IIKAlAl'IOS

once more, however, that there is absolutely nothing in Plato's text about any theorem of Theaetetus. (3) Not only do I have to rejeot individuol IlB8ertions like theso, : but I also believe that some other historical reconstructions, which up to now were more or less plausible, will have to be reexamined in \ the light of the development given above. Let me give just one example of tlus here. I:Van der Waerden,/who today is thlmost prominent histori~ of Greek mathematics, published in 1947/49 an important pape~ which WIl8 the "result of a fruitful disoUBBion with Reidemeister". In this work he says "there was not just one theory of squared rational straight lines," but two wMoh led to the same results by quite different methods".

---+6.....

Van dor Waerdon referred to the one theory as .Ae, since it WIl8 ~ Hupl'osed to go back to Archytas; the othor one was denoted by 8e hecause ita founder wus supposed to be Plato's Theaetetus. First of 0.11, I should emphasize that I usod to find this historical reconstruction o.eccpt~bl.e. I~ particula~ I foun~ the way in whioh the two theories ( were dlstmgUlshed eBl)Oclally enhghtening. Only after I had como to \ IIl1del"Htand the mathomatical concept of dynamia better, did I first IJcgin.!o havo_ doubt!! about it. I would like to give 0. brief account of these doubts hore, so lot us see bow van dor Wo.orden distinguished betwcon the tbcorioa Ae and 8e. About Ae ho says "it proceeds from the question of the oxistence of 0. geometric mean between two lines or numbers, and its mot hod is bused on tho arithmetio of goomotrio Herics ILnd tho related theory of similar plane and solid numbers". On the other hand, 8e "does not me the concept 0/ geometric naean, (,ut employs the 8quaring 0/8traight lines, etc." " So the two theories are distinguished by wbethor or not thoy u~o geomotric means. Before trying to test the validity of this observation, the modorn term 'sCJuaring of straight lines' should be translated back into Greek. If I am not mistaken, botb 'square" rational straight lines' and 'tllJlIlLfing (If strnight Iin~' corn!Hpond simply to tho Greek concel)t.s II Soo tho proviOUB rootnotu. "s . h)' • quared ra t'IOna1 IIt ralK t InCII .. trarullatca tHera, dVlldpra tWppneo, of. n"fillltion X. 2 of thu Elemerau.

PART 1. TDK KAlLLY DI8TOllY OM TUB 'l'DUOllY 01" IIl11A'l'IONAI.II

75

dynamia and dynamei aymmelros." Furthermore, these concepts are

used by Plu.to's Theo.etetus, as well as by the so-called Theory 8e, Dut the following elosoly related facts about dynami8 have already been established above: (a.) The scientific term dynamia means 'the value of the sCJuare of a rectangle' . (b) Tho vu.lue of tho square of 0. rectangle was obtained by tetragonismos (transforming it into 0. square of the same area). (c) According to Aristotle, the nature of tetragoniamos was the finding of the geomotrio mean. Thus the concept dynamia, wlUch is paraphrased us the '8quaring~ 8traight lines', cannot be separated from the concept of geometric mean. In other woms, the concept of geometric mean is used by Theo.otetus (in the diu.loglle) or by tho Theory ee, just as much as it is by the Theory At! which is attributed to Archytua. But this obliteru.tes tho distinction between tho two theories of "squared rational Htrnight lines" which "led to tho same results by quite difforent mothods". I no Ion er beliovo that thero was more than ono thcoj)' , ILncl cu.n seo no argument which could he UHC ~upport HIWh II. hiHwrieu.l reconstruction.

1.11

"'UK SO-OAf.I.K» 'TIIKAK'I'I~... t1S I'.LOBI.EM'

del~ls

/ ' The investigation presented in Part 1 of this book with the elLrly hiHwry of tho Greek theory of imLtiolllLIH. Uy tho 'Thell.otetuH problem' is meant tho question of whether Thea.otetus really made a significant contribution to mathematics, u.nd whieh scientific discovericli clm be attrilllltc.!ti to him, This problem is only of sccondll,ry importance to us. Nonetholess I feel compelled to considor it here n,t loust in pBBBing, hccaU80 tho passage from Plato discussed abovo, which in~~ is u.n importu.nt historicu.l source ~or tho t.heory of irrationals (notico thu.t tlus opinion doos not dopend 10 any way on tho flLCt'" thlLt tho mu.thomlLticilLns ThcmloruH and Thcllctctus aro montioned therein), also provides 'evidence ofThcaetetus' rolo in the hiHtory of scionce'. ]i'urthormoro, tho mathematical section of Plato's

Ii

01

SOli n, 64

Ilhoy".

76

r

tllI:AlI(): l'UK lIJ£UUUUlWtI 0 .. URDK AlATUlWATICS

l'Altl' 1. 'l'UU lCAIU.\' 1II11'l'01lY l>1f 'l'UR TUUOUY 01' II111A'l'lONAI.li

dialogue cannot be satisfactorily interpreted without considering tills whole complex of questions. Therefore, I have inserted here some rcma.rl(s about the 'Thoaetctus problem'. I ho.vo ruready shown that there are no new mo.themo.ticru concepts (or names for them) and still less any theorems which can be attributed to IlJu.to's Theaetetus on the basis of the text we have been anruyzing (T"eaelelus 147o-148b).M We po88e88 two ancient SOurces which soom to contradict this. They support 0. tradition according to whieh concreto discoveries connected with the theory of irrationo.Js were attributed to Theaetetus even in antiquity. Sinee both these sources, a. scholium to llroposition Din Dook X of tho ElemenU5' and a report which probably stems from Pappus' commentary on Dook X and survives only in an Arabic translation,r.a are very closely related to the dialogue The4elelus, wo will have to oxamino them both in detail. This will also enable us to throw light on the mathematical section of Plato's diruogue from IL new anglo. An accurate tro.nslation of tho scholium in question is as follows:'"

To facilitato disoussion of the scholium, Proposition X.D is reproduced here as well:

"This theorem is 0. discovery ofTheaetetus. It is ruso mentioned by Plato in the dio.Jogue Theaelelus, except that only a. special case is Ktuted thoro, 1\8 ol'}lotl(!d to tho gonel'lLI formullLtion givon hero."

Tho text from Plato which was quoted abovc concludes with the statemont. 'J'heuotetus and his frinncl ('young Socmtt!8') attempted a similar ducripliun 0/ /lolid No furthur dutuils or thitl ot.her 'ditlcovury' aro given in the text itselr. Furthunnoro, modern rwaurcbers have not beon able to inrer nnything moro dufinite rrom t.his statement.. lIunco we need not. concern our. KulvUII with It. hun,. t? Euclldcs, Elemenla (ed. J. L. HeiLurg, Lipaiao 1883-1916), Vol V, p. 450, line 16ft. (Sec n. 69, below, for the Orock text.) NOn tlllll 8ttU T. r,. HUII!.h, ItJ.,clitl'" lClemtnJU, Vol. 3, 1)(1 3-4. Tho toxL or thill Arubic commentary, edited by O. Junge and W. Thompson was published with un English translation 118 Vol. 8 of tho Hnrvard Semitio Series (Cambridgo, MII"It. I ~30). IIullth UIIull UlIIIIlI'!iur IlIlItlon or thi" oommentary (IIIJit.cld by'VuII,,dtu,!'arUl 1856). l'hll text which is quotod in English below, appeal'll also in von l<'rit.z'.. article on Thoaotetus in Realencycloplfdu. "Tel OCW(!"I'o ~oiho 8,omfnuw lam. dJe'1/lo "01 I'1",.,,~a, a"~oiJ d n}.d~w" Iv 8,o,~~Tt", cLU'Qef pb pCel"WTt(!CW lyH"Ta" lnoiHJa di JUJD610v otc. (cr. n. 57 ubove). II

!.lUll,

",,,,,ber,,.

77

"The squares on linearly commensurable straight lines have the same ratios to eo.ch other as square numbers have to squo.re numbers; and squares which have the same ratios to each other as square numbers have to square numbers, must o.Jso have linearly commensuro.blo sides; but squares on linearly incommensurable straight lines do not have the same ratios to each other as square numbers have to square numbors; and squares whioh do not ha.ve the samo ratios to each other DB squaro numbers have to square numbers, cannot ho.vo linearly commensurable sides,"

If this theorem is compared with the scholium quoted above, then the following can be established about the latter. Clearly the scholiast is referring to the plLBSo.gO from Plato whioh wo have beon discussing. Be claims that Proposition X.9 is mentioned there. This lL8SOrtion, of course, is inaccurato. Plato's Thenotetus did not state any real theo· rems, he only cllL88ified squares (and even then, he did not do so in 0. novcl way) as follows: "Tho sidcs of squaros whose areas aro whole numbers, o.re of two kindti, those which o.re commensurable in length, and those which are only commensurable in square." The scholium is correct only in so fILl' I.L8 Proposition X.U cuutu.iUti tho Kamo 'cilLtIlIification of squlLres' whioh Theaetet.u8 proposes in tho dialogue. Dut. the scholiast.'s remark clearly derives from Plato's text, honce it. cannot be regarded as indopendent evidenco of a. mathoma.tical discovory by the young Thea.etotus. It is far more likely that Tho.er is correct, whon ho writoH:1o "'J~11I.l tru.clition which ILttrihutcs thiH proposition (X.9) to Thenototus is suspect. because it. probably arose from 0. passago in Plato's dialogue of the samo name. Doth Vogt., and before him Hankel, concluded (lorroetiy from this I'ILllltILgO thlLt tho proposit.iun (X.9) wo.s already known at least to Theodorus , ..". So it. is clear thlLt. tho HcholillHt wlLntocl to eredit ThClII.ototus with diRcovering Proposition X.D, just on tho basis of this plLBSn.ge.

00

C. Thaor, Die Eillmenle lIOn Euclid, Part 4, Loipzig 1936, p. 108.

78

SZAlIO: 'J'U.I>: lIKUJNNJNIl8 011 UJtKKK MATUJWATJes

, A similu.r rema~k holc.hJ t.rue for t.he ot.her 8Ourco, t.he report ment.lOned above whICh survives only in an Aro.bio translation. This sketches Theaetetus' mat.hematico.1 o.ohievements in greater detail. For the aa.k~ of com~)let.ene88,. the most important part of it is repro~uced here In an easdy o.cceaalble English translo.tion.el In the follow~ng quotation, I have italicized those words whioh clearly reveu.l that It was based on Plato's dialogue. / : .. t.he theory of irrational magnitudes had ita origin in the school of , P!thagoro.s. It was considerably developed by Theaetetus tho AtlIOm~, who gave proof, in this part of mathematies, as in others, of ability which has been justly admired. He was one of the most h.appily endowed of men, and gave himself up, with 0. fine enthuSJasm, to the invcstigation of the truths contained in these sciences, as Plato bear8 witness lor hi7n in the work wl,ich he called alter his na~. A8 lor the exact distil&c1ion ollhe above-named magnitudes and

t~e rtgor~ demonstration 01 the propositions 10 which tI,is theON) gave

r~8e, I beheve that they were chiefly established by this mathemat.iClan, and later, the great Apollonius, whoso genius touchcd the highest point of exce))ence in mathematics, added to these discovories II. number of remu.rl,able theorios u.fter many efforts and much labour. For Theaelelus I,ad distinguished 'quare roots ('puisaa.nces' must be the 6""&pE'~ of tho Platonic p&88o.ge) commensurable in lengtl'lrom th086 wl,ich are illC07mnen8urable,GZ and had divided tho ;rvell-kn~wn ~pe~ies of irrational lines after different means, &88ignmg the medial to gcometry, the 'binomial' to arithmetie and tho •apo to me' t armony, o has'IS stated by Eudemus the Peripatetic, , etc. Actually this quotation is noteworthy only bocnUBe of the l!Utt half of the last sentence, where it is maintained that o.ocording to Eudemus tho threefold division of irro.tionallinoa (p't111, bl 6vo &vopaTClJV and In(JTOp~) should o.lHo be attributed to 'l'hcaetetus. It mUHt be loft for future researchers to find out whet1ter an atdhentic and historically -. See n. 68 above. U The ~onnan Lranslation by von Frit.z (soo n. 68 above) is superior horo: ••.Th8&1te~, ~elcbor untel'llChiooon hal. Quadrate (dyrtam8is), welehe kom. mcnsurab61 Sind In dar Llingo von den niehL.konunensurablon 01.0. II II

l'Al1T I. TUB BARL\, Ultll'OIlY 01"

1'Ul~

TUBOltY 01" lIUlATIONAl.ll

7U

reliable tradition underlies this one point wl,ich is independent 01 Plato's dialO(11U. The two p&88a.ges quoted above arc interesting for the following reason. As stated above, the Arabic pa.sso.ge attributes the threefold division of irrational lines (meae, ek dyo onomaton and apotome) to Then.otetus. A critical examination oftbis claim lies outside tho bounds of this investigation into the early history of the theory of irrationals. Both sources, however, relying on the mathematico.1 section of the Theaelelu8, also attribute scientific discoveries to Then.otetus which could not possibly be due to bim. This co.n only be described as a false tradition. If these two reports did not betray the source of their information, there would not be a.ny way of examining their credibility. However, since Plato's dialogue is explicitly mentioned as 0. source in both en.ses, their statementa co.n immediately be checked. Without doubt Plato's dio.logue has been misunderstood. From our interpretation of the text, it is clear that Then.otetus could neither have introduced any new definitions of irrational quantities, nor have stated a.ny now mathemo.tico.1 theorems. The only remaining question is how this false tradition aroso and whether the origins of tho legond can be further explained. In my opinion thero ia some serious textual evidonco in favor of tho fo.lse tradition. One could almost say that Pln.to misleads the reador (both ancient and modern) by the way in whicb he presents his material. Deforo inquiring into hiH motiveH for doing HO, let Uli tako 0. hard look at tIm evidence which sooms at first sight to justify completely the false tradition. Once before I I)osed the question of whether Thcodorus, when he enumerated and explained the dynameis botweon 3 and 17 in front of his studenta, was from the outsot delibcnLtely preparing the way (perhaps even asking) for the comprehensive cllLBSification which thoy then carried out, or whothor they o.ccomplished this on their own. I beliovo that this (Iuestion can bo answered with cortrunty. Undoubtedly Tbeodonls must have expected this 'indopendent work' right from the start. Later I will attempt to establish this view more firmly, but for the moment I just want to point out that Plato's text, at least on 0. first reading, gives very much the opposite impression. One just has to read Thoa.ct.etus' speech to get the impression that he is talking about his own scientifio discovery in which at most his

su

II~AU():

'1'1110: bXUUUUHUIi 011' Ol1XKK ld.ATUlI:ld.ATICII

friend ('young Socrates') shared. There is nothing in the text to indico.te that the ground was co.rofully laid for tllia discovery by Theodorus. It docs not say that Theodorus chaUenged his students to find a way of c111.ssifying the dynameis, which he had shown them, from a common l,oint of view. The text merely says that "for some reo.son Theodorus stopped" when he reached the dynamis 17. In this way Theaetetus cmphuizea more strongly his own initiative. It is not just that at the beginning of his speeeh the young man says to Socrates, "You are asking something like what your n4muake the other 8ocra1e8 and myself were di8CU88ing here recently", but later on o.s welJ he empho.sizes his own initiative in the matter, by the words "and now it occurred to 118". Furthermore, 'us' certainly does not include Theodorus, just Theactetus himself and 'young Socrates'. Plato's Thea.etetus is undoubtedly convinced that he is ta.lking about his own scientific discovery. This cannot be an accident. It is clear that Plato deliberately made PheaetetU8 apeaTe in 'hia way. It is not surprising, therefore, that even in antiquity readers of this passage thought it had to be interpreted aa meaning that Thea.etetus wo.s talking about his own independent discovery. Now the question is what did Plato hope to accomplish by this CUl·jous portrayoJ. He could hardly have been ignorant of the fact that everything which the youth spoke of as his own scientific 'discovery' waa known to Theodorus as weU, since this waa mathematico.l Imowledge which had boon in existenco for a long time. One could argue that he wanted to deele out his 'hero' Theaotetus in borrowed plumes, but I think that this would be obviously false and misleading. It implies that he must have wanted to make Theaetetus appear ridiculous; and this is clearly impossible. On the other hand, 'Theaetotus actually decks himself out in borrowed plumes' when right "t the beginning of Ilia speech, ho clwms that he and his friend discovered the distinction between squo.re and reotangular numbel'B, even though - - this discovery could not possibly have been due to them. Frajeseu is correct, whon he writes on this problom,

L

Questa presentazione dol Teeteto platonico come vero en/ani prodige spiegu. o.ncho if f"tto ehe iI giovinotto "ttribuisco. a so stesso (0 1\1

up....,,,,., M...........,...

Bori.lV,,'8

PAllT I. TUX I::AI~LY UIIITOIlY 011' TUX TUKOIlV Olr llULATIONALli

81

coli ega) nientemeno che la ripa.rtizione dei numeri ~n qu~t~ e ret~ngol"ri, con Is. nomenclatura relativa: cosa lDconceplbile. Si tratta invece, secondo noi, della bo.ldo.nza, e anche della s.c~rsa conoscenza ad csso. colleg"to., propria del giovanissimo che rlbeno d'u.vere scoporto Ii dove o.ltri han no trovato prima di lui •.. ". Not only docs Thcaetetus ingenuously depict old knowledgc aa 'his own discovery', but he is not even reproved for doing so. No ~ne suggests to him that he is serving up o.s his own discovery something which must have already been known to his tea.cher. Instead of s. reproof, the youth eo.rns praise from Soorates; "Well done, lads I I think that Theodorus was right to praise you." Furthermore, these words are undoubtedly meant seriously. They should not be t"ken o.s mockery or irony. , Now in my opinion it is by no means easy to fin~ out Plato s true intentions in this CIUle, but thero is no reo.son why It should ~~. ~he Mono. Lisa's smile is not easy to understand. The worthy Pluhstme, for tho most pu.rt, does not understand it at all. He just shlL~es his hend, for ambiguities and subtlo nuances are simply not ac:cesluble to him. But thon there is no reo.son to expect that great artist PI"to to be any less enigmatic than the creator of the Mon~ LislL. . Perhaps we can come a little closer to Plato s true meanmg by obsorving closely tho way in which he portmys the young Thea.ote~us. At the beginning of the diu.logue, Thoaotetus is described in the loft~est tones o.s xaM, TB xal ayaD&,; he is brilliant and ho.s great mathemn.bcal aptitude.&« Never before haa Theodorus come acroBB su~h " youth. "lIe approaches learning and research like a stream of .n~~le88 flow· ing oil, and yet he is very tranquil and modest about ~t. O?O ~uld go on like this for" long time, just repeating the IlrU.IHO which IS 1:10 Iibel'o.lly bestowed on the youth in Plato's dialogue. . . One thing, however, is usuoJly forgotten in modern diseUSS1~D8 of Plato's portrayal ofTheaetetus, namely that the. po~t~it itscl~ IS not completely unambiguous. There o.ro inconsistenClOB 10 It and If these aro ignored, one runs the risk of misunderstanding Plu.to.. Let us begin by observing that Theaetetus is not only a gl~ted and mathemu.tically very tu.lonted youth, but "tHO u. very nawe one.

tJ

.. cr. n.

1.. von ,Inr \Vlulrdnn. /Crwadulnde JViaaen8CIUl/t, p. 271ff.

6 SaW

'. "

....,

82

I'ART I, TUb; KA1U.Y IIIl1'rOltY 01.<'

IIZAlIO: TUX IJUUIHHIHUII OF UIlKKK laIATJlKMATI08

It. would IICOm thu.t somotimes ho OWl only givo fu.lao u.nd evon siml,lo (not to mention stupid) 8.Jl8wors to tho questions which u.re nakod him. For example, to Soomtes' question "Whu.t. is knowledge f", ho who "approo.ches learning ... mee 0. stream of noiselo88 flowing oil" replies without any idea. of how difficult and complicated this question really is, 1Iy giving first. of 11.11 0. surprisingly fn.cile enumeration of the different kinds of knowledgo. Socru.tes can only respond with conccrued irony, "You are very generous and liberal, my friend,lor you were only aslced for a Bingle thing and Set, you immediately give us many things 01 diOereni kinds" (146d). But the young man does not understand this irony and asks with touchingsimplicit.y, "What do youmean by thal,Socrates 1" SocrlLtcs, unable to answer such 0. naive question, prefers to abandon irony for the time being ILnclsn.ys, "I don't mean anything by it, or porhaps I do ... ". '1'0 think that simplicity and naivety are ~ncompatible with brilliant tll.l~nt is sur:ely mistu.lcon£4s a mll.t.ter of fiLC"b it is just this unaffected nalvet.y whIch lends espeCial charm to t.~giftcd youth and makes him so sympathetic to both Plato and tho reo.der. There is no question of 'fhou.etetus being siml,le-minded or stul,id. He immediately gives n COJlSl,icuous display ofintelligenco by his quicl, and u.ccurato nasooiation of idena. No sooner hu.s So orates oxplained tho bo.nal and oversimplified example of the different 'kinds of clay' to him, than the youth is reminded of 0. related problem on a much higher level, which lies in the field of mathematics. Socrates' 'Idnds of clay' and Theodorus' dynameis lie rather far I"part. An ordinary youth would not have seen tho connection botwoon the two so readily, but Thou.etetus' capacity for abstract thought seems to have functioned quickly. In I" f1nsh he piclts out wlmt they have in common and at that moment he also establishes real conto.ct with Socrates. In answer to Socmtes' question, the quiet and unBBSuming Thou.etutus tells of his mathematicuJ 'discovery'. Yot he is II. little nertIOU8 and excited 68 be does so. Wo experionce his oxcitemont with him, I~ mnmory clLrriccl him blLcle to t\e time ILt which he mado his beautiful discovery. I am reo.1l~mo.zed~ the simplicity of the teohniques which enable Plato to portray Thea.etetus' mental state while he is HI'IIILldng. This stlLt.o iH IIl1Lrlwtl by prolixity on t.he ono }",ncl ILIIII impatience and ha.ste on tho other. As I have tried to point out above, '1'1"\lWWtU8 8omotim08 lIKOH mlLthomat.innl oxpl'CBHiollJl whioh are rodun-

'l'lI~

l'IIb;OILY Olr IIl1LA1'lOHAI.II

83

dant 'Lnel plconu.stio, and ILl. other timca he UII08 onoa which are exCCBsively abbreviated, rumost to the point of incoherence. Furthermore, it is by no means uninteresting to observe at which points he is longwinded and pleonu.stio, and at which points he is terse and concise. He docs not refer to the trivial concept of 'square' or that of 'square numbCl" by tho simplo name n;,reaYo)lI~ or TeTeayO)vd" T£ xallddnMv(!o", but ILdds quite superfluously that it is also in/nElJo". This ovcrzelLlousnOM would bo ludicrous in 0. trained mathematician, but 'I'hco.ctetus is not. 0. trained mathematician. Ho is just a young o.nd gifted, but still inexperienced scholar who is o.nxious not to be misunderstood. Hc is st.riving for precision in his words, yet this endeavour which in itself is laudable, misleads him to a simple I'lconu.sm in the simplest eMe of 11.11. nut this minute exo.ctness and striving for precision is ILblLndoned us soon u.s he reo.ches thc point which he supposes to be the most import",nt - tho correct dOKcription of sidcs of squares us falling into two clo.sses, those which are mekei Sfpmetros and those which are dynamei Sfpmetros, whioh he considers his most significa.nt 'discovery'. 1~hC8e arc tho very conoopts which have to he exphLincd with precision, for thoy aro not so f'LmililLr to non-mathematicians as the trivial concept of tetragonon. Yet in explaining thorn, tho youth falls victim to hll8ty impatience o.nd the joy of discovery. From tho gmmmatica.l point of view, ho describcs them in a completely incoherent way. The one is referred to lIy tho singullLr mekos, the other by the plural dynameis." It is hardly surprising that the content of those ungrammatical 'Lud im}lrccillO expressions could only he understood in 0. goneral way. (As 0. matter of fact, insofar as they led one to believe that dynamis could be called n correet mathematicnJ term as well u.s dynmnei synunetrolJ, \hoy woro not properly understood ILt 'Lll.) .. Tho incolUlilll.ont. U80 of llingulur amI pluml ('4;pc~ uml 6tmi,4fl,) ill lIurprill· IlIg hocau80 a fuw lincs lul.or (148b 0-7) Thuootot.us docs not. ropcat. this 1I1I1I,,"llIthon. 'J'hunl ho 11[14,"111' of nce' Toil p~,,01" Te "u' "lII'apew,. l\fonlovur I'rof. G. Calogero, who WIIB Idnd enough to inform mo of this incolUlilil.oncy, hoa pointod out that Theool.ot.us' U80 of the oxpl'C88ion delCeafJa, is wlgraml1lllti. Lnt 11K conKltlnr thn following nxamplu of Plut.o'lIllllnoflangulIgn: T'p dyaO,p rl} .. danl}lI dl!lC~ao.l& (t.o dufillu vil'LuCl In Lunlllt of goutl ulUl CIvil), It requires only two IIlight ull.orat.ions to repair tho grammar of our toxt and mnk" it. conronn to thiR nxmnpln. Wo nood only clll"'g" 14;PC~ to p~JU4 in 148h 7 CIII,

",,' flii """'ii

IS~AUc): , ....: Ull:U1NN.NUIS 01' UILl!:ll:K UATItIWATJCS

If one now rc1lects on the subtle nuances of Plato's portrait, the following interpretation of the mathematical part of the TheaeJelll8 hecomes more plausible. I need not emphasize once more that in reality Plato's Theaotetus did not discover anything new. NonetheleBS Plato makes the youth speak u.s though he were discuBSing bis 'own discovery'. In fu.ct, Thell.etetus seems to be of the opinion that he made a new scientific ditlcovery hy his own efforts, with at most some assistance from 'young Socru.tes'. Indeed he does not even give an account of the fact that the very concepts (square number, rectangular number, mdei syrnme/ros and dynamei symmelr08) which he uses to describe his alleged new discovery, just relJl"Csent traditional, old ideas. But the eUBe with whioh the gifted youth deceives himself, indicntes that 'J~heodorus soored a great pedagogic SUCCC88. Like evel'y true mathematics teacher, Tbeodorus probably wanted his students to learn to think for themselves. Sinco he was a skillful teacher, he seclllS to have prepared them well for this. Doth his students succeedcd brilliantly in giving the answer which must have been expected of thcm. On the other hand, in ten.ehing his students to think for thelllselvcs, he 1~lso seems to IULVe been very tactful. He u.sked no direllt 111I~tltionsj nor did ho just tell them to find a llI~me for the d!J1u1 meis ~vlllch he had shown thorn - a name which reflocted what they h,ut III co~nmon. Such a demand wu.s not necessary, for bu.sod on pn.:oJt, c.xpertence. ~e could rely on his studenbi to gueBS his unspoken intenlions on their own. It wu.s more important to him that the nctual llcdagogic strategy ronULincd unnoticed. He succeeded so well in this that it remained a mystery as to why he stopped his demonstratio~ I~t, the dynamis 17. All tlmt ~rheaetetns Hays is, "For somo reMon ho tltoJll'cd l~t this dynamis". Clearly the teu.cher unobtrusively fell silent ILl, the "l'l'rol'rilLte timo 118 if unnble to continue the train of thought UII.I roplucu dlTFd/le~ hy dUPUII" in 148B1; thon B comploto pBrnphrosu ofthu 1'''"lIlIgo \vollllllN,: Batll IU'V y(!tI/l/lal Tdv ladnAw(!ov Itol bcbrWOII d(!I{)111w nr(!tIl· ... ~{'.JI" ... /11)"'" (or all/,/,lrl!u",) wl!uld/uOu, BUill "" J.h, drtf.lOllrJH'J (UI' deIO,I,)" rtr('uywp(Coua.), dUFcipe. (01' aU/IIl"f!OtJ' weuIC1pe{)a) oto. Although this is noL only 1111 Irnlpronl!lmh1u. buL ulllCl u vury olugBnt. (lUmphl'WlO of tho pllll8Ogo I clo nnL think thuL iL IH absulutuly IIllcllllllUry to omulld Llau t..'UtliLIullul tuxt.: WILlaouL ~Iturution, thu moaning of tho tuxt is complutuly cloar and unambiguoWl; ancl II. nllnain8 80 wit.h thu altumtionB.

"AltT 1. Tilt.: KAllI.Y 1I1t1'1'OIlY 01' TUK ,'Ull:OItY O¥ IItllA'l'WNAI.lI

85

any further, he let his students go on to find out lor tlremselves what he had deliberately, carefully and yet inconspicuously prepared for them. Thus they found 'independently' tho solution which their teachor hlul been oxp~cting from them right from the start. Tho abovo interpretation, however, might suggest that Theodorus would subsequently have had to clear up the youths' illusion. I am firmly convinced that nowooays no one would omit to do this, yet it scems to me that Theodorus Wtl.8 a better pedagogue for not doing so. Furthermore, I am not in the least bit surprised hy the fo.ct that Socrates Wll8 so pletl.8ed about the youths' discovery and did not find it necessary to disabuse them. After all, from his point of view, their little self-deception could only have 'been of 8econdanJ importance. In view of what ono might call their independent rediscovery of some interesting mathomatical facts, it would hu.ve been ludicrous and pedantic to inform them that these facts had 1~lready been discovered by others_ So now it does not scem very surprising that the later scholit~t took Theo.etetus' words about 'his own discovery' at face value.

1.12 'l'lIK IlISCOVEItY 011' INCO&UmNSURABILITY

Tho most important historical conclusionH which may be drl~wn from the foregoing interpretation of Plato Cwl be summarized u.s follows: Theodorus of Cyrone did not begin a new chapter in tho history of the Greek theory of irrationals. The text Plreaelelua 147c-148b furnishe8 no support for the viow thnt Theodoru8 or Thenotetus contributed lmmething new to this theory, whothor it be a concept, the name of a concept, a theorem or a proof. As 0. matter of fu.ct the cOllcollt dy,aamia (tho vlLlue of lhe "'lUILn) (If IL rcctlLnglo) which is uscd 110 clutlllLlly in the same text shows that the theory of irrationals must lu~vealrcu.dy boon 'luite well devoloped in pre-Plntonio times. 'fho valuo of tho

HIIIIILI'O of I~ I'mltlLlIglu' '~ILIIIIUL

ho

HUl'nl'l~tc"l fl'lIl11 "'lrtlC/cllti"m""

(tho transformation of a rectangle into a square of the samo arel~). Furtlmrrnoro. lLC(lOrclillJ; t.u Arist.otle, tho (lHHOnCO (If t(,./rtl!1fJuislIIfJ.'I consists in finding u. mOlLn proportionu.J botween two l~rLitrlLry line segmonts. Of course oJl three of these concepts (dynamis, telragonislIIos.

4 \

I

I

86

8:tABO: ru~ BKUIHHIHUII Olf UllUK )lATUIWATICS

l'All'1' I, 'l'UX XAILLY llltiTOItY OM '1'UX '1'UJo;OILY lit' IIUtAl'h)I'IAI,:'

I thinl, that the following point is worth emphasizing from this quotation. Aristotle is fully aware of the fact that incommensurability, by whioh he seems to mean only linear incommensurability (hence his example of tho diagonal and sido of a square), cannot bo a no.ivo concel,t obtained directly by reflection. "Everyono finds it natonishing that thero exists something which oa.nnot bo mcnaured by the smo.llcst common measure ••. ". This astonishment marks the beginning of the scientifio way of thinking; at a naive pro-scientific stage, one would probably not even bo awo.ro of tho p088ibility that two straight lines could bo linearly incommensurablo. Also a naive way of thinking, which is concerned first and foremost with practiooJ matters, does not recognize the existence of incommensurable magnitudes. In practice a 'common mcasure' can ultimately be found even for tho diagonal and side of a square. Ono nced only choose 0. menaure which for all practical purposes is sufficiently smoJl to obscure the incommensurability of the two quantities. Thus incommensurability is a concept whoso origin is tl,eorelical mthcr than }Irilcticill. If Aristotle is right in slloying that "o1&e tvonder8 ••• all/,e incomme1&BUrability of the diagonal and side of a square", then it may also bo conjectured that to begin with, peoplo could not cWmly I\OlI coolly accept tWs newly discovered and surprising fact without further udo. It seems much more likely that they initially made attempts to 1188ign 0. preeise ratio to the lengths of tho two lines in question. There is in fact a pnasage from Plato wWch, amongst other things, seems to bear this out. I am referring to the famous pll88age about tho 'wcdding numbor' in the Republic. (Book VIII, M6cff.) A full interpretation of this mathematiooJly interesting pll88lLge would take us too far afield.eB It suffices here, however, just to draw attention to tho irnportllnt nmthomatical term which makes its first appearance in this pll88age. According to the Iiterature,Ge "Plato calls the number 7, the rational dia!Jo7lai bolonging to tho tlido 5." This USHortion rnul~na that if one 1188igns a longth of five units to the sido of an arbitrary squaro, thon

and finding a mean proportionoJ between two line segments) antedaw Hippocrates of Chios, because the construotion of a mean proportional (Proposition 13, Elemenl8, Book X) must have been known to him.ss As we c:a~ see, a ~toriooJ explanation of the mathematiooJ con(.'Opt of dY~71"~ IS suffiolent to ol>on up new p088ibilitics for reconstruoting the lustorloa.l development of the theory of irrationals. For this reason I will discu88 quite fully the most important and best-known terms which refer to irrationals in Greek mathematics, using the same method which has been demonstrated above for the example of the concept dynalni8.

Tllo moa' o;om,?on !m.i.?f con..:", wb;cb .ro duo t.u '100 cliscovo.y of mathematlooJ Irratlonahty are ooJled in Greek aVPpeT(!CW and aatipItET(!OV (meaaurable and 1wn-mta8urable). As Euclid states in Definition I, Eleme1lls, Doul, X. '''J.'hrule magnitudC8 which are melUitll'cll by tho same measure are said to be commensurable ((/VppETe a pEyi{}q) '~nd those for which thcre is no common mewmre, are said to he tncommen.surable (daVppETea lJl. • • •J." Undoubtedly the concepts commensurable and incommensurable originated witWn Greck mathematics itself. Tho existenco of incommcnsurability could nevor have been discovered outside the scientific and deductive framework of Greek mathematics. As Aristotle pertinently remarked:'?

b.

~

" ... all begin by wondering thatthings should really be na they are .•. Thus one wonders. •• at the incommensurability of the diagonnl and side of a square. ]for at first everyone finds it natonishing that there exists something which cannot be menaurcd by the smallest common mOlLBure. FinlLlly however one comea to a different conelusion ••• provided that olle is informed about the matter. For nothing would surpriso a geometer more than if the diagona.l should Hluldonly bocome COllllllonHurILhlo."

Ie 011 t.hu intorprul.at.ion of t.his J1IlIi8I1So sou A. Ahlvul'8, 'Zabl Wld Klung hoi Pluton' (NocUs Romanae, For8Mungm Uber die KullUl' der AntiJ:B, cd • l·rofulQlor Y. \Viii, No. U). Unrn-8LuLLgW'L 1062. II L. VIUl dor Wuorden, EJ'UXlClumde lViB.llnBcll4/I, p. 206ft.

.. Suu n. 37 BbovtI.

n.

"Jlelaphy.ics A 2.083aI3".

I

l'AUT 1. TUg Jl:AULY BIll'fOllY OF ·.rUJo: TUgOllY Olf Uu\A1'10NALli

lIZAUc): 1'UJl: Ub:UINNINOll Olf UllXKK. MATUXldAT1CS

the ltmgth of the diagonal of that 8fJUUorO is approximately soven units. It is easy to 800 this by using Pythagoras' thoorem. Ha repl"08Onts tho side of the square and d repl"08enta its diagonal, then the theorem IItalcs that tP = 2a2• So for a = 5, rP = 50 and d = V50 1">4 7. This explains Plato's terminology. For the moment, however, we are only intel"08ted in the expression 'rational diagonal' itself, not in the above-mentioned fact. In fact the two mathematical terms usod by Plato in this connection (~'rrov and 6et!fJTOV) are usually translated lloB 'rational' and 'irrational'. However, it is worthwhile recalling that ~fJTOV originally meant 'that which can be expressed' and llet!fJTov meant 'that which cannot be CXlll'csscd'. If one tries to explain the origin of the latter expreBBion, the question arises lloB to why the diagonal of a square should be (!I\lled ll(!t!fJTov. It can only be becauso, having 8oB8igned 0. number to the side of 0. given squo.re, one wlUlted to do the same for its diagonal, nnd when it WllB realised that this WlloB not possible, the diagonal in qUCHtion received the name ll(!f!1JTov (that which cannot be expressed). The fact that in this passage Plato explicitly mentions not only 6trJpa(!oc; ll(!(!fJToC; (the unexpressable diagonal), but also 6,apa(!oc; I?fJTlt (the expressable diagonal), shows that ll(!(!fJToV cannot be conIItrued in tlus context lloB meaning something forbidden. Undoubtedly mystieuJ-religious 6ef!fJTa70 were concerned with tllings which should not. be expreBBed lloB well. Nonetheless the diagonal of 0. squo.re wn,B not called lJ.(!f!1JTOV for this reason, but just because 0. number could not be ll.8Bigned to its length (assuming of course that the length of thc sides of the same square had been llBSigned 0. number). The number 7, on the other hand, being approximately equal to the length of the diagonal of a square with side 5, can be deseribed as 6,apBT(!oc; efJn1, 1.110 'oxprclIHII.blo (ratiollnJ) diu.gonlLl', bocll.Ul:IO it is u.otually 0. numbor. This short explanation belies the tradition which views the discovery ILIUI even more so the public discuBBion of mathematical irrationality IL'! 'l:lacrilego'. It seoms that the tru.dition is just a naive lcgend which spmng up later. This discovery WlloB most probably never 0. 'scandal' I tl 1I11l.thomlLticians. Moreover the phrllBe in Plato (Republic, Book VIII, 546c4-5) wllich mentions both concepta (expre8sable and unexpre8sable diagonoJ) in 70

Herodotus 6.83; Xenophon, Bellenica 6,3,4, Euripides, Helm 13,23, etc.

89

one breath, roads as follows: l"aTdv ,uv Oe,{)pwv dxd 6,aJd'r(!mv 6fJTciw neJUC&6o~, 6eoplvrov bdc; l"ddTrov, de(!JjTrov 6d c5voiv. •• • Instead of a translation, we shall content ourselves in this case with an exact paraphrase 'of the text. As we know, the Greek words express the same number (4800) in two different ways, namely lloB 'lOO squares on the e:Epre8sable diagonal of 5, eu.ch suoh square diminished by I' (i.e. 100 X (7'-1) = 100 X 48 = (800) and also lloB 'lOO squares on the unexpre8sable diagono.1 of 5, each suoh square diminished by 2' (i.e. 100 X (50-2) = 100X48 = (800).71 Thus the diagonal of a square whose sides have 0. length of five units can be described both lloB 'unexprcssable' and as 'exprcBBable'. The 'cxpresso.blo' diagonal is 7 in this case. So we can say that in Greek 'exprcsso.ble diagona.l' is 0. locution for the approximate value of the length of the diagonal. Furt.hermore, the length of the unexpreBBable diagonal is writ.ten lloB V50 in modern notation. The Greeks, however, instead of using this notation, preferred to mcasuro the square which could be constructed on the incommensurable straight line in question rather than its length. (We have already discussed this point fully in connection with the Theaetetus.) They then said that this diagonal, when 'measured by its square' (i.e. when 'squared'), was of 50 (square) units. We know from Proclus72 as we)) as from other ancient sources73 that the Pythagoroans developed the method by which onc can generate side and (exllreBBable) diagonal numbers. The pair 5 lLlld 7 mentioned by Plato form the tlurd. term of this infinite sequence of numbers. Of course the sequence can also be viewed as an ancient method of approximating If one calculates the ratio of each diagonal to its side (d : a) ILa a deoimal fmction, then the intorest.ing soquenco obL1Liucd wnds towlLrdll the limit r2~7. Not only are the Pytho.goreans supposed to have discovered side (and diagonal) numbers, but nowadays it is gcnerally agreed (and

V2:

71 Sou n. 68 abovo. A. Ahlvel'8, op. cit.., 11. 12, n. 4: "In tho language of mathematiCll, d{},OI,d, dnd ••• ml!lInt. 8tJUare 01 ...... 11 Proclus Dhulochus, In Plaloni.llem publ. Gomm. (I!:d. W. Kroll, Li(llliae) 1901, H, 23, pp. 24-5. 71 Buo tho BourcOA quoted by van dllr Wuortlon in Erwacl,ende lVi8.en.cha/" pp. 206ft. Cf. E. Stamatca, EuJ:lidou Oeometria, Vul. 2, Athens 1963, pp. 9ft.

7.

uo

8:1.A1I0: l'UII: IIl>:llINNIl\Ult 0.,. UiI":KK UATIUWAT1Cli

rightly 80) that they alac> gave the filllt scientific proof of the incommensurubility of the side and diagono.l of a square.?6 This proof is givcn in the proposition to be found in the ElemenL9, Book X, Appendix 27 (t.he reference here is to Heiberg's edition of the text; the proposition does not appear in Heath's tra.nslation of Euelid, but is summo.rized there in the introduction to Book X). It is ba.sed on a reductio ad abaurdum argument which shows tho.t if the side and dio.gono.l of squo.re are commensurable, then the same number must be both odd and even. But it is worth noting that there is a slight difference in terminology betwcen the text from Plato which has just been quoted and the Pytho.gorean theorem found in Euclid. Plato speaks of the 'unexpressIlhle diagonal' (6,apST(!O, l1efl7JTO') whereas in Euclid the diagonal of 0. square is sa.id to be 'linearly incommensurable' with ita sido (dmiPP£TflO' If1T1, ••• p~)ts,). Of course both these expressions refer to tho same fact which we would describe as the irrationality of V2 (if the side of the square is chosen to have unit length). Nonetheless IL hist.orical conjecture is suggested by tho fact that t.ho Greeks had two ways of describing this phenomenon, o.lthough it must be admitted that there is no further evidence to confirm the conjecture. 'eltt) cOlljoot.uro ill t.hlLt 'unoxpre88ablo diagonal' is tho older of tho t.wo descriptions. First of all tho Greeks attempted to assign numbors to the dio.gono.ls of squares whose sides had o.lso been assigned numlIeni. When t.hoy fOalitwd t.hat this could not bo dono, thoy oX(lroHSed this fact by describing the diagonals concerned as 'unexpresso.ble quantities' (liefl7JTO). Thus thoy were o.lready very close to the creation of completely new mathematico.l concept; but the new concopt tinnily made ita appearance only after the expression 'incommensumhlo' (ddVPPST(!OV) had beon coinod ta deseribo it (more precisely, only after the expression 'incommensurable in length', p~)te, dmSPPST(]OV, had been coined). '!'ho question remains JUt to bow the llroblem of the diagonal of tho square ever arose, and wbether there is 80me way of reconstructing

,. cr. O. Boeker, 'Die Leino vom Ooraden und UngeradeD 1m Deunwn Buch dur Euklidiechen E1emenw', Quellsn und 8tudisn air Ouohidate dar Math. et.o. B. 3 (1936), pp. 633-63 (Reprinted in Zur OflBchichte dar grlechiachm Mathe· matik. ed. O. Beoker, Darmstadt (1966).)

l'Altl' 1.

'l'U~

I>AIU.\ 111:.1<111\ "I' 1111\ ...................... .

at least. in part. tbo procoss of matbematical thought. wl~ch led to the discovery of incommensurability. At first t.hia sou.nds lako ,an odd question. On the one hand. it cannot. be answered WIth certamty on the hasis of the facta which bavo been handed down to us. On the other hand. the problem of t.he diagono.l of the square seems to bo so simple as to make one believo at first sight that one would eventuo.lly arrive at it almost by oneself. So t.here scems to bo no se~se in sea.rching out further historical connections when tbe matter b?mg investigated is 80 simple and stmightforwa.rd. Neve~hel~. ~ beheve that the following conject.ures ca.n claim some h1Storlco.l mterest, even though they are just conjectures.

1.13 THB rRODLBM Oli' DOUDLING THB SQUAUB

;

1 i

My conject.ure is t.hat tbe problem of tbe diagonal of the square &.rOso in connection wit.h doubling the square. 1 have two pioccs of evidonco in favor of this view. Firstl~ ~t. is known that tbe doubling l)roblem occupicd Greek mathomatlClo.ns over 0. long poriod of timo. To substantiate this ~laim, it. suffices ,to note here t.ho imaginativo proposal ma.do by Hlppoorates of CblOS for atto.cking the famous "Dolian problem" (t.ho doubling of t.ho cubo).1a Alt.hough it is true that be was not able to carry through his own propOlIU.I. nonet.helCHH snon after him ArchytM of Taro.s fou~d the first 8Olution to this problem. a solution which ma.de UIW of HIppocrntcs' proposal. Furtbermore, after Archytas tbere were even more mathematicians who concerned themselvcs with this same problem and found other solutions to it whic~ did not depe~l~ ,on Archyto.s' work. Thus it may be conjectured. With some p~auslbdlty that in an earlier time (I mean of courso a tlmo before Hippocrates of Chios) doubling the square was just as interesting 0. probl~m to mathematicians o.s doubling tho cube became at a later st.age m tbe development of mathematics. One just has to think about this possibilit.y to realise immcdio.tely that there is 0. famous literary po.ssn.ge which shows. amongst other tbings, exactly how the problom of

,I !

"cr.

O. Becker, DaB mathematiBc/1ll Denken der AnHkfl. OUttingon 1!l57, pp. 76 fl.

'!

U2

IIZAUc); TU~ llXUIHHIHUIi Off OIl~K MATUIWATICH

doubling the Hquare WWl treated in early Greek mathematics instruction.n This brings me to my socond piece of evidence. AH we know, there is a pusao.ge in Plato's Meno where Socrates ILsks a simple, uneducated slave to tell him how the area of a square whose sides are two feet long, can be doubled without altering its shape. So &8 not to be misunderBtood, Socra.tes immediately draws tho square whose sides are each supposed to represent a length of two feet. (Thus the square which ho draws consists of four smaller squares.) Ho then asks how a square with twice the area co.n be constructed. Since Soorates explicitly mentioned that the sid08 of tho originnl square are two feet long, tho slave's first idea is that the required s(lulI.re mUHt havo sides which are twice this length, i.e. which are fOllr feet long. Soorates' response is to draw this new square, byextending one sido of the original squILre to twice its length and then conIltructing II. square on this new side. It can be seen immediately from his drawing that tho area of tho original square has not been doubled, hut quadrupled. The slave inovitably realises that his first attempt ILt an answer to Socrates' question was incorreot. If tho length of tho Rilles is double(l, the area obtILinod is four, not twice tho originILl. Tho slave now thinks l\8 follows: a square whose area is twice thlLt of tho original will clearly Imvo longer sides as woll, so the Hillos of tho squaro hoing Hought, will havo to bo longer tllan two feel. 1lowever, they cannot be four feet long, because a square with four feet long sides has an area which is four times thlLt of the origilllLl. 'rhus tbo required length must be IU8 than four feet. Now three lies hetween two and four, so perhaps a square which has sides three feet ill length will have twice the area of the original square. Orice again Socrates responds with a diagrum. Ho extends ono side of the original Hqllll.re by a single unit and constructs a larger square on this now side. Again it is obvious from the diagram that the aroa has not been doubled, for tho now square consists of nine smaller oncs, wherell.8 it Hhould only consist of eight. Thus tho slave is forced to admit that yet again bis answer is falso. "0. Buc\(ur (Gntndlagen der Mathematik in !1fl8cl&ichllicher EnlwiclduRf1, Frol· IlIIrl;-l\funich 1064, p. 100) corructJy dWlcrlbod this pBIIIIIIgu from thu ./Ihno, WI "a living portrait of 11 lesson in elumuntary geometry, 88 it. would have boon l.I\u,;ht at tlmt timo".

PART I. '1'1I~ .KAlI.LY UlIrrOUl{ Olr TUK TIIXOlLY Off IIUlA·rIOHA1.8

93

Tho problem is finally resolved when Socrates draws 0. new di~~m and shows that if the sides are doubled in length, then the orlgmal squo.re co.n be quadrupled. (This is the construction which the slave first wo.nted to carry out.) Now, however, the diagonals of theso four new squares also form a square (see Fig. 4) whose area is made up of four equol triangles. Since tho original square is only mado up of two such triangles, Socrates has succeeded in drawing 0. square whose area is twico that of tho original, by moans of the diagonals.

Z'ttt

Fig. 4

'J~IIO l'UBSU.ge jUHt summarized is OIll'CCilLlly instructive not only because it shows that the way in which 0. naive and uneducated person attempted to solve this problem WIUI first of all to givo a numerical value to the sido of the roquirc1l8(lUILI'6. hut ILIHO becauHe of some vcry informativo remarks made by SocrnteH. After the slave's second unsucccll8ful attempt, Socrates says "Now then, try and tell us exactly how long the required sides will bo, and if you do not uxmt 10 exprU8 thi8 as a number, then just show us." (83e n£,eiiJ IJl'iv dntiv dXf!,{J~' xoi d I'~ (Jov).e, D.e'{}I'e.i", dlla 6tiEov.) Theso words are n. quiet hint to the informed reader tho.t the length of the Hides of tho squaro with twice tho original area (i.o. tho longth oCtho diagonal of the original square) cannot be given Ilo numerical vo.luo, because it is an /ief!71TOV. Without 0. doubt the problem of doubling tho square - tho attempt to find tho number which, when ll.88igned to the length of the sides, would yield 0. square with twice the n.rea - led to the problem of the diagonal o/the 8quare o.nd thence to the problem of linear incommensurlLbility. Indeed ono would almost lilto to llostulate 0. direct connection between the mo.thematical problom in the M eno, o.nd tho Pythagorean proof that tho side and diagonal of a square are incom-

1)4

IIZAIlc): 1'UII: II"UIHNINUII 01" UItKJo:K AlAl'UJWATICS

l'AllT 1. TUK KAILLV U1S'I'OILV Olf TUK TllKOIL\' UI' IlIllATIOHAJ.lI

lIIolUmru.ble which is fuund in Euclid (or mt.hor in tho I'ppondix to J~uclid cited above). On tho one hand, it is indicated in tho Meno that

once ~ numerical value has been assigned to the side of a square, then the diagonal of that square cannot be expressed o.s a. number. On the other hand, the Pythagorean proof is purely number·theoretie and ta\<es t.he form: if the side a.nd diagonal were commensurable, then the same number would have to be both even and odd, which is irnposllible. I believe that the evidence just quoted (the long.standing interest in tho problem of doubling the cubo and the po.ssage from the MenD) goes some way towards confirming my conjeoture about tbe discovery of linear incommensurability. If we accept tbat tbe Greeks discovered ~ho li~ear incommensurability of tho sido and diagonal of tho square lJl tholr attempts to double the square, then there is 0. sonae in which tho usual interpretations of this discovery will have to be revised. Up to now the historical significance of tho discovory of mathomntical irru.tionality WM interproted in tho following way. Tho earliest Pythagoreu.ns could bo said to have "worshipped numbors".78 For them, numbers wore "t.ho roele bottom of tho ontiro universe", the world was formed "by imitation of numbers" and the hOlW?IlS were "harmony and number". According to Aristotle (]Ifeta. pl'YSICS A5), they arrived at this view because they were so much occupied with mn.thomn.tics. The old teaching, however, is supposcd tu hnvo bcon s~II'I(6n by tho discovory tlu,t tho side and diagonn.l of II. squu.re were Incommonsurable. The diagonal of a square having no common mOMure with its sides meant that if the length of the sidcs is tu.ken as the unit, thon the length oCthe diagonal cannot be measured i.e. it cannot be expressed either as a whole number or as a fraction: Thus something which wo.s "not a number" but just an 11ef!1JTOV (something which could not be expressed) had been discovered. It is "Jleged that this disconcorting discovery led to what used to be called a crisis in lite foundations 0/ Greek matl,ematica.7e 11

7e

cr. B. L. van dar Waorclen, Erwacllmde lJ'is8enschajt, p. 204. cr. H. HIUIIIO and H. 80hol7., Die GntftdlaumJ.:rim der griecAilIcAm Mallie.

'1

~

":

'""

I

I,

95

Alt.hough tho nowor toxtbooks hardly mention this 'foundlLtional crisis' any more, the historical interpretation based on such 0. 'crillis' (or at least on 0. 'critical turning· point') still survives, albeit in 0. dis· guised form. Van der Waerden, for example, writes ~hat the Gre~ks "turned away from numbers" in the course of theIr mathematical development, just because of the discovery of incommensurability, and that subsequently a geometrization or geometric 'formulation' of mathematics took place.80 In his own words:

t,

'I ~;

In the domain of numbers, the equation x'l. = 2 ea.nnot be solved, not even in tha.t of ratios of numbers. Dut it is solva.ble in the domain of segments; indeed the diagonal of the unit square is the solution. Consequently, in order to obtain exact solutions of quad· ratic equations, we havo to plL88 from the domain of numbers to that of geometric magnitudes. Thero aro two points in the interpretation sketched above with whieh I disagree. li'irstly I do not believo that the new discovery required a revision of tho anciont Pythagorean doctrino ('ovorything is number'). Secondly I cannot accept the conjecture that the dis· covcry of irmtionlLlity led tho Greeks to favor geometry at the expense of arithmetic which up to then had been esteemed more highly. (In principlo, arithmetio was always prized more highly than geom· otry by tho Greeks; of. Chl'ptor 3.27 of this book.) Let us first considor the extent to which the ancient l>ythagorean doctrine 'everything is number' was shaken by the incommensurable. Even if the side of the square is ehoson as the unit of length, it is misleading to say that the diagonal is really 'not 0. number'. According to the Meno, tho Greeks originally sought the line segment which would enablo them to double the area of 0. given square (whose sides had been assigned a. numerical value). Thus the problem which led to the discovery of incommensurability was a geometrical one. They could only establish that the line segment being sought was 'not meo.surable in lengll,' (i.e. that its length was 'not 0. number'), after they had realised that it wo.s the diagonal of the original square. Since,

"''''ik,

l'un.Dllohtll'tll, Borlin 1928. For an oJlJlOlllng viow, 80tI B. L. van dor Waurden, 'Zenon und die Orundlagenkrisa dor griuchischen Mathematik', Malh. Ann, 117 (1940) pp. 141-61.

10 On this toplo soo the chapter entit.led 'Why the geometrio formulation t' In Enom:llcwlll Willsmsr./IlII', pp. 204-6.

DIl

PAnT 1. THK KAu.LY lIlBTOll.V OM TlIK TUKOJl.!{ OM lJUtATIONALS

97

HUlIe): 'l'UJ!: lIKUINNJNOIl OJ!' tutJ!:KK KATHJWATIOS

however, this line yielded a squll.re of twice the originul area, the ideo. of measuring linearly incommensurable segments by their 8fJ114ru mus~ ~ave suggested itself immediately. This is the reason why Plato (POliltCU8 266b3) ou.lls the diagonal of the unit square dvvapB' dlnov, ("two feet when measured by the square constructed on it"). , We co.uld 8&y that the diagonal of a square was recognized to be Imearly mcommensurable ae the 8ame time a.s it wa.s recognized to be commensurable in square. Hence it is unlikely that this magnitude was ever thought of a.s "ab80lutely indeterminate from a numerical point of view". The diagonal of a square can be given 0. numerical value (when measured by its square). This ca.se is similar to the introduction of the new mathematica.l concept of dynami8. The value of the square of 0. given rectangle whose sides were not 'similar plane numbers', could be given 0. precise numerical vruue a.s an area even though tho sides of the square concerned wore incommensurabl~ with the u.nit of length. PhU8 tile Pyi/lagorean doctrine 'everything i8 number' ttY'S tn no way shaken by the diBcovery 01 linear incommensurability. Although ~he length, which could not be 8.8IIigned 0. numericru value, ~VILS d~scrlbed at first as an l1(!etJTov, the initial surprise gave way IIllnlcdlately to the reruisation that the lino whose length could not ho given 0. numerical value should be measured by il8 square. . Si~ilar~y I disagree with those who maintain that the discovery of JrmtJonahty led tho Grooks to "turn away from numbers" or "to ptlSS from the doma.in of numbers to that of geometric magnitudes". It is of courso true that some earlier mathematics must have been reformulated. after the existence of incommensurability was recognized. (In partIcular the notion of 'ratio',ldyo,", which until then had applied only to numbers, would have had to bo redefincd so as to inolude incommensurable mugnitudes.) nut the interpretation outlined abovo ~the Greeks turned away from numbers, etc.} would only be wammted If one could prove that tho non-geometric problem of the irrationruity of V2occurred to the Greeks before the geometric solution to this samo I'l"Oblem made its appearance. As fu.r ns I ClLn seo howovor such 1m itlOl\ is not su})portod by auy sources. On the oont~o.ry, all ~ur Greok sources speak unequivocally in favor of tho view that doubling tbo H~IIRro Rnd tho mellsurn.hility or nc\n-mcusurILhility of tho diagoOlLI of 1\ sCJuare wero authentic geometrical problems right from the start, and that the problem of V2 is just n. modern equivalent of these prob-

lems which hILS been removed from thooriginru, purely geometriC framework. In my opinion, therefore, there is no justification (at lelLStin this case) for speaking about a subsequent 'geometrica.l formulation'.81

1.14

DOUBLING TIlE SQUARE AND THE MEAN PROPORTIONAL

My previous conjecture that in all likelihood the problem of doubling the 8fJ114re led to the problem of the diagonal 01 the 8rJua.re, enabl~ us to discern some othor historicruly interesting connectiOns. We lust havo to realize that in the minds of the ancients, the problom of doubling the square was identica.l to another problem. This can be seen from the following. In his historical trea.tment of the problem of doubling tbe cube, Becker remarks,a "In this form, the ta.sk seemed intractable to ancient mathematicians. Therefore Hippocrates of Chios transformed the problem of doubling the cube into another problem, namely to find two mean proporlionals x and y between a (tho edge of the cube) and 2a." It ca.n oU8ily bo soen that Hippocrates' prop080.1 does load to a solution of the problem. For if a : x = x : y = y: 2a then

ay

=:r-

and

xy =

2

24

bence

i.e. Now the question is how a Greek mathematioian of the 5th century chanced upon tho happy idea. that the problem of doubling the cube could be reduced to the problem of 'two mean proportionrus', even though he wa.s not in 0. position to carry out his proposaJ. My opinion is that Hippocrates just used an argument by analor!Y to obtain his pl'oposal. He wanted to troat an unsolved problom 111 solid geometry after the manner of a problem in plane geometry, I. I do not. wunt. t.u go iuto t.hu prttblmn of t.hu IIII·mallnel 'goomuLrio algubna'

of the Greeks in this book n Cf. n. 76 abovo.

7slUlb4

(BOO

howovor the appendix).

U8

II~AUO: 1'IIK U~(JIHHIHtlll 01' (J1t""K 1JA'IIlKUATIClt

which had been solved 0. long time before. He may have started out with tho ideo. that 'square' and 'oube' are, in 0. SORBO, like figures. Furthermore, 0. square oan be doubled by finding one mean proportional between its side (a) and twice this side (20), i.e. by finding an x such that a : x = x : 20. Honce one will be able to double 0. culm by finding tw~ mean prolJOrtionals betweon its edge and twice this edge. This argument of Hippocmtes 1en.dH me to conjecture that tho ancients considered doubling a square and finding a mean proportional between a number and twice tlmt number to be synonymous. Theso two prohlems are not just tho su.me in essence, hut are vory clos61y related to each other from the historical point of view 11.8 well. I think this conjecturo is worth noting for the following reo.son. Aristotle's o.ssertion to tho effect that the essence of telragoni8mos (tho transformation of 0. reotanglo into a square of the sarno area) conaiswd in finding a mean prollortioDlLl (betweon two arbitrury lino scgments) ho.s already been quoted in an earlier challter. Furthermore, it WlUI indicated thoro that tho problom of telragoni81nos may itself havo led to the now mathemlLtical concept of dynamis and honce to tho recognition of linonr incommensurability. 1.'he ideo. that tho problem of 'doubling tl,e 8quare', or 'finding a mean proportional between tl 1w/IIbcr (111(1 tlMee tlmt number' mo.y ho.vo occo.sionod tho discovery (If incommcnsul'ability fits in very well with these oarlicr conjoctures. In the Ill,st analysis, 'doubling the squo.ro' is nothing but a specilLI cl\Se of tctragonisnl08. In both 01\SC8 one ho.s to find 0. mean proportionlLl bctwccn two numbers (or magnitudes). The only difference is that in the first co.se a mean llroportiono.1 botween a number (mo.gnitudo) and twice that number is sought, whereo.s telrago"ism08 requires ono to find 0. mean proportional between any two numbers. This insight suggests tho conjecture that the discovery of lincar incommensurability, 11.8 well 11.8 the realization that linearly incommenHumhlo straight lines woro commonliurable in squo.ro, wo.s brought about by attempts to find 0. mean proportional between two straight lincK (mIL,.;nitudes).1:ho problom of tho 'monn proportional' wo.s very impurtant in ellorly Greok mo.thomatics. Thus it o.plleo.1'9 that tho problem of irro.tionulity originated in tho theory of proportions. I·'or LhiK I'tllLKUn I will ILttoml't to CliHOUHH tho ou.rly history of tho Greek theory of proportions in somewhat greater detail. Such 0. discussion is contained in Part 2 of this boole.

2. THE PRE-EUCLIDEAN THEOUY

OF PROPORTIONS

2.1

INTRODUCTION

To tho extent that it ho.s been clarified by previous research, tho early history of the theory of proportions found in Euclid can be summarized roughly u.s follows: ; Euclid begins his discussion of the theory in Book V of tho Elements. 'fho fundamentul definition of this book is Definition 5: "Magnitudes nrc slLid to be in the sarno ratio (b T(P mrrtp ).Oycp), the first to the second 1\8 the third to tho fourth, if any equal multiples of the fimt n,nd third lLre alike, greater than, equal to or less than o.ny equal Illultiples of the second and fourth taken in tho corresponding order." From this definition it follows that tho equality a : b = c : d holds, if nnd only if for any whole numbers m ILnd n, 100

> nb

1M

= nb

ma

<

and

implies me implies me

nb implies

1M

> nd

= nd <

nd.

This ingenious definition, whose completeness ho.s often been 0. source of o.stonishment,l is usuully attributed to Eudoxu8, 0. younger contemporary of Plo.to. According to a scholiuml which is unfortunately of unknown authorship, the wholo of Book V is essontio.lly the work of Eudoxus. 3 At all events, the definition could only have been formullLLcd at IL timo when incommonsurable magnitudes woro lmown to exist and an effort was being mado to ensure that tho theory of proportions, which up to then had only boon applied to numbers, could bo uxtmulud to incuUllIlonHurlLltlu mll,gnitucluK. , - ! , 1

J !

cr. n. •cr.

1 J•• van dllr Waerdon, Scimu Awakening (Oxford Univol'8it.y Prc811 1061),1'1" 176-0. Euclides, ElemenlB (ad. J. L. lloibclrg, Lipsiuo 181111-1916), Vol. V, p. 2110. IT. J•. Heat.h, Euclid'lJ Elements, Vol. 2, p, 112.

7· ,

)

, 100

SZAUc): TUK Ua,:UINHUW8 011 UllKRK alAl'IUUIA1'ICS

This immediately suggests the question 88 to whu.t definition W88 used be~oro tho discovery ?f incommensurability, when the thcory of proportIOns W88 onl~ applied to numbers. This question could easily bo a~wered by reforrmg to another dofinition of proportionality which wMBlmplortho.n tho one mentioned abovo and applicu.blo only to numbers, namely Elemenla Book VII, Definition 21: "Numbers stand in tho samo ratio' when tho first is the same multiple, the same part, or the same numbor of I,arts, of tho socond 11.8 tho third is of tho fourth." ~o these two definitions could be said to mark two different epochs i~ the historical developmont of the theory of proportions. The more highly developed theory whieh could handle incommonsurable magnitudes 88 well 88 numbel'8 W88 made possible by Eudoxus' definition (V.5) and this definition must undoubtedly have been of later origin than the other ono (VII.21). This simple chronology may indeed. seem illuminating at first glance, but the historical problems which it raises and leaves unsolved c~nnot bo concealed. As lI.n example, let us consider Euclid's propositIon VI.13 - the construction of 0. mean proportional betwoon two o.~bitmry line segments. A full proof of this proposition can only be glve~ ?'& the. b~8i8 0/ Eudo:cwJ' definition 0/ proportionality. Tho other defirutlon fo.ds III this oase, because it co.n only be applied to numbel'8, and the longth of the moan proportional botween two line segments whose lengths cannot bo written Il8 similu.r plano numbers (of. Definition VII. 22) is itself an incommensurable magnitude. TI~us one is led to conjocture that the construction of a. mean proportional betwoon t'"':o o.rbitru.ry lino sogmonts only became pOllHiblo after E~doxus had laId down his definition. This conjecture, howover, (~()ntr8.(hc~ tho well-documented historical fo.ct that the geomotrical construct.lOn of 0. moan proportional WII.8 familiar to the Pythagorel1.ns bofore tho timo of Eudoxus, and must even have been known to Hippocrates of Chios.' Tho question is how this construction could havo boen provod at that time. Any answer to suoh 0. question must nocc:sso.rily bo conject.ural. In the presont CllSe, the following threo (~UIIJoctUros SUGGC8t thellll!l)lv~:

-,

,-

II:

';-! !' -j

t

I,

I ~

-

.

"

,-,--

0;'

-

.

'. :

,

.

.~

~_

TUE "UK-KUCI.JIJKAH TUHOItY 010'

"ltol'OJlTlOHII

101

(I) Originally, before tho timo of Hippocrates of Chios, 0. full proof of Proposition VI.13 could not be given. At this time tho 'intuitive validity' of the construction had to suffice instead of 0. proof. Even if tho construction of the mean proportional were in fact older thu.n the proof, the plausibility of this conject.ure is lessened by the fo.ct that tho sUlndards of rigour domanded of proofs in 5th century Greek mathematics were alreooy very high.' In my opinion it is almost unthinleablo tlmt Hippocrates, for oxample, would have been satisfied with 0. so-cu.llod 'naivo eonoo}Jt of proportion'. (2) Another possibility is to conjecture that Definition V.5 is in reality much older than hM been believed. It should not bo forgotten that there is no ancient sourco which o.ctually credits Eudoxus with this definition. It has only been attributed to him because of the abovo-mentioned scholium which ll88erts that tho substanoo of Euclid's fifth book is tho work of Eudoxus. (3) Finally, one could imagine that there was 0. pre-Eudoxian definition of proportionality which Euclid fails to mention. This definition allowed 0. preciso treatment of irrational IlS well IlS rational proport.ions, and mado possible 0. full proof of the geomet.ric construction of the mean proportional. In the 1118t few docades scholl1.1'8 bolieved that they had in fu.et rediscovered just such 0. definition. Aristotlo mcntions somewhere (Topica VIJJ.:l, p. 15RL 29-35) 1m intero.'iting definition of sameness of mtio (thu so-cu.llcd I&nl/'ypllairetic or anlanairetic definition) which runs IlS follows: "ThOle magniludu are in the same ralio wl,ose aJul'gpllaireais is Il,e same." Tho oarlior interpretation of this pu.ssago WII8 thl1.t Aristotle is pointing out the difficulties which, in the o.bsonco of a suitable definition, the discovery of incommensurability raised fur rnllthollllLticlll prollf. 'rllO nnthyphairetic dofinition iH HUPI'OHct! to havo holped in overcoming these difficulties.' I showed some years agos that this interpretation of Aristotle's text (Topica VIII.3, p. 158b 29-35) is completely misto.Jeon. Aristotlo

I:

. i

.I

: On t.hbs t.rwudut.ion of t.ho tlXI'NSllioll dN.wyCW, 8UU Pt». 148-67ff. ~ thll papor by VWl dar W88rden in Zur GucJaichle thr griecJavcJaen MalIza. tnoliJ:, p. 226, n. 28.

'

"AU1'

',-i

• JIJill. pp. 216-6 . 'Cf. O. Beoker, 'Eudoxos-St.udion I' in Qtullkn und Studien zur Gaac/.icJlla dar Mollumllllik, 01.0. n, i (1933), p. 311-33. • A. S7.ab6, 'Ein Bulug mr diu vOnludoxillOho l'roporl.ionon.Luhnlf' Arclliv lilt' Begri8sfJucJlichle (Bonn) 9 (1964), pp. 161-71.

1u:.!

II:1:AUc); 'l'U"

UI>UIHNUWtl

OJ!' IHlKJ!:K laIATHJo:UATICtI

"AhT ~.l·JI~

~Ii ccr.tainly not. t.u.lking about t.ho difficult.ies of mathematical proof I~l t.~us pll88age, nor docs he intend to asoribe any suoh historical slgmficance to the anthyphairet.io definition as the modern interpretillion at.tril.IUt.es to it. Not.withstanding my objeotions, the fact remains that Arbtotle docs /--- quote t.his definition. The only question is whether the Greeks them/ solves ~id ~own tile definition in tllis form with iJU:Ommen8Urable magni( tlld~ In (as Beoker and, following him, some other soholars \. heheved), or whether they applied it only to commensurable magnitude8 (us Reidemoister conjeotured).10 As far as I can see, a deoision betwcen t.hese alternatives cannot be made with any certainty on the basis o~ t.he .argumen~ u.d~anced up to now. (Of course I hope that the thsCuSSlOn cont.amed m the ohapters which follow will cast some ncw light. on. this. whole qucl:ltion. My only reason for not returning later to t.ho hmtorlcal problem which is concerned with the different definitions of proportionality is that it is not central to tho subject of my invcstigations.) Thus it can bo seon that up to now rOl:lolLroh into tho histol'illlLl d~velopm~nt of the theory of pl'Oportions has been concerned mainly With the mfluence which tho discovery of incommensurability hnd ~.II tho flll·ther.devolopment of the theory. Hence there wll81css interOl:lt 10 the early lustory of proportions. Only passing reference was mado to tho fa~t th~t the terms which Euclid uscs for this theory revclLl [ IL Ctm_ll(}ctIO'!~~.!~!!!~~thcory of m u s i c . l I - - I would now like to efucidat6tliedovolopment of the pre-Euclidean theory of proportions by giving 11. history of these terms. In so dOing, J Khall draw extensively on the results of my latest publication on t.his subject. 12 ____ .__ _

:nnd

~

• cr.

O. Ducknr, Do, Mall,etnalUlcl,e DenI:en der Anlil:a (OUttingon 1957), p, 103, n. 26. '0 K. Ituiclullluiator, Dcu NzaI.:la Denken der Oriedum llumbu.... 1949 ,,,, II S Ii ' . ., , I)· "-• . eo, or exwnple, the paper 'Die dexoi in dar griuchischon Mathomatik' Ardnv JUr RtJuri88(JBlcliichle (Bonn) 1 (1055) pp. 13-103 • IA S ...... , , • . , • zu",:", Diu rrtlhgrlochlaohu Pmportlonun-l.ahru imSpiogullhrerTormlno_ 10gIO • Arc/llve lor lIi,tory olltJrEac' Science, 2 (1006), pp. 197-270.

G

.·n . .:-,l3;lJl:J•.l., ... A,\

JU&s.U",

va.

to .... , . . . . . . . . . . . . . . . .

2.2 A SURVEY OF TUE 1IOS1.' IMPORTAN'l' TEIUlS

It would be wrong to believe that a history of the tenns whioh were used in tho mathematical theory of proportions will be able to throw light on the very beginnings of this theory. Such II. history has much more limited objectives in this case. So as to' bo clear from the outset about what may realistically be expected from an investigation of this sort, let us recallj,he kind of things which we were n.blo to learn about the theory of irrationa.ls by investigating the history of a mathematical term> --~~---.-. It was shown in Part 1 of this book that the following facts about a mo.themn.tical term like dynamei BtJ1nmetros (measurable with respect to the square const~ucted on it) can be established with considerablo certainty. (Tho term is introduced in Definition X.2 of the Elements.) /'The exprcsSion dynamis was tatum over into mathomn.tics from tho L1anb'llage of fino.n~. Just as when converting one currency into another, 'having tbo so.mo vn.lue' was expressed by tho verb dYJUlstitai or by t.ho nOlln dynamis, 80 tho 'squlLro having tho same arelL 118 11. given I·ect.anglo' was {JeBol'ibcd by thcso Bame t.wo words in gcomot.ry. Thus the word dynamis came to have tho special meaning of 'the vlLlue of t.ho squlLre of a rectangle' or'squaro' in tho t.cchnico.llu.nguago of geometry. However, since the transformlLtion of such rectangles with sides which wcre exprcsscd as numbers often resulted in squares whoso sides were not mcasural~te in length, it was desired to mell8ure these same sides by their sqU\Lfes and not by tho lengt/llt. '1'hl\.t is tho origin of the expression dynamei aymmelros. In this ease, therefore, tho hitd,ory of 110 term could bo said to hu.vo clucicllLte
l-.

-··-tl

It

l'.-H. I\liohul, De l'y"llIf/ure d I';uclide, PIUiIl 19GO, pp. 305ft.

IU4

II:LAUc); l'UI> llJ,;UINNINOS 011' OIlIl:KK MATUIWATICII

1

It woul~ 00 a hopeless undortaking, in my opinion, to attempt an oXlllanatlon based on tcrminologicul considerations of how the idea of proI)ortion actuaJly came into being, espeeiaJly since this idea is clcarly a very ancient one whoso beginnings can bo traced back to the most primitive human thought.

S? the P~?t investigations will not concern themselves with the earliest ~gtnrungs of a mathematical theory of proportions. They are ?nly tn~nd~ to throw light on the origins of the tbeory of prop~~lOns w}u~h 18 presented in the Elemenl8 and which undoubtedly orlgma~ wltb the Greeks. It is my view that in all likelihood this pre-Euclidean proceBB can be aceurately reconstruoted by investigating th.ose e~re88ions wbich wero lator adopted permanently into Greek sOlenco/_Let me therefo~tart out by listing)be moat important of thollO terms whose origins aro to be investiga.ted more fully in the sequel.

•

t

~~c!id~s

oustomary way of referring to 'ratio' and 'sameness of ratIO IS Illustrated by tt~e following po.ssage: Elemenla VII.ll: "II a to/lole (say A = a b) y to another wlwle (say B = c d) cu the subtracted parI 01 lhe one i8 to the subtracted part 01 the other (A : B .. = a : c) -ldv ~ w~ 8Ao, n(lo, 8Aov, oihCIJ, dlpat{!d)el, ned, dtpat(ltOtvra II,en tI,e ren.winder 01 tlte one i8 also 10 the remainder 01 the other cu the one toltole y 10 tlte other (6 : d = A : B) -Halo 1omd, ned, Tdv 1omo" lUTa w~ 8).0, ned, 8)'ov. "As we can sco, Euclid just U8CS tho preposition ,~: (1r(!O,) to express 'ratio' (8).o~ ned,8J.ov and ).omd, ned, ).omo,,} whereas ho .U808 a so-called adverbhLI clause of comparison •• for '8a;"enes8 01 rat:o' (w~: .. oiJ~w~ •• •). l'hill is 0. very common way of exprelllling thelle notIOns. It IS a]so to bo found in an archaio-sounding fragment of Archytas, where ho disollll808 the so-called geometrio mean_u

+

+

---.----

"UIJJlETetHQ

M, 61tHa lCIJVTt 010, 0

1r(lCOTO, (or 6eo,) nOTI Tdv clM£(IOV

Hal 0 droT£eo, nOTI Tdv Te/TOv.

'

I

; I

/,

A geometrio mean is present, when the first term is to the Becond as the seoond is to the third.

II R. Killmer and B. Gorth, AW/Uhrliche Grammatil.: der grUchi,chenS l vol. 2, p. 400; 4th cdn., Hunover 1966. prac 'UI, .1 n. Diols and lV. Kranz, Ji'ragmonlo (Ier Vor,okraliker vol 1 p 436 (A h tad B!!). ' ',. rc y_

The only grammatical difference between theso two quotations is that in the 8Ccond, B8.mencss of ratio i8 expressed by a so-called adjeotival clause of comparison (olo~ .•• Hal • •• ). This way of expressing 'ratio' and Isameness of ratio' is hardly very illuminating from 0. hiBtorical l)oint of view. The only thing whi~h can be said is tha.t this usc of the preposition ned' (to desoribe 0. ratIO) may well have been a spocio.lized one in mat~tiCs;Thertf'hqll'o1>: ./-o:bTy no other way of explaining how Aristotle WlLB able to change ( the meaning of this mathematical expression BO that it roferred to '--his category of 're]o.tion' {TO neo, n}. .,. , More interesting is the fact that the four-place re]atlon proportion (or 'so,meness of ro,tio', as it is ulluo.lly oalled in this book), which is expressed by the formu]o. a :. b = c :~, ,WDB denoted by. dvaloy~a ~n the terminology of mathematICS. Euchd s only U86 of tillS term IS 111 Definition V.8 of the Elemenl8: "The least proportion consists 01 three tenns" Avaloyla ~ l~ax/~ IUT/,,}. Because this definition is completely redundant DB far lLB Euclid's propositions are COIlcerned, it hlLll ofton been regarded lLII 0. 'later interpolation'.IG Howover, this qucstion need not concern us here. The word d"aAoyla (in tho senso of 'geometric sameneBB of ratio') is undoubtedly an old mathematical term. On one occlLllion Aristotle oxplo.inB the formuhl. a : b = = c : d as 'geometric analogia', in the following words: 17 lAs term a ill to term b, so is term c to term d' (lUTQt Ilea w,o Cl 8eo, neo, To,,~, olJTCIJ, d Y 1r(lO~ TOV &). Theru arc also quotationll from Euclid which show that the word avaloylp is an old mathematical term. Euclid never UKeS solely the words" ~d,_ M"5_ to moan 'the 8Ilm~mtio' (sce for oxaml~!o, Dofinitio...-v.5)i Homet.ill1OH ho lImploYK !LIHO the arciiiLlc"expreiision QVCPO"O" (for 'in tho same ratio'), lLII for eXILmple in Definiti(Jh VII.21 , which WlLB quoted above. . /--1 intend to devoto lIomo Iu.tor olllLJLters to II. thorough inv08tigl~ti()n , / of tho fundament8rexPreMion'~ -~f ti~~·t.lleory of proportions (in < particuhLr lOyo" dJ.dloy01' and dvaloyla). However, there is another ' - mathematioa] expression which is the key to the following historical discussions, and I would like to start out by concentrating on it.

itT. J•. Hoath, Eau:lid', ElerMnt8, Vol. 2, p. 131. n Nirornadlfllln EtJlia, 11:11 hr..

luu

:;~AIIt); 'UII,

~n

IIIo:UII'I h!I'IOS 011 Oltl.:KK MA'l'UIWAT1Cli

Ol'cok a 'term' in a pro}lOrtion (8Il.moncss of ratio) is caUed Beat;. TIlls word occurs only once in the Elements, namely in Definition V.S: -"1.'he least proportion consists 0/ three terms" { ... b Te«11v B(!O,t;}. AglLin,'llowever, there is a great ae-afoI8viaenee-whlchsupportS -the view that this word (Beat; in the senso of 'term of 0. proportion') WIlS ILctually 0. well-known mathematical term. InoidentlLlly the definition Htates that the leo.st proportion hIlS three terms, i.e. a: b = b : c. Aristotle expresses this same foot by saying that everrProportion' has at leaHt lour terms. According to him, even the least analogia (the so-called dva.to"la UVVBmt;) actually has/our terms; it is just that in ~his CaHe the middle term (6) is counted twice. In explaining this,ll Aristotle uses the word oeOt; to denote Ierm8 of a proportion, just 88 Euclid does in the definition mentioned above. There are many other OXlLm}lles which could be given of this same usage of the word Baot; in mathematics. I t is my opinion that a historical explanation of the origins and development of the .£!!l-Euclidean theory of proportions haH to start from the word Beat;_ My reasoning is as follows: Although the term Beat; is used only once hy Euclid to mean the 'term. of a proportion', he uses this same word on other occaHions in two completely different senses: / (I) "Oeo, ill mOllt fl'tlC)Uontly uacd in the 1Illemenl8 to refer to the ! dcfinitionH. This usage undoubtedly comes from the language of phi( IUHIlJlhy, which I will diK(ltIHS more fnlly in PILrt :1 of this hoole. Tho nllO ",~f 0e0t; to moon 'definition' haH nothing to do with the theory of lroportions. (2) Tho other meaning of Oeot; ill more im}lortant to us in this connection. In many 11ll.H8ages Euclid uses tho Greek word Beat; (masculine) in its everydlLY meaning of 'houndllorY' or 'endpoint'. This lIIeaning wus elearly tal(en over into geometry:--For-example, Detinit~on I.l~ of the Elements states that O(!Ot; la-clv, 0 T,,,ot; laTe nie~, winch HOlberg tmDIIIu.tes us "Perminus est, quod aliouius rei extremum ellt". Now I conjecture that this latter meaning of oeo," (terminus or endpoint) was also taken over into the theory of proportions. After ILII. wO still talk ahout the terms of IL proportion. nut tho 80nse in "Ibid, 1131 a31-b5.

l'Au"r ~, 'l'IlK l'l1h,I;lIt'I,IIIKAI'I l'IIt:.)IlY .... 1·lt.)I'·)".... tll\,.

which u. proportion can be slLid to have

~~~....hll.S

.". yot to be

explained. We h.we to ILnswer sueh questions aH why the Greeks spoke of endpoints in connection with proportions and mtio~; why they m~in­ tu.ined that every 'samenCSII of ratio' {dvaAoyla} had four endpo1Ot:s ami ovory 'ratio' (M"ot;) hlLd two; and whother this no~oncla~uro IS significant. I do not boliove that tillS ean he Il.Ccompltshed lust by conHidering the two terms M"ot; and dVaAoyla, for they mako t~e na,mo ! Oeo, seem obscure and meaningless. ~f, however, we bear 10 mmd 1 /that in tho mlLthomatico.l theory of music developed by the Pytha- I goreans, and also in the 8edio canonis which has come do~n to, us aH 0. work of Euclid, the 'ratio of two numbers to each other (winch lu.ter received the nlLme M"ot; in arithmetic and geometry) WaH called . ~,dC1T11lla thon we 0.1'0 well on tho way to answering theso questions, c:::> In th~ theory of musio dUlcmJpa hall two m~ani~gs. On the one hILnd it means 'musical interval' (and also the r.-.tlo botween. numbers' which expressed this interval), and on tho other hand It had itH I cvery
II

Il88igned numbers. , (2) Ry ILlI ILP)llllLt'IL",~, tho 'nLtio of two numbers to each other received the name M"ot; somowhat lalor. A '\&"ot; ilj also Haid t:o hlL~O two Beo, (end points) because originally (in tho theory of musIc) tillS

1""1

H:tAUO; ,.,..; UJ.:UINNINUIi Olf Ult"b:K IIATIIIUIATICli

I'AILT II. TIIK l'ILK·KUC1.WKAH 'rUXOltY Uif 1'lUII'OIlTIUNIi

word m~ant somothing simiJlLr to diaal As in geometry soon came to mean the a.matter?f flLCt, M)'o, of music. me OB dUUJIema In the theory

=,00.

It seems, however, that these names are of relatively late origin.

They presuppose that the strings were numbered consecutively in 0. way whieh was customary in musico.l practice and wruch is still reflected in our names for these intervals (ociave [8], filii. [5] and lourth [4]). However, this was not 80 in the earliest times. Originally the strings were not numbered--;-Jnawad"l!ltCln5ile had ita own name. For oxample, tho topmost string (wluch W8.8 the longest, and produced tho lowest note) was cu.lledWiaT'1 (the highest) and th~ bottommost string (wruch was the shortest, and produced the rug hoot note) was called ~. The remaining strings lay between these two outermost ones; of these, the only ones whose namoo need be mentioned here are the pea", and tho :IICJ(!apeUf/. 20 Hence tho concordant intervals could be described simply by thenamoo of the strings concerned (i.e. thoBO wruch produced the tones in quootion). So, for example, the octave W8B known as the concord of the hypale and nm.S1 -~"-----W;know of some earlier names for these threo consonancoa, wluch are even more interesting. The lourth used to be called crvllapd; this word actually means 'holding together', i.o. 'holding together the first and last strings of 0. tetrachord'. (These two strings were I.eld to(Jether and sounded at the same time, or directly after one another; in this wa.y a. fourth WOB produced..) Similarly Nicomo.chus reports that amongst the Pythagorea.ns tho filil. was called dloEtia (also written d,' tJEEtiiv or d,' tJESIWV).22 This name clearly comoo from the fact that to produce 0. fifth, two tetro.chords wero joined togethor and the note from each one which WI18 sounded was usuo.lly desribed as the 'sharp' (i.e. the '.!iJIJ!) note tJEEia. Finally the ocIatJe was just called depovia, 'joining together' (i.e. the 'joining to(Jether 01 two tetrachordB'),23 beco.use the two outermost strings of two tetrachords joined together were 80unded to produce an oetavo.

2.3 CONSONANOKS AND INTERVALS

My principal aim now will be to elucidate h . heclLllsO tho history of this sciontific te ~ e musical term dldC1T7Jpa, tion about the Whole re-E I'd I'm glVOO great deal of inform a118 about the origin o/the uc I ea~ t~eory of proportions OB well starting out, however, it is w:~1=:~ 8e:,~f an analogia'. Deforo about tho theory of music in antiquity. re some well-known facts CA) diastema

OB

IOU

consonance

In the theory of music d,d . confirmed by consult' C1T7Jpa meant Intervo.l. Tlus fact can be Hhalll'cgard an inte~:r s~:; IGreck le~jc~n. For the timo being, wo without wo . p ~ ~ tho distance between two tonOl:l' In antiqu~:~fe::: ::::~~~:~ri tho truo me~njn~ of this word~ concordant inl..... ~l nterestedmo.mly In the so-cuIJed ~. 8 or COnBt»uuu:ea 18 I . aI . JULine for intervals of til' k' 1 " n musIc practice the usual • I S 1Il( was ClVpfMll.,la b tl With the concord (CIV -) T-' , eCRouse lOY had to do '/ltpctwEI" of two sounds In th shall only bo conoorned with the three . . 0 prcaont work wo intervlLls; these are the octave th I ~ost Important concordant namOl:l of th . ' e / ourh and the fillh. The Gl'08k sion dldcm]p~~n~r;~s thro ~ sOlmo light on the origin of the expros.... used In t Ie theory of . I I . . p , .. mUSIC, 80 lave bsted them below. (My tofere TI . nee IS ape s DIctionary 1849) , Ie octave WOB cu.Jled d,anacrw" (act oJI ~ d,a' . tho accord which II88C8 tl u y '/ :IIacrw" xoedwv ClVptpowia /"U0fJ.,)the lourth W8.8 dlC!:'tcr(fdero~~:::~~1 t~ed(:ight) strings). :he name of ~~tho accord which pnase tl I I TEcrcrderov XOedCIWCIV}UIJOJ"la, .. the fillh WI18 called dlthr':n 1roU:~ strings) and correspondingly which Pl188es throug~ 'fivo s:;~.:11 B X°(,Jdwv uvprpruvla (the accord IIU

=

I

cr. Pluto, Ti,e Rel'"blic, Book IV. (43d. 'Puo'IIII,,·Ariatotolus, "De }lobus musioia problomatu'" 1I1u8ici 8criptort:8 (Jraeci, 00. C. Janus (Llpeiuo 1896), probloma 23 and 21.i. n See tho entry under dlOEda in Papo'a Dlotlonary (1849), alao Porphyri08 Kommenlar zur Hannonulehnl du Pro'cmaios, 00. DUring (G6toborg 1932), p. OU.21. .. Seo tho o.rliolo by H. Koller in Museum Helveticum 16 (1060), pp. 238-18; nlan n. J•. van dllr WIUlnlon, Ihld. 17 (lOGO), Jl Iff. 10

II

'--@/I!J I. cr. n. r•. van dur Waerde 'Dl II . 78 (1943), 163-90. n, e annonJoluhro dar Pytbogorcor', Hermes

~------

;1

·h , j

110

8~AIJc):

l'AUT 2, THE 1'1lIN':UUJ.WI!:AN 'rllGOJW Ott l'llOl'OllTl0NIJ

'l'UIll 1J1.:uINIHNUIi o~ UUJU:K lIA'l'UIDIA'l'lCS

pia.no key board, yet it was also known to Il,e Greeks, CUI the term 'interval'

Thcse ancient names (or the three most important. consonances o.ro 11.11 mentioned in 0. fragment of l)hilolo.us from which I would like to quote 0. few wOl'ds hero:2&

6,&CTTflPU indicatea."

11.8

L

interval

In my opinion one jUllt ho.s to survey tho various names and doscriptions of strinRs and cons'2.no.noos which have beon listed up to now, to see that :Without doub£~they all stom directly from musiclLI pmcLice. Ultimately it WM in i;~cal pmctice that strings first received nu.mcs and I!,-.~~_ w~!o. Msigncci ,numbers; also the consonn.nces were «CHCri\)e
I'Dials and KrWlZ, Fragments der Vor8okratiker, vol. I, 44 B6.IOfY. II W. Durl,ort, Weisheit ",ad TVu8mllchll/l, p. 348.

-

/ Whoever wrote this sentence wo.s clearly unaware of the fact that // to the Greeks the musical interval (diastema) WD.8 something 11.8 real and concrete D.8 /1V).).apa, the 'holding together of the two outermost strings of the tetrachord' (i.e. the fourth). It is misleading to talk of ~rea~?l.!~n in terms of straight Ji~~". One should rather point out that the lIluslcii.rintervarTdiiutema} ~wus 0. reat~~L~~~~l!~ str~g~~t limu"lu.9Jl, w:~~~~u~~!?!~!~~~. It is ev~n mon: misleading to read in the same o.uthor20 that ~hC conceptIOn (of mtervo.ls) as straight lines is linked to the name of Aristoxenus, the theory of ( i'-... musical proportions is (L88ocio.ted with the Pythagorco.ns .••". I must confess that I have not succeeded in discovering any kind of rco.sono.ble meaning in this D.88erLion. uThe conception (of intel'vo.ls) 11.8 straight lines" and "the theory of musicn.l proportions',' cannot be contrusted with one another, because the word diastema. (interval) means both 'straight line' and 'numerical ro.tio of 0. musical interval'. -Furi,l~rmoro, it is impo'BSible for this fact to have esclI,ped the notice ( of anyone who ho.s read with sufficient cu.ro evcn 0. Hinglo Honwnco ,--from the 8ectio Oanon.i8 (in the Greek original) .. It is worthwhile recalling here that the falHo (meto.phorical) interpretation of the meo.ning of diastema (musiclLI intorvlLI) WILl! not unknown in antiquity. Let me just indicate here some of these false interpretations, 11.8 found in late clo.ssico.l writers. Cleonides writes thlLt "An interval (dia8tema) is that which is contained (Td n£t!t£XOpeJIov) by 0. low and II, high note".27 Liwrally almost the sarno explanation co.n be found in Gaudentius.28 Nicomo.chus describes 0. musical diaatema D.8 0. 'way' which leads from 0. low note to 0. high one, or vice verso. (&ddv no,av dnd PQ(!I)"Cll TO t; el!; dEVT lITU .J} d"dna.4V}.2D Such learned 'explanations', of course, were possible only because it wo.s known exactly what the word d,oCTTflpa meant in o~ nary uso.go, na.mely 'distance between two points' or 'straight line'. ono reii
"The extent of an octave is o./oorll, and 0. filth (depovlat; de ptydJOt; ian av).).apa xal (),' dEe,aJl). • •• For the distance botween the topmost and the middle string is n, fourth (ICIT' )'Qe dnd MOTa, fnl l,cal1aJl /1V).).apa) and the distance between the middlo and the bottommost is 0. fifth (dnd de pil1aa, lnl VllaTa" d,' dEe,av} ••• ". (I3) diastema.

111

rr

I '\ ., ,

.,

.1 Ibid., p. 349.

., MtUIici IIcriptore8 graeci, ud. C. Janus (LiJllliao 11105). p, 170,11. Ibid., p. 329.23ff. I Ibid., p. 243.2 .

II

112

IIUUO: l'U~ U"tUNN1NOIl Ott OlLX1l:K UA1'UII:UATI08

dealt with here. The two tones of the consono.nce are thought of na two points in space and just 11.8 between two points in spaco there lies a 'straight line', so according to the metaphor, between the two tones thero can be said to lie a diastema. (Thus the fact that the diastema between two tones was ~~ ~ ~~18traiu.hlli~ ~,,~~ is forgotten.) Probably the first person who wanted to explain 'niualcal interval' in 0. metapborico.1 sense (and who by a.ll appearances deliberately wanted to change its meaning), WII.8 Aristoxenus. In the 4th century B.O. he wrote: 30 "An interval is that which is boundcd by two sounds having different ~nsions (i.e.. pitc~es). Thus according to the basio concept an

mterval mamfests Itself both na a-.!!!lLerence in tension (tJ,atpOea T,," Elva, TaaECLlv TO tJ"J.C1T1Jpa) and na a 8pace-·capo.lile of'taking in those tones which are higher than the lower pitch bounding the interval, and lower tho.n the higher one (Tono," "~uu1, tplJtfyyCIW). A differenco of pitch, however, consists in being more or loss taut.". It ill clear that this change in the meaning of diastema, from a ~-p!~~ly concrete to a metaphorical one, is very closely tied to the

basic too.crung of Al'isto"xonuB,'f6r'he' WM an opponent of the Pythagoreans. Vo.n Jan hIlS summarized his teachings na follows::n . "In his harmonics he did not inquiro into the origin of a tone nor he II.8k wl,eJI,er it U1a8 a number or a Bpeed. The ear need 'only hs~n unaffectedly to. the ~ange of tones; it will tell us with certainty wluch tones harmoruze With each other••.. His system is bll.8ed on the fourth and the fifth which aro easily perceived consonances, and without asking 1(thjc/a numeri~Lm{iQ'Lu.11Mr)a..!/_ ~~,..be wna able to determine from them the whole and half tones, ete. tt


As II. matter faot, Aristoxenus' own words also attest to his~i­ ,_r.yl!lggorean tendencies:u .OD'.e uartnOntBC U • h en Fragmente dll4 Arillt0zen06, ed. P. Marquanl, Borlln 1868, pp. 20fT. II C. VUII JUn, 'AritsLoxttnwt' (In lleak'wycloplldifl IU, II." 1067-86). .. Boo P. Marquanl (ed.), Die Harmoniecl&en Fragmenl8 dll4 Aristozeno6 pr. 46-7 (32, IDff, in ltfoibom'8 numbering). '

l'AllT ll. 'l'U" l·lU....:UCI.IIUlAN 'j'UJo:OIL\, UI' l'IlUI'UIl'l'ION:;

11:1

"We are attempting to draw conclusions which are in agreement with the data, na opposed to the theorists who preceded us. Some of them introduced completely foreign viewpoints into the subject and dismissed sensory experience 11.8 imprecise; hence they made up intelligible causes and stated that there were certain ratios beJween number8 and 8peed8 (J.~01J' TA Ttva, de,{lPWJI slJla, "al Tax'1 neo lli'1J.a) on which the pitch of a note depended. These were all speculations which are completely foreign to the subject and absolutely contrary to appearances. Others renounced l'CU8ons and arguments completely and proclaimed their o.ssertions as though they wero oraeulu.r sayings; they likewise paid insuffioient attenti to the data. II , Aristoxenus is undoubtedly talking o.bout the Pytho.goron.ns when . he says that some theorists wanted to account for the pitch of a note by means of "cortain ra.ti~sliotween-inlriiber8". Furthermoro, he was ; obliged_~lterJlro~ the meu.ning of dia.stema in a metaphoricu.l wu.y, I both beco.use he wo.nWtOdrsoourirtllo 'tCaching of thePytho.goreu.ns' ~becau80 heu.ttributcd differonces in pitch to whether the string . wo.s more or less taut. (Wo willsce in the next chapter tho.t it was the Pytho.goreo.ns who coined the musica.l term diasletnIJ and that for them this word denoted a 81raigltt line whoso end poinl8 yielded 0. numerical ratio, namoly the numerical ratio of the consonanco concerned. Aristoxenus had to cho.nge this meo.ning, since he did not want to admit thu.t numericu.l ratios had anything to do with consonances.) This explu.ins his extraordinary interpretation of 0. musical / / interval (diastema) as either a 'difference in tension', or something like '0. space capable of taking in those tones which are higher than the lower pitch bounding the intervruand lower than tho higher one'. It is appar\. ent from this forced explanation that Aristoxenus wanted to deprive the ~y~ho.go~o.n .col)(l9pt_dia8te~,of its o.rigino.l concrotO- mcanmg.-' mu.tter of fact, he WII.8 partially successful in thi8~endca.vour. " According to the original Pythagorean concept, there wore always two numbers lI.tltlooilLwd wiLh IL mUtlielLI diaalC1/ll&. As l)urphyry HtLys in his commentary on Ptolemy's theory ofharmony,3:1 "Most "avovUtol

AS-a:

•• I'nrl'liyrioH, l\",,,,mmllJr :11r l1flf'moJliele/,rc ,lelf P,,,ICOJIIlJiOIf, "cI. DUring (G6t.oborg JD32), 1" 112, :':2-3: Hal Taw HCJ1'CW.HWI' dA Hul1lviJayoeeifUI' 01 1I,uio", Tel ".aaT~l'arll .i/'-rl TWP AdyfUl' Uyovan', 88&Ab6

114

II~AUt): '!'UlI: lIUUUUUNUS 011 UJlb:KK UA'!'Uh:UA1'lU:l

and l)ythngoreans say inlerva18 (dlaan7para) instead of numerical ralios (Uyo,)." These words mean that 'intervo.l' (duit1TfJJlG) and 'numerico.l rutio' (Uyo,) are equivo.lont concepts in the terminology of Pythagorean musical theory. This conclusion is confirmed not only by tho fact that the term duit1TfJpa is consistently used to mean Uyo, in tho Seclio Canonis, but also by another pll.88o.go from Porllhyry. Although the Pytoogoreo.ns are not explioitly mentioned in this latter pa.sso.go, it is cloar that the reference is to t h e V "Some co.Il a numerical ratio between end points, diMlema (TtW U)'o" ~al n}v axial" TW" neo, d.u~Aov, BeCl)'fl TO dlO,t1TfJpa ~al06a,); these could be charucterized in terms of their end points as M)'ot, lUI well 80S dta(fT~para, namely tho fourth would bo epitritoslogos (4 : 3), the fifth would be /,emiolios 10008-(3': 2) and so 00:';- ...... --. This quotation is interesting, not just because it shows that tho t,vo concepts 'musico.l interval' (dia&lema) and 'numerical ratio' (lO{Jos) were equivalent 80S far as the Pythngorea.ns were concerned, but also bOCtLUSe it clearly implies that end points (Beat), which must also havo been numbers, could function both 80S 'end points of diaatemata' and 80S 'end l)oints of logoi'. - .. -._, It now remains for us to show how tho word duit1TfJpa could mean 'tho distance betweon two IJoints', 'the musioo.J interval between two tones' and 'numerica.l ratio' 0.11 0.1. the same time. 2.4 TUE 'J)IAHTEMA' BETWEEN '1'WO NUMBERS

I shall now try to oxplain the genesis of two interesting concopts, 6uiaT'1pa nut! Beal , whioh figure in tho l)ythugorean theory of music. Although as far 88 I know there is no ancient source which doo.Js directl! with the qucstion of how theso concepts camo into boing, thore III a noteworthy eXl,oriment describod by more than ono lato clnssical author, which could o.Imost be said to provide an answer to il., Hufurll (luClting (UIU Clf th(lHlI tLuthnrs in t.rlLnttllLtioll, thoro am two facts whioh should be atated: It IlJld., p. 94, 31ft. My t.rwwat.lon of this puaaugu is provhdonal. Boo aIao that. part of the text. rnarkod by n. 42 below.

l'AII'l' :l, "'IIK "lIJ.:·J.;UCI.IlIl1:AN l'Ul1:UlI\, UI.I "IWI'tl\lTlONIi

115

(1) As hOB already bcen mentioned, the Pythagoreans expreaaed musico.l interva.IB 80S numerico.l ratios. (2) There is a wholo scriM of ovidence which shows that thoy always used tho same fixed numbors for this purposo.3$ Furt.hormoro, the numerical ratios which corresponded to the threo most imlJortant COIIHUlULIlOCS woro tLlways 12: 0 (= 2 : I, tho OCt.ILVO), 12: D (= 4 : 3, tho fourth) and 12 : 8 (= 3 : 2, the fifth). Now Gaudentius tells of the following oxperiment by means of which Pythagoro.s is supposed to have discovered the numericnl ratios corresponding to the three most important musical intervalll. 30

He atrctched a st.l'iu o across u. ruler, a so·called canon, and divided this (I"ulcll') into twelve parts. First of all ho plucked tho whole string and o.lso half of it which comprised six units; ho foullel that tho tone of the wholo string harmonized with that oCthe half (12 : 6) according to tho oc/ave ..• Then ho plucked the whole string onco more and also three I,arts of the whole:n (4 : 3 = 12 : D), and he found that these two tones harmonized u.ccording to the lourth. Finally ho plucked tho whole string and two parts of tho whole (3: 2 = 12 : 8), and found that this timo tho two toncs hn,rmonized according to tho filth ete. The first thing to 1.10 cml'hll8ized is that tho n.coustic experiment describcd in tho above quotation cannot be characterized as 'physically impossible'. It must be admitted that in late antiquity acoustic obtICrvlLtions and experiments which are physically impossiblo were frequontly attributed to Pythagoras,:sBbut experiments with tho 'canon' are to be distinguishcd from theso. Even thoso modern scholars who aro rightly skoptimLI ILiJout Huch HtoriOH ILgI't!O fur tho moat plLrt. thlLt "tho Pythagorean thcory of music can bo vorified to somo extent"SO .. Cf, n. }{olltlr, lUu,euna Hellldicum 16 (1969), :t40-1. GaudenUua lived In tbtl 4t.h century .A.D. Boo MtUici ,criptoru Graeci, 1111. r., ,'1II111H (T.illHilll' I Rflfi), I'. :J41. 13rl., ror t.1I .. (1IInt-ut.inn; cr. "IHO n, T•• VBn dor Wutlrtltln, Sciencll Awakeniny, 1)' Ur;. n'l'ho phrll80 'thrtlo parts of tho wholo' means tllres quarter8: similarly 'two JlUrla or t.ho wholo' In t.ho noxt. sontonco mOllns two t/,iNi8. II W. Durkort, lVei"lceit und IViB8enscl.a/',. JlJI. 364". II Ibid., p. 363, II

8-

llti

l'AIIT 2. 'III}; l'IIJH';Ul:UlJJ.:AN TUEOILY 01.' l'1I01'01t1'IONIi

by meu.na of the CIU10~.. Thill of course sti1l1ea.vea open suoh questiofl8 U.8 whother it is correcC'to o.LLribute any such experiments to PythagorWi himself, whether the 'cu.non' was perhaps just an Uartificial piece uf I~pparu.tus which was devised later" and whether the proportion",1 Ilumuers of the musico.l consonances were really discovered in the COUl1!e of experiments of this kind. However, we mo.y postlwne anawering thellO questiona for the time being, since we ~terested in a diffemnt aspect of the experiment described above.· ~~..,. Tho 'cu.non' montioned by Ga.udontius was a. stretched string (monochord) on a. measuring rod which was divided into twelve parts. The point ofha.ving 0. duodecimalsca.le was clearly to ensure that the ratios betwecn tho lengths of string which were sounded could be easily determined. So we may assume not only that the measuring rod Wl~ divided into twelve po.rts, but also tho.t each part was numbered. The Il.bove pUBBage deo.ls really with three experiments. In each ono the wholo string (0.11 twelve units of the measuring rod) was sounded firtlL I\nd then 0. shorter soction of the string WI~ plucked so lUI to pro· duco 0. tono consonant. with that of whole string. To ca.rry out tho Hccond step of each eXl)Criment, it must ho.ve been necessary to prevent I~ scction of the tltring from vibrating. Although Ga.udentius dOCK lIut 8u.y anything a.bout how tbe string was shortened in llractice, wo tlo have somo information about this from othor sourOO8. A smu.ll bridue (VnaylD)'w~) WI~ moved under the stretched string. tO This is lhe fCll.Hon why I conjocture that tho canon WWl not only divided into twelve po.rts, but was also numbered. This would enable one to see ILt IL gllmeo where (i.e. at which number) tbo vnaylD)'£v~ stood, whon Lhe string wu.s plucked 0. second Lime, o.nd henoo how muoh of the Hlring was vibrating and how much was kept silent. Alt 8.11 oxu.mplo, lot. us lool( more clollOly u.t Gaudontius' aLCcount of how l)ythagoras is supposed to have discovered the filtl,. First he I'ludwd the wholo string o.nd then he shorwned it to 'two parts', i.t •. whlln ho plucked tho IItring II. second timo, a tl,ird of it wu.s kept Kilent and ttoO tl,irtU was sounded. So when he plucked the string for thl! fh"Ht timo, tho bridgo (t\naywyeti,) was positioncd at tho ond of tho 't!ILIIUn' (Lu. at the numbOl' 12), and it WlUi o.t tho number 8 when tho 00

cr. l'orllhyriOll, KOIntmmlar zur lIarlllOlliele/,ro du l'101omai04, uti.

(Goleborg 1932), p. 66, 2411.

DUring

117

string was plucked 0. second time. Tho two tones which were produ~ were consonant according to tho filth and between them lay 0. mUSical interval which could bo detected by the ear. TillS so.mo 'interval', however, was also visible to the eyo, for the bridgo was first positioned at 12 on the canon, and then at 8. Those two numbers are the e1ul poinJs 01 the inJerval of the fifth, or ~eol ToU _~,,:~~aTo~l as thoy ___ " were co.lled by the Greeks. From this we sec Immediately why tho \ Pythagorcans maintained that tho proporLional numbcrH of the fifth ~J were 12 : 8 or 3 : 2. This-interpretation also helps us to understand why Porphyry says in the pu.ssngo which has already been quoted above,': that the ~y­ thagoreans thought of tho diastema (musica.l interval) as 0. 'numenca.1 ratio' ().&yo,) and as 0. 'relationahip between end po!nt' (tlX iat ; TW)' ned~ a.u7jAot.I~ Hew)'}. - I believe that the above satisfactorily explaina the origin of the musical terms diastema lI.nd "oroi, even though neither of them is OXllliciUy mentioned in the text of GaudontiuH which wo haa.vo been discussing. In the acoustic experimonts of tho Pythagorea~s, the w~rd 6,at1TTJpa meant 'straight lino' and referred to that Sedlon 0/ stnng ,.----on the 'canon' wl,icl, was preve.~ted I!!'!!'_'!.~~_r.a.t~nu, when the second tone 0/ a consonance 1MS" proouct4 (i.e. after the whole string had alre~y been sounded) and was in. tl,is way necessary lor tile creation 0/ a mu.ncal inlerval. n Hence 0. word which actually moant 'straight lino' cnmo to have the meaning of 'musica.l interval' as well. Furthermore, sinco tho end points (Heol) of this straight line were numbers on the '~non', the word (S,aCfnlPfl WIl8 o.lso uKcd to desoribe tho 'roltLtionKhip botwcon two numbers' which w(~ exhibited by the proportional numbers of the consonances (12 : 6, 12 : 9, 12 : 8 and so on). I hopo thlLt the l,reviuulI interpretation lu~ thrown lIomo light ?n the gencsis of two important concepts from the thcory of mUSIO. II Accunlillg til l'lllI.o'!I.li"llIglIlI l'leilc:bua (170-.1), UllYlIlIO who WUlltli to 00 UII expert in musical theory hus to know "t.ho interval8 or high and low notes" (Td 6WO'TJtl'aTa • •• T~ tpoWij( d;vnlTd, TIl nle. xal fJaetiT'ITO') and thll "tlnd points or t.h"H" illl."rYIIIH" (XIII fOIl, 6f.>OII(; TI;;I' " ...arrlllllrwl'). II Btlo n. 34 uboYt! • • 1 Thia also onllbltlB us to mako BtlnaO or a rragmnnt by Philolalls (Diols and lCrun:r., l"mlPllenlo der Vur,whilliker, T. ·140 26): TI"~' ""itlT'IllIIix,ikalll' el..", Vn€f?OXq,.. 1'lao VneeoX't ia tho diOeTence ootwoon tho two BtlctiollB or aLring on a 'canon' which produco tho Lwo notos of B consonanco.

...

118

•

SZAUe): TJJK Ul!:lllHHINtlll 011' OJlKKK MATHJWATJC8

'!'hCtiU are lJuif1'n'/pa, 'tho mUllical interval botween two tonoa which

ill expressed Il.II 0. ratio between two numbers' and 8eo" 'the end points of II. dia8tema which are expressed as numbers'; henco 8eo' also came to mean 'tho numbers themllelves in the numerical ratio (of a musical consonanco)'. According to the explanation given above, these two ~ fundamentu.l concopts from the Pythagorean theory of musio only become comprehensible andv-Jneo.ningful whon they u.re thought of in ,~:~nllection with the/~non1 for they were developed in the course of ILcouatio experiments with this instrument. Howevor, there is in. widespread vielV ,~eording to which the monochord and canon of the Pythagorea.ns ~ supposed to be "an artificial pieco of apparatus which wo.s devised Iu.ter"." Indeed it has even heon conjeotured that thoro was no measuring scalo on tho 'canon' until after the timo of Aristoxonus, i.e. not beforo about 300 B.O. U The major pieco of evidenco in favor of this conclusion is the fo.ct that tho 'canon' is not mentioned in tho Afuaical Problems (0. spurious worle of Aristotle) nor in tho Sedio Oa7.onis t' similu.rly it is not montioned hy writers of tho 4th and 5th contury in genem!. 'fho explanu.tion given for this is that most probably no such musical measuring inst,nJmont oxisted at thlLt time. I think t1l1Lt this view is mistal(en J becu.use the ~~~n~rl be conclUSIVely l)rOved-to"iia.;;'cxi8io(rii.~ !~~[.~tfto 'time of Pla~:- Ti,,~etim., .1~t uS'recall 'tJie fonowiilg: Thoro illaiillitoroatiiJg }tILrontheLioo.l romlLrl, mii,(iO'bYPJiLfO,.rwlirct' runK as follows: b ptdcp de ToV IE ned, Ta d(MeHa avvtp'I T& Ts,Wc'&lUuz Hal TIlinIT(],Tov. A correct translation of these words would be:.7 "TM ralio I! lUtelll,e ratio l~arelobe lou,uJin belween (Lho ratio of) 1M 6 to tile 12.'~ Using the terminology of the Pythagoreans, "pro).cOJ' (= 1~ = 3 : 2) 1~l\(t lnIT(!'Tov (= 1~ = 4 : 3) are the proportional numbers 01 the /illh and lourth. So Plato is saying that the proportional numbers of the fifth ILiul the fourth (whioh of course can also be written IlS 12 : 8 and 12 : 0) lie hetwoon 12 ILnd 6 (t.ho Ilroportiono.l numbers of the ooto.vo, 12 : 6). CWo havo already quoted a fra.gment of Philolausu whioh

, V..

-

.. Cf. 1111, 38 "ncl 3D ahov",

.. cr, B. I., van dOl Woerdun, Hermea 78 (104.3). 177. .. Epinomia 091a. ., Thu trWlBlation Wlcl IlItorprutation of t.hia 1)l1IIIIUgo is alBo disousaed by dur Waerdon, Hennea 78 (104.3), 186-7. &I Cf. n, 24. abovo,

VIU1

l'All1' 2, "'JJK J'1Us:,EUOLllJKAN 1'UKOll.Y 01' l'UOl'OllTlONIi

no

states thu.t in musical pro.ctice the octavo consisted of the combination of a fourth and of a fifth. Plato's remark stro880S that tho proportional numbers of the fifth and fourth lie right in between the propor~ional numbers of the octave (12 : 6). This proves amongst other things that attempts had already been m8.d6Tn-the tlieoryofmusio to oxplain ' ., ,J---tlio resoliition"of theoctAve into a fifth and a fourth. This point will I bo taken up later.] -'-, .. - .. -.-~"' .. -TIUnmly-questions now are how it ever occurred to anyone to pla.co tho proportional numbers of the fifth and fourth (which Plato does not write as 'proportional numbers', but in accordance with an o.rchaio convention o.s tho corresponding fro.ctions 1~ and 1~) right 'in belween (tho rati~ of) 1M 6 to the 12' and why 12 and 6 were chosen as tho proportional numbers of the ootave, when this interval can also ( be described IlS 2 : 1. I believo that theao questions cannot bo u.nswered properly unless one bears in mind that the mCllSuring instrument of the Pythagoreans (tho (mnon) which WIlS used to illustrate the pro) portional nllmhers of the consonances was .t,l~icled ~n~g ,~'!lJ.el~e,.parlB,. In other words, Plato'li remark proves convincingly that the canon existed~at that timo. Modem attempts to regard the canon as "an artificiaCpieceofapparatus which WIlS devised later" 11I~yo, n?tbeen ( successful. i

-~

2.5

A mORE88JON ON TIm 'rnRORY 011' MUSIC

Tho prcsont investigation is ohieOy concorned with tho early history of the theory of proportions. It touehes on particular problems in ~noiont musical theory only incidentally. It lIeemB to me thu.t in dis(ou88ing the genesis of the ooncepts dUlt1Tr}pa a.nd 8eo" both of which : wero originally aplllied only to the theory of musie, an CIIIIentiu.l con.'-.tribution hIlS also been .E!~o !Q,~i!~ hi!ltQry o.f ~!!.~~h~_ry ~f p~~r~~~ns' _ As we IULVO 8een, both tho concept of diastema (interval expressed as 0. numericu.l mtio) and thu.t of llOroi (the o.ctuu.l numbers in 0. numerical ratio) were developed in the course of acoustio experiments with tho monoohord ILnd canon . From now on wo could concentra.te our attention on tho Lheory of proportions itself, for thero i8 no longer any need to concern ourselves with I)articulo.r hiliLoricu.l problems in tho theory of musie. Howover. sinco I believe that the method applied abovo ca.n also shed now

120

IIZAIlO: '!'UU lIUOUININU8 011' OllKKK lIlATUKMA1'1(,'11

light on many queations about ancient musical theory, lot me digress here and WsCU88 theao queations, even though they are not strictly relevant to the history of the theory of proportions. Tho queation of how tho Pythagoreans came to express intervals by means of numerical ratios has frequently been discu8lled in earlier reseu.rch." The answer which was given to this queation can be said to havo had two aspects. On the ono hand, it was stressed that the / Pythagoreans could not be held to have "obtained the numerical ( mtios of the consonances solely by observing the various lengths of \~ string", whereas on the other hand it was insisted that "the empirical method used to measure the tonea with numbers Wll.8 a secondary matter for the Pythagoreans" and that "there could be di1Ferent opinions about is". This latter assertion is borne out by Theon of Smyrna who wrote:IiO "Some want to undorsto.nd the (numerical ratios of the) consonances in terms of woights, others in terms of sizes or in terms of movement, and still others in terms of veBBels." It is my opinion that the way in wlifch the question was posed inovitalJly led to the view outlinod abovo (and furthermore that thill view is only partially true). The rather abstract question of how tho Pythagoreans came to express intervals by means of numericl1l mtiuH cl1.n only bo answered by attempting to survoy tho rich vl1.riuty of clll.llSical and late clWl8ico.llitero.ture which deals with this question. These accounts are of uneven worth, BOme being completely unreliablo ILnd if ono looks only n.t thom, thon ono is forced to accept the abovo view. If, however, somo facts about tho scientifio languago of the Greol(s ILre taken into consideration and in addition the previous question is givon 0. more concrete formulation, thon 0. toto.lly di1Ferent conclusion will be ron.chod. - - . Tho linguistio facts to which I am reforring are tho following. : In ancient musical theory, musical tones were usually called~ from tho verb TBl,ru (~o,stretch). Thus 0. musical tono was above all tho tono of a stringed f;;~~J,t. Furthormore, tho consonanOO8 WOI'O \, culled loe6wP uvlltpon'la (concord of strings) in Greek. Theao two facts "Soo E. Frank, Plato und dill 8og. PytMgorur (Halle 1923), pp. 160-1: W. nllrkorL, JVei6heil ur&d lVi88tm4c1uJII, pp. 348-64; also VIU1 dor Waerdtllls' imp()~papor (S08 lIennu 75 (1043». 10 Thoon of Smyrna, 00. E. Hiller (Llpslae 1878), p. liD.

l'AII'r II, TUIIi l'UK·KUCl.IOll:AN TlIKOUY 01" l'Il01'OllT10NK

121

clearly indicate that Greek musico.l theory Wall based predominantly on experiences and experimonts with 8tri71J/e4 ~J:!,Imenta (or with just a single string). If a more concrete form is given to the previous question. i.e. if one 88ka why the Pythagoreans used the names 6"iC1T1Jlla and 8ea, to_ denote musical intervo.la expressed as numerical ratios, then it I)t;comea immediately apparen_t.thn.t these proportiono.l numbers . mUst originally have represented ~_~.t!.~ .l~t.~. These Bam~ . two expressions alBO indica.te how important experimenta must have ~'\ / boon at ono time in Pythagorean science. Pile concepl8 'diastema' a,"! ) ( 'horoi', wl,ich Ilave been analyzed above. could never "ave originaleg./ \..,. without musical experiments usi~ the" canon. . The assertion that the emplncal method used to ascertnm tho numerical ratios of the consono.nces was 0. secondary matter for the Pythagoreans (which is correct as it stands). receives 0. new emphasis in view of tho abovf}o Although the concepta dia8tema and horoi originated in the course of cxperimenta with the canon, Hippasus was already able to establish the most imlJortant consonn.nces by means of bronze. diskf?i. 51 Similar exporiments were 0.180 carried out witlCve8sels containing difforent a.mounta of water and with wind instrumonts. U ClcILrly the Pyt.hugoroana wILnted to ahow thut other kinds of experimonts led to the same prol)OrtionlLl numbers for tho \ consonances as did those original experiments with tho monochord \. and canon. This explains why these sn.mo PythagorcunH could subsequontly hold tho view that the empirical method used to nscertain the proportional numbers of tho consonances was a socondary matter. Tho actual numericn.l ratios of the consonances which hoo been cstILb;( (, n Hshed once and for. all ~.ere important to the Pythugoreans, not th«: I! J\J ~)iri(:lLl mot1)()d.u~~.~ .~~rtn.in them. ~ I o.lao believe that the above account of the concepts diastema and horoi enables us to give a better explanation of many things from tho llutently flLlso tradition which origilllLted in IILte o.nt.iCJui~y. As an example, let us consider the following po.ssn.ge:u II Diels and Kranz, Fragmenteder Vor8oLTaliker, I. 18. 12; cf. also 'V. BurkerL, Wei8heit und Wi88eR8chall, pp. 366-0. II cr. B. f •. VIUl dor Woordon, lIermea 78 (ID4:1), 172. II Ibid., p. 170.

1 ,),) ~~

8ZAUO: TU~ U..OINNUW8 Olf UllJlliK WATUlWATJC8

'''I'here is 0. nice story found in Nichomachus (p. 10 Meibom), Gaudentills (p. 13 Meibom) and Doothius (pp. 10-1 Friedlein) whioh tells how the Pythagoreana came to represent intervals by means of numerical ratios. It cannot p088ibly be true !lowever. According to the story, Pyt~llgo~_~ns plL88ing by ~m?d heard the tones of the falling..!!amme~produoe various m rvals, in particular tho octave, fourth arul"fifth. Ho oarefully weighed. tho hammers ILIl
"1 f one had actually tried to carry out this experiment and measure the numerical ratios (of the consonances) by means of suspended weights, ns Pythagoras is said to have done, then the attempt would inevitably have ended in fi"Uure. For the weights corresponding to the octave are in the ratio 1 : V2. not 1 : 2." So we can see how modern research has uncovered some falsehoods in traditional accounts. In doing so, however, the question is left untouched of how this experiment, whioh oould only have been thought out &nd~er carried out, ever ocouned ~anyon•• Yet il scems to me tho. far as this question is conce~t least a part oftheaboveacco . very infonnativ~. When it is alleged.thatPythagoras "suspended four identioal strings and attached weights to .. Ibid., p. 173.

l'AII.'1' It. 'l'UK l'llll:·KUOI.JI)I~AN TIIKOUY Olf l'llIIl'tJIl'l'lONK

them", it ia clear that he wanted to use the weights to measure the tension of the strings. An eXlloriment of this kind could only have been devised after it was realized that the pitch of a note depended on the tension of the string in question. This, however, is exactly what Aristoxonus, the opponent of the Pythagoreans (see p. 113) who refused to acknowledge the connection ~ween proportional numbera and consonances, succeeded in doins..:Thus I believe that the experiment described above WILB in part ~ '.Pythagorea:n.answer'. to the too.chings of Aristoxenus. The person who mvented this lmp~tl~ experiment and attributed it to Pythagoras, wanted to convmce lua readers that the traditional proportional numbers of the consonances could 0.180 be demonstrated by considering the tension of the string. Of course this cannot be done, yet the so-called experiment is not without importance for the history of soience. In other words, my position is that the late cllLBSical tradition, which is concerned with the science of music and which gave rise to the story quoted above, originated in attempts to vindiclLte the Pythagorean theory and that these at~mpts met with .only pa~tial"", succe88. At first, whon the exprcSllions dwatema and /toro, wore COined, (with reference to the monoohord and oanon), ono did not need to ) worry overmuch about tho tension or thickness of the single stretchcd , / string. At thi~ Hmo no ono ILBked whether tho .tono itsolf WILH ~ speed, a movement of the a.ir, a vibration or somethmg else. Hence It could be maintaincd that tho consonances depended simply on number8. (Tbese numbers, of courso, originally expresscd ratios between tho lengths of individual sections of string on the monocl~ord.) It ~llS only later, when it became neCC88ary to defeml tho anCient teachmg agai!18t i_tII~o~p_()nents, that these same numbers were. also supposed to represent ratios between weigldB (i.e. between variOUS degree8 0/ tension of the string) and between the various t/,iclcne8se8 of the objects which produced the tones (in this case, bronze dUikoi). This also explains the invention of partly nonsenaicalstoricslike the one mentioned -above, which tell of impo88ible observations and impractical experiments. Furthennoro, it oxplains why Theon of Smyrna stressed that it was not important how the proportional numbors of the consonances were obtained,"

1

.. Boo n. 60 abovo.

l~,'

Il~AUO: Til" UIl:UINNINUIl Olf Utl1l:KK lIA'l'UKlIATWII

l'AIL'l'

In ~he aame wu.y, I believe that I can eXlllain why ancient musical tI~corlsta w~re inconsistent in their method of associating a number wlt.h ~he pitch. of a note. Originally_ when the starting point WtlS

,1

cXI.lerlments WIth the canon, a larger number (a longer section of strang) had naturally to bo associated with a deeper note. But thia • method had to be turned around completely when it became necessary to defend the ancient teaching against subsequent objections and take newer speculations into account. As van del' Waerden ~te.1G "It is clear that those who sought to interpret the numerical rati~ in ~rms of ~eight or speed, were obliged to assign a larger number 10 a higher note •••"-----, ~here is yet another interesting tro.nsitional state which is worth notang. I a,? referring to a CllBO in which a newer theory has already been tu.~en mto account, although the original method of associating proportIOnal numbers with intervals is still used. Let me quote onco moro from van del' Woorden's work:11 "A noticeably vacillating position is taken in the 8eclio" Oanonis: wherel1.8 in the introduotion it states that two toncsaro in u. numerical l'aLio to each other because tones aro quantities which are increased wi,en the tone is raised and diminisl.ed when. it is lowered, in the main hmly of the text tones are represented by straight lines lI.nd highor tones correspond to shorter lines."

c{

2,6 END POINTS AND INTBRVAI.8 PICTUllBD AS '8TB.AIODT LINKS'

, I believe that many details of the pre-Eudoxian theory of proportIODS: tlS well as of the Greek mathematical theory of music, can be Hcen m a new light if one bears in mind the explanation for the origins of the concepta diastema and horoi which was disoussed fully in chapter 2.4. So let me recapitUlate here and repeat the etymologies of these t.wu wurds.

~.

'l'Ut: l'II",1l:tfCt.10ICAN 'l'III';Utl\' 01' 1'ltlU'UIl'l'IONlf

In the field of mUBie ~tdf1T1]pa (straigbt line) originoJly meant the piAce of string on a 'canon' which was prevented from vibrating when the second tone of a consonance was produced (i.e. after the entire string had already been plucked). If tlus piece of string had not been kept still. it would not have boon possible to produce the second note of tho desired consonance on tho monochord. The Beot were the end points of this piece of string (the dia8tema) and were shown as two numbors on the cano". It would not be superfluous to indicate here the precise way in which the concept of musical interval (diastema, in the above sense of the word) developed into that of the ratio between. two numbers. AccordingJ;.9 the previous explanation, the diastema was the actual interval tWe6~t 0 tones. It could be detennined acoustically and was a visible a length of string on the 'canon' (namely the length b 'the section of string producing the first note differed from tho one producing the second). This is the only interpretation ,vlucb makes sense of such musical oxpl'ClJ8ions as Beot TOO ~,actn1paTo~ (end p~!.~lJ,jU)f.J\R,inW.rva.!). This length of string -o;-"th~'~~ochora WiiSBllocificd by its two end points (by two nU11l1lcrs on the cILnon). just as a stnLight line is given by its two end points in geometry. These matters. however. can be viewed in ILnother way which differs in pu.rt from the above, A musical interval is chlLracteriz.cd by the consonance of the two notes which bound it. 'rhese two notes harmonize with oach other and form the intorvn.l which is slLid to lie bctweon them. Hence it was uften the CMe in musical practice thu.t the consonance oftwo notes was described by the two strings which produced it. For example. the octavo (as has already boen mcntioned58) was clLlled the concord of the vnanJand Vlln7. This suggesta the idea that in the /'" lIceory of music diastema may also have been construed as the concord I produced on the monochord by two diUerenl (iiigi1i8o/string. instead "-df as 0. piece of string which did not vibrato. Of course theso two lengths of string could still be cho.ra.cterizcd by tho cnd points of the latter pieco. As a matter of fact. this is the viewpoint found in tbe Seclio Carnmia. To 8CO this. wo flood only look ILt tho beginning uf the proof of Thcorem 1 (in the Seclio Oanonis): 1(fT{IJ ~taf1T1]pa TO Dr .... Heiberg translates thi" as "Hit intervallum BO •• ,'-. A reading of

Hilum" 78 (1943), pp. 173 1. n Ihid .• p. 173.

II

. f "

.jI,

1:!5

Boo n. 21 abovo,

126

~

IIZAUc); 'l'UK 1111:01NN1N08 OJ>' OIU~Jo:K. UATUll:MATJCS

the Greek words together with their Latin translation would loud une to think at first that the letters B and 0 (or B and 1) refor to tho elUl poin18 0/ a straigld line (the diastema). This agrees with the explann.tion offered in chapter 2.4 according to whioh the diastema was tho scotion of string which did not vibrate (the straight line upon whioh the musical interval depended), and was specified by its two ond points. SUI'prisingly, however, the diagram whioh aocompanics this plltlsago in the Beclio Oanonis makos it olear that the letters Band l' (two numbers on the canon) represent hoo straight lines 0/ diUerell1 ienu1hs• The only way to explain this fact is to say that by the time tho propositions in the Sectio Oanonis received their final form, diaalema no longer meant simply "the length of string between two numbers on the canon, which did not vibnl,te (and by which the section of string producing the first noto differed from the one producing tho Ilccond)" . Instead it. refOl'red to the two sections o/slring wllic1, produced Ihe notes in question and, since these could be characterized by the same two numbers which had origino.lly been used to specify the uctual intervo.l (the piece of string which did not vibrate), it had como to mean 'll,e ratio between two numbers'. This slight change in outloolt (namely the cho.ra.cterization of a musical interval in terms of the two lengths of string which produced the notes bounding it, instead of as a piece of string whioh did not vibrate) led not. only to the crention of a new concept (ratio between two numbers), but also to a curious anomaly in the use of language. I am referring to the fact that the word ~ulC1T'1pa (in the singular) which originuUy denoted a single straight line in the theory of musie, had to be illustrated by two 8traight lines 0/ diOerent lengtM. These two lines represented the lengths (expressed as numbers) of the seotions of st.ring which produced the notes, and hence could also be regarded as representations of the numbers themselves. Diagrams of thilJ sort become even more interesting if one considers that what were earlier called the end points of the interval' ({leo, TOO a&aan7paTo\) IIIlIst hlLVO 1100u shown ,LIt slraiyltt lines in them. In this way it wso becomes clear why the two straight lines which in the diagram from tho Bedio Oanonis arc supposed to represent thl) two sections of string could eac1. be designated by only one leller. As long as diastema meant 'the piece of string on the monochord,

"AlLT 2. l'UJo: l'llU,KUCJ.lOJ.:AN TU1I:OlLY

O~'

J'UCJl'IHlT10NIJ

127

which did not vibrate', the obvious way to characterize this straight line on the 'canon', wu.s in terms of its two 'end points' {1lOroi}. One end point wu.s usually tho number 12, i.e. the end of the canon "-\ i (because for the most part a note coriSohant with the key note of the whole monochord was sought). The other end point was the number ~ at which the una)'(()'w, stood wh~h~!leCQJ!
J

l

(I) 'rhoro is a tJOri08 of ProI)Ositions (o.g. V II. :1, L2, 13, 14, 16 ILlllI 17) in whioh lines are described by just a single letter. This way of picturing numbers is the same as in the Bedio Oanoni8. (2) In the same book, however, thoro uro Prollositions (VII. I, 2, 7 a.nd 8 amongst others) in which numbers are represented by lines and

12M

H:tAUO: TUK

IIb:UIHHIHIlIl

Ulf 1l1tKKK llATIUWATlC8

t.hese lines a.ro labelled by two letters, one at the beginning and one at the end. This w~~a1I1l!S~!l~l~-'J!o_l!letry. (:') Thero are even some propositions (e.g. VII. 4, 5, 6 and 9) whioh UIlU both the above methods at the same time. In these propositions, ono number is shown u.s a line labelled by a. Bingle leller, whereas another is shown as a. line labelled by two Idler8. So we oa.n see that at Euolid's time a. decision ha.d not yet been made about the way to designate (number) lines in arithmetio, i.e. .nhoutwhether to a.doptthe 'musioal' convention (deriving from experit----.!Jlents with the canon) whioh used 0. single letter, or the more 'geometric' one which usod two.- -

2.7 'D1PLASION', 'UEMIOLION' AND 'EPlTRlTON'

I n tho COUl'tlO of our invtlStigations up to now we have loarned to distinguish two stages in the development of the concept diastema IL'I it applies to the theory of music. (1) Dllring tho first stage diastema (musioal intervw) was the length of Htring on the monochord which did not vibrate, and by which 1ho Hcction of string producing the fil'llt note differed from the one 1'1'(lIllIcing the second. It wu.s ohamcterizeci in terms of its two end 1'0iniH (ilOroi) and these could be read off 8B two numbol'll on the 'canon'. (2) 'L'he I\CCollnt of this concept given in the Seclio Oanonis clearly helongs to a later stage in its development. Here diastema does not refer to tho piece of string on the monochord which was kept from "iunLt.ing, but rather to the two sectiofUI of string which produced the notes. At this time the word unequivocally meant the ratio betwoon (.ho two numbors which OXIII'tIHScti tho longths of thoso two sections of Htl'ing. Now I believe that I OILn ovon l)oint out a third stage whioh I!.receded (1111 IIt.1lur t.WII. 'fhili lltagu might btl IIILitl tu have prolJarod the way '--------for the concept diastema. Since in my opinion it gives us 0. great dew of informn.tion not only about tho theory of music but wso about tho wholo hiliWI'y of Greck lIlathomatiCtJ, I would like to discu88 it mthor more fully. My starting point is the passage from Gaudentius, which bas

l'ART I. THill l'lLK·";UUUlll AM THKOlLY Olf l'lL01'OlLTION8

r

129

been quoted once:1II "Pytha.goms stretched a string across a ruler, a. so-called canon, and divided this (ruler) into twelve parts, etc.". These words indicato that tho numerical ratios of tho threo most important consonances were found only a/ler lhe 'canon' had been divided into twelve paris. The canon with its twelve divisionH WILH the instrument with which the Pythagoreo.ns performed experiments and by whose help they were able to ascertain the numerical ratios of the important musical intervals (12 : 6 for the octave, 12 : 9 for tho fourth and 12 : 8 for the fifth). In faot, on tho basis of Gaudentius' account of theso experiments together with our knowledge of tbo canon, we were able to find out the original (and exact) meanings of the musiCILl terJD8 diastema and horoi. It is time, howevor, to look somewhat more critically at Gaudentius' account. It seems unlikoly that the experiment ho outlines, was originally carried out by first dividing tl.e canon into twelve parts. This would not have boon absolutely neC0888.ry, so long as one merely wanted to exhibit tho proportional numbers of the throe most important consonances (the octave, fourth I~nd fifth) separately on the monochord, without regard to the connections between them. To obtain the ootave, it is only neoesso.ry to IUJlve (2 : 1) t.ho strotchcd string. The fifth is obtained by dividing in into three parts. (The wholo string and two thirds of it produce notes which aro 0. fifth apart, 3 : 2.) Similarly, it lIufficcs to divide the string intoquarlers to obtain a fourth (4: 3). Even the words which Gaudentius chose to describe these experiments are such as to enable the attentive rea.der to establish for himsolf that the oanon must originwly havo hcon dividcd into two, three and four parts, before it was divided into twelve. TlnUJ it can hardly be the case tiUlt the original diviBion 0/ the mU8ica1 canon wa8 into twelve parts. In other words, I construo Gaudentius' account as an outline of how t.hCHO lIluHitllLl oX(lOI'imonl,H wore [lorforrntlll ILt II. rell,Uvoly hLte stage. Originn.uy thoy wore cl\rricd out simply with a monochord. The canon W/UI only introduced ",ter, when it WILlI discovered that there were good rolLliOfUI fur dividing tho JnOlLlluring rud bonOlLt.h t.ho lIt.rowhcd string into twelve parts.

II

800 n. 36 abovo.

GSub6

130

SZAUO: l'U'" lJUU1HHUIOil OM' OIlXW( AlA'l'UXUATIUII

I hope to be able to show that the above is a correct reconstruction of tho process oC development, by explaining the moonings oC some terms Crom tho theory of music which olearly came into being before tl,e introdUclion oflile canon. The terms to whioh I am referring are the oldOllt known names for tho octave, fourlh and fiflh. In tho Pythagorean theory oC musicO o the most common nn.mo Cor tho numorical ratio of tho oclave wu.s 6mAaa,ov 6,acn'1pa or, in aCragmont of Philolaus,01 I5mAOOJI. This exprOllSion is translated by the formula 2 : lor, bearing in mind tho twelve divisions on the canon, by 12 : 6. However, one should ILlso attempt to translate it literally and thon consider it in connection with the monochord. The teohnical term I5mAaatov 6taO'T'1l'a means literally 'doubled line'. It is olear tbat tho octave received this name because o.Cter the stretched string had been plucked once, it WaH shortened to halC its length and tho shorter Hcction oC string which rOIIultccl produced a note lying an octavo abovo tho ono produccd by the wholo string. This stjJJ leavea tho question of what the word dill8lema itself melLns in thill context (i.o. in tbo exprOllSion 6mAaa,ov 6tacn'1Jla 'doubled line'). It clearly docs not havo tho mooning which I oxplained earlier, Iliunely "the picco of string on the monoohord which WUB kept Crom vibrating". When used as a nn.mo Cor tho numerical ratio oCthe octave, the exprOIISion 6mAaO'cov 6uiO'T1lpa can only be referring to the wltole monochord. In my opinion tho only conceivable way to account for this narno is the following. 'I'ho hruC of the stretched string (i.o. tho section of string which WI1.8 plucked to l'roduoo tho second noto of ILn ootave)OS was regarded 11.8 tho 'unit'. Relativo to tbis unit, the wholo monoobord becomes a :cloubled lino'. So if the monoohord is treated as 0. 'doubled line', i.o. If the wholo longth oC string is plucked first and tben the unit ch080n for the present case (half of the monochord) is plucked, tho consonance (If tho octlLVO is obtained. 10

Uf, Nicomnchi Nnolairidion. 00, O.

J,ln\lS

2liO. 22ft; also PI "to's Ti",,,e.

t18,30, II

1:0

At first glu.nce this eXl,lanation is ru.ther surprising and may not scom to make much 8OJl8O. Nonetheless I must also insist that there is no possibility of explaining the ancient Pythagorean names for tho proportional numbers of tho fiflh and fourlh except in this manner. On the monochord, II. fifth is obtained by first plucking the wholo string u.nd then holding II. third of it still, 80 that only two-thirds of tho whole monochord are sounded tho second time (3 : 2 = 12 : 8). 'rho ILncient }>yrho.gorean namo for this in1crvn.1 wu.s til'toltov 6tacn'1pa63 i.e. 'line 1~ (units) in length'. This dCBcription also refers to the whole monochord, for in this co.so IUf well, tho scction of string which produced the second sound (tho length '0-8' on the co.non) WIUf tho unit. This WIUf regarded as 0. 'wholo' (8.tov) and in rolation to it, the COII\1,lete monochord (i.e. tho length '0-12' on the canon) WIUf 'one whole together with half of tho wholo', namoly 0. 'lino 1~ (units) in length' (tip,o).tOV lJtaO'T1lpa). So if tho monochord is treated IUf a 'line 1~ (units) in longth', i.e. if the whole string is pluokod first, followed by tho unit (two. thirds of the string) relu.tive to which tho whole string is 0. 'line 1~ (unita) in length', then ono obtuinll 0. fifth. The Pythagorean namo for the numerical rutio of the fourth (4 : 3), lnlT(!&Tov 6taO'T'1l'a," co.n be explained in 0. similur way. On the monochord 0. fourth is obtaincd by kooping u quarter of the string from vibruting when it is plucked for the second time. ThUll tho second noto is produced by three quarters of the string (12 : 9). If this length (0-9) is reglLnled IUf tho unit, then tho whole monochord (0-12) becomes 0. 'line l~ (units) in length', brlT(!&TOV 6tat1nJpa. (The Grecle word briTetTOJI mco.ns 'a third in addition to it', i.o. 'in addition to tho unit'.) -----~-~- .. - - ~---~--~-~ .. -~ ,'., Sul'},riaing confirmation Cor the above eXl,lanation of tho terms til'tOAIOV o.nd briTetTOV 6taO'T1lpa is to be found in Theorem 8 of tho Seelio Canonis. To prove its correctness, however, it would bo ncces8IlJ'Y to fCI,rint here almost the entire Greek text of this thcorem together with it.s u(l('.()mI'ILIlying dingmms. (Tho only thing which should nut be Corgotlun when interpreting this ProllOsition [8eclio Canonis 8], is thlLt o.di(J8lema, the ratio hotwoon two numbcl'H, is always illustrated

Soo n. 24. "bovo.

~' Th., othur possibility is that tho muted aedion or string

umt, but t.his 1'0"

1',U"I' 2, l'Ut.: 1'lIIC,ll:UCl.IUUAH l'UI,I)II\' til' I'IIUI'''1I1'IOHIi

dldl11''Ipa.

BOOmB

W88 regarded 88 tho unlikely in viow or t.he related terms "JUcSAtcw and Inlrq,-

., SIlO n. 21 above (problom 4.1) or Plato's Tirnaeua, 36, II cr. Plato, TirnaelU, 36; or PhilolaU8, rrogrnont 0 (in Dillis Wld Kranz, Fragrnenu dill' Vor8okralib,.).

132

l'AllT 2.. TUK 1'1CG·";UC!.JlJRAtf TUb:OllV 01' 1'llOl'OIlTIONli

II:tAlIil: TUK lIJ.:UINtfUIOIi Olr OJtKKK: IlATUKUATIu:J

hy two 8traigl.t linea in tho Bedia OanoniB. Hence tbe whole monochord ILnu the piece of IItring which produces the second noto, the unit, 11.1'0 ropresented by two distinct straight linu.} 'fhe etymological investigation conducted above hIlS led us to two concl usions: (1) The introduction of the terms dipla8ion, hemiolion and epitriloll I " diastema into the theory of music ajlt.edates lhe division 0/ the canoll f' --0n1o twelve parts, although tho pfuportionOl numbers of the three most important consonances woro known a.l!eady at tho oarlier time -, [:.? anu_.!~~__ g~ve~ as !_a..tios belween lenutll8. The-quest.ion 'of-whliUitir /WlClie ratios were' diScovere
I

133

specificd on the canon, immodiately roveallt tho ingenious simplification which was brought about by dividing tho 'canon' into twolve parts. While tho names diplasion, hemiolion ~nd epitrit~n diastema wore current, it was necessary to seck three d'Uerent untts oLl~~!~ for measuring the monochord. In addition, this system of measuremont led to the U80 of If(ldi~'ntwo cases (ltemialion and epitrilon). I However, when a. mew>urin I' hich had already bcen divided into I twelve plLrts WIlB laid under monocbord, the wholo process suddenly becamo much simpler. It was no longoI" neCeHSll.ry to find 0. new unit ) of longth for each of tho three consono.nccs, nor to express the length of tho samo monochord in various different units. Instead of this, the length of the piece 0/ strinu wl,ich 10a8 kept from 1Jibratinu could simply be specified by two numbers on tho canon. Thus tho introduction of a numbered CILnon seoms to have brought with it the following important innovations: (0.) The word difl8lellla no longer refencd to tho whole monochord ~__ in diplasion. I,emiolion and epitrj~~ diCl.!tema), but just to tho soctlon of-~tring on tho monochord which WILlI Iwpt from vibrating. Only through this did 0. dia8tema come to be a musical interval in our scnse of the words, i.e. an interval between two successive sounds. (b) "Oeo" lite end points 0/ a,e piece of Btring 1vMch did not vibrate, expressed fl8 Itoo numbers did not enter into the theory of music until after tho introduction of tho numbered ell.nun. (Of courso the concept of lO(Joa, in tho sonlto of tho 'nLtio hotwcclI twoniiii,l,cn.:; __'~~~(~~~i~ not existliCfore'Uielntrodiictionoftlleca.noJl. On thislopic ·sco chlLpter 2.18, 'Tho creation oftho-mathcmatl00Jcoiiccpt of Myo,'). Whenthe teCh~cal termawe-liavolicon ,)iacuilSirig,vero firat introduced into mu~ic, there wore as yet no such things as I,oroi bccnuso at thnt, timo tho wholo monochord, not tho H(lOtion of strin~ which did not vibrate, wu.s treated u.s tho diasletn(& . ..----(c) After the introduction of a numbered CI~non, the fro.ctiona which ... / had been used earlier in the theory of music (I~ and 11) became ;~unda.nt. The fifth and fourth were from now on described not as Jines I~ or I} units in longth, but simply in terms of the numbers which gave tho end points of 0. certain section of string on tho canon (nll.moly that aoction which did not vibrate whon tho 8Ocond noto of tho consonance in question was produced). ~enco they c~~~~.~l!_sp~~j_fi~!!_ in terms 0/ whole numbers. Of course tho OJ(I names (for tho fractions)

i

----

.-

134

8~Au41: TUH BlCOlNKIKOli Olr OILKKK KATUIWATIC8

(b) The shorter line was considered' to ~~~be unit i~ t~ and was subtracled from the longer one. This loft tho piece of string which had been kept from vibrating as a remainder. (c) To ascertain the length of the remainder, one tried to see how many times it could be subtracted Irom the shorler 01 tl,e two linu. This could be done twice in the case of the fifth o.nd three times in the case of the fourth. because in the former CB.se the piece of string which did _!lOt !!l?flLte wa.(1UiJr~it ~_~!t~_~n~~bi~h p~~ced the secondn~!...,whereas in the latter,. it was only a third. o.s long.. .. ..Thus the shorter line was first subtro.cted from the longer one and then the remainder was subtracted from the shorter, until in tho end nothing was left over. The method described above is well known to readers of Euclid. It is the 8O-called 'Euclideo.n algorithm', or method of suuessive subtraction, as it is termed in the more recent Iitera.ture on the history of Greek mathematics.OJ In the Elemen18 there are two contexts in which it occurs. The first is in Book VII whore it is usod to find the g~atest common divisor of two numbors, The method is applied to o.nswer the question of whether or not two numbcl'B are relatively prime. As Proposition VII. 1 states: "Given two unequal numbers, if one subtra.cta successively [i.e. the l)rocess is repeo.ted with tho subtrahencls and remlLindors] the smaller from the grao.ter and no remainder meo.sures the preceding number exo.ctly, until the unit is reachod, then tho original numbers are relatively prime...aa On the other hand, in Book X successive subtro.ction is used to answer the question of whether two quantities o.re commensurable. If this method is applied to two incommensurable magnitudes, then the process of successive 8ubtro.ction will not terminate (cf. Elements

-CIlBl.

were still retained, lI!1~_~eae were no longer understood as referrin , to fractiollll, ~ ~!lelJI~~o~~:,~~~~i'beiween' whole 'numbeJ. Hence tho anCient term ;r,."iJAtov1,ilOT1i,."i, (orexample,W1i1ohor1giila1Iy mennt just a. 'lins I! (unite) in length', ca.me in classioal times to mCILIl 3 : 2 as well as 12 : 8 or 9 : 6 (all proportional numbers of the fifth). 2.8 TIIB EUCLID BAN ALOOBITIIH

'~'h.e et~mology of tho musical terms dipla8ion, hemiolion and

' I

epltrt~on duutema has led us bo.ck to the anoient period, during which

exptmments to ascertain the numerical ratios of the three most imp(~rtn.nt consonances were made with only a stretched string and wlt.hout a canon. It is wortwhile to take 0. ~loser look at the method ~vll1ch was em~loyed in these experimenta, since it also played an Important role In Greek arithmetic. 'l'he problem was to find a note which, together with the tone produccd by the whole stretohed string, would make 0. fifth or a fourth. L Wll.l:l already known at this time (from eXl)erience with windIIIstruments) tt~o.t the object which I)roduced the sounds (in this Cll.80 IL stretched Htrlng) would have to be diminished in ordor to ob... · . ....111 t I10 dCIIlred consono.ncej the only question was, by how much. Of c(~tlrse ~ho answer to this question could only be obtained after much trll.LI and orror. Whon it wo.s finally discovered how long a pieco of string had to lie ltopt from vibra.ting so that the remainder would produce a note whioh was 0. fourth or,a fifth above the one produced lIy the whole string, the section of string which yielded the second noto of the consono.nce was regarded as the unit. The last step wo.s ~ eXI~1'C88 the length of the piece of string which was kept from Vibrating as a fraction of the unit which had been ohosen. In tho caso of the fourth it was a third and in that of the fifth a half: ~rhe squence of steps which made up this method ~ay be ~ummarized lIS follows: .(a) On discovering the shorter seotion of string which, together WIth the whole string, yielded the desired consonance, one was o.ctunlly flu:ccl with tlOO linu 01 diUerenl lengtlul. The one wo.s tho wholu monochord and the other was the section which produced the second

!

IInt.o.

laa

I'ART 2. TIIK l·UK·KUCLll>KAN TtlKOItY 01' l'IIOl'OU1'IONIi

X.2). As far as I know, the origin of this method has ne~n el).!cidated.

The only thing whioh wo.s known about it

..

(from~~

work),

"cr.

•, ,; ~

~,

B. L. van dor Waeroen, EnDtJehenda JViuenscha/I, pp. 20S and 236. .. H""t.h's Lruns)"Llnn • ... net Kut. Dana," Y~(IC:RlI~!_ ~c:hA;. SIcri/ter, 'lilt. UII (VeIl vul. Raekku I, 191'7, pp. 199-3SI. - ... --

.IIUt/•.

H."""",,,,),

-

136

BUUO: TUK UKOINNINOB Olr OJl~1t UATIlJUUTI08

was that it was known already to Aristotle.- As Junge once wrote": "The method may well be very a.ncient; one is inevitably led to it, if one wants to ascertain the ratio between the lengths of two ~nes." The explanation presented above of the musical terms hemiolion and epitriton diastema suggests that the Euclidean algorithm (auccessive subtraction) may well have originated in the YytbagoreaD theory of music. More precisely, this method W88 developed in the courao of experiments with the monoohoi'd a.nd W88 used originally to ascertain the ratio between the lengths of two seotiona on the monochord. In other words, successive subtractio~~~_fi!at developed in the muaioo.l theory of proportiona. This conjecture fits iJlverj--well with-Ule other known foots about the method. Let. me recall some of these here. Euclid described the operation of successive subtraotion with tho verb dvDtltpaledv.7°The noun derived from this is d"o"rpaleSC1l,:11 Aristotle mentions this word once71 and at the same time alludes to an a.ncient definition of sament88 of ratio which Euclid did not include in the Elemenl8. This definition is quoted in the commentary of Alexa.nder of Apilrodisias as well as in the Sudo. (soo under dvcUO)'ov) and runs IlS follows: dv010)'0, 1'B)'Hh] ned, lli'1"a,
'xs,

\I

II

Aristotle, TopiotJ 3, p. J68b 29-36; of. also nn. 7, 8 and 9 above.

.. O. Jungo, 'Von HlllpiUllld bill PhlloloUB, "C1Wl8lou. at. Modl&wvalla" " RfMle Danoi4e de Philologie eI d'Hi41oire 19 (11168), pp. 41-72.

ElemmU VII. 1. 11 AluXlllldul' AphrodlalunBla (Vol. 2 of OOtlltnonlclricJ in Arl,,'o'eletrt OMOM, ed. M. Wallies. Berlin 1802). p. 645. ft See n. 68 above.

I'AUT 2. l'UK l'!lB,Jo:UCUIJKAN 'l'UXOll\' 011 l'IIIII'UU1"IONll

137

ing to do with the theory of music. It must have boon immediately evident to them that in establishing the proportional numbers of the fifth, for example, it was all the same whether 0. monochord with just threo divisiona, or 0. canon with twelve, W08 used. The numerical ratios S: 2 and 12: 8 (or evon 0: 6 for that matter) were equal to eooh other. Furthormore. in all these C08e8 successive subtraction applied' to the two quantities concerned yielded t.he same result.]

-----

"

2.0 TnB CANON

In chapter 2.7 [points (a), (b) and (c) on p. 133] I summlLrizcd the principal ways in which oxperimcnts on tho monochord woro fu,cilit.ILted by the introduction of 0. numbered measuring rod. Now. however, I would like to talk about some other advantages it had. It was well known from musical practice thlLt. if 0. pair of tetrn,chords were joined togother, the two outermost strings (the .melT'] and the produced an octave.73 Furthermore. it WM recognized thn,t tho octave W88 compoRed of two smallor musical intervals. Aa PhiloltLus wrote in the fragment quoted previously:7t "The extent of an octave is 0. fourth and 0. fifth ••• For the distance botween tho topmost (tman,) and the middle string (plCJ'Cla) is 0. fourth a.nd tho distance between tho middle and the bottommost (bti "eoTav) is 0. fifth. OJ //' Now the question is whethor these same fo.ots can also be iIIustmted ( on 0. monochord, i.e. whether it can be shown on 0. monoohord that ! the diastemata of the fourth o.nd fifth together make up tho dillstema \~of the octave. Of course it was impossiblo to give 0. satiltfu.ctory answor to thia question 08 long as three different units (depending on whether an octavo, fourth or fifth W08 being considered) were used to measur~ t.he length of the monochord. When 0. mOllSuring rod divided into four eqUal parts W88 placed beneath the monochord. however, it Immocliatoly booomo much OlUtier to provido l~n 1~lUtWor. Tho dilulenu, (the piece of string which was kept from vibrating) of the octave

'*")

70

,. 8ee n. 21 above. fa See n. 24 above.

13M

could now be characterized in terms of the two end points of the string by the numbers 4 and 2 on the measuring rod (4: 2). So it WIU! olUlY to show that this diastema WIUI in fact composed of two slMller ditl8temata, namely the fourth (4: 8) and the fifth (3: 2). An octave is a ratio composed of 0. fourth a.nd a fifth,11 more precisely, on the monochord the length of an octave (4: 2) is composed of the lell!!th of 0. fourth (4 : 8) plus the length of 0. fifth (8 : 2). 'el1Us experiments with the monochord and numbered meaauring !'C!d revealed the samo faots aa had been learned from musion.l practice. Suoh experiments with 0. oanon, which perhaps waa originally divided into only four llo.rts, may ruso have helped to give Tere~the almost mythical meaning whioh it had for the Pythagoreana.18 It should not be forgotten, however, that there a.re no sources which mention 0. on.non divided into four parts. Although the invention of the canon is traditionally attributed to Pythagoras (and whatever onc may think of the half legendary figure of Pythagoras, this dating ill not to be doubted), the sources for this tradition either do not discuss the way in which the oo.non was divided, or they only ment.iqn that it waa divided into twelve part8. Hence the existence of 0. on.non with four division can only be conjectured. Nevertheless it is quite IL 1'1ILIlsibie conjecture, if one bell.rs in mind the llroport.iono.l numbers of the consonances (4 : 3, 3 : 2 and, putting these two together, 4 : 2) ILIIiI t.he Pythagorean tetraklys. On closer inspection the disadvantages of a canon with four divisions (if I\n instrument of this kind was ever really used by the Pythagoreans) 1\9 opposed to one with twelve, soon become apparent. As an example, let us consider the following case. Of course in musical practice it was immaterial whether, given 0. 1I0to, one which differed from it by a fourth W(\8 found first, fullowed by I\nother differing by a fifth from the latter, or whether the order WI\9 ,·everscd. In either OU.8e the combination of the two intervals l'esulted in a tone whioh diffel'ed from the originru one by an ootave, It is impossible, howover, to illustrate this fact (that tho order in \Vhi(~h tho two int.orv(~ls al'U found ml~y bo roversed) by moans of a moon with lour divisions. In this case it is only possible to combine

,. n. L.

1'AUT ~. TUK l'llK-KUCJ.lIJI;AN 1'1I~:Ull\' 01'

ZI%AJlcl: TUK JlKUUUUN08 01' UIlKI.:I" UATlllWATICS

VWl der Waerden, Erwacl,ende iVi886f&8chal'. p. 183. ,. Ibid., p. 167.

l'llOl'OIl1'ION:i

139

tho two ditulemala if tho fourth precedes the fifth (4 : 3 and 3 : 2 together make 4: 2). On the other hand, a canon with twelve divisions ontLblcs ono to choose 12: 9 (ll. fourth) followed hy 0 : (} (IL fifth), or alternatively 12: 8 (a fifth) followed by 8: 6 (0. fourth). In either case an octave (12: 6) results from combining the two diastemata. '.rhe reason for dividing the measuring rod into twelve pll.rLs ca.n now be apprecil\ted lUI well. We nced only bear in mind that originally tho length of tho monochord WIUI measured in tl"ee diOerent units corresponding to the three most important consonances. In the case of an octave, tho monochord was divided into two parts; in the case of a fourth it was divided into lour. and in the case of a fifth, it was divided in.:o t"ree. To illustrate these three consonances on a single monochord all at tho sl\me time, it wu.<J first necessary to find the least common multiple of 2, 3 and 4, which is exactly 12. Apart from n.\lowing a fourth and 1\ fifth to be c~mbillc? in a.ny order 1\ canon with twelvc diviHiollH ILIHO mlLlces it posslhlo to Illustra.te another interesting conneeLioll between the numerical mtioll of the consonances. I n.m referring til tho fILCt. t1mL inlLllciont HIIIlI'CCS Lho int.crvlLlllf thc fifth is frequently descrihed lUI being urf!iller thll.n UllLL of Uw fourLh. Let me remark here in passing that thill 11.<Jsertion by i~self nlready roveals tho wll.y in which it WIUI CUHtOUIILI'Y to mell.<Jure JIltervll.ls Oil { ! ]the canon. Tho longer section of string. or the wholo monochord, ~must always havo heen plucked bofore the shorter ono. Henco the pnirs of numhers oht.ained were 3 : 2 ll.ml ·1 : 3. The way in whieh " the ancients thought precluded the possibility of reversing t.he order ( of either pu.ir (i.e. of It.s.'iociating 2 : 3 with the fifth, or 3 : 4 with the ~ fuurth).77 Jf tJUch lL ClllLllgO Imll heen IIerlllitt.cd, the n..'lHllrtioll thlLL the fifth was greater_~h~n_1b(,'.19.\'l!·~.b ~o~!~_~er~l~i~\~y_~ot~l~~~.~~~.:?.:r:~t. (j\lthougl1!lli-> 4/3, 2/3 < 3/4.) Now this ClLl1 lio shown very easilY on a cl\non with twelve divisions. In thiH cu.
'--

14U

tlZAUc): TIJ~ UKU1NN1NOti Olr OItKKK UATUlWAT1C8

l'AnT %. 'l'UK 1·1l1l:·1l:Ul.a.1JJEAN '1'l11l:UllY 01' I·UUI·Ull'l'lOr.1I

II,e fililt 111a8 greater liUlI/, tlle fourtlt, namely by tlte lengll' between ti,e numbers 9 and 8 on tlte canon. As Philolo.us wrote: '8 " ••• the fifth exceeds tho fourth hy an bro,,6oov (9 : 8)".

2.10

ARITHMETICAL OPERATIONS ON TIlE CANON

r

It is worthwhile to take 0. cloKCr look at Philolaus' assertion thiLt tho fifth oxceeds the fourth by an e.J!E{J!l.Ofm". Of course ancient literature on the theory of music is full of such statements. For exam. pIc, there is a passage from Ptolemy's Theory 0/ Harmony whh:h discusses the subject matter of Philolaus' remark in somewhat moru dctail.7t This IJassnge is of interest to us chiefly because it talks about t he proportional numbers Itemioli08 log08 (3: 2) and epitrit08 log08 (4 : 3), instead of fifths and fourllts lLnd goes on to say that the 'difference' (VllEeOx~) between these two ratios is an epogdooslog08 (9 : 8). Tho mothod one would use to olltrun this difference (between tho fl'/Lctions 3/2 and 4/3) is obviously division; for 3/2: 4/3 = 9/8. Yet the Greek expression for 'difference' used here (WrE(!ori) customarily !'Cfers t:o .IL .di~erence o~/ained by subtraction, not to a 'quoticnt resulting from dl VISion. Thcre IS, for example, 0. pu.ssage in Porphyry's commen. tury on Ptolemy's Plteon} 0/ Harmony in which it is explicitly stated t1mt nlthough the pn.irs 6, 3 n.tld 2, 1 hlLVO tho samo log08 (6 : 3 = 2 : I). theil' differences are not equal (al 6'VxE(!OXall1vUloCj.80 In the one CMO the difference is 3 (6-3 = 3) IUld in the other it is 1 (2-1 = 1). So tho wurd V1lf(!OX,1, jUllt IiI(O ita OPI,ollite, 1llcI"c" is undoubtedly I~ teclmicul term which hod to do with subtraction' both words denote

~

Ule difference.

U

\~.

,i

•••

.

Tho c)uUlltion now ill why tho qtlolieul of tho ratioll :I : 2 and 4 : :I (i.e. 9 : 8) WD.8 called a difference in tho theory of proportions, oven t,hough the smallor of thCKC two ratios (4 : 3) did not represent 1\ Huhtrahend hut a divisor. I beliovo that this question can be eMily 7. Run tllII (lruvimlH 1I0t.,. '1'hiH f"IISIIIIIIII. of I'hiloluuB ill rurthur uvitlunclI rur th .. ,uel. thut tho canon with its t.wulvo divisions was already in existenco ut his time. ,. Die IIaNtlol&iele/lrlJ des KlaudiOiI ]'lolemBioll, ed. r. DUring, Berlin (1030). 1'ho whole p08llllge is translated in VIlQ dor WlI6roen, Hermes 78 (1943), IIlOrr. 10 Dilring'8 edition, p. 92.lff.

,

l,

',--

141

answored on tho bMis of Homo facts about tho canon which I explained above. The proportional numbers of the fifth (12 : 8, Ilemiolios IO(Jos) nnd fourth (12 : 9, epitritoslO(Joa) both referred to lines on tho ClLIlon, tho former to tho piece of string between tho numbers 12 ILnd 8, and the latter to the piece botween 12 and 9. In order to find out how much smaller tho ratio of tho fourth (12 : 9) wus thll.n that of the fifth (12 : 8) WD.8 on tho sarno canon, it WIIS neCC8So.ry to subtTlLct tho smlloller line from the greo.tcr. This left IL 'differeneo' (VllE(!OX11), IlILmely tlle line between tho numbers 9 and 8 oii'£Iiecarion. ... '" -Thu~:ifi-;'the-GreclC: -tIl~~y~f p~~p~l:tion~ t-fie-qtiotiont of two rlLtios is deseribcd D.8 a difference, the historicn.l cxphmation for this remarkable fact is tllJ~t tho operation which toda.y is undcn!tooll IL<; the division of ono fraction by a.nother, wus originally conceivcd as ILl! ol,cmtioll on 0. canon with twelve diviHionH !mci WIL'i Himply the

subtraction

0/ a 8/lOrter line from a lonoer one.

Of course it is not only tho term WrEeoX~ which testifics to tho flLct lhlLt IIscel'to.ining tho 'difference' botween tho l'atioH 3 : 2 allli .. : :, >WII.s originally not a matter of division for tho 01'001"", hut (Mincc thiH , operation wus carried out directly on IL 'canon') of subtl·ll.ction. ~Indeed. it even scoms that the operatiun which wo would dcnote by :1/2 : 4/3 could only bo described in tcrl~1l of liUblitiCiiCm' in Ul'eol(.TJilsexplo.inS PropoHitioll 8 of tho Seclio Oanonis .;- iliv d3t;r-~.u,o,l{ov c5uurn/paToe; illIT(!LTov"uloT'Ipa o'rpal(!el}jj, T,IAtl/1ft)" )ttlTalf:lll£Ttu 11l,iytS()o" (i.e. :1/2: 4/3 = IJf8) which Heiborg tmllldlLlcll u.s: uSi ILh intcl'vlLllu IJImqueo.ltoro intervallum scsquetertium IUlferiur, quod reiiJufltitllr IICHflueoutlLvum CHt." Tho vorh formH ,i9""I!/:I'}ii ILIUI )taTuA";1lf:TtU ILl'(: characteristic of 8ubtraction, not division. (Compu.re this with A'lemenflJ Doole IX, Proposition 27: leiv cind llE(!tl1omi ,If!t{}pov lienol; a'pull!E'Orj, II Allmu!: :1tl!t!tctlfue; 1UTu,; "Ili lL 111111101'0 illll'lLI'i lllLI' .·mblmhitur, ,,"'i'll/Wi implLr erit.") ~rh()Ho terms from th~ Groek thcory of I'rol'OI't.ionH whic:h Ilclieriho tho multiplication of one riLtio by anothor u.s "dditi01I, eu.1l he explaincd in the 811.me way (as originating from opcrlLtionH on 0. canon). I hl1 ve ----J.LlroILCly montiulJ",l tllILt. tho octlLVO WIUi (:IIIIHitll,mll to 1111 IL t:olll/ltlltUti interval, dUlt1T1]pa uvvr:dJiv. If it is borne ill mind that the picco of string whkh had to bo Impt stm in orelor to produco the second note of ILn octavo (i.e. the lengll' betweon tho numbel'S 4 and 2 Oil I~ CILIlOIl with four divisions) WD.8 actuwly compoundcd of the lengtl' of I~ fuurth

142

, .'.;x---

l'AIL't :t 'rUG l'llK,KUCI.umAN 'l'UKOllY 01'

IlZAlI41: '1'lIl1: 1I1l:01NNU11I1i 0 .. UJll,;t,;K )lATUJo;UA1'1C8

:"j/(tho piece lying botweon 4 and 3) and that of 0. fifth (tho pieco lying I ( between 3 and 2), thon this use of 0. term for 'addition' immediately 1\ ~ecomes comprehensible. The verb (IvvnOI"at is nothing but tho tech'-''1lical expression for adding. Pol Obviously the two ratios 4 : 3 and !J : 2 (or oven the fr"ctions 4/3 and 3/2) conaidered by themselves must 110 1nuliiplied together, not added, in order to produce the ratio of ILn ()ctlwo (4 : 2 or 2 : 1). Tho unoxJlected usa of 0. term for o.dtlitiun, instead of ono for multiplication (nollcmAaaldCetv), is explo.ined lIy refl'ronco to tho original operation on a ell,non. The addition and subtrnction oflengths on 0. canon had far-relwhillg effects not only on musical terminology, but also on the terms ulletl in the Greek theory of proportions. Even Euclid in the Elemeliis / frequently emilloys the expression avyxe{pEtlo, ldyo!;, whore the verh i t1vyxeiaOat doea not have its uSUlLI meaning of addition hut refcl'" instead to multiplication. Let me give aome examples hero to iIIuah'l\to , this: l!Jiemenls VI.23: "J~C)uilmguhLr panLllelob'Tams have to ono ILlltlther the ratio compounded (MYo" lXEt TOV avyxelPEtlov) of tho ratiull IIf their sides." Elements VIII.5: "Plane numbers have to ono another the rcllio compounded (loyov lXova,,, TOV (JVyxelpevov) of the ratios of their sides." Another eXIl.ml'le ia furnished by Definition 5 of nook VI, which is pmh/Lhly ILn interIloIaLt.ion: "A rat.io is siud to he compounded ofnLt.ius (Myo, Ix MyCLIv avyxEiaIJat ).iyaal) if it is formed by multiplyi"g I o~cther tho mlLgnitudca of rILtios."n '1'ho expression (/t)yxelp£vo!; ).dyO!; (compound ratio) derives originlLlly f!'Om an oporntion on ,the canon, WI docs the related technical term IknAmr/(dv Alf)'a, (,iOllbit't1 mtio) whi"h "pptllLI'!! in Dofinition V.n of thll J~lelllents. Concerning the latter, van der Wll.Ordon hll.H written:u .. Dnuhling n. mt·jo clerLrly corrcspomls to multiplying a fmotion lIy iIH,·lf, II'ur if" : II -' b: rl, thou II/I: (II/b) , (/I/C) = (11//1)3. 'I:ho Ul'tlolut

=

I' li!ucli,1 u""!llly ulIIl,lnyK tllII vurl'" f1t",,,OII'"' IlIIII attyKdaOru to mUlln IIUtl. fur uxuIIIJllu, 1~lulllllllta

n,"'.

IX.21 ami 22. II It. is worthwhilu to rocall hero Thoera' rumark (to be found in his truOIIlu. t,ion Die Ekmlenleoon Euclid, part I. n. 60): "Dulinition VI. 6 is pl'ObublYBJlurio\1s. but it muy Willi bll llUrt. of a th~ory of proportions which IlIItodut.ca Dook V." .1 Math. Ann. 120 (1947/9), 133-4.

l'IICH'Oln'It)N:I

were certainly no less able to mUltiply frn.ctiolls then we arc."l\Ioreover. Tannery bad already tru.ced tho expression 'doubled mtio' blLck to the theory of music.8' Tho only thing which I IULve t.o ILdd is that IL)though it is of COUnlO Il.ppropriate to refer to tho theory of music ill this context, it is not by itsclf sufficient. The technical t.orms we luwe hecn diseuasing (including the expreasion 'douhled ratio') remain unexl'laLincd unlO89 mention is ILlsu mudo of tho l\l'iLhmetiClLI ul'cmlionH which were carried out on the canon.55 'fhere ILre two other lloints worth malting in cOllnection with these ol'erntiolla: «1) Frank was already aware of the fawt tlmt tho Grecl<s used t.he worclH 'nddition' and 'subtraction' in the musical theory of proportions to mOILIl 'multiIllicl~tion' and 'division' rcspcctivoly.80 Ono CILll hardly hlaLmc him for being puzzled by this. Yet instead o(mlLldng a sCI'ious I\ttuml't to oxplain this surl'riHing fact, he hit upon tho fl'ivulous idca thlLt 'eomllounding l".I.tios' was too complex an operation for the Grecl<s to luLVo heen lI,hlo to perform it prior to tho 4th century. VlLn der \Vlwrtlen's responso to thiH wu.'i: 87 "Compounding raliull ill this WILy is such an olemento.ry and obvious operation that I do not understlLnd why }i'raLnk insists that the Greeks did not IULve /~ clclLr idea ahout if heforo the time of ArchytlLH and Eudox~1tJ." Van ~ Waerdcn WII.'i completely corrcct in rejecting Frank's t:t!.volouH vio~on the other lumd, however, he fn.i1ed to understand what Fru.nf, found 'pl'Ol>lenmt.ic'. (110 did not IUlo tlmt l!'l'ILnk WILH UlULhiu lo l'cHolvc tho confusion which 8eemed to exist in the Greelc theory of proportions between lLCidition /LIIcl 1n1tlliplimlinn on the one aitln. ILIl(I IIlIhtra.(:tiun 1",,1 division on tho other. ~o neither t.ho philologiHL nur t.hl! IIULtlwllIILlichLII lIucceeded in expltLining the fact whieh had puzzled l!'rlLnl,.) / (2) 'l~hCl uUll1r fiLUl, whic:h I\lUi t.u 1m mnphn..'ii",llti hlll'l\ iH 1,1111 rClIIClwill~: /' In tho Crtltll< theory of proportions the WOI'cis lill' additioll ILlltl sllh· trlLcticm !Lro indeed uHe(1 t.o mOI~1l multipliClLtiulIlLllcI division mSJlectivcly. 't:hiK IIHILgO llClIIIUH 1'1'11111 I~atl l,illlU whma !'ILI,illn WI'I'CI nl,ill clllkullLl.c'cl (liroctly by lengUlB on a e~ If, however, 0110 considors tho theory

c....

Van ,Iur Wutlrclull' HUU 1.1", I'ruviuulI IIOW, Durkurt, lVei8l,m und lI'ilUJBnscillJ/I, p. 348, II, :1. I""'" tim \Vlml 'Str.,c1wnvoralollung' in thiH (lonnnCJI.iun. II 1'/010 urul die 80g. Pyt""goreer, pp, 168-61, OH)Jllciully p, 16M, II, I, "IIennes 78 (1943). p, l7D, N N

,

"

8:1:AnO: l'UK UIl:U1NNINUit OJf UltKKK AIATUW4ATI08

uf 1'I"Oport.ioll8 us preaontutl in Euclid, ono BOWl that almost tho sarno expressio~ ~or ad~tion and subtraction ocour in its historically later parts, tlus time wlth the meaning of real addition and 8ubtraction. In Definitions V.14 and 15 of tho Elemenl8, for oxample, the words 11".'0£(111; Myov or dU1letCl'~ Myov, and refer to o.otual addition a,nd subtraclion of the terms of M)'o" not to tho addition (i.o. multiplicat.lon) or subtraction (i.o. division) of two M)'o,. Of course this later ul:Iago III~ nothing to do with oporatioll8 on the canon.

.meeon

2.11 TIlE TEOBNIOAL TERM FOB. 'RATIO' IN GBOMETRY

[n tho last chapter wo disctlS8ed some arithmetical operations on tho 'canon' ('addition' and 'subtraction' of diastemata) which had 0. u.sting effect on the terminology used subsequently in the theory of proportionB. Such important expressions as 'compound ratio' 'doubled . ralio', 'diDerence between. two ratio8' etc., whioh in reality' refer to multiplication and divi8ion, would have to be regarded simply o.s "'- IinguiKtio onigmo.s ~eaoh of them could not bo traced baok to 0. part.icul~r opo~ation on t!lO canon. We shall have to return to tho subjoct of Imthmotlcal oporatlons on the oanon in a IILter ohaptor, but for the t.imo being let us intorrupt our d1soussiorf oftlilfmUsioo.l theory of proportions and devote tho next few ohapters (2.12-2.16) to the "lInllicloration of somo other geometrical terms. It. is o.stonishing that the musical term d~pa (the ratio between t.WCl numbers) is never used in geometry. T.ho word d,dcfn]pa in geomotry means 'line' and never anything ~e~or example, Euolid's third postulate, which has already been quoted once, states that a drclo can be drawn with any oonter a.nd ILny line 8egment _ nanl dlal1T~par&.) In geometry the concept of ',alio between two number8 or qu(mtities' was not expressed by d,dcfn]pu, but by the word Myo~. Tho (.'Onjecture that these two terms (M)'o, and d,dcfnJpa) must IlILvo something to do with eo.oh other is confirmed by the following two fn.cts: (I) As wo.s mentioned above,88 M)'o, and d'OcfnJpa were equivalent concept8 in the Pythagorean theory of musio.

~

(2) Just. 1'8 a musical diaatefna had two 'ond points' (oeo,), su in geomet.ry ono referred to the oeo, of n.lo"o, (us is shown by Definit.ion V.S of the Element8, for example). These two facts, however, do not tell us anything about the origin of the conool't. Mro, (a: b). In Ilarticular, we do not know whet.her ---, this term originated within the theory of musio, perhaps as 0. synonym ' for 0. musicILI d,acfnJpa, or whether it existed ea.rlier in geometry Imd was only subsequently equated with 0. 'musical ,stacfnJpa. To be able to answer these questionB, it will be necessary to conduct 0. more thol'Ough invesLigu.t.ion .of the geometric term lo)'o, and its etymology. It'or the time boing, homer;' outpoint-of departure will not be the term My~ il6elf; but the Greek name for 'proportion', i.e. the word ci)'aloyl~. I have already mentioned (in chapter 2.2) that the four-place relation of proportion (or sameness of ratio) WILS called civalorla in Greek geometry. Our first to.sk will bo to invcstignte how the word civaAoyla acquired this remarkable meaning.

I

--P

III

Ruo p, 113-4.; also nn. 33 and 34.

2,12 'Al'aAoyla

AS 'OEOM~10 PRorOR'flON'

This invest.igILtion gILins 1I0mo n.dded interest fl'OlJl tho fact Umt t11e word-~ st.ill in current uso, although its meaning ho.s changed , somewlll~t.. Furt.hermoro, wo frequently speak uf 1m 'amLlogy', "y '-wliicllwe mean something other t.hu.n 0. 'geometric proportion'. Let us now tako I~ loolt ILl. the history of this Greek word. Modern vlnianw uf t.ho word ci)'aAo),ia are to btl fuunt! in alll~ur(j"clLn languages. Moreover, they all have roughly tho sU,lIle meaning. 'Ana.lugy' mmmH HimiIlLrit.y, conformity, relationship, or tho extension of 0. rulo to similILr CIJ.l:lCH. 'fheso meanings alroudy nmlm it clear thlLt the modern word dorives from the Greek !/rannnatical term (by way of tho Lu.t.in word analogitl of course). Grook gr-d.mmariILr18 have used this word with its present day meaning since HelleniKt.io times. It is Ie&:! well known thILt this mmo word WlUJ not originally 0. grammatical or linguistic term, but 0. mathematical one. The root of the word civalo)tla is obviously lOyo" whioh in mathematics meant the 'ratio' bet'wClon two numbers or quantities (a: b). ·Al'aAoyla itself 10

BIAW

1·16

SUue); TUK UK01NN1NUli Ott UllK"lt r.tATllKr.tAT1CII

I'AUT 2. 'l'Ut:: l'lU·;·KUCJ.IlJKAN TUJ::OllY 01' l'ltOl·OII.TIONIS

•

147 I.j

dl~He .. iht!tl

IL 'plLir of rn.tios'. li'ollowing Cicero,8' it was translated into LILtin hy proIWrtio (a: b = c : d). 'fho Gruel, grammarians of Hellenistic times undoubtedly borrowed their term d"aloyla from the lan(. gUllge of mnt.hematics. So in the l88t analysis wo are indebted to Greok mathemlLties for our word 'analogy'. Ollr principal concern now is to find out tho original and precise mennillg of the mathematica.l term d"aloyla. Before attempting to do so, howcver, there are two important fn.cts which havo to be stated first. ( I) 'l'ho expression d"aAoyia did not figure in the everyday language 01 the areeks, i.e. it WBS not 0. part of the common vocabulary of Gred( ILa it WBS spoken in classical times. Thore are, it is truo, some pa.sslLges from PlatollO which almost lead ono to believo that at his t imo tho word o"aloyla had already been adopted into everyday speech, hut in reality all tho evidence (including these same passages) points to tho fact that a"aloyla WBS actually an artificial eoinago, lIOl .~at'allt. Moreover, it originally hnd no moaning outsido mnthoIII1LLicH, Imvi'!.~ beon coined by matliomlLticians to oxpress n matholIlatical notion:--.subsoquently the word was takon over from mntholIlal.it,s illlo-~ducated speech and In.ter still it WBS transformed into 0. It!cllllit!ILi Ltlrm of ILJlothor branch of ICILrning, IlILmoly grammar. 'fho flLct l.hlLt it became part of the common vocabulary of Greele shows t.ho PllormOUR influonce which the mlLthemn.ticR of tho 6th century oxcrlt·tl till tho whole of oreo I, thought. As wo shlLllsoon BOO, tho term (j,·a).oy/u WILa coined in the 6th century and by Plato's time it was alre'lId), Ifuil ..\ n. com mOil worel in tho RIHlelch (If tho edllca~l. (2) Jt. would appear that the compilers of those Greek lexicons, "'hic!h ILrtl most widely used nowadays, did not know tho exn.ct and lruo IIIl'lLllillg of tho word d"aloyla. AILhough thoy record tho meaning of thiH word in a mannor which initially satisfies tho uncritico.l reader. Uu'y mvmLI un other oCClLSions t1l1Lt thoy havo not really understood it. Let me illustrate this point by quoting some BlLmple passages from thcso dictionaries.

C;

.. 7'irna6US S6U de Uniwrso," 112: "Id optimo 0888quit.ur, quao Oraece tbaAoy/a T.utill" (.Ulliolldum cst. enim, qlloninm hooa prinmm 0. nobis novnnLur) compm'Olio proJlOrtiulJe dici poteaL... 80 Stili, for example, Plato's Politicus 267b. or Aristotle, Nicornachean Ethics, 0. :I.

In PILSSOW'y dictionlLry (1841) dvaAoyla is oxplained as follows: "entspreehendcs oder richtiges Verhii.ltnis, Proportion, AIlILlogie ete." [corresponding or correct ratio, proportion, analogy]. Similarly, Pape's dictionary (1849) gives: "diLS richtige Verhii.ltnis, Proportion, 'Obereinstimmung etc." [the correct ratio, proportion, conformity]. Finally, Liddel and ScoLt (1948) list under d"aAoyla: "mu.themILtical proportion, analogy etc." (ILnd thon goon to quote tho most important authorities). Naturally there is for tho most part no difficulty in applying the meanings given above to the vo.rious ancient texts)n which the word occurs. NonethelC88,~ _S_~!l!u~ono attemptH~ find ~the dorivati
ova

-II Tho aarno holdll for Diols Rnd Kranz, Fragmmte der Vorsokraliker, 1, p. 436, whore t.bo JlbrCUIU d ..d AOyUII ill t.llo IlUcom) Arcbyt.us·rrugmunL ill t.rwullut.ud by t.ho Gennan word ·unalog'. I havo no idoo. what. the translator had in mind wbon ho wroto t.hIH.

'.I'i 'j ,1:

.'1

!!;

II'

I!

l.j Ii

:;;LAIIIl: l'UI' IIKU1HH1NtJII Oll UIUmK MU'UKMATICII

An exact undorst.u.nding of tho mathematical term d"alo)'la C;Ul IInly Lo gaincd Ly st.u.rting out from the following, rather obvioulI, flLct. Although the noun d"aAoyla sooms to be a direct derivativo of tho Ildjcctivo dvaAo),o" this latter is not the root word whose explalllL' tioll will lead us to the solution of our problem (tho meaning of Ilvu).o),ia). It, would appear that the adjective d"dAoyo, itself WILa derivcd from tho adverLial oxpression o"d.lOy01l, i.e. originally thcro WU8 only the llhmse d"d. AO),O", It is written as two words not only in /:Iomo parts of Plato (e.g. in the PIUledo, 1100.: dva TOV at5Tov .l&yov), Lut also in a fragment of Archytas which "reprcsonts perhaps tho only surviving, authentic mathematical text from the time before AUlulycus and Euclid"." From this phrase thereoa.meon the one IULlltl the familiar expression dvdAoy01l (see for example Elements, Dofinitiollll V,U, VII.21, etc.), and on tho other hand the adjectivo dvaAOYo, IL'I well iL'l the noun civa.toyla. So lot ua now try to find out tho meanillJ; of (j,'UAO),OV or d"ci lOyo".

l'AII'r 2, 'l'II}: I'JII':-}:UCLlOJ';AH TlIXOIt\' Ul' l'JltJl'IlIl'l'lOl'i::l

"

In tho Elements there are two different usages of the word dvaAOYo" which havo to be cleurly distinguished. Let me give an example of each of these here. (In both eMes the Greek text is quoted together with Timer's German translation.)" An oXILmplo of tho first usago is Definition V.O: t:

ra

TO"

aUro"

tl,,&.to)'OJI

l1.01ITa AO)'01I JUYiIh] xaktal}{J).

Die dll886lbe Verhiiltnis habenden GrOssen sollen i" Proporli07' 81ehend heisson. [Magnitudes having tho same ra· tio are said to be in proportion.]

Tho other uaago can bo illustrated by the st"toment of Proposition VI.12:

re"jj"

60iJelaiiJv £VI)elwv

dvdAoyo" neoaeueeiv. 2.13 'A"dAo)'o"

14U

TacleT71"

Zu drei gegebenen Strecken dio viertc Proporlionale zu finden. [To threo given str"ight lines to find u fourth proportional.]

The fil'l:lt thing which IULS to ho empluLSized is that our knowlClIJ.:o til' lhe expre8l:lion civdAo),o" comes exclusively from the language uf lIlathcm!Lti(~.v3 It. is tl'110 t1mt on ono (lccMion PItLto (Pluzedo, 110(1) IISl'S (IVa Mrov !L1Il1 d"a rl;v ).O)'OJI in contexts which al'6notoLv juullly

avro"

IIllLthematicnl, none tho 1068 a closer inspection of the 1'l188ages concerned SIIIIII revolLls thnt l>ItLto is in fact using a mathematical term. In !Lny cvent, the meaning of this exprossion can only Le explained by refer. ('lIl'C to lite lanyuage 0/ mallu!111alica. .. R"u O. 1'opliLz, Quelle" urad Stwlien .tur Gescl,idlle der MallU!malil:, dtl, n, 2 (19:12), 288, n. 6. Suo also 11(. Timpanllro,Cllrdini, Pitagorici, Testimonillll:e e [·'rtlllllllellti, lillie, 2 (Flonlilflu 1962, p. :172), U Bllrl(ert. (JVeisl,eil "rill Wusmsclm/l, pp. 414ft.), who bolioves that th" IIl1ltlu.maUcul wrm ~ WU8 derived from the languago of finanoo wrilA.'11 miI41.'u.lingly: "Whun SIlvorul quantities 'oorreapond to thu sarno umo~t' thuy wcru Iklid to bo til'li Uyo,,". (Tho quantities wure coins of difltlrent denominatioll IIr "urruunillll, J I fail to 8et! how he can make such an lISIIurtlon without eiting lilly IIvidullOU fur it.. As fur U8 ( Imow, t.heru III no unolent tuxt where tho I'hrulIlI ill wred with this meaning, In Cuct there i8 no evidence that dvd Adyoro wos UBod lit nil ouuide rnatlllllllalics,

As we can soo, in our first ex "mIlle tho word dvdAoyo" refers to "II thoso magnitudes which constitute the proportion, and is tmnsl"ted by Tluulr Jut "i" Proportion sle"e"d", whereas in uur secund eXILml'Ie the word refers to only one magnitude (TaUer" dvtUoyo,,) and hence is transl"ted as "Proportionale". The latter example would suggest that oach ono of tho magnitudes which goes to mlLke up tho proportion is itself an dvaAoyov. However, wo are left with tho question of what exactly an tl"dAOY01l is, and why the word does not decline. 1.'he connec· tion between these two usages is by no means settled at the moment. It turns out that the usago which occurs in tho second quotation is tho later of tho two (Le. it belongs to alu.ter stlLgoof linguistio devel· opment). Hence we shall return to it in u suLsequent chapter Ilnd deal with the other ono fil'Ht. Our starting IJoint will be Definition V.O; moreover, we will leave tho word QvaAo)'ov untmnsl"ted fur tho timo being. Thus our provision!Ll "C. Thllllr, Die Mlelllellle von Euklid, I..oipzig 1933-7.

tlZAIlO: 'l'UJJ: UJJ:UIH1UHtltl Olf 0IUU~K UATIlIWA'J'ICli

l'AIlT 2, Till!: l'IIJJ:·J.:UtJl.UH>AH

'1'Il~on\'

0 ... 1'1\01'OIl'1'10HII

151

int""I'I'I'Ot.ILliun of t.ho dofinit.ion rUIlB u.s fullows: "JIagniltuk8 having tile slime m/iu (),oyut;) are 8aid to be dvUAoyov." I t ill necessary to know what a mathematical 'ratio' is, in order to ulldon~tand this dofinition. For the moment, wo will accept the Eucliclcan conception of Myot;, ILCcording to which tho lOyot; between two numLcrs or magnitudes (a and b) is just a : b (8.8 wo would write it nowll.dILYS). I should perhaps mention right away that the Euclidcan (!OIwcl'tion of mathomatical mtio (lOyot;) is by no means the originlLI OliO. We shlLlIsce later that in pre-Euclidean times lOyot; W8.8 a somewlmt mOl'O comprehensive notion; but this fact need not concern us jUlit now. 'rhe Euclidean concept of lOyOt; is sufficient for our present I'UI'P()1I0 of discovering the meaning of civtUoyov. If we know wlmL the IIl1Lthematicu.1 mtio (J.oYOt;) of two magnitudes is, then the interpretation of Definition V.6 poses no problem. The definition clearly stlLtC8 that if the mtio between two magnitudes a and b is a : b, and if the ratio betwcon two other magnitudCll c and d is the 81Lme 8.8 that hctwcen a and b (in our notation, if a: b = c : d), then these lour magnitudes (a, b, c and d) collectively are said to be civdloyov. Thill interpretation also u.ccords with another definition in tho Elements, which is arithmetical in nature. I am referring to Definition

to each othor o.s the other two. Wo cannot apply either of the preceding definitions and speak of d"dAoyov unlcss there ILro at loust four numbers or magnitudes present. all Of course this interpretation does not contradict Definition V.S of tho J~lements, which expressly states that "the le8.8t analogia conllists of t',ree terms". In IL C1LSe of this kind (the ci"aloyla t1VV£x~t;, 8.8 it is called) tho middle term is taken twice (a : b = b : c), so that all in all thero are four terms, not throe. Aristotle h8.8 explained this point in greater detail.81 The other important fact which follows from the preceding definitions (V.6 and VII. 21) is that the word ci"&.loyo,, describes sameness 01 ratio (lOyot;) betwccn two pail'S of numbert! or magnitudes (i.e. it describes the relation which holds between two pairs of numbers or magnitudes each having the same mtio - ro" aVro" la,,~!,): Its meaning would be expressed in English by tne phraso lin the same mtio'_ Aristotlooa also attributes this meaning to tho word civalorla. He writes: "Analogia is 8amene&8 of ratios, i.e. lOyot."

VJ1.21:

Our next task is a purely linguistic one, namoly to explain how the word d,,&.loyov is able to express the notion of 'being in till! same ratio'. The word lOyOt; clearly means ratio in this context, so our task reduces to explaining how t!lJUOlily between ratios can be expressed by pre~-., fixing tll~prepOBitiQI) a)-a: 'I-have already mentioned that the adverb ci"&AOY01l W8.8 at one time written as two words (dva AOyOV). It.)akea this {orm in the second fragment of Arehyt8.8. We now want:.~ find ou~xa.ctly what the preposition d,,& means in tho expression d",uf?)'ot; and d"alar1a. First, however, lot me just su.y thu.y~ fu.r as I kn~vb.his question h8.8 never before been cleu.rly formulatCll. Nonetheless, it has received an answer which is both erroneous and widely accepted. In dictionaries (see chapter 2.12) tho word avdlnyot; is explained 8.8 follows: "dem MyOt; enlsprcc/umd, ge",asll, vorhli.ltniHmilssig" or "acr.ordi7'f1 to a duo l&yot;." These explanations oft'or a translation of the proposition d"d 8.8 pa.rt

ded}/loi dvdAuyov da,,,, liTa" d nl]wTIl'; Till; J£vT/euv "al d TeiTUt; TOO T£TU{!TOV lad",t; 11 nollanAdatot; I} TIl lIl~nl"If!nt; "1 ra avra ,'/e1] WO'tV.

Analogon sind die ZahlenlS wenn dio ort!te von der zwoiten Gleichvielfaches oder derselbe Teil oder diesel he Menge von Teilen ist, wie die dritte von der vierten. [Numbers are analogon when the firt!t is the sa.me mUltiple, tho saUlO part, or thosamenumberofparts, of the second 8.8 the third is of the fourth.]

But.ta thUdo dufinitions Hhow that tho word avc.Uoyov dCllcribcs at 100LSt four numbers or magnitudes, two of whieh have the same ratio 81

I.IuRth trlUUllatea dJldAo)'OJi in this contoxt by proportional, whilo Thaor

U8COS III

1Jroporlion.

............~ TIIB

I

':

l'RRl'OSITION

av&

.. Arlst.ot.lu, NiclJmao/ICt'" E'/,il~. II :llIl31: " Ylle III'IIAO)'/" ll1liff" Na' lv dnaealV lAaxlaro".

" Ibid.

lad Adywv,

PAllT 2. TUX ,'UK·l!:UCUORAN TURon v ot" l'UOI'ORTIIlNIi

S~AIJt): 1'111> JU>OJNNJNOII Oa.' OJU':l!:K. AJATIIl!:AJATICS

of the compound cbdAoyo" yet it would 866m to be a mistaken one. I l~anll()L Lelieve that the original use of the preposition ol'd wus to oX)lmlltl 'cnt~pl'tlchond', 'gemlitiH' or 'according to', for quito different prel'ollitiollB were uBually employcd in such cases. As an example, Id 11M l'oll!lilitH' tho preposition xUI'd. XenoI,hon (Oyropaedia, 8.6, 11) writeli: HUTa ;.orov Tij, 61wdp£w, (in proportion to their power). Similarly, ill t.ho works of other (Lutllors we encounter suoh I,hra.ses 118 xaTa Tdv 1It1T1h' A'/)'III' or Tl!OnOV (Herodotus, I. 182), xaTa Toih'ov Tdv .wrov (Pln.to, Pro/a!l0rtUj, 3240), XaTa Myov Tdv dxom (Plato, Timaetl8, 30b), xaTa niv AO)'()J' TcJV aUTov (Aristotle, Nicomachean Ethics, 1131b30) and HUTel ni,' civaAoriav (lllo.to, Politictt8, 257b). In 0.11 these instances just tlmt fUIIC:tioll of the preposition tiv& whioh one thought to find in tIm cOlllpuunds dvdAoyO)', and cbaloyla, is performed by the other I'l'Ol'ul!it ion XUTU. Evon Horuclitu!!, when ho wants to say 'according to tho IIlLmo llatul'e (or proportion)', does not use O"aj instead he writes: £1, TOV aUTov lOyO)'.88 Thesc mmmpics Imggest the following two conjcctures: (IL) It. is unlilwly that ava (in the expression ava Mrov) had originally the SI\IlHl meaning ILS xaTa and £I, in the above examples. It is espccitLlly unlilwly that uva and xaTa sliOuld have been synonymous. These two prcpusition!!, ILt ICILSt originally, had directly opposite meanings. '..t",i IIWlUlt 'upwards' or 'up', whereus XaTa meant 'downwards' or .down' . :-----------., (h) J\II we slmllsoon see, a groat deal of hiHtorical information can be obtained once wo have established exactly what tho preposition ava means in tho phrlLBo a,'a AOyOV. ~onethele88) want to emphusize here t Imt tho exphLlIation of ava.toy(d"'Whic)i Tam going to propose holds good or~ly for the meaning of this mathe~!l.tical term at a very early period.I'L wuuld be wrong to think!tha~lis word h(\d precisely the Immo rnmrning in later texts, such las th080 of Plato. The quotation fl'om tho PoliticU8 given above shows that Plato uscs the word avaAoyla in 1\ sCllse which differs from tho ones actually listed in the dictionary. '1'11(1 phru.'le XaTa T~V ava.l.oylav indicates that by Plato's time the original IIwlminJ,: of Ih'll in the word avaArI),la must already havo been forgotten. (This ill the only way to explain the fact that a new adverbial phrlL80 "lIl1ld I,,· flll'llll'll from dvaAoyla hy prefixing tho proposition xaTd, ovon VI R,.o frngll1ml!. 31 in Diolll and Krllnz, FmgllltlnU ,Ier Vor8oL-raliker.

1

, ~l

t

I I1

though ava.l.oyla itself wus just an adverb which had been converted into a noun.) The same is also true of avd).0roJl. In educated speech this word undoubtedly camo (evon in classical times) to mlllLll roughly the same as our adjective 'analogously'. Proolus,eg for examplo, writes: XaTa I'd avdAoyov, which SohlinborgerlOO quite corrcctly translates as "in analoger Weise" [in an analogous way]. This, however, is not the meaning which interests us here. Our present concern is with the original meaning of this mathematic!l.1 term, i.e. with the meaning whioh was probably current in the 6th century D.C. and tmccs of which aro to bo found in many of Euclid's propositions (since the propositions concerned date from a muoh earlier, pre-l1latonic period). In my opinion it is not difficult to aseertain what ava means in the expression ava ,lOY"", and hence to disco~er the originalI and c~act meaning of the mathematical term avuloy,a. Amongst t 10 VILrlOUS meanings of the preposition ava, oJl the major Greek lexicons list a socalled distributive ono which occurs espccinJly in numerical contexts. For example, in Xenophon we read: ava nme. nCJ(!ac1ayya, Tij, TJPie a , (at the rate 0/ five para.sangsa day - Annbasis4, 6, 4) !l.nd: ava IxaTov l1roea, ("a hundred men apiece" - ibid., 3. 4. 21). SimiltLrly Diophantus (Arithmetica IV 20) writes: dva aVo (two at a time). Tho examples which the dictionarics quote to illustrate tllis particular meaning of ava aU have to do with numbers. We can correct the impression which this leaves, howevcr, by adding the remark that Ol'a was used with this distributive meaning not only in numerical contexts, but also occ(\8iona.lly before nouns having some connection with the concept of 'number' or with that of 'division'. An example of this usage, tho expression ava pieo"IOl is quoted in KUhnOl"s GrammarlOl where it is translated as "wechselweiso" (by turns). •A"a is being uscd distributively in this phruse, since the meaning 'by turns' clearly derives from the original Iiteml meaning of ova piea" wllich was 'a piece at a time' or '(taken) in parts'. .. Proeli DWdocl.j in prim"'" Euclidi8 Elementorum Libn,," commenlarii, ed. O. Frludluin (I.ipllilW 18'13), 11 '1. 2. 100 ProklUB DiadochUB, Konunefllar %Um eraten 11ucl. wn Euldula }tJlelllcliten, translated by P. L. Schllnool'(,'Or, cd. M. Stelcl', Halle 1945. 101 Fourl"I,!,,", '1'/,,, 1'/,fIlmi"i,." Wmlllm, .t'1K. 101 n. KUhnor-ll. Oorth, .A IlIJ/W.rlicl,1l (1r",,,,,mlik dcr uriec/lilfc/,c" .~prac/le. SatzlehrtJ 1. Teil, 4th edition, Hc~nnover 1055, p. 4'14.

Nuw Lcliovo t.hat. tho phruao dva Myov h~ to be understood in the IIIL1ne WILy; for just. UB dva pieot; meant '(taken) in paris', so dva .IM),m· melLllt. '(taken) in louoi.'. There are two very important factlt which support. this explanation. (1) The llictionaries tell us that the preposition dva had its 'distribuLi \'e meaning' especially in numerical contexts. Bearing in mind the oXl'r('s:-;ioll uva pieo" this remark has to be supplemented by the obsorvntioll that ava could also be used distributively before nouns whioh wero I'cilLtcd to tho concept of 'division'. Alieot; (part) and 'division' (tlislribuere in Latin) aro very closoly related concepts. On the other hand, the phrase civa .tOyov shows thatciva could be used in its distributive sense before nouns which (unlike 1CWrB, f,ear:.w and 660 in tho above examples) were not themselves numernls, but which at one timo eould not be separated from the concept of 'number'. lAuos WII..Ii originally just the 'ratio betwoon two numbers'. At. the very least, these considerations make it plausible that the pn'plIsition in the phrase civa Myov is being used distributively. (2) A second important argument in favor of this view i!-pr.ovided hy t.he Wll.y in which the expression ava lOyov was used in the language of nmLhelllll.t.ics. Two l)o.irs of numbers wore said to be proportional whenever the members of one pair were in the same ratio as those of tho ot.her. lIenee t.ho terms of the relation (of proportionality) were first distribuled for the sake of contrast, and then compared. 'l'here is no doubt that the exact meaning of tho expression ava lO"o" WIlS '(tILI{cn) in louoi'.

2.15

'l'JIE ELLlI'T1O EXPRESSION

dva lOyov wo, or wa which WUB subllc()ucnt.ly a~bl'eviated to. the standard expression ava l&yov (or avaAoyov, as It was sometimes written). It must be u.dmitted that the more complete version of this phrase does not occur in any of the ancient texts which have been handed down to us. As far as I know, they contain only the elliptic form dva Adyov (or avalo"ov). Nonethele88, thore is no doubt tha.t .we hav~ succceded in correctly reconstructing the complete expre88lOn. EVldence for this is furnished by the two definitions from l~uclid (Elements VII.21 and V.O) which have already been quoted above. To interpret and transiate these satisfaotorily, they have to be supplemented (or at least it has to be kept in mind that they need to be supplemented) in the following way.

Definition VII. 21: aetDpol dvalo"ov (iO'ot) slatv, 8Tav o 1CewTOt; TOU 6evTieou "al 0 TetTo~ TOU TtTae-cou ldmc't; ~ nollan1adtot; 11 TO mno pieot; II Ta a-na pieri

rod,V.

Numbers are equal (when taken) in logoi, if tho first is the same multiple, the same part, or the BUJ1le number of parts of tho second, os the third is of the fourth.

Definition V. 6.' Ta To.. mnov lxona 1&,,01' p6)lilhj avaloy01'
a"a .to"ov

The ILbove considerations o.ro simplo, yet thoy establish conclusively that tho proposition ava has 0. distributive meaning in the phrase Ih'li AII},!}V, nlld take us 0. stop further in our investigation. Aristotlo omphusizes that "analogia is sameness oflogoi". So, in order to establish that fOIII' numbers or quantities wero ava lO"ov, one had to show that they wero 'equal (when taken) in lauoi'. 'fhis implies that ava lOyov ([tnlton] in logoi) actually meant 'equal (when taken) in louoi', or illllt.lWl" wordH UtILI, ava Myav WILlI I~Ho-cu.llod defective or elliptic ox(lI'68sion. The phrase used. initially in Greek mathematics must ha.ve boon

100

PAnT 2. 'IIIK l'UE,BUCLlyll:AN TllEOU\! 01' l'IL01'OU-nUN:I

I11.AUO; '1'IIb: lIh:OINNINOS Ol" on£EK )IATHIUIATICS

Magnitudes which have the samo logos are said to be equal [when taken] in logoi. 103

Of course there are many similar l'UBSlLges from Euclid which need to be complemented in exactly tbo same way as the above. Lot mO mention IL few of these here. In tho Elements Book VIII, Definition 22, the Pythagorean concept of 'similar plane' or 'solid numbers' is described 118 follows: "similar plane or solid numbers are thoso whose sides (i.o. factors) are equal (when taken) in louoi" (ol civaAoy01'
For oxample, Definition VI. 1 and Proposition VI. 6.

1M

Ii:l:AUO: 'l'1I1>: UJl:UINN1NUIi 01' 01l~~I!:K UATIlKUATICII

l'wl'llliiliun V1.5 of tho Elemenl8 is concerned wit.h triangles whoso Hides aro 'prol'ort.ionu.l', i.e. 'equal [when taken] in logoi' (10." Tel)'ruva ni; nAfu(lu; civ&;'o)'ov lxn). Furthermore, in Buch propositions l\8 Niement8 VIII. I, 3, 6 lI.nd 7, the Greek phro.se la'll ~O't" onoaOlOVv a(!d}poll~ij, a"aAoyo" •.. which Tho.or translates o.s "Hat man belicbig vielo ZILhicll in geomct.rische Rcihe ... " (given arbitrarily many numbers which form a geometrical progression • .. ), is rendered equally well by t.ho more literal trn.nslo.tion: "given arbitrarily many numbers /\\Ihich tn.lum in succession, are equal in logoi ..• ". (/ J t is eo.sy to understand how t.he elliptic expression dva 10,,0'11 co.me , into being. In compo.ring lO(Joi, one wo.s usually concerned with whether or not. they were equal. Henoo the phro.se '[taken] in logoi' wo.s used for the most part in conjunction with the word 'equal'. This meant t1ULt ILfter IL while the word ;0'0101' raa, which expressed equality, could he omitted I.!ince it wna implied, so to speak, by the rest of the phro.se. Thus the tnmco.ted phrnae ' ... [talcen] in logoi' co.mo to have the meaning 'eqltal [whon taken] in logoi'. The fact that dva 10,,0'11 soon (~ILJIle to ho written o.s one word is independent evidence for the thesis •_ UULt thiH dliptie oxpression doveloped into 0. new word with its own mcaning. The new word of course Wl\8 the lLh-aabcenacccpted in mathematical language, that it wo.s possible to derive the noun ,iJ'flA01'{fI from it. •Avaloylu wo.s the mathematical property possessed lIy lIullI\I('I'd or quu.ntitic8 which were o.vciAoyov. Inotherwords,dvalo),la WIL.'l 'sameness of ratio'. We canllot bo certain lI.bout when mlLthemo.tieio.ns coined the tcchniclLI torm dva).o)'o" ([ta.ken] in louoi) nor about when this adverb gave ri!!u to t.he noun dvalo),la (sameness of mHo), for no mathematical Itlxlli IIIL\,t' KIII'viVtltl frtllll t.hia l'oriOtl. lluwevOl', t.heso developmonts certlLinly took place beforo Plato's time. From his time on, the o.rcho.io t!XI'I'('!lHicllI ,b'n ld)'ov (equal [when to.l(en] in 10(loi) wo.s hlLrdly used !Lny IIlllm, except by l~ucJjd. On the other hand, even Plato no longer lIsed dvaloyla with just its mathematical mooning. Although ii, WIl.."i tll'iginnlly 1\ Il\Ircly mlLthemo.tico.l term, it seems o.lroo.<1y lIy his timo to have been current in educated speech. The imprl'sHion len. by Aristotle's writings is of dva;'o)'la o.s 0. philosophical term whllSl! IImthomatical origins scom to have bcen almost completely forgotten.

157

PAllT 2. TW!: l'ltK·";UCLlllKAN TIIKOILY 01' l'nol'Olt'ClONS

Thoro are, however. some indications that the arehaie word dvcUoyov survived as 0. mathematico.i term and that its meaning subsequently underwent somo cha.ngcs. These will be discussed in the next chapter.

2.l6 /'

'l'UE SUBSEQUENT lIIS'rORY Oli'

dvcUoyO'l1

AS A MATUEMATIOAL TERM

In the preceding chapter we showed how the phrase dva. M)'~V (an o.bbrevia.tion for d"a U)'ov fO'o,) came to have the meo.mng equal [when taken] in logoi' (or. put more Bi~ply, '~n. lite sa,!"e rati~'), and we established that it llrobably aoqUired tillS meanmg durmg the archa.ic period. The text of Euclid's Elemenl8 is !ull of examples whieh could bo used to iUmltrate this particular meo.nmg of the phrase. Howover, thero are two rcmo.rks which need to be made in this con!lcction' the first is that in clo.ssical times Greek mathematiCians no longer ~d the word dvaloyo" when they wanted to ~ay 'in.th~ sa.me ratio' and the second is that by t.his time tho word Itsclf (m Its role o.s 0. ~o.themlLtical term) hlL
..

1M

~)\AIJ'): 'I'1Ib: IJ~:UI

"All'r 2. 'I'll" ,'IW·J.:UCI.III'.AN 'J:IIIl:UllY tll' "llUl'IJlI'I'IONlI

NNINUII Ol<' t1I1UIl:K lIATIIIU1A'l'1t.:8

'1'1"11(1111'1 itll IILI , . For thiu rel"':lon it heclLmo l'ollHihlo to speale of a T(!lT'I Ih',IAU)'O" or a peaTJ dJld.toyov, III« and even of pedo, dJld.loyoJl de,{)por; allli/dam !l,,(lAoyov U(!'{}/lol.JO&'l'hislattor men.ning (which originn.ted in I hu III't'·Platunic period) is of COUl'HO different from tho archaic ono alld postdates it. Furthermore, it scoms to have l,repn.rcd the way fllr till! c:rcntion of tho hyhrid u.djective dvcUoyor;. (It should bo menI iOIll"I , howover, thlLt thill u.djectivo never figured in mlLthomn.ticnl

lall~uagt~.)

Ali we ClLn 8ee, in the course of time the word dvd.loyov came to have two men.ningll in mathomn.tics. It rotn.ined its original meaning of 'e(!~!1L1 [when tuleen] in 1000oi' and n.l80 came to mean 0. 'p~portional'. This ambiguity scems to have been partly responsihle for tho fact I hat it latcr hecamo more uSllal to employ II. differont exprellHion for I lit! Jlotion of hcing 'in tho SILme mlio'. I n.m referring to 0 aVTO~ ),II"ul; ('the samo logos' or 'the samo rutio'), which WIl8 the phrn.so IIst,cl mUlit frequently in Greole mlLthomn.tics to express sn.mencs.~ of mtio. Actually Euclid ILnd the mlLthomlLticin.ns who followed him 1'lIIployCl1 this exprC88ion in two slightly dinerent wn,ys. 'J~hoy sILil1 I IlILt l'ail'H of numhers (or (Iun.ntitites) '/uJCl the same rutio' (To,.. aVTuv Ali)'IIJ' 'xow, - IIl~e, for oXl\mple, Elemell/s VI r. 17) and nlso tlllLt thoy '.~/fI(ld i II tho HIL1I10 mt.io' (Iv TfP "UTfi' AdyflJ dill" - 1:100 l~lemcn/s VII. 14). IHence, if the phrll8e 0 aUTO, loyo," is included, mathemo.tioin.ns had al. tllt'iI' (lisJlIlIIILI threo WILyS of oXI'I'CHHing t.ho concopt (If hoing 'ill lhe HiLIIIO rut.io'. 'l'heBu Wl!I'O: (I) 'f1\(J ILrchaic expression dJldAoyov - 'oqual [when tn.ken] in loyoi' (2) '1'ho Iwverhinl claullO of comparison oiJTtJ), as used, for example, in Elemen18 VII. 11 (cf. p. 104) (:') '1'ho l'hrnse (i aUrur; Adyo".] Now I.III! t.crm lIHed mUMt. frequently fur 'in t.ho su,mo rlLtio' in JtJ/cmelll8 V (a hook which iB I:IUPI)()Bcd to have lJcon mostly tile work of gW/OXllS) Sl'.cms to he 0 aVnl, A&yor; (the sl\me ratio) nLther than lire lLt'dmic dJlcUoyoJl(equal in 1000oi). It itt tnlo thn.t the word dvdloyov aillo OCt·urs in this hook, hut it n.ppClLrs only in proposition which run III' classified u.s t.he work of elLrlior mat.hcnltLt.icians or 118 now formultLtionll of earlier results. This cn.n he Beon from the following.

w, ...

I~ loa

Suu Propositions VI. 11 Wld 13. ":I'·IIII'lIt.•, vn r. " 11 ... 1 12.

"

15U

In tho fundlLtnontlLl, lIo-clLlled ]~ucloxilLll dofinit.ion (V. 5, quoted above on p. 00), 'samenoss of rat.io' is defincd u.s 0 aunlr; Myo;. However, this definition is followcd immediately hy Definition V. 6: 'Let magnit.uclc·~: which luwo t.ho SILmo rat.io (Tflv aUTo,. lxona lOyu,') be called dvd.loyov (equn.l in logoi).' The.llLtter is clen.rly not II. prol,er mn.themn.ticlLI definition. It lIervcs only to int.l'Oduco ILn alternat.ivo WILy of dcacrilaing t.ho conccl't. (0 aVTOr; lOyol;) defined previously. NOJletheles.~, t.hese two definitioJls, when read in the order in which they occur, Icad one to expcct t.hILt hoth 0 aUTor; Aoyo, Il,nd dJl&Aoyov will bo used to refer to 'samenC8S of ratio' in Boolt V. Let us see how often each of thcBO phru,scs is ullcd. Altogether, the notion of heing 'in tho slLme nLtio' occurs in seventeen of the proposit.iolUl in Boole V. In twelve of thcIIO (V. 4, 7, \), 11, 13, 14, 15, 20, 21, 22, 23, ILlUi 2·1) it is referred to hy tho phrasc 0 aVTC)r; .tayo;. On the other lllLnd, dvcUoyov is used in only five of theso proposit.ionlJ (V. 12, 16, 17, 18 ",nd 25). This statistic is by itself sufficient to suggeHt. tho conjectur~ t1I1LL tho l,ropositions in Uoole V were written down ILl. IL time whcn t.ho UHUILI torm fOl' 'in t.ho slLmo ratio' WILS no 10ngCl' d,,&AOYOV, hut mU,er 0 aUTc}, Ao,..or;. In at lelLSt two ClLSell, however, wo can go furt.her t1ULIl t.his (Lnci eXl'ltLin why t.ho ILrclllLie exprcssion dJl&lOY/Jv WILS usml wit.h t.hill 1Il1llUling in HOllIe V. Bllfut'O liiscUHsilig t.lll'Sll in greater dctlLil, I Imve to mention an interesting peculiarity of Boole V, to whi(~h inHufficient. ILttent.ion IlILs hcen pu.id lip to now. l'rul'OIiit.itln V. HI of t.ho J~le1tlcll/s sLlLtoH tlmL: "If, ILl! IL whole is tu " whole, so is II. part Bubtracted to a part subtmcted, tho remainder will alllo he to t.he remlLintier u.s wholo to wholn." If the above is reu.d in the origintLl Greek, then one is ustoniHhed to find tlllLt it is word for word tho SILmo I\'~ a thcorem to be founel in noole VII, IlILllIcly, Pnll'osit.i(ln vn. 11. 'rho t.WO vtll"llions diffl'l' (lnly in that tho adjectives 8).0" dtpalf!r:t?£lr; and ).omor;, whose form is neut.cr in Boole V, ILro put into tho mn.sculino in Boole VU. Thill difference, of courso, "rises siml,ly from tho fiLet that 'JlumLror' in Greeli is IL miL'!' culino noun (o dt!,{}po,), whoreas 'mngnitude' is neuter {TO pEyI:OO.;).11lG The Prol)()sition in Boole V ILpplics to arbitrary 'lIIlLgnit.udC8'; in Huok VII, however, it is formullLted only for 'numbers'. The differenco of gender also reveals which of the two versions is the earlier one. There 101

IJuuth (Kurlill'lI 1i:lernenlll, VitI. 2, fl. 312) I1lIIku/l 1.1111 anmn point.

HiU

H:tAU(); 'l'UI!: UJ.:uINNUW8 Olf UIII!:I(K. MA'l'lUUIA'l'lW

iii no doubt tha.t the proposition could originally bo formulated only fill' IIIIJllhCl'S, as it ill in Book VII. (Quite apart. from t.his, thero aro othcr conlliderationll which havo led seholars to conjeeturo that tho Jlmpositions in Book VII antedate those in Book V.107) Only aftor I Ill' tlwory of proportions had boon extendcd to cover incommensu· mlJle magnitudes as well, could this sarno proposition (with jUllt a slight grammatical change in its formulation) be incorporated illto Book V. Of course, when this took plu.ce a new proof was aillo required. The old one, wbieh held for the C880 of 'numbers', wue 1I0t valid for arbitrary 'magnitudes'. Thus from 0. mathematical point of view llroposition V. 19 is completely differont from VII. II, even though it appeo.rs to be simI,ly 0. minor linguistie variant of tho IILtle." III Illy opinion, what we have just loarned from compu.ring thcso l\\'o propollitions (V. 19 and VII. 11) also provides us with an eXI,lanatioll of why tho urchaic term dvdAoyov is used for the notion of being 'ill the !!lLme ratio' in somo of tho propollitions in Book V. Let us look lIH1rO du!!ely at tho following two propollitionll: V. 12: "If Il.ny number of l1launilude.s bo proportional (dvtUoyo,,), as olle of the antecedents is to one of the consequonts, 80 will all tho anlet'celunt!! bo to all the conllequont!!." V. )I): "If four maunitudea bo proI,ortionw (dvo).0ro,,), they will also be proportional alternately (xall"alldE d"tUoyo" laTa,)." [I.t'., if(,:b=c:d, t.hon a:c=b:d.] If wo confine our attention to the statement of these two propo!liI illJl!;, thcn they aro seen to reappear in Book VII formulated almost. word for word in tbe same way. Correspondingly to V.12, thero is: VII. 12: "If thero be as many numbers as we plell.8O in proportion, (ch'(ilo)'ov), then, as one of the antecedents is to one of the consequents, !ill arc 1111 the antecedents to all the consequents." And corresponding Itl V. 16 wo have: VII. 1:J: "If four numbers 110 proportional (dvdAoyov), they will ILIHO 110 proportional alternately (xallvall~ d"tUoyov laovTlu)." AI! WlI CIUI sec, l'roIlOBitiolla 12 and 1:J in nook VII apply only to ml1llbcrs, whereas the colTCltponding onca (12 and 16) in Book V aro furlllllllLlt!ti for InbitrlLry 1IIaU1litude.s. No doubt those propositions 107

B.

r.,

van der Wlterden, Erwachetule lYissen8c/UJ/', pp. 182ft,

c

PAllT II. Till' PllK·KUOLlDHAN TIlItOI1Y Olr 1'1lOl'OIn'10NII

HH

could originally be stated and proved only for numbers, and were subsequently generalized so as to apply to magnitudes o.s woll. Whon they wero modified, however, their wording was left unchanged, except that 'number' (de,fJpdt;) wo.s replo.ced throughout by the newly defined concept of 'magnitudo' (l,lydJot;). ." It is immediately apparent from this why the archaiC term tiVtUoyOVIS used in V. 12 and V. 16 to oxpress 'in the sarno ratio'. It is 0. relic of the earlier versions (VII.12 and 13) of these propositions. Furthermoro, it is precisely because of the use of this earlier terminol~gy that the so-co.llcd Eudoxian definition of proportionality (V.5), whioh employs the phruae d O.)TOt; UrOt;, had to ~ supplemented by o.n~ther (improper) one (V.6: "Let magnitudes which have the same ratio be called clvalorov"). It is not only in Book V, however, that cJ amot; lorot; ocours more frequently than tho archaic dvtUoyo". This is also true of Book VII, even though tho propositions in this latter book are probably of an earlier date than those in the former. It is interesting that tiva).oyov (with tho meaning 'in the flame ratio') appen.rs in lwo of the definitions (21 and 22) in Book VII, whoroo.s cJ amdt; AOyOt;, a phrase of more recent origin, does not oocur in any of them. (This fact is 0. further indication of the groat age of those definitiolls.) Yet there are altogether only three propositions (VII. 12, 13 and 19) in this book whioh mention dvdAoyov (equal [when taken] in lOUoi), o.s opposed to 8ix (V~I. 14. 17, 18, 20, 21, and 22) in which. cJ amelt; Uro.t; occurs. Thus,~n tho entire text of Euclid, the expression tivd)'oyov (m the senso of m the so.me ratio') appears less frequently than the phrase cJ amelt; AOyot; whioh has the same meaning.

2.17 OUTS OF TaB OANON AND MUSICAL HBANS

In the lo.st five ehapters (2.12-2.16) our historical investigation of the mathemo.tico.l terms dvdAoyov and dvdoyla ho.s been confined to linguist.ic matteI'S. Becu.UHU of tWa. we have not yet considered how it came about t.ha.t matliematicio.ns formed UrO& (or ratios) out of pairs of numbers, nor why they compo.rod numbors '[taken] in louoi'. In order to formulate those questions in 0. more concrete manner, we must resume our discussion of musical subjects. II 816W

\

,

162

l'AllT 2, 'l'BE PUE,EUVUllKAN 'l'UKOllY 011 l'UOl'OIlTIONIi

IIZAlI\); 'l'UI>: lIKOINNINOS 011' OIUlliIt MATUIUIA1'1C8

Although we have already devoted one chapter to an account of Lhe (:nlcuIlLLions (addit.ion and lIubtrootion of lengths) whioh were performed on the canon, the Greek name for these operations has not yet bcon mentioned. The.Pythagoreana called them cul8 o/lhe canon. 'fher" ill, fur example, a 1)ll8Bago in Ptolemy's Theory olIIarmonylOl where he sketches the procedure followed by certain of those who performcd experiments in the field of musio. It runa as follows:

ca.n be shown that these t.hree were known in the fifth century B.C. nnd l)robably even earlier. Thcy are the arilhmdic, geomdric and harynonic (or subconlrary) meam. Before we look more closely at the text of ~e earliest account of theac which has been handed down to us, it!.s worthwhile to recall what. the significance of the arithmotic and harmonic means was in the study of music;lt hiLS been written that: 1ot , ..-

"'rhoy do not. obtain the divisions (of the string - TO, HaTaTOp~) by calculation, but by stretching the string and then moving the little bridge until each of the notes being sought after is heard. They flIlLrk tho required cut (C11JpeloWrm T~ TOp~) at those 1)lo.ccs, ILnd do not concern themselvclI about the ratios."

The arithmetic and harm~Jlic means effect the division of an interval into unequal suuintervals. The subintervals are the same in both cnsea excel)t that they occur in the reverse order .... The classic example, mentioned by most authors, is the division of the octave. The numbers 12 and 6 stand in the ratio 2 : 1. Their arithmetic mean is 9, and their harmonic mean is 8. The former divides the octave into a fourth and II. fifth; the hitter divides it into a fifth and a fourth.

We IiCO from thia account that the word TOp~ or HaTaTop~ (cul) ill uSl'd to describo the result of trying to find out how long a lIection of atring on tho monochord should be kept from vibrating in ordcr to pr(l(luce a cOll8onance. (Such attempts have been diHcU88ed more than OIICO in previous chapters.) Clearly the Pythagoreana differed from tho 'Experimentalists' in that the former took numerico.l riLtioH into account ns well, whereas the latter completely ignored them. Figumtively speaking, the whole string (the monochord) was 'cul' at the pllLco on the 'co.non' where the little bridge (the WrayW)'w,) eventually stood, i.e. at the point sepo.rn.ting the length of string which WiLt! plucleed to produce the second n~te from the section which wns kept from vibrating. The words TOp~ and HaTaTO~ a.ro synonymous with clLeh other. They are technico.l ter~from the theory of music lLud were quito commonly used as sueh.~ is also worth noting thaS:> the earliest surviving work of the Pythagoreana on the theory of musio (111ll 1(lxt of which bears Euclid's name) is cn.Iled HaTaTOp!} HOv&ro, or Sectio Omlo'l1s. - - .-'-----The Cllls 01 Ihe canon are especially important beco.use it is clear Lhat I.heso oporatioll8 mUllt also have led to the so-called 'mwical means'. 'l'he later Pythagoreans dealt with many different kinds of 1I1t'1L1I; of UIllSO, however, t.here are only throo which interest \lS. It lOS

'.

163

,! I

I!

The interesting connection between arithmetic and harmonic means can be better understood if the CILIlon with its t.wolve divisions il5 kept in mind when reading the above., An arithmetic mean is obtained by outting the canon at the number 9, because this divides the diastema of the octave, t.he section of the string betwcen the numbers 12 and 6, into two shorter sectiona, en.ch having the same Icngth. Thi,!. cut is sit.uated exn.ctly at the millpoint between 12 and 6, ILOd henea is at 1.110' so.me diHtance from each of the two endpoints of the original diastema. Yet the two equal lengths (12-9 anel 0-6) induced hy the cut reprcsent two distinct musico.l intervals (numerico.l ratios). Furthermore, since 2 19 < the larger numbers form the smaller interval (12 : 9, a fourth)

i'

and the smaller numbers form the larger one (9 : 6, '" fifth). The harmonic mean, on the othcr lmnd, which is the subcontnLry of the arithmetic, ill obtained by mILking the cut at the number 8. The fILet. that 12 >! meana that in this CWlO the longer of tho resulting scetions 8 6 (that between the numbers 12 and 8) corresponds tu the greater ratio I"Tho following, which is not. a literal translation, is taken from van tier WacnJun's paJlor, /lermes 78 (1943), 182.

nit! l1arlllonielehre des Klaudi08 Ptolemai06, ed. Dilring (Berlin 1030) 66,28 ft'

I.

•

J

~

.

"

"

:

PAll'!' I!. Tilt.: "llJ!:·J!:UUl.IlJ1l:AN 'l'ltJ!:OILY 0 ..' l'lUll'OIl"llUNIi

164

II~AUt):

expl'C88Ctl o.s 0. fmction of the first, is the samo as tho amount by which the second exceeds the third, when expressed as 0. fraction of the third. 1l1 In this cnse of equo.lity [when taken] in lO{Joi, the ratio [inwrvo.l] of the larger numbers is the greater and that of the smlLllor numbers is tho 2 lesser

(0. fifth), while the shorter one (that botween 8 and 6) corresponds to

the IUluLller (a fourth). The connection between the three means is the subject of Archytns' 2nd fragment, which is quoted hero mainly because of the interesting terminology used in it.no

Il iaa , M, bn Te~ Tfi. Ilovatxfi., ilia ,Ih. dl!u'IlFlTl"d, 6wr1ea 6~ a Yllru1't:T(!,,,d, TelTa 6'vn£1lavTia, c2v "oAlOVT, Oellov,,,dv. Cl(!,OI'FlTI"el Illv, 8""a lruvTl Tell~ (J(!1lL "aTel TelV Tolav vnlleoxav dva ).(~yov· cp neiiTo~ 6wrleov .mlle1xll', TfII'tlP dt:VTEeO~ Te/TOV .mlle1XIlI, "al Iv TaUT? Tfi. dvaAo"{~ (1VJUlinTe, 1'11'0' Til Taw /lEIC&V(IJV 8em" 6,0.11T111,a Ildov, TO 6A reov I'£'&vruv IleeCov. a YEWIlIlTe'''cl 6/, 8""a lrun, (Il()~ (J 1fl!dTO~ nOTl TO" 6eVTEflOV, "al IS c5t:vl"£eo~ nod TOV Te{TO". TOVTm" 6'01 IlIlICOV£t; fao1' no,oma, TO 6111fITFJIla "al olll£{OVt;. (; 6'v1f£vanla, c2v "aAoVIl£1l &e1l0"IXUV, 8x"a lrunl (TOea" t[J> d neliTil; 8eo:; .mllelx£' TOV 6svTleov UVl"aVTOU 1l1e£', TO.nqJ cJ Jdaot; TOV TelTOv imee1xe, TOV TelToo pleel' "{,'nal d'lv TavTIIl Tfi. dvaAoyi'll TO TI'IV Ilt:lCOVQ)V 8eru1' 6,df1TFJpa pel/;ov, nj IIt~ TCu" 1l£'&vru1' Ill! eov.

In musio there are three means: the first is the arithmetio, tho, second is the geometrio and tho third is the sub contrary , which is sometimes ca.lled the harmonic. The arithmetio mean is present if the throo numbers differ, [when taken] in louoi, as follows: tho 8ocond numbor oxceods tho third by as much as the first exceeds t.ho second (12-9 = 0-6). l!'urthormore this equality [when takon] in lO!Joi moans that the ratio [interval] of the larger numbers is the smo.llor ono, and that tho ratio of the smaller numbers is the larger

[1: < ~] .

The geo-

metric meo.n is present if the first number is to the second, as the second is to the third, In this oase both the larger and the smaller numbers have the same ratio [4: b = b : c, where a > b > c]. The suboontrary or harmonio mean is present if the numbers n.re suoh that tho amount by which the first exceeds tho second,

Diuls and Kranz, FragmtlnU clsr Vor80l:ratim, 1', p. 436. Reidumoistor ("Dus Bxlikto Dunku" dur Griuohon". Hamburg 1949, p. 27) olaims that this fragment is not authontio. In my opinion, howuvor, his argwnont is not convinoing. 110

16G

TUR DKOINNINOS Olr OaREIt MATIlIWATICS

[18 >~] ,

As wo ca.n sco, the terminology employed in this fragment is drawn from tho Pythagorean theory of music. There is no need to enter into any further discu88ion of the terms 8eo, and 6,aC1T11lla here, but the use of dFa lo"ov and d1'aAoyla is worthy of spocial notice. Evon though these words can be translated o.s '(taken) in lO{Joi' und 'cqlllLlity [whon tal(on] in 10yoi', they are undoubtedly being used in the fragment quoted above with 0. meaning which differs from all the other o~es which we havo conllidered up to now. In this frogmcnt, tho pairs of numbers whioh go to make up the arithmetic mean (12. 0 and 9, 6), are compared dFdloyOV, and the arithmetic mean itself (12- 0 = 0-6) is said to be dvaAoyla. Henco this usage suggests that lOyot; refers to tho rolatiorUthip ootw(len two numoors which iii (Il((lmplificd hy 12 . 9 and 9-6. [Nowadays wewouldrogard '12-0' and '9'- 6' o.s terms which denote numbers (resulting from the application of an operation to other numbers) rather than as expressing 0. relationship between tlVO numbers. This remark I\pplics to tho word 'ratio' (and tho notation 'a : b) as woll; although it is used l)roporly to refor to n. relation, the common

practice now is to speale 01 ",e ratio between two numbers as il it were itsell a number (such notation as 'a : b = c : d' ill evidence of this). 'l'hcse

1.0" in thu caso of 0. hurmonio moan tho rolationllhip between tho threo a = b-o' c So' numOOrs (a, b ami 0) is 8XProBlllld by thu fonnula a-b ,an pur,,'ICU' III

12 6 lar, 8 is thu hnnnonio mean betweon 12 QIld 6 sincll 12-8 = 3 "'" 8-6'

Hit;

UlilLgClt, uf COUI'BO, u.ro ~ reflection of modern id60.8 about mathematics (in particular about the closure of the real numbers under the operations of lIubtraction and division) which it would be wrong 10 allribule 10 tile Greeks. Hence it is necessary to abandon the modern terminology when discussing these concepts. The relation referred to in the precedin~ pl\rILgraph would Le denoted in English by the now obsolete phraso 'aritltmetical ralio'. ]tlla

Thus the conclusions which I draw from Archyt&a' second fragment nrc iLS follows. In tho theory of musio, the word Myo~ originally mcant any kind of 'relation between two numbers', and not just tho 'ralio between two numbers' as it did in geometry. Consequently tho wort! umAoyla meant any kind of 'equality between pairs 0/ numbers'.m It was only later that lOyo, took on the narrower meaning of that I'cIlLt.ion hotwecn two numbers which we would refer to as the 'ralio' (a : b). Furthermore, this change resulted from the use of the word in geomctry, not in tho theory of music. The corresponding remark holds for d"aloyla, namely that in mathematics it came to have tho IiJlccilLlizcd men.ning of 'sameness 0/ ralio' (a: b = c : d) iJUlteo.d (If referring in general to pairs of numbers whose members bore the same r('llLtion to eu.ch other. Apart from the fl"~ment of Archytu.s which we have just been cunsidering, I know of no othor ancient text which uses the matheIIl1Lt,i(',nl term lOyo, in this old sense (i.e. with the meaning 'any kiml Clf I'elatiun Letwecn two numbers'). In Euclid, lOyo, is always used to melLn 'ratio' (a: b, us we would write it). Nonetheless, Aristotle himself t!on·oborn.tes my' conclusion that d"aloyla originally had II. wider meaning in mathematics. On one occasion be writes that tho propurtion a : b = c : d is what "mu.thomo.tieio.ns refer to Il.8 the geoIIIl1t.ril: tUuzlogia"1I3, while he remlLrlcs on another occasion, about the notion of piacw, that "if ten is many and two is few, then six ot:cllpiclI the mean .. ,' for it is as muoh greater than the ono Il.8 it is HIIIILllcI' thu.n the other, and this is the moan according to the arithmetic ILnl\logia" (ToVTO de piacw lad" xaTci n)v de'(}JlfTfudtv d"a..toy/mr).1IC ,.10 'J'rlllllllllto"'11 now, III filet Archytaa also refura to the Itannonlo mean f"OI/:II1"nl. qUllt.,.1 "hovu. III Nicomachean EthiCll, 1131b, 12-3. IIIThid., 1108&33. III

08

analogia in tho

Uri

l'AnT Ill. rUle l'IlK·KUCUDKAN 1'UHOllY 01' l'lUlI'OH,1'lONg

IiZ-AUc); TUhl Bhl01NN1N08 Olr OJUU:K. lIlATDIWATlal

So the original meaning of d"aloyla must in fact have been 'B8.meness of any kind of relation between two numbers'.1Ii

2.18 THE OREATION OF THE MATHEMATIOAL OONOEPT OF lOyo,

We hu.ve come ILcroM two different meanings of the mathematical term Myo, which have to be clearly distinguished from each other: (I) In Euclid, and in the mathematical literature genemlly, lOyo, denotes the relation which we would call ratio (a : b). (2) In the fragment from Archytaa discusSed in the previous chapter the word lOyo, had another meaning. Archytll.8, when talking about the o.rithmetio mean, compares pairs of numbers d"ci ..t&YOl'. Furthermore, he refers to the arithmetic meo.nitself(12-0 = 0-6),o.ndalso to the ho.rmolli(~ mean (12 •.. 8 ••. 6), DB d"aAoyla. Hence he must Le using the word lOyo, to denote any kind 0/ relalion between two numbers.

----.

.-

.

'"

"My conjecture is that the first ofthtlsu meanings (mtio) WII8 ohtll.incd by II. process of specio.lizo.tion from the second, and is bo.sed ulmn tho following reasoning: (a) In the terminology of the thcory of music, the word for 'ratio' was du1C1T71l'a. The way in which it acquired this meaning can be described roughly as follows. Initially it meant 0. 'straight line' or 'length'. It then camo to mean that 'scction of string which WILS kupt from vibrating' (to produce the second note of 0. consonance on the monochord), and finally it WIl.8 used to describe the 'ratio betwcen (the) two numLors' (ussociated with the end points of this picce of string). A ditulema had of course two 8eo, (end points or numbers). When tho various calculations (i.e. cuts>. were first mlt.do on the canon, and it was desired to compo.ro different line segments contained in an interval, the term lOyo," WDB introduced~ It wus used to denote any pair of numhers whieh formed tho end puint8 of I~ diastenul. It goes without B8.ying that according to this ancient musical usage This is why tho lator Pythugorcans wore ublo to refer to tho remaining WI dWJMyla, On this subjoet, BOO P.·H. Michul, J)e l'ytlUlf/ore d Euclide, l'uriH IUIiO, PJI. 361i-411, utHO Lho paper of mine liBtod in n. 12 above, p. 102, III

mean (pladl''lTe,), which will not. be discusaod here,

SZABO: TUK UL:IlINIUNOti Olf all"!>:'" IalATUlWAT1CII

(I.ho tmoos of whioh are to be found in our fragment of Arohytas) the words loyo, and 6uiCfn]pa are not synonymous. (b) I t was only when the theory of prollortions, whioh had initizLlly b~l confined to l1!uBio, began to be used in a.rithm~Tc'anagcometry 1\8 wclitliatthe-mo8.iung of Myo, ohanged from any 'relation botween two numbers' to 'ratio'. The decision to 888ign this meaning to Myo, may have been prompted at least in pa.rt by the fact that the ambiguity of the term 6uiCfnJpa (straight line and ratio) must have been espeoially confusing in geometry. It was advisable, therefore, to introduce an unambiguous name for 'ratio' and reserve 6"iCfn]pa (at. least in geometry) for 'straight line'. After the new meaning of Myo, (u.s a mathematioal term) had been Settled once and for all, a. corrcsponding change naturally took place in the meaning of d"CLloy!o,

~iCH:h::~r::::::.~::::IS::~'o:::::o:~ . .

The next question which presents itself is how the word .loyo, acquired its meaning of 'relation botween two numbers' and, in I'Brticulll.r, 'ratio betweon two numbol'H'. This meaning was peouliar to music u.nd mathematics. In everyday spceoh .loyo, was used only with such meanings as 'word', 'speeoh', 'conversation', 'narrative', 'ILccount' and 'reason'. I believe that this question ca.n be answered only if the following is kept in mind: Since "OTcUoyO, meant an 'enumeration', in cl888ioaJ times the word Myo, must have had amongst ita other everyday meanings one like 'number', 'quantity', a 'sories or tota.lity of certain things helonging together'. In Herodotus (lII.120), for example, one RndlJ tho worda: lou y«it? b d"6ecii,, Mytp - "Are you in the ranks (or amongllt. the number) of men ttl [i.e. 'Are you worthy of being regarded as a ml\n ,'] Similarly he writes on anotherocoa.sion {11I.125}: b dmean06w, Mycp 1IOI£15POO, - "he added tbem to tbe totality (or to tho number) of his slavcs". In this usage Myo, scoms to be a concept similar to ciC!,Op&,. The pbl'UBO oiJ~' b M"cp, oh' b decOplp,lIl whioh is quoted in Papa's lexicon, supports this conolusion. Rather than oiting a hoat III

Seo tho entry under Ad,Io,.

J.U-'-

of other examl'lCIJ, lot me just quote 0. passage from Aristotle (Nicomachean Ethics, 1181b20), which runs IL8 follows: l" d"aOov"ae U,,(P yl)'VETO' TO llano" ,,~O,. neo' TO pdCo" "a"&". - "For a lesser evil compared to 0. greater one, counts as a good (or is counted in the ranks of goodness)." It seems to me that the diIJtinotiv6 meaning of Myo, as a musical term is derived from the usage of the word which is discussed in tbe preceding partl.brraph. Since one of tbe everyday meanings of .loyo, Was 'a 8eriea of things' or 'a combination of several objecta or numbers', it could be given the technioal meaning of 'a combination of only two number8 which were related to one another, i.e. a 'relation between two number8'. (Of course, tbis would have required tbe conscious creation of a new concept.) 'In addition, there is an interesting ambiguity about the word ..w"o~ as it was used by the Pytbagoreans. In musio and matbematics they used it to denote 0. 'relation between two numbers' (a-b) or (a: b), yet at the same time they followed tho general philosophical pmctice ~.of using it to me!Ln 'urulellltLLllding', 't~o~g1.at', (lr 'rolwon'. Thore uro innumerable passages from the literature on the thoory of music whioh could be quoted to show that on Bome ocoasions it is not at ILII easy t.o dilltinguillh IIlunply botween tholle two moanings (lo"o, 1\8 'numerical ru.tio' !Lnd Myo, 1\8 'undel'Htanding' or 'reason'). This al'plies for the moat part to passages whioh are not difficult to understand, !Lnd whose meaning ilJ clear and unambiguous. Nonetheless it is dimcult, wben one reads them, to know whether Myo, should bo transln.ted u.s 'reason', 'understanding' 'reflection' or 'numerical ratio'. Of the many examplcs which jJJustn~te this n.mbiguity, I want to mention only two here. We have alre!Ldy discuB80d 0. p888age from Ptolemy ll7 in which he describCH thu procedures omlJloyod by cortlLin uf thollo whu CUIIdueted musical experiments DB follows: "They do not obtain the divisions of the string by calculation, but by stretching the string and thon moving the little bridge until eMh of the notes being Imught after is heard, etc." \ The moaning of thill pWIBage is not prohlematic ILl. all. If, howover, it is read in tho original Greek (oM' 8Ac.J, h, TtP M"cp 1IotOVVTOt Ta, "'Soo n. 108.

.::.:.'

1 ;0

I'AUT 2. TU" 1'ltK-KUCl.IUKAH l'UKtllty 01' 1'llOl'tHt'l'lIIr;lI

IIZAU!): l'UK UK1I1HIHHOli Olf Ollh:KK. UATIlKKATIClII

xaTaTO/la,), then it is not clear whether lO)'o, is supposed to mean 'roflection', 'understanding' or 'numericaJ ratio', since any of these senses would fit the context. (The word is translated by 'calculation' in our English version of ,the po.ssage.) There is 0. similar pBS8age among the writings of Aristoxenus, the opponent of the Pythagoreans. 1l8 In it he maintains that only our senses (in particular, hearing) should be called upon in deciding about musica.l tones, since they alone are competent in this matter (., a~tll, itITtv ., TOVT06 ~ela). He then goes on to add that 'our senses are not in need of any help from logos' (oodA l&"ov delTa,). As the context shows, this is another case in which Adyo, can be taken to mean 'numerical ratio' as well as 'rational thought'. In the last analysis, Aristoxenus is objeoting not only to the fact that the Pythagoreans called upon rea.son ().Oro,) instead of relying exclusively on sense perception, but also to the fact that they wanted to make musical interva.ls, which were supposed only to be heard, dependant upon numerical ralios (1&,,0')' This ambiguity in the meaning of the word Adyo, (relation between two numbers, and rational thought or reason) was naturally of con· siderable importance to those Pythagoreans who wanted to view the 'rationale' of the universe in terms of number.

2.20

TIIB APPLIOATION OJ!' THE THEORY OJ!' PROPORTIONS TO ARITHMETIO AND GEOMETRY

Tho evidence presented up to now is sufficient, I hope, to show that 11.11 the important terms of the theory of proportions have their origins in the theory of music. Furthermore, we have seen that this holds not only for the terms themselves (such as d,dtln1pa or l&yo, and Beot), but also for the concepl8 to which they refer. Since Greek mathe· maties without these terms and concepts would be inconceivable, ono's first inclination is to think that the original theory of proportions WIUl concerned exclusively with !pusio and that it was subsequently applicd to arithmetic and geometry. This conjecture is borne out particularly by the existence of certain geometrical terms ('doubled

'i

1 'j

~'

... Harm. BIoich, 63.1311.

,~

~

,I

, I

',I

.., .

1, 1

ratio', 'compound ratio'. 'diJl'erence betwoon two ratios', etc.) which oan only boexplu.ined. at lell8t from 0. linguistic point of viow, by ref· erence to the calculations performed in the canon (cf. chapter 2.10). How~ver, the application of these concepts in arithmetic and geom· etry '-o.~as far from being~ mechanical procedure. It seems ru.tI~er that nothing more tho.n'& start was made in the theory of mUSIC, and that the development of what we would understand by the theory of proportions actually took place within arithmetic and geometry after the bll8ic concepts had been taken over from musical theory. Indeed, their application to a new field even brought about some changes in the concepts themselves. To a certain extent, this should already be evident from our discUBBion up to now. Although those Pythagoreans who concerned themselves with tho theory of music came later on to regard 8,atIT7]pa and Uyo, as equivu.lent concepts, the fragment from Arohytas makes it clear thu.t A&"o, did not origina.lly mean 'geometric ratio'. and hence that d"a.toyla did not mean 'geometrio proportion' but merely 'equality [when taken] in logoi' (where logos was some unspecified relation). It seems thu.t on the whole the cumpari80n of quantities [taken] in logoi was much less importunt in the theory of music than it was in arithmetic ".nd geometry. It is true that pairs of numbers (the constituents of an u.rithmetic mean) are compa.red !iva ).0)'0" by Arohytll8, nonetheless the notion of 'equality [when taken] in logoi· was not of any great impor· tance (from the point of view of furthcr developments) until it bocame featured in the geometrical arithmetic of tho l>ythu.gorea.ns. 'ro soe this, one should look at the Pythagorea.n definition of similar plane numbers (Element8 VII. Definition 21). rAs was mentioned in Pu.rt 1, the Pythu.goreans thought ttmt l1.ny ! number, when decomposed into two f!l.ctors, could be regarded o.s II. .: plane number. (Ii'or example, 3 was represollted by 0. rectanglc with .,.,. sides 1 and 3.) The definition (VII. 21) stu.ws: "Similu.r plane .,. numhers (8/loto, bdnedo& df!,O/lol) 0.1'0 those which have their sides proportiona.l (01 dvdloyov (i'cta,) lxone, Td, nAwe d,)." , So 3 and 12, for example, would be similar plu.ne numbers fLccord~ng to this definition, since they can be represented by II. l)u.ir of similar rectangles with sides (factors) 1. 3 and 2. 6 respectively. It is obvious that tho logoi furmod from tho corl'081)onding sidos of thoso rot:tu.nglcK are equal (1 : 2 = 3 : 6). Furthermore, each of them is composed of 0.

,

...

O"AUV; £III. 111"'''11'1'''''''01. U¥ UIUI:Ila.. "AI·U~A·l·'t.'It

PART 2. TUK l'l1K·KUCLllHUN TIlKOUY

I

\

pa.ir of line segments, just 88 a dia8lema (an intervw expressed 88 0. numcrical ratio) in the theory of musio waa produced by two sectiona ~I 8triJ~ which differed in length. The concept of 'equality [when taken] In logo, , however, waa much more important in geometrical arithmetio (i.e. in the arithmetic of plane and solid numbers) than it waa in the thcory of musio. In the latter its most important use W88 in the fomlUlation o~ suc~ results aa that the proportional numbers of a filth were 1~lway8 ulentlcal, knowledge of which did not lead to any further discoveries. On the other hand when applied to the corresponding sides (factors) of plane numbers, this same notion revealed something of great importunco. The relation of similarity between reotangles could be characteri,zed in terms of it. O~ c:ourse even before the introduction of the concept a"aloyla, mathematlolans must have had an intuitive idea of what it meant for two rectangles to be similar. However, it waa only after they ~alised that 'geometrical similarity' (between rectangles in the first. Instance: and. then between rectilinear figures in general) wna ~reCl?ely the equ~,hty [when taken] in logoi' (d"aM"la) of correspondmg sldc8, that thlS concept become a truly soientifio one. So we seo that a concept which our etymological investigations have led us to conclude originated in the field of music contributed to a better undorHt,mding of geomelriCtlI similarity when it wus applied in geometrical arithmetio, ~!fJlilarily waa soon to be the 'analogia of sides'. This explains how the word d"aM"la came to be used by the eduea.ted in non·mathematical contexta to mean 'similarity' (dp0WrrJ'), (As is well known, Aristotle frequently uses the phrase -rd c:WdloyOJl with just this meaning,)111 It is worth emphaaizing at this point that there is another indubita.ble evidonce to show that thesoientifio concept of geometrica.lsimilarity nHlHt )mve boon an invention of tho Greeks. 1'ho following definition makes this plain: Element8 VI.I: "Similar rectilineal figures are such as have their n.nglea severally equal (6po,a ampaTa eMJVyeappd. iar"" 6aa -rd., Tt .... MeI4pIaYIlic8, N. 6.1093b17ft. For dsoaAoyla in tho IICIN18 of 6pol6n'1'. 880 PIJilodemulI: On MelAotU o/lta/~ A Stud•• 0/ AncWnt Em • • • P H L d E ' " p&rICUm, , • Bey an " A. Lacy, Philadelphia 1041. Similarly Alexander Aphrodislenais (vol. 2 ofOommmllJria in Arilltolelom GracetJ ed. M. WalUes, Berlin 1892) p. 646: Iff, explains d.cf.toyOP 88 6polOll, On this latter, see my papor listed in n. 8 above,

or

1'1l0I'ou:nON:I

173

),l1Wla,.raa, lXE' XaTa plav) and the sides about equal angles propor· tionol (xal Tci, xeel Tci, raa, )'ClJJlla, xAeveci, uJla.toyoJl scil. laa, lXEt}." Thus similarity is defined as having equal angles, and sides in the 8ame ratio (d"OlO)'0"). This presupposes both the concept of 'anglo', whioh was d~veloped during the oo.rly Ionian poriod,120 and that of d"dloyOJl, which came from the Pythagorean theory of music. The difference between the above definition of similarity for rectilinear figures and the definition of similar plano numoors is immediately , 'apparent, Plane numbers were always thought of as reclanglu in l...... geometrical arithmetic. (The Pythagoreans also considered othor Idnds _ •. J- lof plane numbers, such as triangular and pentagonal ones, but these are never discuased in Euclid's account of arithmetic.) In the CIl.'5e of 8imilar rectangles. however, there was no need to insiHt on their angles being equal, sinoo a rectangle by ita very nature has only right angles, Furthermore, the sides of these rectangles always represented integeraiThe musica.l theory of proportions dealt only with integers and never with incommen.mrable magnitudu; so we have an additional ''explanation for the OBSO with which concepts from this theory could be applied in geometrical arithmetic, It was a different story, however, when mathematicians turned to the similarity of rectilinoar figu1"C8 in genernl. ThiH prohlem t1CKIH lIut always yield to the simple arithmetical techniques which work so well in the case of rectangles. Of course they were no loss certain in their own minds that the sides of similar reotilinear figures stood in the same ratio that they were about the sides (factors) of similn.r plane numbers. Yet it was often impossible to prove this rigorously (in the CBSO of triangles, for example) so long as the theory of propor· tions (as in a~thmetio and the theory of musio) applied only 10 number8 . and not to incommensurable magnitudes as well. It WM convonient, therefore. to base the similarity of rectilinear figures in the first pla.co 'on the equality 01 their anglu, The fact that corresponding sides stood in the same ratio came out of thil!..,,:,Lm.Est u.s a consequence. My conjeoture is that the concepta of tho musical theory of proportions were applied first of all in arithmetio. Their application WIl.8 facilitated by the fact that numbers in arithmetio, just as in the theory

I:,

t

ll

. L/

o.

Bookor, GrundlcJgm der Mallaemalill in guchiclallic1aer Enlwic/dung, Freiburg-Munich 1064, p. 27. 1.0

, " I I

'I "

'Ii,

I'

Ii; ,

, .

!; t. !,

J

'I:

i '

I

,

I

.i

:1 I

I!,

.I' I

',':,':',

174

B~AUO: THE IIEOUIN1N08 01' OIlKJ.lK AlATHJWA'rlC8

I'AIlT 2. TBK l'll.E·KUCLIDJIlAN TBKOllY 01' l'n(Jl'Oll.'1'1ON8

of music, were represented by straight lines. (Hence t.he product of two numbers was a rectangle, and product of three factors was a solid.) Furthermore, the application of this theory to geometrical l\I'ithmetio contributed towards an understanding of the problem of \ \ geometrica.lsimilarity, and this problem in turn soon led to the problem of linear incommensurability. I even believe that the theory of music --.....'ll prepared the way for the discovery of linear incommensurability. \.! This is the point I wish to take up in the next oba.pter. . "

.....~--~

2.21 THE MEAN l'ROl'ORTIONAL IN HOSIO,

; \j

-

ARITBMETIO AND GEOMETRY

In a previous chapter (see p. 164) we discussed a fragment from Archytas dealing with the tltree means in muaic, and pointed out the significance of two of these (the arithmetic and the harmonic) for the theory of music. [As has been remarked, these two means effect the division of a musical interval into 'UntlJual subintervals, thc subintervals being the same in both cases except that they occur in the reverse order. The classic example is the division of an octave (12 : 6) into a fourth and a fifth (12 : 0 and D : 6) by the arithmetio mean, and into a fifth and a fourth (12 : 8 and 8 : 6) by the ho.rmonio ono. ] The so-oalled geometric mean lies in between these two, but the role which it played lsless olear. According to Arohytas, the geometrio mean is that 'end point' (8(!ot;) between two other 'end points' ("numbers") which divides the original interval (say a : c) into two ~C!-lsubintervals. (i.e. in such a way that a : b = b : c). In the case of an octave:-ilie-geometrio mean would bo tho numbor x satisfying the equation 12 : x = x : 6.' Even if fractions are allowed as well as whole numbers, there is obviously no such 'number'. (Certainly there is none in the sense of Euclid's (,dcfinition of number.) .An octave cannot be divided into If,uo equal ! !subintervals by a number. This was not just a practical difficulty for f i the Pythagoreans, but 0. theoretical one as well .. According to their \ ltheory, any musical interval could be expressed as a ratio of whole \ n 11111 hortl. 1\8

!

y

It is plain that the above appHes to fourths (4 : 3) and fifths (3 : 2) wcll; as with the octave, there is no number .which divides them

175

into equal subintervals, nor is there a (numorico.l) geometric mean between their proportional numbers. This leads us to the question of why Archytas counts the geometrio mean as n. musical one, even though no such mean can exist in music. As far as I can see, no liglit is shed on this question by the context in which his words appear. Hence any answer to it ca.n only be conjectural. My own view is that an explanation is provided by the following. /-It" is clear that the geometric mean did not receive its name until after mathematicians learned how to construct it in a geometrical way. Originally it was a source of considerable puzzlement to those ( interested in the theory of musio. The only thing it has in common with the genuine 'musical means' (i.e. with the arithmetical and harmonic ones) is that it too emerged first in the theory of music. We have in fact definite proof that the Pythagoreo.ns attempted to divide musical intervals into equal subintervals. In other words, they tried to find the geometric means of these intervals and came to the conclusion that this could not be done in an arithmetical way. Our evidonce for this is Proposition 3 of the "Sectio Oanonis, which Sta.tes: l21 \ "No t!tol1.n proportioOlL,!)umber ca.n ovor bo found hotwoen two numbers iii a ratio s'Uperparlic'Ularis." To underst.n:n(l t,his proposition, one needs to know that the Grcclcs meant by a 'ratw superparlicularis' (~y~~e!~.v ~tadT'Illa) what we would denote by '(n 1) : 11.'. This kind of ratio is exemplified not only by fourths (4 : 3) and fifths (3 : 2), but also by octaves (2 : 1). The proposition states, therefore, that tIlere is no (numerical) geometric mean between two numbers in a ratio 8uperparlicularis. (It should bo lcopt in mind thu.t all tho important mUHiCiLI intervalH n.ro in fact examples of tIllS kind of ratio.) Furthermore, the Gre,ks were well ,ware of the fn.ct tho.t this result was always true'rAto matter whic~ultiples of the ratio superparticularis wero used to 'test it out.

+

III Most scholal'8 since Tannory's timo havo 08crihcd this proposit.ion t.o Archytos (soo, for exo.mplo, van der 'Woerden, Hermea 78 (1943), 169). Our source, BoothiuB (DIl Muaica III. 11), objects to "a proal of ArchytltB" whlln talking about t.ho propolJit.ion in qUOllt.ion. 1 do not. think t.hut. t.hill by itsulf ill " suffioient reason to attribute the entire proposition to Archytas. In my opinion. 8eaio OanoniB 3 was formuluwcl long Loforo his timo.

176

1'.\ In' :£. 'l'Uil: l'IlK·"'UllL!UUAH 'l'UKOILY 01' l'IlOl'Oll1'IONli

IJZAB/): 'l'UK IJKU1NNIN08 Olf OJlKIolK. UATUIWATI08

This muoh is plain from the proof of the proposition in the Seclio Canonis o.nd from the other propositions whioh are ll88umed in the proof. 1I1 Their lack of success with this problem in the theory of musio prompted the Pythagoreans to pose it in a more general form. They asked under what conditions it was poBBible to find between two numbers a third having the property that it formed the same ratio with both of them; i.e. they wanted to know wheM(mean proportional number existed between two numbers. In Boole VIn of Euolid's Elemen18 there is a whole series of propositions which deal with just this question. Let us recall BOme of them here: VIII.ll: "Between two square numbers there is one mean proportional number, eto." VIII.12: "Between two cube numbers there are two mean proportional numbers, eto." VIII.IS: "Between two similar plane numbers there is one mean proportional number, eto." VIII.20: "If one mean proportional number fo.1ls between two numbers, the numbers Will be similar plane numbers." There is no tradition to confirm that these arithmetical propositions do in faot antedate the discovery of linear incommensurability. However, since it is possible to prove them without suoh knowiedge,1I3 it does not seem unreasonable to regard them as baving paved the way for the discovery of how to construct a mean proportional to any two straight lines in a geometric way (cf. Elements VI.IS). Thus an insoluble problem in the theory of music (to divide the interval associated with 0. consonance into two equal subintervals or, in other words, to find 0. numerical geometric mean between the two

numbers associated with a consonance) may have led, by way of arithmetic, first to the problem of geometric similarity (for rectilinear figures) and then to the geometrical construotion of 0. mean proportional to any two straight lines. In a. sense, however, the lattor assumes 0. knowledge of linear incommensurability. For the mean proportional between two straight lines whose lengths are not similar plane numbers is itself 0. straight line incommensurable in length (cf. Part I of the present work).

'>

111 As far 88 thesa other proposit.ions are concerned, I caanot. but. agree with van der Waerden's analysis (see the previous note). However, I would like to dissociate myself from the view that. they are all due to ArobytBB bimB8lf. lla Incidentally, I agree with van der W80rden's remark (S88 Zur GucAicAU der uNchiachen MaIAemalUc, p. 224) to the efloot thBt. eM problema in Boo" VIII are wry clo.talll connedcd willa eM I1a«wy 01 irrational6. 'l'hesa problems act.ually prepared the way for the theory of irrationals. This Is one more l"888OD wby nook VIII oould not havo boon written during tho time of Arohytaa. Thothoory of irrationals, 88 we can 888 from the ooncopt. dynamu, dates back much earlier than tbis.

177

----.-

2.22

THE CONSTRUCTION 011' THE IlEAN PROPORTIONAL

~ The construction of a mean proportional to two given line segments -.2s discussed by Euclid in Proposition VI.IS of the Elements. We know tha.t Hippocrates of Chios, the famous 5th century geometer, must already bave been familiar with it.lIt The ancients, however, have not left us any information about the age of the proposition (VI.13). or about the steps which led up to the discovery of this construction. The following attempt to reconstruot these foots could thorefore bo dismissed right from the start 118 mere conjecture. Yet it scems to me that it is still worth making, because the whole point of stUdying the history of mathematics is to try to understand how tho development of science has led from one problem to another. The first thing to remember about Proposition VI.IS is tlmt 8imilar right-angled triangles (Lro used in its proof. 'I'ho problom of tho mean proportional is of course inseparable from the problem of 8imilarity for geomotrico.l figures. It was 0. more thorough invOHtigl~ti()n of geometric similarity which provided the solution to the former problem. Furthermore, their experience with arithmetic must have suggested h to the Grcolcs that this was the right way to approaoh it. For they already 'li' knew that 0. mean proportional number existed between two similar plane numbers, and were also able to co.lculate it (see pp. 51-3). It sooms likely that they were familiar with 0. special CI1.86 of the construction hofore heing ablo to oarry it out in its full generality (as is done in ElemenU VI.IS). One case in particular strikes me as especially simple and obvious.

I

au Sco n. 37 to Part. I abovo. 12

5zab6 •

., " .


176

II"AUO: TUII: UKUIHH1HIJII 01/ UII.Kti:K. AlATUII:UAT10ll

Uccu.ll that at the end of Part 1 of this book, we were led to conjeoture that the problem 0/ doubling a sqtUlre Wil8 originally equivalent to the problem 0/ finding a mean proportional between one number and a second wl,ich wtJ8 twice as large. In fact, when the Greeks discovered

,

"

that the diagonal of a square was the line segment nooded to construot a lIecond square having~i~ tht) ~~ of the original one, they must at the same time have realized tbat it represented the mean proportional between one side 0/ theslj1lare and a line which was twice as long as this side. The first fact would bave been visually evident to them (cf. p. 93), wherellB they may havo become convinced of tho second in the following way. ._.- ' . -- ) (a) It must havo been self-evidenVto anyone that the two squares had to be similar. (This would have been so, whethor or not thoy also took into account the fact that if a number was lIB8igned, as a length, to the sides of the original squo.ro, the same could not be dono for the sides of the second one.) Furthermore, it must have been equally obvious that this WllB also true of the right-angled triangles obtained by halving the two squares. (';[0 begin with, of course, the eviden~ for this was only visual and intuitive.)

l'AlI',' 2, '1'1111: l'IUN':UCl.IOKAH 1'UJ>OIlY 0'" "ltul'Oll.'flOHH

from the theory of music. I am of courso reforring to tho problem of. halving tI,e octave (2 : I), i.e. of finding tho mean proportional between

two lengths of string which produce an octave.] The argument sketched above is not valid unless the concepts of Mro~ and waAO)'/a are taken u.s applying to magniludes as well u.s to numbers. ThuH they havo to be undemtood in the senae of the so-called Eudoxian definition which is found at tho beginning of Book V of t.ho Elements. It. ill an open qUCHt.ion whether the definitions of theso concepts were extended at the same time it WtLS rcalized that tho diagonal of a square WIUl also a melLn l,roportional, or whother this cjid not take place until later. " In any event, tho way in which mathematicians discovered that tho diagonal of u. square wu.s the men,n },roportional between ono of its sides and 0. line twice tlmt length also taught them a great deal about how to eonHtnwt. u. moan prolmrt.iomLl to two arbitrary line segmonts. . (1) The first. thing which must have been clear to them WI\S tluit the simplest way to ohtlLin a moan proportional WI\S by using aimilar rigld-angled triaJlgles.

d

II

....

170

)

I. i

I

I'~ I

I :

I

I,

I

f

I~

·Ii ,1/

Y

Fig. 6 a

Fig. 6

(

',,-

(b) It. WU8 now an ellBY matter to form ono logos from tho corresponding sides of the two squares and anothor from their diagonals, just as WllB done in the ClIBO of similar plane numbers.}:!ence tho first log08 was composed of the shorter sides of tho two~triangles (II: d), i whereas the second one was made up of their hypotenuses (d: 2a). -=" Tbeao logoi were evidently equal (a: d = d : 2a), just like the ones formed from the corresponding sides of "similar plane numbers". In this way the dJagonal of t1~~ .o~iginal__squ~ was clearly shown [These considerations also providto be the required ed 0. solution to the geometrical oquivalent of an insoluble problem

mean proiwTiiOnal.

(2) They must also havo roali~ed that the strnight. line in question was the required moan proportional precisely because it played two diOerent roles lUi far lUi tho two triangles were concerned. (In Fig. 5 the line segment d is adjacont to tho right angle in the larger triangle, whilst forming tho hypotenuse of tho smaller one.) The question flOW WIUl IIOW to utilize this information; how, for example. were they to construct a pair of similar right-angled triangles. If they started out with a right-angled isosccles triangles (one half of a square), then this latter qucstion was easy to UnHwer. Two similar (in fact, two congruem) , right-angled isosccles triangles can be got. from such a triangle by dropping a Jlerpcndicular from the right angle to the hypotenuse. It must hlLVO beon immediately apparent that tho

.I 1

I

, I. I

180

"A itT 2, 'l'UK l'llK,lI:UULlUKAN TIIKORY 011 l'J\Ol'OIl'l'IONIi

SZABO: TUlC UlCUJNNIN08 01>' Ou.RKK MATUKHATICS

which states that the angle in a semicircle is always a right one. What is needed, however, is 0. right triangle in which the perpendicular from the right anglo to tho hypotenuse divides tho latter into two segmonts, one of length % and the other oflength y. We have to draw (in the upper semioirclo, say) IL line d perpendioular to the hypotenuse, whioh intersects it at the point where % and y meet. '1'he point ILt which d interseots the circumference will then be the third vertex of the rc(luil'cd triangle, and d itself will be the mean proportional between % and y. We see, therefore, tha.t this construction ca.n be carried out using only some very elomentary facts about similarity together with 0. theorem of Thales pertaining to tho angle in a semicircle. As in the Ilrevious case, a definition of 'being in tIle same ratio' (dmlorla or ($ ).Oro,) which applies to general quantities is in. dispensablo for a rigorous proof of the above. In my opinion, however, we do not have sufficient knowledge at present to decide whether such a definition WlLB aVILilablo to tbe inventors of this construction, or whother they wcre content to base it on intuitive evidence .

,triangles obtained in this way were similar; but unfortunately they were not suited to the purpose at hand. However, the special case in which similarity coincides with congruence may have been of some lIHe in pointing out the way to the more general one. If (1\8 in Fig. 6) a perpendioular is dropped from the right angle of IL non-isosceles right triangle, two right-angled triangles are again obtained. It could not have taken the Greeks long to realize that these too were similar, since their corresponding angles were equal (<<. = fJ and" = 6). The perpendicular d (see Fig. 6) plays a double role of the 80rt discuased in (2) above. It. is the longer of the two adjacent sides in the smaller triangle, and the shorter of the two in the larger one. They could now obtain a mean proportional (% : d = d : y) between the lengths % and y simply by forming logai from corresponding adja.ccnt sides (of. the porism to Elemenl8 VI.8).

•

181

mno,

'I

2.23

CONOLUSION

Fig. 7

The construction of the mean proportional in Elements VI.13 is actually nothing other than the procedure sketched above in rever8e. In the preceding paragraph a right-angled triangle was given, and a porpondioular WIUI drawn from the right angle to tho hypotenuse. 'l'his meant that the porpendioular itself wo.s the mean proportional hetween the two sogments into whioh it divided the hypotenuse. Suppose that we are given instead two line segments (% and y) and wl\nt to find the mean proportional betwoon them. Our first step would bo to construot a right triangle of the sort given in Fig. 7. The two line segments should together (% + y) form its hypotenuse. ('fhis muoh is obvious from the preceding disou88ion.) If we regard the length % y 88 the diamder 0/ a circle, infinitely many right-angled trin.ngles can be construoted upon it, for there is a theorem ofThales1u

+

II. Bee B. L. van dor WserdoR. B~

Wium6cAa/I, p. 145.

I

~

At the end of PILrt 1 of this boole we CILmo to tho conclusion t1mt tho existence of linear incommensurability may have been discovered in I the course of attempts to find a mean proportional between two numbers :I (exprcBScd 88 straight lines) or quo.ntities. From a historical point , of view, the Greeks originally thought of tho problem of irrationality , lUI belonging to tI,e t!leory 0/ proportions. I hope that Part 2 hWJ shown I .. --.. -_. how the theory of proportions, whose initio.l development took plo.ce in the Pythagorean theory of musio, may in fact have led by way ) of arithmetio to the problem of (Jeomdrical similarity and thence to problem of linear incommensurability. Thero is ono thing, howover, which should not bo forgotten. We have up to now confined our investigation to the way in which Greek mathematicians mlLy havo como cl080 to discovoring that Iinel\rly incommensurable straight lines existed. We have not WJ yet shown how they reached tho stago of being able to prove the existence of linear incommonlmrai.Jility in 0. rigorous mu.nnor. Althougb we can suppose that they knew how to double tho area of a. square by means ~

!

L

~

.~

l'Al41 1:. 'I UlI: l·lI ... •... UCl.JUAAH

, J

of it.8 diagonal, and even t.hat. t.hey were convinced by the intuitive evidence in favor of t.he diagonal being for t.his reason t.he mean proportiono.1 bet.ween one side and a line t.wice its lengt.h, more is needed ( to cllto.blish that the side and diagonal of 0. square o.re incommensurable . in length. A simila.r remark applies to the const.ruction of t.he mean proportional to two arbitrary line segments (Elements VI.lS), as well as to the creat.ion of the concept dynami8. For example, the Greeks may well hnve known that a rectangle S units wide and 5 units long could be tru.nttformed into a square of the same area by construoting the mean proportional to two ofiui sides. Furthermore, their knowledge of arith-metio may well have told "them that a mean I,roport.iono.l number /, existed only between two similar plane numbel'8 (cf. Elements VIII.lS ILnd 20). All this, however,~ would-have ledt-hem to the reo.iiza.tion that the sides of a dynamia having an area of 15 were linearly incommensurable (with the bll8io unit of length), only after the existence of such segments WIl8 no longer in douht. Similarly the definition of proportionality could not IULve been extended from numhel'S to generuJ quantities untH after t.he existence of incommensurahilit.y WIl8 discovored. Furthermore, incommensurahility itself is not something which- \ can be known with certainty in an empirical way. Knowledge of it can ) ~ only be obtained by a Bystematic process of reflection. ',,' For this reason Part 3 of tillS book is devoted to the problem of how the Greeks transformed mathematics from a pro.ctical and empirical body of knowledge into a theoretical and deduotive science. After nil, the existence of incommensurability can be proved rigorously only by the methods of the latter, not by those of the former. I do not want to conclude Part 2, however, without briefly expressing my opinion about the import of some of our conolusions for future research. In the lu.st few decades there havo appeared sovero.l imI,ortant works \ on the history of science which concern themselves with pre-Greek I mathematics and wit.h that of the Babylonians and Egyptians in I particular. 121 This researeh hu.s led not onJy to a better understanding of the place of mathematics in pre-Greek culture, but o.1so to a feeling t.hnt. our earlier pioture of t.he Grock worM and of t.he beginnings of

-.s::---

:.'\vl?/.' "

•

r

III Tho moat. important. ruaulta of t.heae invost.fgat.lons Aro oollootod togot.hor ill tho fil"8t throo chaptel"8 of van dor Woordon's book, BcUmce Awakening.

.

"1'UlI:Ull~

u .... l'llUl'Oll'1'lUI\l>

science needs to be thoroughly revised in the light of t.his new historical knowlec1w!. As long· u.s nothing was known about Egypt.ian and Bo.byloniu.n mu.themu.t.ics, the Greeks could rightly or wrongly he regarded as t.ho originn,tors and founders pf science. The sit.uation changed suddenly. however, when it. turned out t.ho.t. many important mathomaticu.l discoveries which had previously been u.ttributed to t.ho 5t.h ( and 6th cent.ury Greeks, had boon made cent.uries earlier by t.he Bo.by"-'onians o.nd Egyptians. ~e Greeks seemed to have been robbed of much of their old importance in the history of science by this new knowledge of pre-Greek cultu~ Otto Neugebauer, 0. dist.inguished authority on the history of t.lllS earlier mathematics, felt. called upon to write t.hat. booo.use we now know sometlllng not. onJy about the t.wo and 0. half thousand years whioh have passed since the classical age, but. nlso about the t.wo and 0. ho.1f thousand or so which preceded it. the Grceks could no longer bo regarded 88 the prime origino.tol'8 and foundel'S of science. m It. seemed that their Illstorico.l position wos no longer at. the beginning of science, but. somewhere in the middle. 1u AttoJlll't.8 wore oven nuulo t.u prove that. t.he Grooles did little more than carryon t.he science of t.heir predecessol'8, and it. wos said t.hat l28 uThe Bu.byloniu.n tradit.ion provided the mo.toriu.ls out. of which the Greoles, 01' more preciBely the llyt.hagorcans, built their mathenULt.ics." . It is clear that students in tho futu~ win continue to osk what the ~. Greeles might have learncd from their predccessol'8. In my opinion, however, theh' research will be given 0. new tum by t.he mu.terial presented in the first. two parts of this book. Although t.hey will still asl, to what. extent. Greek science is just. 0. continuat.ion of its pre-Greek counterpart. o.nd to what. extent. it is 0. completely originu.l creation. it 8cems t.o me tho.t. t.his question hu.s become muoh morc clearly defined than it wu.s before. This is because in tho preceding chu.ptel'8 wo IlILVO como u.crolltl 0. wholo series of important rnu.themu.t.icuJ COIl, 117 0, NUlIguhllllor, Stlllliun 7.lIr Oosohichto dur IlIltikor Algobru In. Quellen .' wul Btudien zur OucMclde der Math. ota. n. 3 (1936) 246-69. Ho goes on to My: "Tho Bubat.nnCO of elementary geomotry, tho olementary theory of proportions And 1.1111 t.huory or nqllotiOnB 18 to btl fOllnl1 in t.htl nllbylonilln mut.htlmut.it:8 --- upon which tht! Qrooks built. thoir own. In ovory C8B8 tho connection bot.wecn tho t.wo iB obvioUB and undoniablo." lit n. I., Villi IllIr \Vuurtlon. Mat/I, Ann. 120 (1!l47/!l) 123. Itt 11. r•. vun dllr \Vuurt.len, Erwachende IYu8ensc/la/t, p. 204 •

1M

BY-Aile): Tille IIIUIIHHIHUK 01'

ClllImK

alATIlKAIATI(''lI

eOI'll! (difl.'Jlcma, lwroi, 1000.9, analooon, a7lalogia, 'C011tlJOuml ralio' nnd

3. 'l'J1J~ CONSl'ltUU'I'ION 01" I\IA'I'IlJ!:MA'l'tC,.,

l'ch~Lcd not.ions, '0. cut. of liho canon', 'mon.n llroporliionlLl', 'geomotrical HimillLrity dofined 1\8 oqun.lit.y of angles lI.nd analogia of sides', dyna",is, .~y"''''clros, mekei asymmelro8, dyoo'1lci symmelros, etc.) which, at len..'Jt Oil t.hc fn.oo of iii, scem 11.8 tbough thoy must be complelely original and rWlltelllically Greek i,tvcnliolJ,8. So thc question is trn.nsformcd into one :d)()ut. t.ho extent to wbich theso SILmo conccpts or porhn.ps similar oncs figure in pre-Grcel( mathonll\liics. 1 h:mUy t.hilll, Uu,t. a clt!l~r and slLt.iHflLctory answol' cnll bo givon to this 'luest.ion on t.ho bu.sis of our pre.<Je,u IUlowledgo of pro-Grook scionce. Furt.hermoro, t.he time for malting conjectures about. WIULt. tho Groelts might. have borrowed from other cultures will come "ftor wo havo aClluircd a bot.ter undomt.anding of t.ho Grool(s and thoir science.

Wl'l'IIIN A DmDUCTIVI~ ]"ltAMEWOltK

3.1

'rnOOl" IN ommR MA'I'Jll':IIIA'l'ICS

At. t.ho ollcl til' 1'1LI't. 2 wo discuHHm\ t.hl! (:UIIIWI:LiuIIA whic:h vlLl'iomulChlllIUS IULvc t.ricd t.o establish betwcen Gred, nmt.hemn.t.ic8 alld pl'o-IIt'Ilenic scienco. As wo sn.w, somo of them cven lIuLinLuincd t.haL Greell sciollc:e WI\8 buill. upon tho lLehiovomenUt of t.he Hahylonians,'ILIld Umt. it. could ho mglLrtlod 1l..'J IL dil'm:t. cont.illlmt.iull uf its 11101'0 uncient., Orientn.l counwrl'lLrt., (t. is problLhly no accident UlILt those schohLl'8 who t.ried t.1I csllLLlish connocLions LoLwoen the Groel, world unci Lho ILnciont. OrienL did not find it nocelUULry to point O\lt tho vory 8ultstlLnt.ial differences between UlI'SO twn cultures. Anyono wlLIlt.ing t.o lolml about. such differcnce.s would Jmvc IIILII t.o consult t.ho worl'8 "f ;LnnUler gruup of III:holtLllI, Mcmbl'l'8 of t.his lut.tcr group, while noL dcnying tlmt Uto UI'IlI'lu; \Vel'll aLle to Imun IL grclLt deal from their predecc8sol1l (or indeed that they act.ulLlly did su, which seelUs very liltcly t·" luLVC Leen t.ho Clum), did not fOl'get tho cssontially now felLLures of Grocl, (ILS opposed t.o pl'o-Hellenie) 8cience. To iIIust.rate this, lot mo quoto tho following from ILn import/LIlt articlo Ly von Fritz:! "Research ulldcrt.n.lmn during the IML few decades hl\8 shown that. nabylonian mathematics of t.ho pro-Hellenic period WM much mom highly dovl!!ol'ed than hnd beon t.houghL previously, Tho n"hylonilLns woro able to find quito good lLppl'OxiulILt.iull8 to t.ho solutions (If rolativoly complicated mlLthomaticlLI I'l'tIbloms ILhout 1\ thuusand YOlLI'8 hoforn Lho heginnillgs of Grcclt nml.hellmt.ics, It'urLhol'lIllll'o it has bccn shown Umt. t.he GreckR t.oul, ove,' ILnd learnt IL grent dea.l from t.t1O Bnt.ylllnilLlIs,"

,

I

I

Ro,,,

I

K.

II.

ge,c/,j~",c

J,

127 t .. I'"" .. 2 "I",v,-. 1.'ril.7., • Dill ,lexn{ i IIcl" .. grim'hilu'h"n 1\1 nllll'lIIl1l.iI,', A rI'''i" /iir Jlegrig.,(Bunn), VIII. J (IOuG) 1:1 .. 103.

Vttll

180

II?A lit): 1'1I1~ II KII I HHI HfllI 01' 1111 Jo:I(K &lA1'lmasATICII

Tho o.bove might leo.d ono to think tho.t tho dcductive feo.tures of Greek mo.themo.tics o.lso hud their roots in pre-Hellenie science. Yet von Frit7. emphasizes: 3 "There is o.bsolutely no evidence to suggest that the HILhyloni",ns, not to mention the Egyptians, over tried to deduco nllLthomo.ticu.1 theorema rigorously from first llrinciples." Furthermore, the flLCt that no tmccs of deductive mathemo.tics h",vo J,c(~n found in o.ny of tho pre-Hellenic cultures cannot be attributed simply to our In.clt of hiatorical knowledge. For t.he previous quotation continues: "Everything which we know about o.ncient OrienbLI mlLthclIl",tics speo.ks against the ideo. tho.t the go.ps in our historicu.l knowledge are rcsponsible for our fo.illlre to detect the existcnce of deductive mn.themo.ties in the pro-Hellenic period." Tho most substlmtiu.1 difforonce between Groek o.nd Orionto.lseicnces is tlmt tho former is o.n ingenious system of Imowlcdge built up ",ccording to the method of logico.l deduction, whereas the latter is nothing more tho.n a collection of instructions o.nd rules of thumb, often o.cCOIllpnnicd by ext&m1,1es, having to do with how some po.rticular mlLthemo.ticlLI l,u81(s ILre to bo cluried out. \Vhethor Huch fundlLnumbLI scientific conccl'ts us 'theorem', 'proof', 'dcduction', 'definition', 'postulo.te', 'nxiolll', etc, wero Imown to Egypt.io.n o.nd Bo.bylonian mathemo.ticitLns of the pro-Hellenie 1'00'iud, rcmn.ins o.n opell question. We do, however, Im()w the following:· " It is not even oorbl.in tlmt the Bo.byJolliauslmew how to formuh1.tc g(!Ilcml theoreJlls. It is truo tlmt in the sn.cred geometry of ILUciont IlIdilL, Imown Il.8 the m1,e rflles, certain gcometrie fo.cts aro stated in tho furm of sulrf18 OJ" ILphorisms, but theso ILre never provcd. Proofs do not "l'l'eo.r in any of the o.ncient Oriento.l texts Imown to us. At best one sometim('s comes o.cro88 some numericlLl verifications."

Tho impn.rtial reader must by now have reached tho conclusion UULt t.J1O mosL sLrildllg foo.turo of Gt'ook mlLthomu.tiCH, which disLinguishes it fmm its Oriento.l counLol'pu.r\;, is the preseneo of proofs, Greek Boienco is (~uncm'n(ld not only with atn.Ling 1'l'oposiLionR, hut wiLh })l'Uvi(lillg genuine 1"00/8 for them as well. The role played by proofs is no h'HS J

I'AIIT

:I. '1'11'(

I:lIIf:4'rlllll:I'II1N

III' M.\TII~:MATII~

un

illll'urtn.nt in Ol'eok LImn it is in cont.ollll'"m,',\· II1ILthomlLUcs; ill flLct, it iR the 8l1.me. Furthennoro, li:uclid's proof.. 1"'1l fur tho lIIost part motlds of their Idnd ILlul ho.ve 80t the stlLluln.rdR uf IIlILthelllll.tien.l rigou!' for gencmtions. NutwiLhntlLllCling the flLoL 1.lmL lIIodel'll JIIn.themILLit:iILllR RmJl(~I.illles find Lhom wILllting in ono ur IL'WUUW rCRl'cct, the proofR ill tho J!Jiemcnl., arc hy IUHI IILl'go exoml'llLl',V' Thu questioll whieh nnw mises itself is whlLt. fil'8t gn.\,u Greek mo.thenmtit'iILns the ide:L LhlLt I1l1Lthematieal prul'oRiLions necd to be pro\'cII, ILIIII how diel \.JUly Imcollle nu sllilleu ILL pl'oving Lhcm Umt their pl'lIul'n 11.1'0 sLilI Iwmil'ccl tmll1.Y ... - A further I'I'0hll'II1 iH whet.her IlllLthctnn.l.idILIIK worc origi,mlly "ILLisficu with mom 1'l'imiLivc ILn(llcRS cotnpleLo proof8 tlllLll Lhc oncs found in Euclid, ILnd if so, how did Lcchniclucs of prout: h(l(~"I\lC more highly dovclollcd In.tm' OIl. It iK dislI.ppoilltillg thn.t 80 litLle ILtt.cntion has hcen 1'ILid t~) Lh('!;c ()llCsl.ions in clLrlicr r('scarch, Although it lIIlIHL hc ,ulmil tcd I.ImL 0111' hiHluriC:1.1 sourCCR do 1I0L scom to providu allY ILIIswcr to thcm, wc sLiII Imvll ILlwicnL lHILLhollml.ic;1.1 t(lx\..q 1..1 CIIIIHUIt., SIIIlII\ IIf UtCRO (~UIIl;LiIl tcdtllit!1L1 wrlllK whi(:h (:Imnot 110 mtwh "lcI"I' Limn tho LexLti thmllHclvcn. I hopo Lhorefore I.Imt, hy ll'l'l'lying the IIIC!thIllIR IIHCU in lite firKt two plLrlH of this boolt, I will 110 l1.ble to give ILIl ILIllLlysis of the mathellllLtical lel'tnR found in Euclid (I1.nd of Lhe conccptH Lo which t.hey refer), which will Khcd somo now light. un \.ItO ILhovo '11H'HtioIlH. Lei. 1111 hcgin Ollr illvcsLigl1.tion hy cnnsillc'l'ill~ lim IIl1\Lhellml,icl~1 WI'lli f(ll' IlToo/alld to T,,'olle, 1I11.lllcly t.ho Grenlt wllrd de/XI'IIPl, JI, is IIHctl very fl'(~lJlJ(m\.ly hy ]~lIdid, ('Hpe(:ia1ly ILl, tho 1',111 of l\Il1.l.hellmt.il~al 11'~lIIOII­ AI.mLionR, HiH 1.l'(ll1.tnl(;lIt of lItem'omK ILlwlL,\'H (~lIl1forllls til Lho followill~ I"LltA'rn. Ji'irst hc SbLtcs Lhc prol't1Kil,iull itself, 1LI1111.1t('1l ;1. IULrlicllltLl' i/ll~LILI\CC tlf it whidt is t.ILI(01l to Ito typical. NexL followH II. (lroof of Lhe (ll'CIposition for Lhis illstlLllce. 'fit;, original proposition (whi('h is now l'(\pcll.wd W01'1I fill' Will''') i:; limn ilirurrml frolH the proof, ILIlt! tho <1isl:llllllion iN ClIlldlleleel wil.h tho phl'lL/iO: I/m:€! ")1:, IleiEu( - "("I~illg) wlmL iL WII... I't!Iluin'elllI provo". Thll fl1."L thaI. UtiH I'lli'lL/it! iH nnv(w t11l1il.II't1,6 .. hll\vH '''t~ILrly UIILI. 1.11" achml denwlullrcdimt is Ule most esscntial part of the whole c1iRCIIS,qioll.

lhid" I'P, 13-4.

• 0, lleclwr, 'FrilhgriochiBChu ltIuthl!III1ll.ik IIncl lilURikh,hro', Arcl&iu fUr JlflllJilcwi6lJenlJclulfl rrro&aingrn), Vol. 14 (1967), 167.

, II. Rhoul.l h .. 1II"lIl.illll"tI \.1",1. \.111'110-('1111,·,1 l,ml,lc"", .111 III1L nlwlI}'R elltU'I ... I" wiU. tllI'RIl Will'll", AR 1',,()dUK r"lI1nrkB ill hi" ,,1II11111'·I\I.nry (I', RI, /)-22 ill I'·ri ... l·

188

S7.Au6: Tille III~IIINNINIIH 01' OIlI~I~K MA... JmAJATJCS

l'Alll' :I.

nm

C:ClN!I"'IlIlC~I'JIIN

"1' MATIII'.MATI("oq

IR!I

'I'lao vcrb ~t:iea, sums U1' and pIJ~CC8 oml,luUlis upon tho demonstration. Of courso the meaning and otymology of the Greek word ~£{1C11VIl' Imvo nover been problematic. Furthermore, ita uso 8.8 a mathomatical term Ims been so taken for granted that tho question of how it came to bo UIIO htLS not been givcn much attontion. o Thore are, in fact, at 168.81, two int.cr08ting I'hl\SCS ill tho dovel0l'montof thill oXl'reNlion 8.8 0. mathonmtiC1\1 term. Defore discussing those, howevcr, lot us considor tho nunIImthematical u8I\ge of thill word. In most dictionaries, tho mCl\nings uf the wurd ~El"vvll' ure split up into threo goul'sl\S follows: (1) to show, to point out; (2) to tlUlke /mown, (c,']Jecinlly by word8) ,to explain " and (3) to slw,,,,lo prove - [or: (1) ,flire floir, tnonlrer,. (2) laire connaitre par In parole, expliquer,. Md (3) dCmOlllrer, prouver]. On tho bl\sis of thif, listing, it would secm tJmt the first mOlLning ('to show', 'to point out' in tho most Iiteml sonso of tho words) is the original one, sinco it is tho most concrete of tho thrce. ] tis clearly the 0110 which Plato ho.d in the mind when ho wrote:? TO ~F.je", .uyCd £I, T~V nuv drp{}aAllwv alc1{}7Jc1'v 1(araaTiJc1a, (by 'showing', I underIIt.'LIld 'to put before tho cyCII'). Yot wo also find ~El1(VVllt usod with the moaning 'to make known by words' in texts n.s oarly o.H tho Ody.yscy (12.25). 'fhero are in fact somo parts of tho Iliad (e.g., 19. 322) I~nd Odyssey (10.33) where it is not certuin, whether the word is boing used in the sel180 of 'to point out with 0. finger' or 'to point (Jut with words'. The LILtin verb dico, which comes from tho mme root as ~£lxvvll' is "Iso worth noting in this connection. Although it is related to digitus (,~ 'fingor' or 'pointer'), it mOl.ns 'to Ray' or 'to teU' (i.o., 'to point out with words'). All things considered, it seoms most likely that right from

LIlli Htm·t tho On!"I, vcrb ~d1("1J11C menllt 'tu poinL nut' ill hoth a figumLi \'0 and a liteml flcnllO, and henco also 'to OXpIILill'. We now Wl~nt to find out how thifl word CILIIlO to be used a..'1IL m"thc· IlllLticlLI t.crm, ILlld what the earliest Grcel, llu~thcnuLlicians me{~nt by it. ThcKO '1ucfltillllH wnuld probahly luwo to I'OlIlILill ullo.nswered, if wo hILt! Iluthing hut tilt! wurd ILnd itA etym(llogy to go on. It iH, hClwcvtlr, interesting to note t1mt 80ll1e flcholu.rs, withuutmo.ltillg ILIly referont:e to tho verb ~£{1("UIl" hll.voRtrcssed the importa.nce uf ui.91wl euidence in Cluly Greek mathematics.s ThiR suggests the ide" that Lhe elLrliesL 'pl,(JtlfH' nmy IlILve involved 80me kind of 'pointing out' or 'making vi.'ii/JIe' (If the flwt.s; inother wordR, ,)E.{XVVIIC nllLy havo hocome II. technics,1 term fur 'proof' in mu.theml\Li('S hecauRe 'to prove' mcunt (Jl'igiOlLlly 'to 1l11~ltU the truth (01' flLlsity) (If I~ mathcmu.ticILI stlLto/llOnt villible in somo WILY', Thill idea cJLIlnoL bo supported by ILlly cxn.mples from Euclid, for tho 'proofR' in tho l~l(!t"cltl•., aro ILlrcJ\dy ill u. polished form and no longor I!lLVO ILnyLhing tu clo with limiting fJLcLK villi hie. In PILI·t 1 of this bool(, hClwevm', wo clllutmllLll (lXI~llIl'l" of IIllLthollllLl.ic:1J to:Lching in antiquity" whit:ll dOCR floom tu support our conjectul'll. Tho eXILmplo comes frum PllLtO'H /If ella (8211-85e) antI i8 of interest to UK here beca.use it toUt'\lCR upon the problom of lJl1~theml~tieal proof. I IlILVO in mind Lho plUlllago where SOCI'ILU'R 11.'41(11 II. Rla-vc how the JUelL of IL SqUILI'O with Ride8 two units long cnn 110 clouhled without ILltoJ'ing iLK fllmpll. lIo thcm dl'/LWR II. dil~gmlJl [RIlU (I) in I~i~. 8] .'11wIIJing the flqlllLro which iH to ho dUII),ltlc!. AfLo .. Lhe RIILVO ILIIRWCrs thlLL tho 8CI'mre will pnrlmps be d(Jublc~ by d()u~,lillg tho longth uf itA sides, SocmlR.s dmwH IL second diagram (2) to slww tlmt n. sqUlnc whuse flidcsare twice as lung

Inin'8 odiLlon): ..... there is a dilltlnction bot.w,,-en a problom and a thllorem. 1·hnt. Euclid'8 Ele,MnU contains bot.h probloms and theorems will be ovidont. rmm t.ho individual proposit.ions and from hl8 pmct.lce of placing at. tho !llld or hi8 demonst.rat.iol18 somot.imoa 'Thift 1ft whnt. was to be dono' (67Uf! ld,. noujaa.) Ulld at. ot.hor t.imes 'Thill i8 what. WII8 to bo proved' (6meld" dd~a,). Tho lutl.cr I'hmso i8 t.ho mark of a thcorem ...... (Tho translat.ion is by OIenn R. Morrow, /'ror.lll' -.Ii Oo,mne,IlanJ on &lUJ Fir,t Rook 0/ E"clid'lJ ElmnflnU, Princut.on 11110.) NOllot.hollllJll t.ho Lorm dtila, do("I occur In t.ho proof of 80mo probloms. It. 18 u8ed in suoh phr08c8 08 1fflO)C£{aOOI dd~a" 6ra ICrA. (Ollr t.II8k 18 to lJhow that. •.• ). • or. A. S7.RIl6, 'Deilcnymi, n.18 mn.t.homn.tlsohol· 'l'ormlnu8 rur boweiun', Mnin N. S. 10 (1968), pp. 106-31. 'Ornlyl", 430 o.

• K. VUII Frit.7., Itl'. cit.. (II. 2 "bovu), p. !l4, 8t.nl~·8 Umt. t.ho nllumlnnt WlO ",hieh WIlli mndn of till! method 0/ 61"HlrptJ•• ilitJn ill pro· gudi,lt!nll mn.t.Iwrnnt.iC'fl inclien~'A Umt. vi.• ual flvitkul'.e IIUUlt. illit.illl1y Imvo ('Inycd n. not. incol18idcrn.blo roill. (011 I.his mnUIOlll4Il11 Oommon Notwn" in tI.., Ele"lfml., 111111 HonUl'R not.o 011 it. (Vol. r, 1'1'. 224-1i ill hill odit.ioll, CIIpocinlly hiR rnmnrlts nholll. t.ho vrrb i'1""(!I,O'lIl'). n.. gOf'8 on 1.0 point. out. t.hn.t. Euolid mlUlll cOIUlidnruh ..... rrorl.8 to n.voill t.his llIot.hml 1111,1 Uuat., ""illg IIIm.,l., ~) .. limil"'!.o i .. 1!f1l1l/,lfI\... ly, lin lit.t.nllll''''·'' \.0 ~iv" 1111 nxiomut.ic rOlllulnt.ioll for t.hu.t. fllirt. of it. whioh cOlild not. bo dispc/lf1fld ",it.h nUll t.o OOIlOIlUIIIll rnr 1111 flORlliblo itRolllplrion.lolmmot.or. Following TtI,illl·1I10Ist.r,.·, Villi Ir,·it.7. ollllnh"I"" "h"L th .. ,H"t.iIlKUildlill" rllnLIII'n of Oro.. le IImUIIIIIIRt.irR i" t.ho t.um whioh It. tlKlle Ilw"y fl'OlII tho villlflll n,,11 Lown.nls UIII abstract. • Rill! II. 71 t.o Pnrt. I.

,

",.il'

I!III

1'1. liT :I. '1'1111,

--rn rn

r--,---,

I

I

I I

I

I

-+I

111

i

r,-,-,

I

' : '-1 ffi

I

_+ ___ J

I,

I

___ .J

121

I

_J

Il'

I~ig.

I"

R

a.'! Lhm;c o[Lho origi nn.1 ono, has n.n ILrOIt. fo II T ti mcs ita Iliw. He then goC!; 0 n to elmw IL thjru diagram (3) sllowing Llmt a square whoso sides arc three

uniLs long, hn.slLIl arcn.of nino sqU:t.l'C unita, and hence is too In.rge, canllot. he t.he <1ollhlooftho origilu"l ono.li'irULlly in hjs fourth cUn.gram (4) hc.'lltollJs I,lmt Lhc IIqulLro on Lhe diaLgOluLI JiltS eXILcLly twice Lho originll.l al'Oo., 1 Lhinlt LhnL Lhis plL8Sn.gO from PI'Lto proviucs an oxcellont illUStl'lI.t.ioll of the WILy in which stu,tomcnts wore vorified at an eluly stage in Lho dl~\'('lopm(ll1t of Grcel( IImthollmLics, i.u., it tolls us how propositiOlIR were 'provcd' in ILn n.l'elmic SCl1110 of this wOI'd, 'tho proposition which ha~ b('on i1hlllLmlcd ILnd 'I"'{lveu' ILhovc, cJ'n bo formul!Llcd n.s follows: "A KqlllLrtl I:OI1I1Ll'lwwti on tho dingOlllLl of ILlloLhc!', IIIl.H Lwice the nUllll"S :LI'(:IL." It. \Va..'! I.his pl'OJlositioll which Socrn.LoR put in t.IlO form of n. qucRtiull 1.0 all uucduCILu:d HlltVC. DilLgmlllR (2) ILlUl (:l) rcfuLo Lho Hilwa's firsL Lwo ILI1SW(lrR n.l)(lnm\w thcir ucfects visible. Diagmm (4), on tho other IULnu, noL ollly showR the correcL ILnswcr (i.o. lIot only formullLtes the pr0l'usi. I.iuu in qucRLiou), but is at Lhe sll.me timo 1I, visible 'proof' of its corrcct· n(,88. Although this plI..'JSllgO conLILins only one oceul'renco of dE.lXVVJll (8:lu: Hul elll~ (JOIU.r.t 11edJJ.leil', ,1.tM "ei€m, . , • ), in Illy opinion it giveR us a gOIll} inl\iClLtion of why t.his vo1'1I RubscfjucnLly cl~mo to mOlLn 'I'l'nVC' in lIlathcnmLics. It would bo wrong, of eOUl'8e, to chtim that thero WR.CJ ever 0. ti 1110 wlum mathcllIILtic:nl 1,mo/.., were nothing more Limn m.aking lIte /ac/., fl;sible. Even afwr IL flLct lutH been Rhown or mn.do visible, o.s in tholLhove examplc, something 1001'0 is nccdeu for it to IULVe bcon proved, A l)rool ill ohl.o.incu hy reflocting upon whll.t hll.8 been Beon, n.nu hy drawing the (:()I"l'Cct conchlRion from it, It is rcflection whjch tranRformR what we sco inlo visiiJle, eml,iriwl evidence. By Ilmldng a fact villiLJlc, wo provicitl only (·he COl'O of IL proof, oven if 'pl'Oof' ill understoou in ita most !Lncien!. H(~Il!!P..

l;lIN!I'I'III1(~I'ION

01' M ATIII':M ATIC!!!

l!1L

Looldng now ILl. Lhe proofR ill Lim IClemenl." iL would ILpl'clLr tlmt lIuLny of them have evolved from ucmonlltmtions Kimiln.r in form to the one in the .Mcno. They nre frequonLly blL.'led on Rimp\c visual ILrgumcnt.'J whic~h clm be rccunlltl'uclcu fl'ClIU l~uclic.l's own words. I luwo M IL !IULt· t.er of flLct ILrgued c\Kcwhcl'olo tlmt tho proof of Proposition I. 47 in l~uclid's Elcmcnts (usulLlly cILlIllll Pytlmgol'a,'l' thcol'Cm) is of Lhis \
l)fUI

exn"'.' 1)"111,,'11 .1,'" (1r"idll'", J/1l111111lrl-t 10·'0,

1112

I'A IlT B, Til K (!OIfRTnUUJ'JOH r)ll r.IATIII'.AIATICII

147.AIII): Till'. 1I1'.IIIIfIfIIf(l8 ClI' (1II1'.I~K AlAl'IIICAIA'I'IClt

I L is clolLr LhlLt Euclid thought of thoBo numbors ILl! lino sogmonLs, Binco ho dono Led oach of them by a pair of lotters (AB, no, oto.). HilllillLrly, the fact t.hat ho called their sum AE indicu.t.cs t.hat ho must Imvo imagined it 011 being obtnined by laying t.hoso sogmonLs duwn one ILflAlr ILnother in t.ho wn.y pictured below.

I - - - - l 1--1 1-1- - - - - - - - 1 1 A 0 C 0

t-t

liahed t.hat those sn.mo propositions woro origilllLlly s"own (or 'made visible') by tho mot hod of V"1IJ!O'1'Oeia.12 'I'his invol ved representing numhem by groups of pohblcs ('I'jjtpo,) in such a way that even numbers cont.ll.ined ILB IIl1Lny bll\ck ILB whiLo pobhles, while odd numbcl'S were obtILined from even ones by adding or ImlJt.racting a pobble of either colour. So l)wIKlsit.ion IX. 21, fur example, (:oulcl be shown by 'I"1'1'orpo,· eta in the fulluwing way: Lot AB =
E

111 0 0 . .

"

,

'

I':

UI3

The proof now continues: "For sinco oneh of t.ho nurnlmrs AB, no, OD, lJE is even, it hu.s 0. half part.; so t.ho.t. t.ho whole AE also hu.s 0. half part. nllt an even number is that wM.ch i8 divi8ible into two equal parts. Therefore A E is even. Q. E. D." (The words in it.alies o.ro exo.ct.lyt.hedofinition of 'evell numbor' givon in Dook VII.) It should be apparent that t.his proof hu.s nothing to do with tho Idnd of t.jsllolizability which wo discussed earlier. It. ean hardly bo claimed \.Iln.t the line segmonts AB, BO, et.c., ILro I\.ccuro.to visullol represent.at.ions of even numbers which SIIOW us t.hat t.hoir sum, t.ho scglIIent AE, hu.s to represent anoth~r oven number. It. is nonsenso to sugge14t t.hILt IL segment of any kind can he t.he visual representation of an cIJcn number. After all, in t.he proof of tho noxt proposit.ion (IX. 22) the HILllle segments (denot.cd by the same 10tLors) are supposed to represont otftlnumbers. FurthermOl'o, to mateo tho confusion completo, t.he sum of l.hcRI! odd numbcrs is n.1l even one which, just ILa in tho proof of IX. 21, iH II(,Ki~IIILlcd hy A E. There is in flLct no WILy to disLinguiHh betw(lOlI odd ILl II I cvcn numhem by using lino scgmonts, for such segments can ILl WILyS bn l:Ilt ill h:Llf. By d(lfinition, howovor, mlclnumbol"H CILllIlut till Imlvcd, Idl.lloUgh evclI numbers can. There is no doubt that Euclid's proofs of t.ho abovo propositions are nllt "isllalizable. Although he routinely observes the convention ofillustrllting numbers by line segments, his use of the word 6d~a, (to show, IAI point out) is merely figurntive. He tlLlks of lJointing things old, but. bis l~r~ll1I\llIILH ILnt (lHIlnnt.ilLlly ILhst.rlult, ILnll Llmir "(.('II's clLnnot lto vi,'llallurl. It. is l'erlmps COllier t.o sce how far )~uclid fallB short of mnlting t.he truM. of these simplo propoBit.ions viBihle (i. e., of 'proving' them, in tho lIlost mdimentnry sonse of the word), if his mot.hod of domonBt.mLing t.Imm is cOlllpll.rod with tho original one. Beclter hM conclusively ostn.b-

000...

00000.....

O.

III 00 • • 000 • • • 00000., • • • 0. 00000000000

••••••••••• The givon oVlln numbers aro first ILrro.nged in IL I'OW (1) (LIld then nddcd by combining them together (2). If their sum is relmILnged (3), we can seo immediat.cly that it. contains Il.CJ m(Lny blaclt pebbles n.CJ whito oneB. lIenco il, hILS been shown tQ he even. This is 0. rat.hor primitive tochnique of proof, but it. docs make t.he essential conLont of t.he theorem visible. We need only gln.nce n.t tho I'OWS of pebblro..8 to convince ourselves thlLt the proposit.ion is true. Thoy enablo UB t.o see wlll1t.ilOr or not II. numbcr is even. Furthermore, we can act.u:Llly RC'.o how I\.(lding or tlLldllg aWILY an odd pebblo converts an odd numhcr into ILII Ilvon aile. In the proof of ProllOsition IX. 22 Euclid HILyH: "'i'ur "ilwO (lad. (llf tho lIumlllll'S) A n, no, en, DE is oclcl, if (In unit be ,'lUiJlrar.l,..a from cae/I, cae" of Ille remaiuders will be even." Yet, althuugh ho BlLyS this, ho dOOH nut flhow tho roa.dor anything. It is not. possible for him to iIIustfILLo the suhtmction of 0. unit from 0. line segment which is supposed to st.and for an arbitrary odd number. In order to do 80, he would nced to int.roduce 0. shorter scgmont to represent. the unit. 'l'ho ot.her lino segments would thon 110 spccific multiplcs of this unit and henco could no longer represent :Lrbitmry (oven or odd) numbers. Thill, huwnvI1I', is oXI~cLly wlllLt )~lIclicl WILIlt.cd t.u ILVoid. It ia "ruo II O. lloukor, 'Diu J.uhru yom Oormlnn Ulld UlIgorBdoJl in nOWlton Dueh dor Eulclidiuuhcn RlmJlonto', (SilO Purl. I, n. 'lli Bhoyu.) 13

SUl1r6

J!l4

,I

'I

,• ,I :1'1

,.

;I

I17.AIII): Tille IIKOIHHINCJ9 Oil OIUmK lIIA·I'IIln.fATICS

UlI\t his first sLop ill the proof of IX, 21 is t.o stato n. t.ypical instlmce of this proposition fOl' the numbers AB, BC, CD and AE, but theso lino ~cglllon is 0.1'0 not int.ondcd to stand for fixed even numbers. The instanco is typico.l in the sense that specific even numbers cnn be substitut.od for AB, BC, ctc, The proof which he gives, is valid for any such substitutions: This is in contn~t to the mothod of'l"ltpoq1O(!la, which ollly mmhlcs olle t() provo pl~l'ticuh~r CU.8es of tho pl'OpoHition, It sccrns that Euclid abandoned visUl~1 domonatmtiufls becl~uso ho wantcd hiH pronf to bo vlI,lid for nlll'ollHihlo Cl~es, Ho turncd til lugical argulllcnts in his sen.l'cb for greater generality, This is why his pl'Oof pl'(J(:eclis hy J'm:n.l1ing thll.t an evon numher, hy definitioll, is ono which CI~II be divided into two equal pm'ts. F,'om this it ohviously follows Uud, Lhe sum of even numhea'8 is oven; if tho halvcs of such numhc,'slLI'O a.dded sepamtely, Lheil' sum can bo ohtaillcd directly u.s tho sum of two cquILI po.rts, Thc idco. that Euclid wu.s ullahle to iIlustrato evon and odd numbcrs in an ILccuru.te manner hccause ho WaR striving for generality hrings 1.0 mimi an int.eresting pMsage from PIlLto. I am referring to the 0110 in which Socrll.tes expln.ins that, as f"r 1m ILritiunotic is conccrned, number., C([.IIIlOt /utlle visible or tangible bodies,l:1 t.hey are mcrely ideal elemellt., which cannot be grasped except by mean.'J 0/ pure llwU(Jltl.u So wo seo thll,t. Greek mathematics, as exemplified by Euclid's pl'Oofs with t.hoir ILvoidunce and suppre88ion of the visual, also endeavoured to LrelLt its 8uhject. mutter as something which belonged exclusively t.o the donmin of 1,ure thought, This trond in the development of scionce wa.." rcsl'oJlsihle for tho most remarkahlo EuclidelLn proofs, Let me give ILII CXlLm pic of one of these here. Pl'Oposit.ioll VII. 31 of the Element" estahlishcs an important property of composite numbers. 13eforo oxamining it more closely, however, lot 119 rocull three dofinitions which are indispensable for its underst.lI.mling, 'I'hcse are 1m follows, J)efinition VII. 2: "A number is a (finite) multitudo comllUsCiI nf units," 1'1,11.0, 'l'JIC RCJlUblic, 626 oUdfJJlii dnoot.xoJlevol' ~c11' TI' uurlj ot/aru " dnfl; Exol'm, 111,1,0,,0.1, nflOf£II'O,'I'I'''!: "lfI,v)''lfal, I. lhiel., 6211: wlI,),UI'O'IO,jIlIJI Ildl'oll iyxwed, dUw, d' oMrJjllU, Il£rrlxrrelCw,'lm Ii

f1.r'llIIrrI

6'I1'
l'AltT :I. Till'. f:IlNlITltUUI'ION III'

MA'I'III'.~IATiCS

Jm;

i)r/inilion VII, 11: "A prime number is that whidl is fllClmUl'ed lIy ILII unit. ILlone." Definition VII. J 3:""A compo.'Jite number (i.e., one which is not prime) iH t.hll.t which is 1II00mllred hy somo nUllIbOl'." Tho t.heorcm (VIr, 31) states t1mt "anycol1tl10sile number is mr.(lsW'Cll "!lsomc prime number", l~lIc:litl's proof of iL J'Uns lUi follows: Let aLe Imy CClllllJUMitc numher. We Imve tURhnw tllILtsome pJ'imo /I 11 IJI her diviues (£ (I.his if; whn.L is 1I1l"I1.nt. hy sl~ying that 'is mel~ul'cd hy some primo IIlllllhcl"), Since (t is cOlllposite, Lhct·c mUllt. ho ILl. IClL'it OIlC numbol', HILy II, whieh divides it. (hy DcfiniLion VII. 1:J), Now" is eithcr primo or c:ulllpoflite, for DefinitiullS V J I. J 11\nd 13 exdude ILlly other pO!l.~ihility. In t.he fOl'mer ('·1\.'10 the Lhoorem is proved, since b divides a I\nd b is prilll(!, J11 t.he /;LtU.ll' c:lUI(l thoro must 1m SOIllO 1111111 her, III\y r., whidl flividc.q b (again hy Definition VB. 13), Nnw c ohviously divides a lUI well, 110 if c is prime the thoOl'em is provcd. If, on the other hand, c is cOlHposite, there must. hc sOllie numher which dh'idcs it, et.e" etc, '.I'ho proof oml'lumizes tllILL this l'rocc.'l.'I IIlIISt. ovenLul~l1y lcrmin"te; oLherwise we would ohtlLin I\n illfillito docl'ClI.Hing sequence of (compofIite) divisors of a, (According to the Gmcl(s, n. lIumher was "lways grol~ler than any of its divisors.) Tho existence of lIuch a sequence for lLny numbor a, howevCl', would contmdict Dcfinition VII, 2. Henco a must bo divided hy some prime number and the theorem is proved. Not.hing is 'shown' or 'mn.de visihlc' ill the proof outlined ILbove; ill this rCApect it fl1l1dILment.I~lIy differs f('Om t.he ILl'gument found in tho AI ellO, It is truo that Euclid uses lille sogmcnts to ilIustm.Le flo typicILI illsLILnce of his ilropositi()n (for ho reprcsellts Itn aruiLl'ILl'y coml'osit.e 111111111<:1' by 0110 such segmont and its divisor hy ",nothOl', shOl'Let' one), but.l.his iildiclLtcs nothing mol'O tlmn his ohsel'vltncc of 1\ tmdit.ioJULI and outmoded convention. His clmin of ILI'gum(!nts simply CI\nllot, he visualized, If we w"nt t.o undtlrst.ILn(l it, t.here is no point, in looking at tho IIl1l1lbel'R (n.'J l'O[II'Cf4Cnwd hy line segmolll.H); wo necd illMtellAl to l(Cep in mind their de/inili01UJ, since tho proof cllll't!lICIH solely UpOIl theRe. l~lIdiel dO(18 lIuL eVflll givn IL III~rnll 1.0 1.111\ prilllil 1I111111IUI' whir:ll is llCilig sOl1ght., The I'l'Occdul'O described in his I"'fmf IIl1Ly tcrmi'lILto with b, r., d or soma othea' numher. Wo CILn only he c:m·I.ILill that it will eventlllLlly u'rlllilllLto, Hince IIIl 1111111 hili' t:lLII hn.V
f'

lUG

81.AO(); TIIK JlKIJIHNINOS OF OllKKK NATIIY.atATIC!I

I IlOliove thlLt the nHLin conclusions of this chapLer can be summcd up 11.8 fullows: (1) A strilting difference hot.ween Greok mat.homatics and ancient. Orimllal scienoo is that propositions aro proved in the former, whereas there ILre no proofs to bo found in t.he mtLt.homat.ictLI Loxts of pre-lIelIcnic cuILure.'I. ("It. is not. ovon certain t.haL t.ho Uahylonians knew how to furlllulaLo general t.heoroms. ")15 (2) In Lhe ottrliest. mat.homat.ico.l proofs of Lho GI'eelts the t.rut.h of lL stlLtement. WII.II shown or made vi8ible. 'this o.ssert.ion is supported by thl'Cll pieces of evidence; t.ho first is t.ho ol"iginal meaning of tho verb (~f;'(I'vIU (to prove); tho second is t.ho eXlLlnplo from t.he ],1 C1l0 which WII.II disl!IIRBed above, and tho t.hird is the fu.ct. Llmt. we can froquently reconst.ruct t.he original visulLI n.rgulllents undol'lying Lho proufs in t.he Elemllnts on the basis of Euclid's own words (especially in t.ho case or his geometric t.heorems). It. is poasible t.hat t.hoso foaturos of GI'oelt mo.t.honllLt.il!s which n.ro mentioned in (1) and (2) ahove, developed uut. of pro-Hellenic scill/we. The pmctical inst.ruct.ions (or 'rules of tllUm b') for solving mILt.hemat.ical prohlmlls, which wero known to tho peolJlea of t.he ancient. Orient., lIlay well have been transformed at n. laLor slage into genuine propositions Imving wider applicabilit.y. It must be admitted t.hat. no one 11IL8 yet. IJis('uvea'cd any sll(!h proposit.ions in I.he lIlut.hellllLt.iClLI texta of Lhe 1~~YI'Lians or UabylonitLlls, but t.his fact. Imrdly just.ifies t.ho claim LlmL I.hm·n ILro fundn.lllentlLI (lifferonees botween Gmolt mId Oriental JIlILt.hcIII1Ltics. MallY of Euclid's proposiLiollR could ho roglLrdcd simply 11.8 mum refined versiolls of ancient calculation rules, while the origilllLI pronrR of t.ho Gmelu~ (Le., t.heir 'visual urguments') do not diffcr in princ:iple from t.he kind of 'numerical veriflcn.tion' found in Ba.bylonian maLhematics. ls Indeed, it. could even be argued that 'numerical verificn.t.iulI' PI'ClJILI'cd tho WII.y for t.ho primit.ivo mothods of proof wWch wcre employed by t.he carliest Groek mathematicians. In short, there st.ill remains the possibilit.y t.ho.t Greek mathemat.ic.'I WILIt noLhing 1lI0ro t.han a direct continuat.ion of pre-Hollenic science. (I) ILllli (2) nhovo 110 not. t.ouch upon t.ho mORt. significant lliffol'Cnl!o 110LW(1cn theso two tradit.ions. I t is therefore neccaso.ry to add 0. third point.

"er. II

Rill!

II. II,

I'AIIT 3. TIIK (''OH!>'TllU(.'TIOH 01' MATIIK&IATIC-'I

197

(3) At some timo during t.ho l're-l~uclidcILn period, Greek mathemat.ics underwent. a remll.rltn.hlo tmnRfurnllLt.ioll. ViaulLI ILrgument.s were no longer accepted 11..'1 proofs; insl.cad t.ho GreekR sought to 'show' the correctncas of their mat.hemat.icILIsLn.Lomcnta in an entirely different munner. J think tlmt t.his 'Lmnsfm·IllILt.ion' can heat. 110 dcscribed 1\8 anli-cmpiricaland anli-visual. It is cx(>mplifie,l in t.he proofs of l)roposit.ions IX. 21 and 22. The simplest. WILy for Euclid to huve dOJllollst.mted t.ho COl·rect.ness of t.hcso propositiolls, would IIlLVe heoll t.o iIIust.mte them wit.h the hell' of pebblcs. However, it. WM (Illite illll'oasihie for him to verify ILI'it.hmeticlLI proposit.iolls ompirkally, since ho uscd line segments to ropresent. all sorts of numhers. In differollt propositiolls the RaJllO line RogmentH stand fur odd, evon, l'rilllllami composite numbers; somet.imes t.hey aro even suppose(l to iIIust.mtc t.ho unil.There is a closo connection between this 't.rnnsformll.tion' of llIut.hellllLt.ics and t.he idea oXl'roased by l}lo.t.o when ho wroLoIl tlmt. lite ,mbjccl maller 0/ arithmetic IfLY within lilt! domain 0/ pure llwtlU"I. OLhm' pll.llsn.gcs from PIILt.n indicaLo t.hat tho Greclts hn.d II. similar vicw of geometry. I do not WILnt to qlloLo UIlY of theso hero, but. I do want. to point out. thtLL Euclid clearly tried to ILvoid vi8tudizabililY even whcru gcomet.ry WM concerned. Of cuumo t.his was IL more difficult. t.Mlt tlmn avoiding it. in t.hc case of nrithmet.ic. Nonet.helcss, he (lid his hest not to ompluUlize tho visual pl'OporLil's of gcomutriclLI figures. (This topic will be delLIL wiLh more fully in Clmpt.cm a.27-3.29 heluw.) 11. seellls t.hILt. new Idnlla of prunf I~PI"'ILI'I~d lLt. Lim t\1\IllCl l.illlnlL.'! Grcalt IllILt.ium1:Ltil'll WILIt hccoming ILIlI.i-empirical ami ILIlti-visulLl. W() havo yet. to examine how such proofs, which {Limed ILt. providing somcthing Illore t.ho.n visual evidence, ctLmC into heing. Beforc doing so, howevel', 1 would like to }mint. out a notewort.hy fClLturo of the purLicultLr proof diacu88cd ILbove. J~uclid proves hiB theorem (Eleme"I.~ V11. 31: Of Any composite number is mell8urcd by somo primo nUII1I.(>r") lIy nrguing t.hat ita nCWLtion lends to IL cont.rlldietiun. If t.he t.hefJrcm is fl~lso, t.IlCro is sumo number which has all infini/e dccrell.lling SCI lUll nco of divisors; hut •. his cannot. 1m, since (L number is IJy (lofinition a (finile) Illult.it.ud,: D.llnposed of units. Anyone who bolioves that Greelt IllILthcmlLLieR iR nothing Illore than IL furthor (lovcinpmollt. nf pro-Hollo II ill HC!iOlH'O, lI1usL find I~ucli(l's

4 I1lmvO. 4 "lIl1vn, 17

R••"

1111.

1:J mill 14 "h"vIt,

HI8

l'A1l1' 3.

8ZAn(); TIIB IIY.OINNJN09 01' OIUmK )IATIlIUIATH:1I

of proal by conlrruiicliolt u.s inexplicu.ble u.s hill tendoncy t.o n,voic1 empirica.l ",\(1 ViSuILIILrgumcnts. 1 do not. thinl{ tha.t. oit.hor of these two phcnomcna. havo t.heir roots in anciont Oricntal culturo. So t.hc next few chal't.crs will Lo clovot.cd to ILn invC8tiglLt.ion of how thcy m'osc a,q plLrL of Lhe develuplllent of Greelt mILLhomat.ics. J [owe vcr, Lho pruof of Propollit.ion V ll. 31 hWi another felLLurc, which I would like Lo discuRS hero becauso it is of somo hisLoriclL1 interest. As WlLa ah'eady noted, the proof stu,t.cs tluLL "if Lhe invest.iglLtion be l'UIlLinucd in this WILY, some prime numbor will ho found which will IIlClLaure a. jt'lIr, if it is noL found, an infinite sories of number'S will mCIL'iUre the numLer a, each of which is less LImn Lhe other: w},icl, i.s impossible in1lumbrr.~" (o:rrC(! Imil' dMI'UTOl' II' ded}/loi~). The WOl'(.Is ill iLlLlit!s (":hich 1 Imvo also quoted in tho original Greek) obviuUldy refcr Lo the definition of 'number': cl(!u?/uj~ Tel Ix /lov&dwJ' avyxc{/lCJ'Ol' :rr).'iOo~. III I~nglish this definiLiolll'Oads: "A number is a (finite) multitude composed of units." (Tho Gl'eel, text contlLins 110 wOl'd for 'finite', so this ill nllL IL liteml t.l'ILnIlIILtion. NonoLholclI8, it is Lruo to tho spirit of Euclid, Hinee he delLls only wiLh finite seta in his al'ithmet.ic.) I havo iLILlicizl!d t.his }llLrt of LIm ClllOtlLlion for t.ho following rClUJOIl: Although there is noLhing surl'l'illing about J~uclid's cllLim tlmt a Pl'OCC8S of t.he SOl't cllltlC:l'ihcd clmnut. go Oil Lo infinit.y, it. docB stl'jJW mo n.s St.I'Il.IlHO \.lULL ho qua.lifiecl it hy lulding t,ho l'hrWlo 'in·IULlIIbcrs'. Ho HOCIIIH allllOl;t t.o bo irnplyill~ tlmt. Lllo l'CIIIILrI, (~oultl be flLIMe in SOIllO "Lllci' c101ll1Lill. Thelie (:oIlHirlm'atioIlB !Iring 10 mincl ZOIlU of 1~lcIL. It. was he who ILr~lIecllR LhILt. IL hotly ill moLiun Jmcl first. Lo covel' hlLlf tho distancc to its c1t!fiIiIllLLiou; Lefuro doing so, howover. iL would neeel to covcr half of l.hiM IIIL1f, ILlUi HO un to infiniLy. Zonll WILlI Lrying Lo nmlte Lho l'"illt. tlIILt. a hCllly ill lIIol.ion t.mvolll IL JIILt.h II1luln tip of infinitel!! many cli.~'tLlu:~.,. CILch ono Bhorter than tho IlLSt. (Ii'or t.his roason ho thought tlmt 1£0 llIol.ion W1UJ pClRHihlo - or rlLtllor, 'cuncoivahlo' - Imel t.hlLt tho vory (~,"­ c:cl't. of lIloLion WIL'4 pIL1'luloxicnl.) TllClargulllont. can ILlso uo inl.e!rl'ret.cd 1U4 laHHel'tillg LIIILt Il.ny HLmight Iino, A 11, can uo split. up into infinitely IlllLlly I;cglllcllts, eaeh olle shorter than the IILSt.; hut t.his is just another WILy of sa.ying that An has an infil~ile decre.a.sillD sequence 0/ divisor.'!. JL would nppOlLl', t.herofore, tlULt. t.ho proof of VII. :11 bears t.mcC8 of

I.' :1 , .1

I'.1

!J I

.1 ,I

10

ArisLo\.ln. 1'''YlIit:R 2!J.2391J!HT. Ilnll 2.:1:11121.

(:ONII'l'lltl<:l'ION Ul'

~tATJU~MA'l·J('.'S

100

ZOIlU'S influcnco. Ita /I,uthor wenL out. of his way Lo elllplllLlJizo Llmt tho proof was only vlLlid for' numborR, and his mot.ivo for doing RO WWl .,rohlLhly tlULt. ho wlmted to exeludo tho eMO considered hy ZCIlO.

tllIO

II

TIJJ~

:1.2 'flIR PROOl' OF INCOiltMENSUIlAlJIL1TY ~lIr

l

lIUJI{ now is to invest.iglLto bow iL came aLout LluLt Greek lIluLhcIIlILLics WILS trlLnsformed into ILn allli-emlJirical a.nd allti-vi.mal Bcit'ne(1 \ which gavo uirLh Lo a nero uolion- 0/ proof- I hope I.e, show tlmt. lhis development WILlI connected with the discovery of linenr illcommen- I 8ltrabilily. So let. mo IJOgin by sUlllnuLl'izing our earlicr dis(!IJ!J8ion ILbuuL I t.ho origins of t.his discovery. '1 Tho Groolts wero led ovont.ually to lL Imowledgoofillcollllncnsumhilil.y by t.heir ILlt.cmpLs to di vido t.ho mURt. important music,t'- intel'vlLla '. (the oct.avo, fourth [Uld fifLh) into crjlili1stiliililcrvi':ls. This problom, \vliic)nLroRefimt. inilw J1nr.~icalJ~e~~y_ OllJTo1JOrtwu..'1, IIlvulvcd finding LI\I~.:~~lIeILll prClI'CII'Lional (ur gcolllet.l'i~ m(Jal~ot.wcen t.he lellgthK of /' RLring (oxIH'CHSCU ILa I'fCll'OrtlOlllLl nlll\l~ which wero requirecl til proc\uco t.ho inlervlLIs in tlucstion. AR IL l'(',Bult of LlleRo iIlVCIILigILt.iUlIB, t.he Ureelta IIIUSt. Iuwo t.'omo Lo mali",o Llmt. no meltll prol'orLionalnumhel' ('xiHLcd Letweon l.wClllulIll1(l1'll inlL mlio .mlJerl'arlicuiari... (cf.I'P' 17·1 fT,). NUllet.hclclIII, t.ho ILl'l'lictLt.ion uf Ilew ClIlICCl'ta frolll I.ho JllIISi(:ILI t.hc.'ClI'Y of l'fClpnrLiuns I.e) geolllnLriclLl ILl'ithlllel.ic (c:f.I'I" 177 ff.) JlJ'OlIlo\.ecl fnrther rC8clLrch in two djfferent dircctiOllH.tont.he ono hlLllcl, a nCCCH.~ary ILlIClllufficient. cOllcliLiuJI fur t.wo JlIl1nh~~ t.o IIILve IL IIICILll I'TUl'ortiulULI II lit! Ildwcull t.I~I:U1 WILlI diHcuvurcd wil.h tho h('11' "f ,,"LIlI) JlUmhCl'H; Oil IIltl 1I1.llOr hmul, t.ho nul.ion of ~eollwtri(:n.l Rimil:Lrity (fnr rcct.ilinOlLr figul'es) WILlI given IL l'l'l!(:iso dcfillit.ioll ill Lerms of dl'u).oy{a (ccluaJity uf mt.ioR bot.wccn corrcspunding sides). OIlCC tho CrecleB huu st.:Lrted to worl, wit.h simillLr reet.ilinclLl· figuf'cs, it. could noL IULVO t.:LlcCIl thom long t.1I find Ollt. huw t.o cunRtruc:t. IL mCILII l'rol'of't.iurml t.o two arhiLl'ILl'y R(>glllenLs (pwposit.ioll VI. 13), even if they were not. yet aWILro Llml; this t:onatrllctiuJI almoRt ahvlLys yielded an incommel/.8urable 8f.'(Jmenl. fo>imillLl'ly, thoy may IlILvo ItIlown LlIILI, the ILrOIL of I~ a(IIIILre could ho douhled hy 1lI00LIIS tlf its dilLgollal, ILllli oven tlmt. Lho diagollal WIJ.H t.hel'ofure Lho mon.11 l'ru\,orLiOluLI hotwc!(m tho side, a, of tho Rqllll.rO ancl

V

J

I

I I

r

200

i

.,.I

"AllT 8. TUK CONSTRUL"1'ION 01' MATllt:MATIC!\

BunO: TIlR DltOlNN1NOS 0(1' OI\EKK !IATIIB&lAT1CS

a lino of length 2a, without realizing that tho side a.nd dia.gona.l wero incommensurable in longth. In order to soo how tho linear incommensurability of two line seg( \ ments ca.n be established, lot us reClLIl Definition X. 1 of the Elements ", which states: "Those nULgnitudes are sn.id to be commensurable (avllP£Te a PEY~) which are lIletl8ured by the same measure, and thOBO ! incommensurable (aavppEf(la 6l ••• ) which ClLnnot have any common LmeMuro." So wo see that the exi.'Ilence or non-exi8tence of a common meMuro deLcrmines whether or noL two line segments are commensurable. II. is, however, by no means obvious that this quesLion can be decidcd by emllirical met/!~ Lot us loolt more closely at how tho Orool,s uscd to (;Kf;J~I~I~ existence of II. common meu.sure for two line segmonts. We have already seen one example of the method whieh they employed in the theory of music.

(~

I

B

c

A

. I;;, '.

In Pnrt 2 of this book we discussed in some detail the origin of the torm Inlf(l'Tov 6"lanlllu (a line l~ unils in length) which WlLS used to denoLc n fourth (4:3). 'I'his interval WfLB produced on a monoehol·d by plucldng first tho wholo string, AB, aud then a shorter length, AC. Empirical methods were used to deLcrmine how much of the string WIl.S to ho Itoptstill cn, when tho socond nuto wMsounded.Nmt.J,beJellg~h AC WM l'~~.!~le!t~ the 'unit' in this eXJ)~~!J_~e_nt~ The na.me IniT(!'Tov, utTi'ird in additi01i~io-it-(Cc~:-ihe u'lft) comes from tho fa.ct tlmt AO C1LIl in IL SOIlSO htl suhtmoLcd from thtl whole monochord (AD) IOlwillg a romainder (0 B) which goes into it e~~y three times~f course the Ilmgt.\l AC need not havo beon talten lUI the unit. The whole opomUon would IwtulLlly htl.vo Leon simplified, if on had boon chosen insLcad. 'rho whole monochord (A II) would thcn have boon .( units long, and

201

the soction of string producing tho second noto would have been mado up of 3 units (1.1 = 4:3). Tho abovo is a reconstruction of how KllccCMivesubtraction (d,'{}vtpal(lEU',) wlLSlI.pplied in the theory of music. (See pp. ] 34 ff. for further discussion of this method.) This same method, of course, can also be used to fi:!.....ul til! g~~l!4~ common-lIU!tl8ure o'-luJOJ~il~~egn~e!..U8. IiI ViCWor-Ufo;'rmt thaTmusicn.1 intcrvws c311alwa.Ys be expressed lLS ratios between whole numbers, tho appliclLtion of anlhYll"aire.<;is to nncient musiclI.I theory is best dl'..8eribcd u.s quasi-geometric. Although it seems to be concerned with line segments (a.nd in IllLI·ticulnr sections of string), theso n.lwlLYs corrcH)lond to wlwle number.'i on tI,e ronOlt,· hence they aro not arbitrary geometrical quantities. As dCKcribcd ILLoVl" BuccC88ivo Rubtmction is all em,lirical mel/uxl, so there is somo question ILbout whether it is reully 8uiLtLhie for UKO in geolllctry. Beforo attempting to denl with thiA mlLtter, let us soo how the method is used in Euclid's arithmetic. Euclid giveK two arithmeticlLI applicationK of uv{)vtpal(lEf1',. They nro to bo found in tho first two propositions of Hook V II. ] I, is hest to begin by eXILmining Proposition V1I. 2, which poses the following problcm: "Given two lIulIlhcm not primo to 0111' another, to find their grclLtC'Rt common mCII.8UI·e. JJ Following his usual pmctico, Euclid URes two line segments, A nand CD, to iJIustmte tho two numbers. CJ) iH thc shortor of tho two !JeClLu8e it is supposcd to represent tho smlLller number. According to him, there fLre two CMes to consider. The first is that Lhe smaller numherOD measureH the grelLtor one; in this CMe CD itself is the required greatest com' mon meMuro. :rhe second elUle is that CD does not meMuro the grcnter number. It thon hlLS to he subtracted succcs.cdvely from A IJ until a numher is obtained which divides tho origilml 01108. This ·n-u-':.;ber will he thoir grelLtcst common meusure. Its cxistence is gUlLmnLced by tho definition of what it mcn.ns for two numbem not to he prime to ono nnuther; so tho onl which tltleclH t.o ho shown iff tltlLt tho mot.hml of uccesluve SUb~Q!f really worl,s. l~ucli(t pl'oves tllILL iL docs, in IL perfllcUy correct ILnd adequate m.LnnOl'. Ilis iIIusLmtionR, howevcr, '1.(lcl Iittlo if anything to tho argument. Our present concern is noL so much wiLh tho proof it.s(llf M with the ext.ent to which tho 8uhtmotionR CILIt 11.(:I.'lILlly ho en.rried (lilt. J n I'nrticul'Lr, itsltould bo noted tlmtsucCC88ivesubtmction will not neceslmrily yicld the greatest Common meMuro of t.he two lino sogments. It CILn

~ I l

unr

~.

202

8tAnO: 1'111-:

lII~flJNNINOS

01'

1I!lV-lel(

I'A 111' 3,

AIATIU:MA1'W9

only ho Il.pplicd t.o thcse i" theory. As fIn ILS J~uclid is cOIwerned, there is II. crucilLI difference between numbers and the line segments whieh represent them; the formcr are composed of uniLs, wheron.a tho latter arc not. Thus successivo 8ubtraction is not a practical aful CO~lCr:) method ex copt whcn applied to whole numbers. We clLn sce this evcn more cloarly by considoring Pl'Ol'osition VII. 1,1 which states: "Two unequal numbers being set out, and tho le8sb,oiug/ eontinuBlly subtrncted in turn from the greater, if the number whieh is lerL nover IIlClLBUrefl tho uno boforo it until II. unit ill left, the origilllLI nllmhers will be pl'imo to ono another." Thero is no dimeulty about this theorem if wo keep in mimi Llmt I~uclid is concorned only with wlwlc numbcr8 and that, according t.o the Urcc\(s, theBe o.ro built up out of units whic1, cannot be divided into (I,,,Y sIIIclller ')arts. S(n;li~ivo subtro.ctiOifor-two iluJribors' which 0.1'0 I'e ILlivcly prime will termilmw onco tho uniL hus bccn reached. Thel'o ia 110 l'oll.<;iIJility of continuing the process ILny further LImn Lhia ill ILrith· IIwl.il~. Whell it comcs to line segment.'i, tho Ilobove methud is unly ILppli. 1~I~hltl to thoHe which, lilto numhers, 11.1'0 mult.iplofl of SUIIlO given unit Tho S('~lIIlHlts which plll.y a 1'010 in thn thcory of IIIUHill nro of thiH Idnd. A IIIl1NiclL1 inlcrvlLI can alwlLys ho expl'cssed IL8 a ratio botween whole 111I1I11,CI1I; hCllce the monochord itself ctLn bo regarded us flO me multiplo of II. ullit, and itlll'l'ccillo IcngLh (in terms ufLhis unil,) clm be determincd ill ILIl OIlIpiri<:nl WILy by HUCt:llssivo Bulll,motion. On Lho oLher Imlld, tho lilll! NI'~l\ImIIH ,iRed hy J~udid to i1hl"tmw numbcrs (~ILllllot he vimvml lUI spccific mull.ipl(~ of sume unit beenuse they ILI'C suppoHed t.o mpl"C8ent flrllilmr!l numltcrs. 'fhis m,!?".IIR that RuccCHllivo Rubtmet.ion, iIlH(![I~r!1.ft it __ _ :L}ll'lil'S,Lo sueb Hegmen\.s, is 1~;;L~~P.·~Ll:W~ltl U1' c.llllpil"imLI lIIetil~~', huL 11.11 ultsLm<:L ILnd idClLI ulle. . ... .,.--,~

1'1I1~

(:()NlI1'ltIIl:I'IIlN

III' M ATIII·:M AT IC!!

2113

We ho.ve yet Lo oxpltLin how it camc ILLout LlULt Iino segmcnLa W('l'e , ,. ulled t.o symbolize numbcrs. Thero is 110 ~uht in my mind Llm~ this ~,j~J!ra.eti~ _Qrigilmted in tho fiei(rofiJitIKfc;-forit \V0.8' Utere tlULt(\vhulc) numbors were identified with IcngLhs of string on a monochord or \ ClLIlon. '1'he theory uf music cliflcus.'1ed in PILrt 2 must thereforo hlLve / r - - yrel!etlctl the arithmotical propositions fuund in tho Elem,/mt8. In f'LCt, " it CVllll influenced tho terminology of arithmetic. Euelid does not tlLlle ILhuul. UIt) 'grOlLumt e!ommOIl divisor' or two numhel1l, hut ILlwlLyS IthllUt their )Cml"h, P£T(!OV (common mca.mrc); furthel"llloro, he usefl tho V91'" • (' to UU!/MUrC , 01'" to b ',I')" . IC ' •'1'1 /ttT(!E:CV e mecrsureu "d mean ,.,. ulVIC • lCHe cxprcR' siems, lilec the terms Id(!or; (pa,.t) anti 1,1(!'1 (1)fJrt8) of a numbel' (scc Dofinitions Vll. 3 ILIUI 4), ILro of ge(JnwLric (or qlllLfli'geometric) origill. 'rhey luu'le lmele to tho timo when Hucccslli ve 8uhLrnction WIL8 ILn ornpirielLI meLhod whioh WILlI ILpplicd 1..) IClIgl.hs of string ILB {ltLrt. of IL lIlufli(:nl ex P(·ri IIIcnt. LIlL us nuw Lurn \.(1 \.110 'I IJ('Kl ill II fir how gild ill flpplicd slIcI:emliivc SII htmdillil ((h'{)vIJ,u1oE:(ftr;) in gellnwl ...y. Thm'c iii ollly one gcolIlllLriciil ~(JIII (X. 2) which lIIontionH Lhis 11101.111111. HClrOI'll 'Iuoting il., J woull1 lilw \.1) ,'ccILII the following tWII rlLd.ll. (I) III t.hll l-hcory uf IIlIlRi(!, U,e ~- . c.ommo" mew;urc (If two lengthH IIf HLring WILX fOlll1l1 hy tl!ling ,h'{)r"/,,,l. (!Ff1C~. (2) )n gcometry, Iinc KCglllenls whieh hael 1111 I:Ol1l1ll01l ",rwwri'. Wel'C s!Lid to 1m linra,.[!1 i,lcoII/.",rl/.",mblr. -. It. fullc)\v8 rl'llllL (I) ILlld (2) thILl, HIWI!C!HHi VU 'i!I!-',L!I!-..c:tio~I~~L I'I'OI'f'~ whidl, whun ILJlplillcl I." l.wlI ille~"IIII11I'II!UlI':~III." ~:gl\l~!n(~,~t'-U~~ IUimttc. ThlA (!UllctuKlon ifl SlIL(""" ill l'rol'lIHitiun X.- 2J which I'llns 1I.'i

J

I'

Ii

11 I

;1

I,

I! I

II I

~lIld I'er'-)s lie entiolled tlmt. Romo scholtLf8 hlLVO t.n.kc.n llXCCp' L~OIl Lu ~uclid' ~urio9. custom of iIIufltmting his arithmetieall'ropo~i.

llOIlS WIth tho lolpot hno segments. Timor, for oxample, romarlwd 11\ his Gcmum tmllslatioll of LIm 1!llemcnts l8 tlmt tho iIIustmtionfl in Uuol(1I V and VJI were similar in IdlllJ, 1mt that thosa in t.he latter boole wero lIut of much help ill undcrHtlLnding tho theorems; ho thoreforo preforred Lo replaco thcm by oneR which made uae 0/ broken lin.e.s and could 1.111114 I'I~Pl'llK""L Kl'oeine nUIIIOI'imLI UXILlJIl'll!H.

1,,1111\\,14:'

IV

~/

l'meU /)in,/oclli in l'rimltlll N,,,;liJ/i.~ ItJ/(!lIIClIlorulII l.it,rlflll OOllllllflllinl'ii

("I. C. JI'rill.1l11i1l (J.il'llilll! un:l) IiO.II: 1111·h".· ... l>i" T,Nr.mr.llt,. IIOIl fi:1I/..ti,I, Pnrt. :I. p. 7:1.

\

'" r, Whllll 1.1", IllII.'. IIf \.WII 1III1'IJlmllllltglliLlltlc.'1 iH ';111 LilllllLlly suhtl'lwLed in tUI"II f""lIl Lhe gl'CILlcl', L1mL whidl ~ Ion neve .. mClLliUl'eS thc OliO IK·rUI'C it, tho nagniLu(lclL wiIU!~dnclIlIIl\lcllflurlLhlo.: . lere is 11.11 obvious senso ill whie:h thiH I.heorem is II. counlcl·IIILrl. lu Lhn "dthmeLic;tl PrullOllition (VI I. I) quuled ILhove. Tho mu.jol' (titTm" (!IWt! between tho two sLemH fl'OlII LIm fae;L \.IIILt thoro l1.ro 110 'lmsl milO' nilrttlca'in geometry .20 Henco fh'r')v'1'''{(!F(flC, whcn ILPl'licel to two lLI·hi· tm..y Iille segments, need nut ILlwlLyS terlllilllLLc; wheren.'1 in ILril,hllldic

rul" '.n,,, . . .

(I' Ylfll/lSf/l{'1

y.l(1 rd I)"ix'''''''' 6J.w,

204

205

R7.An6: 'CIII~ III~flJHHIN(l1I OF OnR~K l>IATIIRAIATICS

il!.nust do so, oven in the cllae of numbers which are relatively prime. PwpoHition X, 2 gives II. criterion of incommensumbility in terms of ~Cliall'fUluhtmclJont-llllomcly.jf tl!!I!J)roc~ go~ on to infi"nity, the two magnitudes to which J~~cirt,g _~pplied are incommensumhle-: Thi,<; criterion, howevor, because it depends upon esLII.l~g-ii~tl~· l'l'rLaill procC&'J ill infinito, is a plLrely tllCorelical OM,' it cannot be al1plied in Imtrtir.e. In my opinion this o.ccounts for the fact that the lI.ncient.'I, :1.'1 hn,'I often been noted, scem never to have made use of Prol'osi· . Lioll X. 2 ill thcir maLhcmn.Lical proofs. 21 Ollt~ pORRible ohjecLion Lo tho ILbove is the following, CcrLain irmLiollal m:~niLudcs, namely Lhollo which are solutions to equa.tions of LlIO forlll bx c = 0 (where a, b, and c are integers), can be wriLlon a.'11)eriCKlic: 'infinite continued fracLions'. Furthermore, for a given nmgnitudo of Lhis sort it is not difficult to verify that the 'fro.ction' in quesLion is periodic. Hence it n.ppcars LlULt ILn infinite proCC88 cu.n sometimes l;I'rvtl M a pract.ical criterion of incommensurability and,~haL is more, tllILt Lho PyLllILgoreans ml\y well have known about t. his.yor on Lhe ono ---., hnnd, n8 Beclter correctly observed (soc PlLrL 2, n. 1ktflO meLhod uHed 10 obluill such 'fract.iolls' is nothing mol'O Limn n. modern version of tho gllclielel\lIl\lgoriLhm (or ci,'Ovtpa{f!£CI,,), and on Lho oLher Lhoy can ellSily "l~ IdlOwn to bo periodic wiLh Lhe help of so-culled 'side-' and 'diauollal1lIlmher.<j'. (1 myself have nrgued above (seo p. 92) Lhat Lhe cnumeraLion of !.heKo numbcm, which WIJ,S dovised by Lhe PyLhagoreans, elLn 110 ______ inhH'l'rdcd ILa ILII lLllcicnL moLhod of ILl'l'roximaLing Lho vlLlue of V2.) Tho following digrCliHion n.boul.sido- ILIlU dingOlULI-numbers is intell
I

ar + +

"fl.

II II. "'l'tIj''''u, Attmvllr3r1 lA' IIturit, tloU" "'''/lIIlIlIli('4 (/tOIll" 11162), 211 liT. S"., UIBO Oli J!llementi di Eudide by A. li'rn.joso nnd L. Maccioni (Turin 1070), I'p. li!l!l-IIOO, whfll'O th" rollowing obaorvllt.ion is mnde: "UJlIl fatt.o strunn: II:; I'rl'lIIIlI J~IIc1ieJn, lie! pro'l!IIO nlLri lIIat.olllllt.ici groci «("'meno pur quanLo (:u 110 ~ glllnto) Hi t.rova 'Il racilo dilllOBtraziono della incommensurabilitA dcllata 0 rI,'lIn elinsullnl" ",,1 C]uaelrat.o Hl'e'ClIlllo it motado d.,Il' Illgorit.mo oucli
dingOlllLl, d, of such a squlLro is lLctun.lly groa(.er tllI\n 7; for by Pythngo\I ,an d so d = rllJl, tl ,corom d Z = 6 Z fill.) My reMon for recalling Lhis po.RRo.go is Lhat it furnishes 0.11 eXILmplo IIf IL lmir of side· o.l1d din.goI1ILI-l1umbers, namely [) and 7. The interesL of tho PyLhngorCfLUs in such lIumbers centered o.round Lwo prohlems. Thcso wcrn: (I) how to ohtlLill S(IUILfC8 with int.cgml sielos whoso diILgol1ILls wero ILJlJlroxillllLLcly O(IUo.l to illtegrn.1 lengLhs; Imd (2) how Lhe 8o-clLlled '"xl'rcHBihlo' nIHI 'illcxpJ"CRRiblo' dingOlmls of n given S(IUILro wero rellLt.cd. There ILre 0. gl'(!ILL mn.IlY Lcxt.s whieh aLt.esL 1.0 Lho fn.cL !.lULL Lho Pyl.lmgOfelLnS wero (~HpedILlly con corned wiLh (2). ProcluH, fl)l' oxalllplo, wl'it.c~ (In Pu,tOl&i., Hem Publiclwl Comlllelllarii, II. 23):

+ [)

V--

Tho PytlULgorcILIIS show wi(.h Lho help of numbers (dui l'W,' UudJ,l(ijJ' 01 nUOaYO(]E'Ot dwcvvouO',,,) tho.L tho [S'lllIlI'CS of the ] expl'C&'Jible dingonalH, Lho counterpart.s of Lho inox}ll"C8I:Ii hie ones (on all'ai< Uuf!'11'OI'; e)capll'(!OI' naea"elpE"at t!'lml) , arc greater or less by 11 uniL !.IlILlI t.wieo [1.110 S(IIIILI'CS 011 LlIO corl'CHl'olJ(lillg sidcR] (,IOVtit), 11£11;°'" EiO'i" '7 lAdHUU, ~m).umoll). (A Iit.cml Lmnsh1(.ioll uf Prod us' worels would falsify their meaning; thoreforu I hllve givon IL p1Lml'hrrl.'Jo of Lhem ILhove.) Proclus iIIusLmlcs his IWlCl·l.ioll wiLh LWII examples; one deals wiLh 1.110 pILiI' (j ILIlIi 7, ILIld 1.110 IILlwr with I.Iw Jlllllllml'a 2 Imd 3. III Lho lil'st CIIJIO tho SqlllLro of tho exprcH.'1iblo dingurml iN 49 ( = 72 ), which is nne les.~ LhlLIl twico Lhe S')UILI"O 011 Lho sido (2 • (liZ) = liO); ill LlIO socolld, th" SCIUllrO on the diUgOOlLI (U = :J2) is one more !.IIILIt Lwice Lho srlUILro 011 Lho Hi,lo (8 = 2 ' 22). • Wo now WILli I. to invcsLiglLLo how th" Pyl.lmgortllLlIs dis,~over(l(1 IL lIleLhud of gelloml.illg plLirs of sido- Imd ILiugmml·Jlullllml's. II, seellls til mo thaI, modern rcselLrch hua c1evot.cd insufficient nLlonl.ion tn this • I.. ,. ' L ' I 1111081.11111. • I"'or LI1011111111. I"LrLs,!huIILl"H IIlL V" 1~()nL(!I"IInwl"l'S .lIIg IIIH",JI"Hm, cd LhCIIH;t,lves wiLli rocording who.LTheoll ufSmynll1nnd Proclus Illul t.n Rll.y ILhuut Mill I1lI1tl.ol'. Tho furmor wl'II!.n (Mxpn..,itio rp.mm mat/If'.1tuuir,arum tld leUC1ulum Platollcm 1tliliullt, cd. E. Hillor, pp. 43-4): "The unit is Lho origin of overy numbor, whother it ho a sido or n dingoIllLI 0110. So wo tu.lto Lwo UI,iLs, a aido-uniL 111111 n. dingonlLl-unil.; lYe Lhen form a new sidn by n.dding Lhe dio.gonni-llnit to Lho side-uniL, and a

206

!l7.AIl6: TIIK IlK(lINNINflS 01'

OlllmK

new diagonal by adding t.wice t.ho side-unit. to t.he din.gonal-unit. ... According to the above, t.he sequence (a/, d/) of side- and dingolllLiIllllllbem hegins with 0. = 1 Imd d. = 1; a z = a l d l = 2 nnd (I z = = 2ft. + d ... a3 = (l2 -I- rl z = {j Itlld d a = 2a z d 2 = 7. nnd in gmwml a/l+ 1 = (til -I- d/l ILIld d/l+ 1 = 2(/,/I -I- d/l' Prod IIH (/,~ PlrUOIW1tl Ilem I'ublir.mll CommellIrlrii, n. 27) mentionH LlwHo salllc fOl'lIlulne for the cnlcuhLLiun of side- n.nd dillgonn.l-numhCl'li. lie wriu's: ,j {)'&/IETeO~ "eO(J).u{loiian n}" 1l).weav, J}, ianv diri/lff(}OC,

+

"h·

[mJ'rli aUJ'TcOEiaa Hai 1l(!oa)'aflovau Tli,' ,'I"','CT(}OI' n)" Emmie; rivEra, dCUJl£TQOe;, nnd gocs on to any tlmL ]~Ilr.licl

)''''l"T1U ll).W(l(I,

1i

+

()£ n).F.I'Qu

I'l'm'I's I·hcso formulae in Hoult II (PmpmfiLioll H. lO) of Lho Ie/ell/e'III,.;. LeI, liS now tl\lta 1\ loolc ILL Lho firsL eighL terms in Lhe sequence of Hidean,l r1ilL~OIml-nlllllherR; t.Il1!lftllLl'C lisu!d in \.110 LILlllo IICluw: Side·flllI/lller·

I II III IV V VI VII Vill

2 Ii 12

2!) 70 1(HI 408

1)U/f/olllll·n"mber

1 :J 7

17 41

no 2:JfI 577

207

I'AILT I. TJlR CONlrfIlUL"l'ION 01' MAl"IIKMATICS

MATIIV-MATICS

Lot us now modiCy tho tablo in a wily which I\nciont mathomat.icians would nover havo thought of doing, hy forming ratios from oo.ch pnir of side- nnd dilLgOlml-numbel'S ILI1II expre88ill~ UteSO I\..'i decimlLI fmctions. l'his will givo us tho following: I II

3:2

111

7:5

IV V VI VII VIII

1 :1

17 : 12 41 : 2U . 99: 70 239 : 169 577 : -108

1.500000 .... 1..100000 .... 1..a 166666 .... 1..1137931 .... IAI42857 •... 1.4142011 ...• 1.41·12156 ....

Wo Hccd unl.v J.(tLnco ILl. tho infiniLo clccillmls ill tho right·hlLlld colulJIll In lice tlmt thoy n.m approximatillns to tho vlLluo of ]~vcn mom intcre!Rting is tho flLct UULt, M wo I'Clul clown tho column, tho odd TUWS tend tOWILr
V2.

(-I- I) (-I) (+ I) (-I) (+ 1) (-1) (+ I) (-1)

(III the right-hnnd COIUIllIl I Imvo indiclLtml whether tho squlLro of (.IIC! nXl'l'clUiihlc dil1gOlULI hns to hc illcrolLlfcd 01' diminished by olle ill unit· .. l.ullhtnin tho squnl'O of tho inoxprcRliiblo di:LgIIIULI, (i.l'. twi(:(! I.ho H'IIIILro 1111 tho correspollding side.) Ollr pre'lINlt inLcrest ill UI(~O numhom stems from tho folJowillg (fll!C Ie:. :-a.IUlIlLlt'H, I~"kleidou Ocom('lrirt, Vol. 2, Atholls 1!-15:J, PI" \lfC.). AK WII rrmcl dowII tho 1,ILlllo, it. is IILviuus UmL tim nutunl dilLgonn.ls IIf LIm III III I U'C'K 1111 (.1111 Ridl'S ill 1!1I11I1lI1I 2, \Vhiuh of 1:lIurHU ILl"O imIClIIIIIII'IIHlIl'ILh'" ill IOllgth, npproxilJln.te more nllt! morc closely to t.he correHpllll(lill~ Ilillgonn.I-J1umhcl'S ill colulI1l1 :1. ]i'urLhcrmorc, tho ILIWiollt J'ythnglll'clLlIH lIIust lul.VO kllowlI this, sillcn our tlLLle docs not cOIILILill ILllyLhillg whi!:h goes Imyoncl thcir IWllwledge oC tho subjoct.

V2.

20M

V2

t.huy nro less than whenever the square of the corresponding dingonu.I·l1umbor is ono less than twice the squnro on the side). If wo now transfol'ln our first tnblo by writing en.ch pair of side- ILnd ding0l1lL1·numbers n.s n/raclion and then expa.nding it u.s 0. conli,uua Imctio7l, tho result is even more interesting. Rows III und IV, fur cXllmple, will loole like this:

III 17

12 = 1

IV

+

1 2+

1

2+2" AR wo move down the columns, we obta.in successively lunger continued fmct.ions, all of which have the same form, [ If the nth such fraction is

I,,,

f''\ 11'1'

81.AnO: 1'IIR III-:tJlNNINOS 01' IlIlEI(K NATIIRIIIATIC8

t.hcn I"'f'

=

1

+ 1 -t~ I" . ]

We are thus led to fLak whether Lho

PyUmgorea.ns knew about this mcLhod of representing tho raLios bet.w('cn side- a.nd dingonnJ-numbers. 1Lahould be remembered thaL, as fn.r UB tho Grccl(s wero conccrned, fml!Lions played no' role in pure mathematies until Lhe time of Archimcd08. (Of course, this is not. to deny that they were used in tho prncticlll apl'liclLtions of mathemo.tics. Furthormore, tho three fractions IIemiulion, epitriton {LIld epogdoon played 0. very important pll.rt in the theory of music.) This suggests tha.t the answor to our question is negati vo. Since thero is no evidence that the Pytlmgoreo.ns made use of ordinary fmctions, it hardly scorns Iileely tho.t they would ha.ve known uhuut. continued ones. However, there is somothing moro which noeds lo be said ubout this matter. As was mentioned above, the EuclidelLll nlgorithm (dvlJvpa4!tact;) is nothing but an ancient version of tho mcthod hy which we calcula.te the rutio between two numbers as 0. continued fmction. Moreover, the hlLaie idea. hohind the method fur genemt.ing the continued fractions in our third tll.hle is easy to grasp, (,,spceifdly if 0110 is familiar with Zeno's argument about 0. body in moLioll (seo Aristotle's Physic8 2D.23{)b2, also p. 198 o.bove). Just (1.8 Zeno's argument proccdes by dividing Ule distance between the body

3, I'll f~ CClNN'l'fI IIC:I'ION Of'

~IATfll':M A"I~

20fl

und iLR destination into Lwo halves, Iml ving ono of these Imlves, ILnd 80 un, CI\ch fmeLion in Lha sequence of sido- Imd dil\gonal-numhers is oiltl1incd by adding 0. half to tho bottommost. denominator of the pree('(ling one (beginning wiLh Lhe unit). All Lhings considered, therefore, it. scerns to 1110 tlmt the cuntinucd fmotions in our third table do corI'('slloml, to SOIllO ex Lent, to the way ill w h ie h the ILllcien ts thoug h t ILbou t sidc- mld diagunal-numbel's. l want now to go bacle to the claim which I mado above, namely, tlmt sucCC88i vo subtru.ction, hecuuse it docs not Lerminato whonlLpplicd two incollllllellsumble mngnitudes, cml only serve M IL Iheoretical .ri/erior& 01 i,u·,om"cer&sura!!ilily. I do not. believe thlLt tllCvnlidity"-6r tT.iH cllLim is .Lncctcd in n.ny WILy hy our diHeuHHion of sido- !LIul din,gonalnumhers. II. should noL ho furgoLLen tlmt. tho claim was made ILI.JOut geOJlleiriccd nmgnitudes, lI.nd for all pmcLical purposes these cannot Le subtmcLed sucCl!ssivelyJ.-.olll olle another more than 0. finite numbcr of timcs,.:..-frldccd, the very n~n- of8Uccc8sive subtrnctiorf gi,ing ontol \in~nity is a theorotic~l~or~e _~hi~h <:anllot.. b~~ali~ed _~~_ pro.ct.icoJ (SmutILi'ly tho infinite sequonco of slde- and dmgontLI-numbers IS part; of a 'hcory which IlIlS only t.ho most tenuous connoction wit.h Lho procticnl ILpplications of IlULthomaties.) My cuncluHion, thcrefore, is that neither HUCCCIiliivo subtmction nor iLR modern vlLrinnt, 'continued fmctiolls', clLn servo llS n. pmcLicnl criLcriun uf irwunllnonsumhility in geometry. ncommensumlJilityiinnlieorelaCiiI conccfit; it-IS no ILn empiriclLI Prol{el'ty of geometricnl qUILlltities. ---.-.---course uc J( IS quito right in usserLing tlULt "II, when tlte less 01 two uneqtud magnitudes is conti7lually subtracted in turn Irom the grealer, thai which is lett never measures tlce one belore ii, llle 11Iag"ittule8 will be incomme'&8urablc", and his 1'1'Oof of this sttLtement is CtltllLlly correct. Neit.her the stnwment nm' Lho proof, howover, holps us to decide in a givon CUBe whether the pruccss ofsuccCHHivo suhtmction, whon applicd tu twu magnitudes, terminalc8 or not. I WILllt to ILrguo MULl. there is no 1mutical method which enables us to rOlLch such IL decision in evcry ~. We might hopo to fincl intuit.ivo evidence and arguments which, nlt.llllugh Lhoy would noL I/O l\ccopLcd IlS conclusivo pmof, would convincc fL nULLhematician tlilLt t.he I'rocCBH W1La never going to terminate. 'fho lIon-IllILthenmticiILII, Oil t.ho ot.hor Imlltl, might. w(lll r(\gILrd these 81LIIle ILrgulIlents as IL rofutlLtioll of the very flLCt which they wcre inlmuled Lo clomunstrnl<,. This cun ho SOOIl from the following:

\~

210

81.AIIO: TIII~ III'.II1NNINII8 01' nUI"ICK ~fA1'III"MA1'1C8

l'All1' :1.1'''1'. C:IINIrI'III1Cn'lUN

r-------...,..,. .

I~ig,

,, "

0

GivellA squAre ABOD (seo Fig, O), we ,vanL to apply tho mcthod of RIH:Cl's. ~ivo suIJLrru:.tjJJll_J,o ils Ride (A JJ) ILlld llin.gonn.! (AO). So leL \18 -"lli~rl~)rrL~t,h A E ( = A 11) ILlolIg AO ILnd ~rmv:Jtho l)erpClldicllhLr, \ It p. from E to the side OD; Lhen EO = It1F (r!.!.0 proof ~his i \ '~~l!f!lIaIiLyja.omiUctlfurJJ.!!!1il.loku.oLhrQ~_it'lJ Hellco ,. = A 11 - (tiC . - AlJ); i.e, CLis-th9JcJlgtJl~)lttAin()~1 bya!1"-'tr~L(lLing the di~cl'cnco ' beLweoll Lho Ride ILIltI dilLgolULI from UIO Rido. Now EP I\nit OP '~l)ec'- . - Lively fm'm tho Ride ILlld din.goIlILf(~TI~~lO;~\re, 10'00. ~l'hcl'Cfnl'e, we elLll I'Cl'C1Lt Lho wholo proceduro for Lhis am"II01' squaI'C (i.e. wo suhLracL IF = EF from OP ILntl dmw Lho ptll'l)clldicullLr f\'Om I to Lho sido /t'O) ILIld ohLain yet ILllother S(IUlLro (wiLh sido 10). Tho sido of this Lhird sqlULro elLn thcn bo subtl'lLet.l!d f!'OlIl il.R dhLgonnl !Llld so Oil. Continuing ill t.his WILy. we goL sllmllol' ILlld slIlILlIor sCJUlLrcs. Thoir sidl'lI. so,;.. howcver, will novel' n1C1uml"O I.heir c1J!LgCIIlILls eXILCU Thus it is clclLr I. 11\ ,IL • \ll1I~h tho rollllLindol'H-fiC~" snll\lIllr "lui RnllLller, LIm prnc('R.'1 of Rucccssivo 8uJ,tmcLioll novel' LorlllilU\U!8. 1I0neo wo may infl'r It.\' J'rol'osition X. 2 th,\t tho two nmgnitutlcs (the side and diagonAl of tho ul'iginnl squnrc) I\ro incnllllllolllUlflLhlo in longth. This 'proof', which R!Ldcmn.chcr fLnd Toplit1. hAvo reconstructed, is not Lo be found in any of tho Ancicn(,-texts-whioh havo como down t.o us,

!

I

=\!!Jt'.

!

I '

I' "

I,:,

------~

'I

,

------- \ -------------_._-----

It H, Rademacher Bnd 0, 'foplit.z.. I'on Z(l/llen 1100 Figllren, Borl i n I !J30, Ii p. 258; cr. (llso O. Topli!.z.. Die E'llwicl.:lllng der In/illitesilllalrwm1lllU, Burlin : I !l49, pp- 2-6, __

r-----

------~.--1

--

"

211

V

~~ o

MA'I'III~&IA'I'IC:S

Jt is deficient in one I'l'Sl'tJClt (luLmely, t.he flLCt thn.t thc.lII'necs.q cnn bo ~tt.inu~ to infinity --.1.!! not p~ve~_ rfgnro..usIYl. hut is novcrthclesS1

IladellllLcher u.nd TopliLz, in Lhcir inLorOHLing IiLtlo book Vo" Zaltlen wul Fiyuren.,22 U8e Proposition X, ~ Lo show UlILt tho sido and d.ill.gonn.l of n squarc Aro Iiiumrly incomlllonRumblo.lfh~ii=)ugumont runll I\S

M~~:

(II'

quit.e convinlling, Eadl IIUll' is 110 cJclLr ILnd sLraightforward UlI\t Rn.deIIIlu~lml' ILlld 'l'iil'litz W('I'll Il'tito wUlIller why no ILllcicnL nllLthenmt.icilLn cver fountla lIimillLr ILrgulllcllt for Lhe exisLcnco nf incomlllensurahility. If Wll :Ll~JtuLnllwcr thill cluesLion fmllla hist.oric"ll'oint of vicw, we will Imvotlt) t"Lcl(~wo SCplLmto but interrclated prolllems; the fil"Rt is to cxpl:iliNlow-~IIf1llte"urlicitU, would inlerpret th" prcceding conRLI'UCtinn, ILIUI Lhc scc:olltl is til give ILII ILeCOllnt. of how 1\ nOIl·/l/Clllte1lw/iciu" wO\lld r(,lwt In iL. ____ , I Lshollltliro streHSctllLt Um ouLHcL, ~{)f COIII'HO, Lhat IIlnthcmfLl.il'nllLlltl 1I01l-lIl1Lthenmlic:lLlpl\(,l(!I'IlK of l'('II.'!Olli·ll~ ILrc md icflll!l diUerell/. Al'islut.lo 1I11L1(cs thill puint ill IL (I1LSlilLgO frulIl Lhe M elllpltysic.'1 which t.ouchl'!! on I ,I t.he proulclll of incolllnulIIsumbiliLy.23 lie rcnllLrl(s \.lULL 1101l-mnLhcII1I\. ! (.iciILlls simply dn lIut Iwlil!ve ill tho oxis\.clI(,c of lengLhs which Imve no i 1\ 1:111111111111 nWILSlII'lllLllII ILI'C t.llcrc.lfol'O pl'rl'lexI:d hy t.ho illcolllrnollllllrlLlrili Ly "f tho Hid" lLlld diagonal of IL SCllIILrc; Oil tho oLher hu.nel, t.hoso who nre 'well vClrsed ill gcomotry' would he C!ven mom 1'Iwpltlxcci if it coulcl 110 shown Llmt tho Ricltl ILIIII Ilil\~nnILI of IL SCIUlll'tl do IlILv" 1\ C'OIllIllCIIl 1Il00l.qlll'll, Tho cliITOI'ellcu hCtWIlClI LlwSIl Lwo WILyll of (.hinlcin~ is w(,11 iIIustmtccl hy the Clirrorolll:e l)(ltwc~pn Lim IIl1LLI"lIIml.iclLlllIIIIIlOn-IIIILUII1' IImt.iC!ILI iIILI1rl'mtiLLiuIIH til' 'i nfillit.(, SIWC!C'HHi Vtl HulJLl'lLl:Lioll', A 1Il1L1.lamlll\.tit!i1L1I wuulcl Jll'OlmlJly l'xl'laill Lhe l'UI1StI'llCliClII slu·Lt:hecl Il.hcrvn ill tho following WILy: (I) Thc fir.'11 .','ep shows tlmt Lim (lin'OrC!lll'1l IlHtW(,CII U'" Ride fLlltl clilLsolIlLl of A.scJtllLro (l1L1I 110 vimvctllUi Lho Hill" (If IL lIew IICIIIILI'C. ~ (2) Tho .'It'cOIul .'Ilel' IIhowII LllILt Mill clilfm'mwtl l'cLw(lCn tho Ride !Lilli llilLgolIlLI uf Lhis lIew IICI"I\l'O is ('Cllml 10 tho 1:0111111, of l'crfol'l1.1illgJ.hll tIf!c'cl!!~_L ClJlm·lL('iCl~\.IU1 I'I'OC:('HH (If RIIC:I!ClH.'!i ve 1111 hl.rn.dicJIl. (:1) 'rho l/,inllllcp Rhows Lhl\t Lhis tIf'I!OIlII dincrOllt:e call he viewl't1 11.'1 the side of yot AnnLher sqllare fLncl Umt. pcrfnrming Lhe next opcmtion ill \.110 pm(:.:HS of Rlic:c:t~ivu tlllht.l'ILc!lioll c'orf'(~l'()lItis Lo su('tmcLins Lhll Ricle of this sCJtllLro from it./! dilLgolIlLl. Thero is 110 need U, C1Ll'ry tho constrllct.iull any further since it. obviowdy CILII (,0 cOlltillUl!cl by repClLting tlte first Lhroe swps in OJ·dcl'. l~urlhcrlllore, UIl'ro is no limit to tho lIumber ofLimes this ClLII lie dOllo.

JI-I·

'I

i I

fi--

212

· r l

As Lim numLm' of SL01'H incroWlcs, of COUl'SO, Lho resulLing SqUlLI'(''''' deCrel1.80 in si~o, 80 that it soon becomes pro.cticu.lly impo88iblo to construct any more of them, JJut this is u. matter of no great concern tOil1J_,sequcnce clenrly not constructible in practice; for the PI'OCess -. I tlf construction nevor terminntcs and, in addition, all the sets encoun. torcd in the extermLI world are /inite,'l4 As mlLthematicians, however, wo / ILrc not interested in tho question of whether this construction ILctually mn be cnrried out. It iH sufficient for our 1>url)oses to Imow as a result of 1//athematical refleclion that wo are faced with 0. CMe of infinite Rucces8ive subtrn.ctiull. TI~~ ~is useful fOIP.'ovillg inC~ml1"l~n.!nll·I~­ ~cly Lecauso it is an i.~_~JOl' tlteorellial) one wInch doos not tcrmlllate after 0. finite mimber of steps, A lion-mathematician, on the other hand, would be inclined to to.lw IL different view ILnd to reMon u.s follows. Successive subtroction is n llf(lrlical device for finding tho common meo.sure of two qunntiticH. Furthcrmore, the appJicntion of 0. l,ractical mcthod Icads to l,rac:liml mUlcr than theorctiCiLI results, Hcnco HtlCCessivo subtraction, evell WIIOII it is ILpplicd to the side "nei dingonlLI of II, square, docs not go 011 t,1) illfinity, By following the l'nJcedure doocl'iLed nbovc, we CILII quite_ ea.sily construct IL sCJun.re which is so smlLlI Umt its side is p,ractically ILIi long us its diu.gonnl; in other words, if we cn.rry out suffiCiently mlmy 8leps, we will eventually find n. length which, for all practical purposc8, is n. 'cummon mensurc' of two incommensurable magnitudes, In my opinion, th~ non-mathematicinn is right to mu.in~tain thlLttho. l'roces.!L~Ll!..ucc.Q8Ri vo suhtmction will nlwn.ys Lorminnte in 1'~C:~~~!l' Thi~ i~ tho reMOII why J.'roposition X. 2 elm (Inly l;j'ov'llo n. ~tiic()reLim~1 (:rit.crion of incommelllmrabiJity, It also expln,ins why no ancient IllILI'I:;jC:IIl1LlicilLn ever used this proposition to l,rove the existence of incommellfHlrILuility- (Wlmt pl'Ompted Euclid to includo such n criterion ill no IIf the Elemenis is ILn interesting question in its own right. To nnswcr

I(suei,u,

~

Ts

j

x

1,

I:

!

\'AIl'r s. TIIR (:ON!o,'UUUClOH ov

S7.AJI(): TII~ ImOINNINGS OF OJUmK ~IATUgMATICS

U 0110 ohollill 1(Ill'p ill lIIilld Ji"rnonlml's t.houghtful romurk (Abstract Sr.t 'l'IIr.oru, Amsterdalll, IDIi3, p. I)): "As 0. rnut.t.ur of fuot., t.ho rccont. rtlsolll'ch in p"YHi"" hllH ill illCII't'lUtilll( 1IIt11I1U1rt1 Il1I1IVi1lflt1t1 110 Llmt. !.lIn oXl'lurntinll c.f IInt.lIl'n l:tlllllOI. 10l1lJ to oiLhor infinlt.uly Im'go or infinitoly smull Int1gnituclos, Thl! 1114ftlllllpLion of Il finiLo oxton!; of tho physical Spll.CO, D8 woll B8 t.ho R88I1IllJll.ioll .... nn finly fillil ••• liviftihilil.y flf II Ult.l.t,r 1lI1t! ""tlrgy (00 tlmt t.h .. omnlhlRt /,lIrI.ioll'ft flf IIIIlLtor 0.1111 onorgy uro fillitu). oOll1)1lul.oly hurmoni7,u wit.h IIxporionl:tI, H thlll( IIIIt'1II8 t.hut thllllxtol'lllli wllrlcl CUll unonl 110 nothing hut. finiLo oota,"

:b

213

MAl'lIF.)\A'I'I~

it, howover, wo would need to undertu.lm IL dotlLiJetl investign.tion of tho uoule in question, II, tMle whioh lie8 outside the 8cope of the llrcsent worlc,) Neverthcless, what Rademu.cher and '!'oplitz hlt,ve succeeded in reconstructing is 0. correct •proof' , Furthermore, it is one lvhich accords· well with the 8pirit of nncient-matllcmil.ti~~ It should be noticed, huwover, that diagrams cannot illustmte tho most importu.nt Il1l.rt of tho ILrgument. Figure 9, for example, shows only the first three steplI of the c:ol1strnctiun; it docs not esttLblish thfLt theso stoJls ClLn be "cpentml infinitely mnny times, Hencl' tho H'oofisl!(I~j,~!~~(}d byy.i~l!I~LQry.Jlhmc5-l _l!!!.~91lIy~by .tl~~~~etJ~~ cOJl8id(Jl:o.~i.()ns! ..< When the Grcel(s discovered the existence of linear incommellsum- ( hilit.y, they found I,henlsclves confronted with n mlLthcmntical f.'t.ct which could not be proved cOllclur;ivcl~_~.l'.'p'meticll.l methoi1s~ Such method~· ~wtl\ldly serve to cstnfiTifiTlUiOexu.ct opposite, nnmcl,L!.ho.t_.!l.n:L.two ~lIlLnLiLlC8 Jiii.v~<~ £C?!"-~no.!,--nl~ILliure. From IL pmctical point of vic,,;; .. PI'IIl'o8ilion X, 2 is uscies8 11.8 IL CI'YLcrioll of illcommclIsuflLbility. IL is not surprising, therefore, Llmt this Cl'iterion WILli nover uscd in antiquity. It WII.8 necc8Hury to turn II.WILY from viHuILI evidence and lo rejcct CIIIrii;iCiHili Tilj~~il{e~~~UC8 before il1c~n;;;;el~s~;r-;;bi;' -~;;gnit';}~;·cO;Wllo proved to exist, -~---~

\ /

• Let us now see how £ll'o-EuclideILn mathclllll.ticiu.lls proved LllILt tho side 1LII(1 djlLgo!lnl of II, squlLre havo no common mCIUlUJ·C. Their proof fClrmel'1y appel~red ILt the ond of no ott X, hut it is relegatccl to ILn nppenclix (27) ill Heiberg's cdition of the Elemenls, As DecleCl' hM remll.rlccd, modern editors (beginning with E, F, Augustin 1829) omitted thc proposition in questiun from tho nmin hody of the lext heclmllo they could not RCO ".I1Y connection betweoll it nlld thll rc.'1t of Boule X, Their action, 1~lthough jusLified, should lIut he 1~II()wc(1 tu OhflClIl'O t.JIC fnct tlmt it is IL 1'l'Uposition of grent ILllti(jllity which WILR cliscov(!J'cci long bcforo tlto l.illlCl or ]~IIc1icl ILIICI wnH I.hollghl. Wlld.1i 1"'Cllltll'villg, if lIul. hy 1~lIdid himKelf, at leust by olle of hiH en,l'licst cilit.0I'R. 25 Wo Iwow th"t lJoth tho II

O. BIII:I" ..... \)i(l l..c· ......

"1,,,\·.·),

VOIII

0" ....,11,11 IIl1tl

Hllg" ..... I",.

• (01'0

n, Iii

81.AIIO; l'UK IIKOINNINOS 01' OIt1mK ~IATIIICMATI(!II

214

propoHiLion n.lld its proof lI.nwdaLo Euclid ueco.uso t.lwy aro menLioned on moJ"C Llu~n one occn.sion in t.ho worlts of Arist.ot.lo.20 Indoed, Decltel' hus ILl'gued quito convincingly t.hat. t.ho so·co.lled 't.heory of oven Imd {)lId' (which Wl~ developed no hLter LImn t.ho middlo of t.ho fift.h cent.ury) WI~ clirect.cd, in part. at. lel~t., Lowo.nls proving t.his t.hoorom." It. Seell18 lil(cly, t.horefore, t.ho.t. we Ino douling wit.h ono of t.ho oldcst. proofa in Greclt mo.t.hemo.t.ics. According t.o ArisLot.le, mo.t.hemo.t.icilLns "prove L1uLI, Lim dingono.l of IL sqUlno is iucommensumulo wit.h its side lIy H!towing t.hllt., if it. is l!.RSuJllcd Lo 1m coJllmensurulllo, odd numbers toill be (''llmi 10 eve,,".28 Decker oUlmrvcd t.hat. overy pUMiule vlLrimlt. of t.he pruof must. fit. Lhis dtlscripLion,2lI 'L"d ho lLt.wllll'ted t.o reconst.ruet. 'L HilllJllifiecl version ofit, ulling only Lho Lheory of ovolIlLlul odd. 'fhe pruof i lI;l' If runs as follows: We Legin hy Mauming that t.he diagollILI of IL squlLI'O iH eOIlHllcnHUmille wit.h its sido, and wo will show t.Il1Lt. Lhis I188UlIll't.ioll cnnnoL possillly 110 Lmo ueca.use iL loads to n ccmtmdict.ion. Hl'JlCO wo will ho aulo to conclude that. t.he Lwu IwgnulIlts in CIUtl.'ILiulI ILro inctJmmonslll'i~lIle. Now the diagonal (d) ILnd Lwo Ildjll(!(lItt.llidcs (ft ILJUI (I.) uf t.ho SI)UILI'O form IL righL-lmglcd iHOH(:llicH t.I'iILlIgle. Ho, hy PyLlmgOl'ILII' t.heorem, wo hlLVIl Hilwo a ILnll d nro collllllellslll"LUlc (by IIssulIlpt.ion), wo 011.11 t.ILI(o t.hem til 1,0 lIumhers; in faeL, we ClLn tlLI(o Lhom to be numoers which aro rollLIi v('I.\' primo. (For, ifthey I~ro not rellLt.i vely prime, we nood merely divide 1.111:111 IIy their common fILet.UI"s.) This melLIIS t.1mt. I~t mosL ono of t.hem c'an 110 oven. The abovo eqlllLtiou shows t.hat. d is even. 'Vo clLn conclude, therefore, Llmt a ",u.sl be an odd number. But if d is evcn, thero is somo numuer '" such t.hILL d = 2m. Sf) our I'I'OVimiR cqtULtion can bo writ.t.on

(2",)2

= 4",2 = 2a2•

Divitling hoth sides hy 2, wo obt.ain

Z. Zf ZA Zt

Part. I, n. 2 abovo. n. n"d"'r, up. ciL., II, 21i "I,tlv... I hil", p. 6014, II. 1 I. /'rior AlialytiC3, I.23.4ln2G nml [,44.60,,37.

RtlO

l'Alll'

S, Tille CONBl'IlUC:rlON 01' )UTlIJmATIL'R

215

'!'his ),LHL equl~t.ioll shows t1mt. a must be an evcn number. Thus ollr iliitil~1 MSumpt.ion hna led liS to t.he eont.rn.ilict.ory conclusion Lhat (I, is boll, even ancl. odd. Therefuro Lho ll.88umpt.ion lIlust bo flLlsc; in oLher words, its negn.Lion musL 110 tru~. As wo elLn sec, tho preceding argulllent. ill I~ reduclio ad absurdum or i"direct proo/. I would IiIw to mention Lhreo flLeLa aUtlnL it., which, in my opiniun, nre imporLnnt from IL hisLoricnl poinL of view. (I) '1'ho prunf IIiLS noLhing Lo do with visual ur cmpirical evidence. Although wo cl~n elLHily dmw I~ pict.uro of UIO t.wo quanLitics wholle ineollllllensum.lli1it.y is to he llruvcd, t.he ILrgnment. is 1\ purely Lheoret.icl\l uno which (Iucs noL «Iepencl UpOI1 Lim villihlo propcrt.ies of sllch I~ mpn:sent.ntion. I L 1L1'1'(mls 10 rca.son (don/!. (2) In view of wlmt. WII.II HILiclILlJOve, iL seolUs lilwly tlmt. Lhe IIIlLt.helIlaLicilLnS of ILIlLilJllity ndul'tl'c1 thiH the()n~t.ie:~lllIot.hod of proof bCClU1SO t.heir earlier lLt.wmpLs ILl. proving t.ho existenco of incommensumuilit.y were not sILfficiclllly clear or convincillg. Tho mORt t.lmL ono clLn hopo Lo esllLblillh by clIIl'iriClLI IIIcthodli is Llmt. 1111 c'uJlunon mea.'Jurc hILS beon found; but t.his does noL rofut.c Lho lmive COJlllnOn-Rcnso opinion according to which ILIly t.WO IImgnit.udB8 ILro cOllllllensumble. To pl'Ove cnnclusively UJlLt. t.ho sido n.ncl ding()JIILI of I~ lI'1UILI'O nrc incolllmcnllumhlo, iL 1I1u8L lmvo !Jeen nOCOSSILI'y Lo tlo\,iRO new lecllllitlllc9 and t.o mjecL 'p1'lLct.iClLI' argumonts in favour of ahst.meL onrs. (3) Tho ILuthors of t.ho proof outlined nbovo 1l1usL (:IOInly havo IIILd 80mo prior noLion of incommeJlsurabilit.y. The creation of t.his conool'L WILS in intself I~ bold step, for Um existence ofineommensumbilit.y is not. a flLot which could havo Imen discClvercd hy ILccident., nor is it. unu which could IIl~ve been ~c(:opt.cd \viLhout. somo 1&.'JLonishmenL. Thero is a consideraulo diffcrcmco uoLwecn ILcknowlcdging t.hat. no common melLSure hILS been found for t.wo given gcomet.dcal mngnit.udcs and esl.a.ulishing t.lmt. no sud, mea.sure em& exist. 'fhe concel'L of incommensurauilit.y, however, ILpplies only Lo t.ho ItLt.ter sit.UI~t.i()n; in n. sonso, Lherefore, iL presupposcs a met.hod of showing t.hlLt incommensumule magnit.udes really exist. In n.ny ovont., 1 helil'vo th,LL t.ho proof we IlILvo ucen disclIRRing illustrates very well Um c1oso connecLion uet.ween tho introduct.ion of now techniques of proof, on t.he ono hand, and Lho roject.ion of empiricism and visunl ovidollt:u Oil t.ho oLher. J uat. lLH Pln.ttl l!Inl'llILHi7.cd Lhat nUIll!JerR du not. IlILvo visiblo or t.lLngiulo bodics ILIllI t.hat. t.hey can only bo

91.AnO: TIIR III~OlNNIN09 01' OItlmK "'ATIIB~IATICS

21G

I'AI~T

comprehended by pure TC(Uon,30 so in the present case it must be stressed that the side and diagonal of 0. square are not lengths which actually can be seen uut are merely ideal entities. Geometrical magni~udcs elm only be incommensurable in theory. It seems, therefol'o;-tlmtltllOaiiJ1:loverY-of linear incommensurability WIIS reln.ted to the development of proof by cont.rnd.ict.ion and 1.0 tho rejection of empiricism in mathematics. \ __ ---

.'j

(

'-------,-----~-'3,3

'rUN ORIOIN Olr ANTI-1~MPlRlCISM AND INDIRECT PROOF'

Let us now turn to t.he question of how Greel{ mat.hematicians arrived at, t.he iden. that numbers and geomet.rical figures were ideal ent.itie8 which had to be treated in 0. completely ILbstrll.ct way. Thoso who are flLllIiliar with mlLthenULtical rcusoning will find this 0. very simple rmd obvious idea, It should not be forgotten, however, thn.t mathellmtics WIlS at one time 0. practical science whose subject maLLer was empiriClLI n.nel whose principles were established by empirical mcillOds. Al~count1\111.8 ILlld surveyom \IIiCd to t;Llm pride in the fn.et that t.heir cu.leulations could ILctually bo verified by means of empirical experiments. In view of ILII thiA, tho Sllddrm turn away from empiricism secms somewlmt

S, Till.. (''ON!O"ILUm'ION 01' MATIIKAfA'l'lCS

217

Now I want to argue that neit.her ILllti,empiricism nor the met.hod of indirect proof could have arisen spontaneously in mathematics, I do not believe, for oXILmple, tlmt mathemu.t.iciILns wore l)fompted solely by their dealing with numbers and geomeLl'icll,1 figures to change mdically their way oft.hinking and to adopt an inLorest.ing new lI1ethodofproof;32 t.hey must havo ueon subject.ed to 801110 influence from outside mathematics. Thore is no other way to explain why u. predominant.ly empirical mathematielLI tradit.ion suddenly and fur no apparent rell.."Ion becamo anLi'crnpiricILllLnd u.nLi-visual. SimillLl'ly Lhel'c is no WII.y to eXl,llLin how Lho dovice of indirect. proof could havc hccn discovered, if we r('strict. our attention to that body of purely prn.ctielLllLnd empirical knowledge which W8.8 at nne t.ime the whole of HlIl.t.hollml,ics. My view is that. Lhcso t.wo features of Greclt mo.themaLies, its rejocl.ion of empiricism and its ehlLl'ILetcristill U80 of indirect proof, ILro aLt.ribut.ablc to Lho dccisive influence of the Eleatic school of philosophy, This book is noL the plrLce for 0. deLldled historical invesLigaLion of Ell'.alic pl,i/O'
HIII'priKing.

Tho ILppOlLmnce or imlirect proofs in mlLthemat.ics is even moro 8urpriHiug. III tho pl'o(JOding clmptcl' we fllLW how ono such proof, which r1irlnot depenclllpoll OInpirieal eonsidcmt.ions or U110n visual cviucrwe, WII..'! UROII t.o l~t.ILblish tho oxistcnco ofinCOlllmOJl8Urn.hility, 0. fnet. whidl could nover ha.vo bcon l,rovcd concluKivuly by cmpiriclLI mon.lIs (/I' hy nU!rc inR(lcct.ioll (If t.ho diagrams uscc! to illustrate construet.ioIlR. It. Khould ho rOllllLrl(cd t.lmt Euclid often uses indirect proofs in tim J~(e­ menta. Indecd, n.s we shall see later, by Plato's time argument.'l of thiR sort 1\lL(1 como to be roglLl'ded us typical of II1Il.themn.tieal proof. 31 nn. 13 nnd 14 uhovo. In tho 'l'hc(ICtCtll9 (1100 1620) Socmtcll t1oJllorofl t.1Il! UflO of voguo nrgulI"'lIt!, hUHI'd on Jllnu"ihilil.y nnd probability t.o 1I01.I.In lJIal.tllrs of importunco ulUl "LilIA,,, 1.11111. R matholllut.icinn "who arguod from probabiliti08and liI{olihoodll in gcum, eLI'y, would nul. bo worth lln aco", flo thlln gOt'S on t.o givo n. vigorouH prollf It)· contradict.ion. I~rom thill I infor t.iI"t t.ho U88 of indirect. proor W88 thought, til be charBc\.t,riRt.io of IIInLhnmlltica in l'lal.o'll tilllll, 30 SOil

31

n I Imve nol. ollon Lllio conjncl.uro stn\.t,,1 dt'llrly nnywhcru, y('t I thinl, tlml. mnthlllllllt.iI:itlllll in J1lLrt.iculnr }mvu hllOIl tliRJlllIIlll1 1.11 viuw tllI'Bn I\chillvmnlllltoR 1\8 om!/! which bolong (lxdullivniy 1.0 nml.llIlInnt.i"II, Thill CUll ho lIOOII rrom ltnill," lIIuiHl.llr'H rllllllLl'ltll (in Iii" hn,IIII, }JIIR c:mktn 1)""k':lllicr Grir.r./II:rt, 1I1111111111'g I!I'I!I, Jlp. lOll) nhoul. ProJHlllit.ion L'c..:IO, IIlLvilig III1L1illlul t.1", I'rnpOIIit.iulI, lin IIHRII .. tH I.1l1ll. il.8 holtlcat 111111111081. originul BI.c.p in LIlli illf"rmlt'n t.hlli. n cnrt.nin IlIImh"r hilA 1.0 bu uvon becalue il cnnnot be odtl. From thill hll c(lIIo1mlc'ft LllIlt tI.e illea 0/ rcnson.

ing coluistcntl!1 wilh concepl4 came abottt as a rcsult 0/ llealing wilh numbers. n In additioll to tho important book by A, Cnlogoro (Studi sull' Elealislllo, nomu }032) I havo t.o montion Panllenules IInrl die Gescllicllle der uriec/lisclicIl .MatllcmatiT.: hy J(, lleinllarrll (2nd odil.ioll, FI'IIIII,rlll't, lUll Main I OliO; lilt ollil.ioll, BOIIII HIIG), I wlllllt! "IBO IiIm 1.11 tinny Lilli r"lld"'ll "tL'·II!.iulI 1.0 foul' of my IIWII papers which, cont.rary to my iniMnl OX(lIlot,,',ionR, 1",1 mil int.o nil iuv(,HtigCltion or tim Ilnrly hinl.ory or Oronlt lnRt.lmlllllt.i,:R. '1'111'1110 IIJlJI"nr in Acl4 A lltiqtta Armlc. !Iliac Scicnliarurn llmt!1fJric(IC (Bllllnpu"t) .uul 111'1' nIl followR: (I) 'Zur Ocachichto dill' griodlischcn Dilllc'l,tiI,' I (11l6:1) J1(1, 377-410; (2) 'Zur OCflchic:hto dllr Dinlckt.ik dns DcnlwlIH' 2 (I ll1i4) 17-1i2; (3) 'ZUIIl VorsUindnio dor RIc'awn' 2 (I !l64) 24:1--119; (4) ']%:lll.iCIl' 3 (1066) '07-103,

218

8ZAJlO: TIIK JlKOINNINCJS OF UIlRKK !IATIIV')IATICS

RiZl'R Llmt truth canllot ho grn.'tped by meulls of 80nso porccpt.ion, which is misleading, but only be reason (Myrp). To get a cloar idea of what ho JIIelLnS by 'renson', let us t.u.lw a look at one of his argument.a; it MSorLs Llmt wluLt is (n} oJ') CILnnot hn.vo come into bei,~u and rUIIs l1.li follows: S"l'l'OHO thn.t what is did (:omo int.o being, Lhon it. could only Imvo como I'Will whn.L is or from wlULt is not .. thero is no Lhird possibility. Nuw if it hacl COIllO from what is, iL would ILlready have been exist.ont before it C:Lllle into being; henco La RILy Lhat it came into being in this WILy would 1I\:LltO no senso. If, 011 tho other hand, t.ho ellLim is ma.do t.hat wlmL is ClLlIlO from what is noL, t.hislon.ds immediately La a conLrn.clietion. What. is (~n.11 novel' Imvo been Lhe opposite of it.aolf, what is noL, and honco ()()UIII not have como inLa being in Lhis way eiLher. I LiR ILpl'n.renL Lhat indirect. lLt'gumcnLs played IL very iml'ort.ILIlt plLrL in 1~ll'atie philosophy. Wit.hout t.hem it would not have been pUMiLle La l'HLILhliHh rmch cent.rlLl ductrin('.8 as LhtLt thero is no ntolioJL, 7to clta"ye, 110 II('collli7lU, '10 "erislLi71f/, 710 s1Jace and no time. Of course, t.hcBo doctrines cOlltradict Lim evidmu:o of our sensos and aro incompaLiLlo with ellll'il'icillm, nOlwLhcleH.'J Lho Elou.Lics, bolstered by their beliof thl\t rmUiOn wu.s tho only guido to truth, u.ocept.cd them. FurLhermoro, tho whule (If ElelLt.ic dilLlecLic is nothing but ILn ingenious ILPl'liclLtion of tho IIIdllllll of indirect prouf, which is why A1'ist.oLle (.'onsitiered Zeno to bo LlIIl in\'cnlor of dialectic.:u I [owe vcr, t.hero ill no rCI~1 diflcrence bctween Z(mll'R dilLlectic ILUd tho lLJ"gument.a of Parmenidcs. Tho most. notewo .... hy fenture of both iH their uso of indirect proof. AN J rcnmrlwd l~boVll, [ beliovo that the influonce of JmolLtie philusIIphy wu.s responsiblo I'm' tho 1'0jccLion of empiricism ILnd vislllLl oviIh'lICO in Grcek nllLthonmticH, ILB well u.s for tho introduction of imliroct 1'1'001'. Indeed, I hopo Lo show in tho following elmpters that tho COI)sLl'twLiul) of Greek mlLt.honllLt.ics ILB 0. deduetivo system ",u.s II. resulL of t.hit! SlLIlIO influonce and tlmt., had it not been for tho philosophy of J"LrlrumidCR Imd Zono, it. would not have beon p088iblo to build up so illWlIlioliH I~ syRwm lIS l~tluli(I'R /illemc".I.'1. I willlLLWIOpL 1,11 giv~ IL IlotlLil(!(1 elllrmlHo of this viow bolow. Fia'flt, howover, I would IiI(O to 8umllllLl'izo th., (1\'illnnco in f,wollr of iL which lum omorged f!'Om otlr diReuHHiun lip lu now. "l,e Fmglllenl.'J 0/ Arilltotlc (rnl. V. H.oso, J.il18iao 1886), rrngmunt 66 (niol;ll",~ l.'II'rLiIlR !l.2(j): '1"lnl d' •AelototfA'I' l" tlji 00'l"OtU eUesn)p Ulhd" (Reil. Z"I',ul'II) "HloO", 6,uA£"Wtik "tA. JI

I'AILT :I. TIIK

CONRTIlUI:rJON

OJ' r.tA'·IIIo:MATIC.'S

210

(I) The CILJ"liCHL ILpl'lictLtion IIf indirect prollf which 1 Imvo uncollnLorcd in my studics of Grook lo.nguago and ullltllro occurs in the did'LCtie I'oom of PILI'Illcnidcs. On the othor Imnd, it iN ngl'eell nowadays that tho tlILI'lil'Rt 100uwn IIll\thoml\tiCiLI II.1'pliUlLtiunH of this technique nro 11.11 without oxccl'tion of II. ilLt.m· dn.Le. This Hilll)llo olJsoI'vlLt.ioJl 1~IJIlIlt. chl'UJlology RlIggeHta tho following conjcotlll'll, U nlcs.CJ the mlltholl of indirect. I'I'00f WILB developed inliepemlclILly in IIIII.LhclJmt.h~'i ILIU1 in ElelLlie philosophy, it. must havo origillll.te(l exclusively ILmong t.ho Elealics. They could Imrdly IU1.ve Iwoptcd it from "Ultlumralics since it doCl! noL fLl'pelLr to Imve been used Lhere until a compnrat.ively II\Lo dlLte. (2) 1 h/Lve nlrendy cmphu.si~ed L1mt t.hero is no WILy Lo ex!,l"in IInw the nll'Lhmlof indirect pl'Oof coullllmve 1)01111 Ilovclnl'ed fmm t.ho I'I'ILOt.icl,1 alld elllpiriclLI knowledge which ILl, Olin Limo nmtle up malhcmat.· ics. Ally lI.ttcml't to show how this Lochnil)llo might IlI1.vo ariscn within IIlILLlwnmt.ies is bound La ond ill flLiluro. Uow(!\'cr, there is no dinicu)t y in ex plrLining how tho tOlLchings of tho l~loILLi.~ Iml to Lho inLrlllhwtion of illllimet proof, fol' it. WILS 11. method dosigllllCl ul'igilmlly t.u rcfut.o thl' cosmugony of AlmximcilCs. 35 (:1) III areel, mILtholl1t1.t.ics tho fil'Ht ILpplil:ILl.illllR of indirect ILrguJ1lcllt. coiru:id(,tl wit.h tho npl'clLl'ILnCC of ILn ILllti'I'lIIl'iric:al ILlld anLi-viRual trend. ThiH is II. striking fact which is ROlllowlmL puzzling fmma ml~t.ho­ IIl1Ltieall'oint of view. Neither or t.hcRn two illnC)\'ILtinm~ would lJo conCdVILlllo ill 1'-llIl'iriCILI OI:~LhelluLtiC8, ILilIl L111!1'C iH III) I'CII....;1I11 why illilirm.'t pl'Oor Bllllulci Imvc 'men ILm~olll)lILllied IIy 1LI1ti-mnl'iricilml, even if mll.thclJIlI1.Lics Imll Lelln IL tholll'etical fll:iencll. AILhough the two I:ILIlnlJt. bo l:Ielll~l'lLted fmm el~ch oLhel' ill tho pJ'Oof thlLt Lho side ILlid (lilLgOlml uf 11. S(IUILI'O 1\1'0 incommclIsul'lLlJlo, IIltLny of tho uLhel' indirect ILrgnlllllllt.s givml hy J~uolid do not result ill eon<:\uHionll which contradict t.he evidence ur our 801lSCS. If wo want to find II. llIelLni ngrullLlld noces,CJILry reitLt.ionship hotwnen indirect pl'Oof ILnrl ant.i·ompil'i<:iRm, wo will hlLvo t.o 1001, IIUhlillo IIl1Lt1almmtics l1.n
220

8MJI!): 'f1lJ( lII~flJNNINOS 01' OJllmK MATIIIUIATICS

not Illwo IJcon ablo to n.cecpt tho validity of tho conclusions which LlIOY rcn.ched by way of indirect argument. Tho connection which they ost.:Lhlisheu bct.ween tho two W(1.8 then carried over into mo.themn.Lics.

3.4 ImCLJD'S

FOtJNJ)A1'lONS

In Lho chnpters which follow I willlLLtempt. La show t.hu.t. tho deuuct.i\'e sYlltem of mat.hemll.Lies pl"Csented in Lho Elc11lcnl.'i eoulu noL havo hecn const.ructed without. ElenLic philosophy. My cln.im is not. only Liln.t. Greel, mll.t.ilemu.Licians in Lho time beforo Euclid adopted nn anticmpiriCflI attitude nnd tho me/hod 0/ imlirer.t 111'00/ from tho Eleo.t.ies, huL also \.lULL Lheir efforts to tmnsform t.heir nULLhomat.ical knowledgc into IL system Luilt up according to ccrtain Lasic principles must. ho.ve becn prulllpwd by tho ten.chings of Lho Elen.Li('.8. I wnnt. to corrobomte Lhis eonjocLuro by examining in detail 0. series of historical problems which hn.vc to do wiLh Euclid'fI worl,. Let us begin, thorefore, hy tu.lting a doser luok ILL Lhe problem of Euclid's foundntions,

• Tho bllOl(s of Euclid's Elcmcll.l8 wore compilcd rLround 300 D.O. Hight. ILt. tho beginning of tho first ono wo find an interesting IisL of unproved n.ssert.ions which are clo.ssified into three groups (1.8 followfI: l. dcfi.I£itioll,e8, 2. postulata, 3, communes animi CON'-epliones.

After IisLing Lhcsc o.sscrtiuns,l~uelid turns hifl o.ttention to tho proposit.ions which go to maIm up tho romnindor of tho book. Ho nover discusHes tho significu.nce of t.ho L1u'Ce groups. Wo nro not told why ho found iL noccHSILry to p);Lce thelll 0.1i tho beginning of his wod(, nor how tho mem1II'I's 111111 gmup dim'I' fl'lllll tlUlIIO of'Lnot.hOl'. All WII lu\()w fl'lllll LlIIl liJlclllcliis is LlmL postulala nnd commune" animi conceptioncs nppear only ILt, f.JUl 1l('~inlling (If BOClI( J, whol'On,q furLhur licfi.nilionell precedo nUllit. of the uthor hoults. VIII, IX, Xll u.nd XlIIILro tho only bool'8 which (!ollt.ILin 1111 nmv definition8, This is clen.rly bocnu80 nil the dcfillition8

()r

I'A ItT 3.

TIII~

CONRTlluc:nON 01'

MA1'IIR~IATICS

221

required for t.ho ILriLhmoticallJooks (VII, VlII and IX) are )jswd at. t.ho lJeginning of Dool, VII, and all tho COIlCOPts used in Dooks XII o.nd X I II 0.1"0 onos which IlILvo beon int.roduced previously. If we wn.nt to find out why Euclid p1'efnccs his treu.tment of Lhe IIIIl.Lhcmn.tiClLI pl'Opositions by listing threo groups of unproved (l.8se1'tiullS, we must Lul':: La tho worle of Proclus Dindochus, who produced an Ildlllir.d.lo commentary on Dook I of tho Elemel&is ill the 5th contury A.D. Proclu8 wriLcs:3G "Sinco this s(:ionco of geomeliry is bo.scd, wo sny, on hypot.hesis (l~

v1to{)luuJ)(; £lvm} nnd proves its lI~ter propoaitiol1s {Tel lrp£~'it; anOdE&xvvvac} from t1otermino.w firsli principles (dna aexwv W(!,C1/tlvCrll') ••• ho who proJlI~rcs nn intl'm]uct.ion to gcolllot.ry should present. sepaJ"lLtely tho pri'lCill1es of Lho scicnce (xClJek plv na(!adoVva, TU, dexu, nj, ImC1nlp'1~) ILIlII tho conclusions which follow from t.he principlcs (Td CTlJlmeerJC1/,um), giving no nrgulllon!. fo .. Lhe principles but. only for Lhe Lheorems Llmt u.re dcrivcd frum thclll, For no scion co demonHtmLes its own firsL principles or prCl;clltH IL rOMon for t.helll; mtller (l;~ch holds LlulJJl ns solf-ovident, LllILt. iH, lUI more ovident thl~n t.heir cOIlRequoncCII, Tho scionce Imows thclII through I, hClllllC Ives, ILIltI the IILter propc>Hitionfl through tholll. ,., Whuovcr thl'Clws into the fI:Lllle )luI. his prindplcII ILllCllhoir COnHeC)UCII('(!S cliHILrr'Lnges his t11I1ICI'H\.lLIllling completely hy mixing up things tlmL do not bclong tog<>ther. For nl,rinci(llo nlltl \Vlmt followR from it ILre Ly ImLure differenL from ouch ot.her." This lluoto.t.ion 1~)Ipen.rs nLIOIUlt Lo tell us why Euclid had t.o stl~ .. L out

lIy IisLing t.hreo groUpR of unproved 1188erl,iIlIlR. These grOUJlS ure dearly cCIIlll'oscd of tho pri,lCi,,/es which l'roclus refers to ns auxal or w(!,ttl,ival ri(!xul, Oil tho conLinulLLion of LIm nbovo quoLILLion 31 ho calls thelll xOIrai a(!Xat ILB wcll. It. is worth noting here \VlmL ProcluR menns by tho word 1
J'ugu 71i.lln. ill "·ri... lll1ill'H udiLiUII fir Lit ..

lIy Glonn J~. Morrow, 11(1. ciL., n Ibid., I' 76. :1-4.

II.

Ii Ilhnvl'.

"1111 III II HI J.nry .

"'ho J. ...",nl"LiulI iK

222

,·1

!l7.AIII'I: Till-: IIlmlNNINOII 01" OIU·:I~K AIATIlKlIA'r"!lI

It. iH generally acccl'Lml that t.he sot. of principles ILRSemhlcel hy J~IIe1iel is neiLher nec('.~.~a,.y nor sufficient for his development of IOIL\,IIOIIIILLicH. Tannery hILS pointed out that. l~uclid's definit.ions (I. 4 ILlld 7) of 's/raiglttlillc' and 'TJlane area' are completely suporfluous u.s far M t.ho hooks of t.he Elcmmt.'1 are concerned. They arc included among t.ho principles !Jut arc nov('r used at 0.11.38 110 also remarltetl thn.t tho prinI:il'lcs ILI'O not. the only unproved assumpt.ions which Euclid mlLlwH UKO of in hiH exposit.ion; in t.ho very first comlLJoucLioll (Proposition 1. 1), it is a.'I.'1l1l11c'!l without. proof tlmt. the t.wo circlelll which ILTe dm,wll thcro will alwaYR intersect. Of COli me, 0. great. nmny similar examples 11.1'1\ tu 110 found in t.he Eleme,tl.'1. 3D Whcn Proclus s1'OILlts of ]~uclid clividing t.ho whole of mlLt.henmt.ic:al Itllowlc:dgo int.o unJlroved ',ypol',eses on t.ho ono hand and consequ.cm:cs del'ivml from thcso on the ot.hcr, t.his has to be understood u.s 0. dCHcript.iun of ILII 'ideal requil'llIlIont.'. The t.hl·co gl'Oups of unproved lUI.'Jert.iunH wcre int.cnded to Bervo ILA a foundlLt.ion for mILt.homo.t.ies; t.hoy WCI'O to he St.ILt('tl at t.ho outset., and everyt.hing clso had to !Je dorived frolll t.hcm. Tho extent. t.o which J~uclid Kuccceded in relLlizing t.his intent.ioll, howClve'r, is n. sel'amto (IUest.ion. Whet.her Euclid's prineil'lcfI are in flLct neccKSary and sufficient. t.o ,,"I.ain Llw whole (If hifl J.tnolllet.ry is /lot. tho only prohl.,.n cone:tlrllillg them which interests UK, We ILlso WILllt to lenow why t.hoy al'O liKt.ecl !lllelm' exactly //trce IiclLtJings, what distinguishes tho principleH in ono lillt. from thoso ill ILnol.hlll·, ILnd, lI.hovel ILII, how the ielelL of cOlllpilill~ suc~h Jillts of prim:ipll'H ILI'OSC in the first 1'11LCC. ()1I0 uf tho Lhillg!l whie:h nml(('S it e1il1icmlL to nnSWI~1' LlII~o qlll'Hl.ioIlH ill Lim fad LlIlLt. LhllHO who wro(.o ahout. IIlILI.llImmtiCfllLllcll'hilosophy in Itlll.illllil.y e1iclnot ILlwlLYN ILgl't~O Upllll 1.110 wILy in which I.ho Greolt words l'ul'I'c'HI'Clllelillg WI 1ms/uin/n., dc{inilimu'•• , oLe. WOI'O 1.0 1m uRud ..Pl'Cle~IUH, flll'I!XlLlllpl(\, quotes IL SClnt.cnce from AI'chimcdCK t.o show t.ho.t the I;Lt.Lcr c'ollllht ILK TJO.~/ui(f/a principles which h(l, ProclUfl, IJcli0vcs wOlllel 110 Ite(.\.tll' c1CHcrihed ILK commrmes animi oorn-.epliOJlRs.· o lIo gOCll un to '" NCI'"" tillS J~t/l(/1!1'I urrrqullli 10 (IR97), I·I-R (PI" 1i40-4 of .MJllloir"R Sdell';' /iql/lJII 2, li40-4).

,. 'Sur l'ILIIU ... nt.iciLti dl'" ILxiollll'" d'F.udicln', IJulleti" del! Sciencc.• fllfl,"J· IIIlIliqltl!lI, 2nd 81'ril'8 8 (1884), 102-76 (reprint.",) in IIfJllloires Scknli/i,/",..• 2, 4R-Il:J). 4ft I'm"'"", of'. ('it.. (II. 2() "hovr.), p. I RI. 111-24.

I'AIIT 3.

TIlI~

C;QN!I'I'lluc:nfIN III' JIA'I'IIV.JIA'I'II'll

o!Jservo t.ho.t. t.ho term which ho UflCfI for cOIIUJume.~ animi co,tccpli.oJlcS is employed !Jy other authors ill I~ wider ILIIII more cOlllprehollKive sense. Therefore it fleemfl that, in post-l~uc:lieIOlLn t.imes at. leu.st, thore wa.'t IlO unanimity ahout how vlLrioufI IIlILthcnmt.iclLll'l'incil'les wero to be CltL!tsified. Indeed, ono modern scholtLru hn.!1 rCllmrked tlmt totul confusion and vagueness I'revlLiled amongst nllLt.hcmmLicio.us o.ftor Arist.otle as far Il.s t.he terminology reln.ting t.o ut!xu.l WILS concerned, and tlmL even the II~ter conllncntl~tOl'8 011 Euclidl\J1(l AJ'chilJll!llcs who sought to intl'oclucn n. preciso Arifltot.clilLn terlllinology were IlImhle to clelLl' it UJl. This confusion ILl so shows in the flLcL Llmt 1>l'Oclus, \Vhcn discus.
UK, von Frit.z, 'Die ~X"l in !Iur griudlillClllI'1I lIIuthnlllnt.U,', Arch,,, fur 1It!!lrig"uc8c/,ir.i,lo, 11111111, 1 (I!llili), 101. U SilO, fo\' uxnlllpJII, p. RI, 20 or I'I'0chlll, "I'. dl .. (II. 20 nlmvlI), U Ibid., p. 'l0. 4-6 nlld)l. 17R, 1-2. .. 8o(t n. 36 nbovo. u PmcluB, op. cit., (n, 20 cahovn), )I. 77, 2: lIC1.u.;)(,~ c5i )(ai lI';l'lII rui/ra (Bcil. lci, ,1(lX.I, 'ik rnltfuf,,'1') )(aAni/nll' ';1100",,.,.

22,1

II?AII!): Tim IIY.IIINNINOII

(H'

I'AIIT S, TUR (''ONII1'IIUm'ION

ClILKKK &IATIII'.MA1'ICII

This ILmbiguiLy in the melLning of the term MollEolI; see IUS to go blloCk lJuito lL long way in mathemlLLics. My relLSon for thinking so is the following. PhLLo sometimes uscs tho vorb MoTiihl,,' in tho sonso of 'to determine' , •LII give lL definition'.co His use of the word in the following quottLLion is pm'lmps even moro intoresting: t7

"I flLlley tlmt you Imow that those who study geometry and clLlcullLLion lLnd similar subjecUt, take as hypolhesc.~ (Mo{}IJUl'ot) llU! odd ILnd the ellen, lLnd (rguru, lLnd tl,ree kinds of angles, ILnd other similar t.hings in each difforent inquiry. 'fhey make them into hypothcscs (11017/110,,£)10' MoOlc1Et~ aVTa) ILS though they knew them, IUld will give no further n.ccount of them either to themselves or to others on the ground tlmt they nre pllLin to everyono. Starting from these, they go ,011 till thoy lLrrivo by agreement ILl. the ol"iginn.1 object of their inquiry ... It. ill I\pplLrent. Lh",t Plu.t.o, when he LILII(s in this pllMtLge ILbout hYl'o(imo{}[o£t(;), is referring to the fundlLmental principlcs which lLre IIHllnlly (~ILlIed arcllai 1Iy Proclus. However, his chuico of eXII.ml'lcs ill lIut \\' it.hollt int.crcst. He ment.ions fi1'8t. t.he conccpts of 'odd' anci 'ovcn', whi(:h ILre defined in Dook VII (Definitions 6 and 7) of the Elements, [Lnll Lhcn the 'three I
'''('.~r.8

to HlllI, li,r IIXlllIIJllu, VIII.rlllitle" I GOll: IIllIl nillo lIige. ., '1'/I/! llt!ll11blic VI.610 c-cl; t.ranslat.ion by A. D. LimlBILY (Evtlrymnn'8 l.ihrnl'Y, I OliO), I. A l'l:laitllc1llt'R, O,lflrfJ ulllnia, mi. J. I., lluiburg, Liplliuu 1910, vol. I, JlI', 24R nllli 21i2,

()II

1oIATIII(lIATICS

225

far II.S t.ho foundatio1l.8 of mnthomlLtics Ino concerned, is notoriollsly inconsisLont,tl 'rho fILet UULt in ancient t.imes t.he snmo word could refer bot.h to definitions n.nd 1.0 the foundations of ml~Lhel1llLtics in geneml is in iUtelf vory revealing. We mlLy conclude from it. \.IULt. t.he distinctions which l~uclid drew between vlLriousltinds of nmLhcmaLicn.1 princi plcs wore not required originnlly. He doubtlC88 hn.d his own relLSons for cllL88ifying his II.'!8umptions under t.hree hendings, buL UICRO necd not. concern UII for Lho preaont. Now, ILS '!'tLllllcry hILS rOllllLrlwd, no dist.inction is IIlILlIo botween defi l1&tio1l.8 and postulates in tho oldesL complete work on Greek lIlat.hemn.t.ics which hILS beon handed down to us (he WILS referring to On a lIIoving Sphere by Autolycus of Piln.ne).GO We enn, however, say something moro "bout this nmtter thnll '!'allnery did. He pointed out thlLt t.he second of t.he t.wo 'dofinit.ions' tu Lo found in Autolyeus' work is relLlIy a 1,o.duiale,&l but fl~i1cd to add tlmL there is no justification for referring to eit.her of the t.wo lL88ortions in C)ucsLion M definit.ions. They are not preflloCecl by nny Auch titlo in Lho origilllLl manuscript. nnd ILre ClLlIed horoi (i.e. 'definitions') only beclLuso t.ho Lext wns embellished in an arhiLr:Lry and superfluous man ncr iJy Cunl'lu.l R:LtlChfu98 (1Llso known U8 Daaypoclius), uno of its elLl'ly etiiw1'8. Unfurt.utmLoly Hultseh, when ho clLmo to prepare hiR own edition (If the wurl(S of AuLolycus, did not. relllove tho occurrences of OeOL which his prcdoCC89or had inserted. 52 l>erhIL}>s to begin with it WILS not considered niJsolutcly necC88ary to identify the u.ssumptiolls which stood ILt tho beginning of n mnLhemntical work by giving Lhom Romo Idnd of tiLie. CertlLinly AuLolycus dues not hlLve IL sl'ecih.1 nallle fol' hiR ILRl:lumpLions; he simply sttLtcS them 0." cit., (n. 41 ubovo). p. G7. l/l,/iaera quail rnovelur lib8r. lJll ortiblu III OCCIJl/WW libri duo, ed. Io'. HultBch, LiJlllillO 1886, ROO t.ho indox 1I11,I"r 81?O': "In""r dcfiniLlonNl ot. axiom"t." Rb Aut.olyco nulhull rliscrimnn fllnt.ulII MIIn doen" p, Tannory, nt.o," .. 'J'lllillury (Mcfmoire" Scielili/iqlle~ 2, liS), "n...r lJuot.ing boUl of t.hum In R )~ronch LnlllRlntion, goo8 on to I1l1y: II CHI. cillir quo ont."" IInconclo d6finit.inn ORt., "n r,inlit.O, tin po,.tutal, Ill. nOll8 "I'P"'II1I1I8 oinai clu'irliliuScliulA'IIII1I1L avnnl. )'!lInlillu, iI n '!St.IliL JIM du r~glu rio dloUnBuor "vile preciRion 1.,8 ddfillit.iOIiB ot.ln" IlxiomCII, quo tout. etolt. oUlm moblR pouvult. ~t.nl runge OOIiB III m~mo ruhrlcllIlI. II y n 11\ un f"lt. cOllolddmhl .. , .. rn HuJt.oell'" IIIIIt.lon (BI'" n. GO IlhovtI) Lh .. IUIIIIlIIIJlt.lon8 nro onUUocl 'eOL in hot.h of Autolycuy' workB. On bot.h occlisiOllR, howovor, wo find tho wordo 'adll. DIl' (1.0, addidit DlIl/ypodiw) ill hi8 orlUcnl npl'RrntUfI.

" I{. VOII

lrrit.7.,

'0 A IItolyci,

16

8 .... 106

])8

226

wit.hout proof at t.he outset.. Furthermore, fundamental mathematical principles can somet.imes bo recognized by the way in which they ILI'O formulated. As Ta.nnery hOB obsorved, if Euclid had omit.tod the headillg alt~pam from his list of IJostuiales, we would still have been able to rccognize them for what t.hey WOI"O because t.he list begins with the expreSHion YnJoiJw. 63 SimillLl'ly in hool(son m[Lt.hcmaticstheciefinitions ILIIII oLher fundamont.ILI principles cCluld have been I'rcflLced by the word ,;noxdo{}cl). This eXI'J'eSHion is to ho found in t.ho worl(s of JILt.er IImt.lIl!nllLticitLlls; ArchimcdOH, for OXlLlnl'lo. uses it. to intl'(l(iuco t.ho principles of fluid meclllLnies in his boult 0" Floating Bodies.r.tlt'urt.hermom, we encounter the SILmo verh UKt,cI in a very similar way in the lIloHL mwient part of Pyt.hu.gorClLII ILrithmetic. t.he 80·called theory of mid ILml even. n I IIILVe in mind tho phmso ME£! OUX tin&x£'Ta, which (tCCIII'R in Lhe proof of Proposit.ion IX. 34 whon a conclusion conLmry to hypothesis 1108 Leen relLched.

3.5 AIUS'l'O·l',.Jo: ANI) l'oIlNDA'I'IONS Olr I\fNl'J1ICIIlA','ICS

1n Lho IJreceding chaptt~r we 8aw t.hat tino{)io£ct; WIl.8 u8ed to denote dl'fi/lit i01LSILS well u.s the unproved/oundations of mathematics, For the time heill·g. we will not look into the threefold oitL88ification of mathellllLt.iC:LI principles which is found in Euclid. We sl1lLII turn our alitention insten.tI to t.he qllo.~tion of how the idea Inoso thlLt a mathematical discuBSion lind Ltl he hlL.~ed on unproved and unprovable foundations, In other words. wo want to discover the origins of tho idea that mlLthemu.Lics Il.8 a dcdllct.ivc science nceded a foundation of definition8 and axioms. Before lLtLempting to answer this question by applying the Bl\lI1e methods which ha.ve been uscd so ofLen in the first two parts of t.his hook, I would like to 8ummarize briefly the approach which recent IlcliultLrs have ta.ken to it,

,

"

I'AUT :I. 1'lIIe t:CJNH1'ltUt:rION

SZAIlO: Tille lIROINNIN08 01' OIUCKK ~IATURMATIC8

UP. Tnnnnry, M41J1oire8 ScUllti/iqll" 2. 68: "II est pnrrnitomont. IlUl"tihln qll'I':lIdido n'niL uullulliont. jU8orit.: alrrJ/1ara 011 ~I.o do 8C8 »OItLUltlt.e gOulI\6Lri. IIUI'II; mnia on tout. CIl8 iI loa B nnLtomont diBt.ingu68 dee dofinit.ionR flrCcctl"ult'A t'U I"A cornmou.""nt. ',}rrJoDw' (qu'jJ Aolt. dt'mnmid)," II IC von li'rit7.. 'Din dexu1ln dor gl"inchlAohlln l\fut.ho"",Uk', Arcl.iv liir IleoriUIfoe.,clticltte (Bolin) 1 (1 Utili), 67. .. Ron n. 12 nhovo,

o~·

AlATlmaIAl'JCS

227

Von Fritz hM observed that. before t.he time of Aristotle 0. lively controversy pl'Ovailed I"Lout. whcther Lllel'O could be such a thing IL.'t 1,roven kllowledge in the stricLest sense of the words; 80ll1e lIlainLaincd Lhat t.he scarch for it would lend to an infinite regress. while others dnimcd that t.his difficult.y could Le overcome by proving the vtLrious (Impositions which' consLiLut.cd a syslt!1I1 of lmowledge fl'OlI1 OliO ILlluLher.r.o AristoLle opl'oHcd IJOLh these views. His doct.rine WILa LlULt. cvery science had Lo bo derived from true hut unprovlLLle fimt P":tl' ciplcH, nlltl he ILLt.crnpLcd tu cstll.hliHh Lhe vlLrillllR properLies which thcMe principlcs woul,l Jll:cd to p088ess, NoneLhelcss. 0. remark which r/'OclUH mtLde;L1.lIut. Al'ollunius of IlerglLU (llIuncly t.hat ApolloniuR nUellll'u-t1 to prove J~uclid'8 first lLxiolll) scelllK Lo iJl(licatc LluLt MIILt.c ILa Lhe t'1I11 of the t.hird (:cnLury futilo ILttempLe were sLilI heing made to prove firlit prillc:iples. Therefure, it would appelL\" Llmt AriKtoLlo's view WIU'I lint. IIl1ivellially acCt'pLctl ovell IL hund-'l ycars nfLer his denLll.r.8 VOII l~l'iLz's view is LlllLt Adsk: J (Iescrvcs n great. delLI of t.he credit. fill' IILying down Lhtl fOlllld'LLiulIll A III1LLlu!IIULLit'8 ill IL sysLonULt.i(J WILY, or nt.lell-st for IlIlLldllg known t.he pOHSibiliLics ILlld lillliLs of such IL foulIdILt.ion. Of CUlllllU hl! doos not wnllt to dOilY that II. considomhle ILlIlount. of nULthematiclLI mlLtorilLI WUH lLvtLillLble tu Aristotle, nor thn.t Al'iswtle ILctUlLlIy mo.do U80 of it. (Thus he thinlts it possible. anti even IiImly. LlmL a stlLrt WtL8 IIUWO on IILying tho fnunda.t.ions of maLhollllLties ill prc-Aristot.elian t.imes.) Yet he insisLs511 thlLt Aristotle's discovCl'ies were t.he one.s which nuwe it possilJle, in priJu;iplc n.t IClLBt. to place IIlILthoIIULLi(~ on IL SCCUI'O fountill.tiull. ILlIll offtll'H t.ho following jllstificlLLion fm' t.his daim -It WIL.'1 ArisLoLlo who ILt.tempLed to show tlmt. every sciencc hlLK t.u Hwnt out from principles which. ILlt.hough unprovablc. ILrc nOllct.heless lJoth true ILlld cerwLin . .I!~urLhermorc, definiLions, post.ullLLcs n.nd axioms cannot Le regarded ria 0. foundatiuJl for mathemlLtics until it n.clmowletlged that theso nced not be provcd; indeed, ulllcs.I\ nne rCILlizCH tllILt. fundamenwLll'l"inciplc8 ILI'O unl'rovlLble, one CILJ,"Ot even conceh'o of IILying a fOllndlLtion for IIlll.thonmties, Anyone who believcs thlLt 11.11 tho various propositioJls which go to mlLI{C up I),scientific system enn he

is

r

k

.. l'o8terior A rafll!lliclJ. 1.:1. 7~1J1i1T., UIlII 1.19-2:1.8:lh, 32-R41\1I. u l'ruflhlA, up. (IiI.. (n. ~o "hClvn), .,. 183. 1:1-4. fa K. VOIl Irrjl.T., op. tilt, (II, /i4 "hovn), 04-1i . I I Ihiel., (I. OR.

"fl.

228

II1.AIH): TIIR IIICOINNINfllI Ill' llIl~I~K MATlllmATI(~

proved from ono o.l1ot.her will not be o.blo to 800 any difforence bot.weon n. theorem and 11. fundo.rnonto.l principlo; ho will rego.rd both M provlI.blo n.nd will not undorsto.nd why mo.tbemat.icio.ns hn.vo to l>refo.ce tho st.nlomenta which t.hoy intend to prove by I~ list of 'fundamental' prol'osi(ions. On t.he ot.her ho.nd. rnathomo.t.ien.l lI.R1mmpt.ions will I~l'p(lar nA arhiLmrily ehoson starting points, o.nd not 118 real foundo.tions I\t 11.11, to (,hose who think tho.t proofs rest on an infinit.c regress. eo As wo co.n sco, thislLeeount ll88igns 11. very importtmt role to Arist.oUc; it. RC!CmS almost. to BUgg08t. t.ho.t ho wU-'J t.ho fimt. person to Bot I~b()ut providing II. t.ruCl foundn.t.ion for mo.t.heml~tics. Yet., if 110 ono bofom ArilltoLie Imcln. clear ul1clCll'8t.allding of wlmt. Willi involved in In.yillg t.heso fUUlIC11\tiolls, t.he significance of definit.ions o.nd axioms in pre-Ari8totelian IlllLthemo.t.ics requires some expll~IlI~t.ion. This ml\tt.er is not. discusscd by von Frit.z, nor does ho give !Lny indieo.tion of how 8Ophist.icaLed tr.cJmiques of proof could havo boen devcloped at. a t.ime when t.he \mRic foullcllLt.ions of mat.hellllLt.ics !tad 1I0t yet been t:lari{ied. I believo tlmt Lhese lcchniques, whieh had been all but perfected hefore Aristotlo's I.ime, prcsuppose 0. 11101'0 or )C88 clear undernt.l~nding of foulld'LLionlLl IlllcsLiClns. Von Ji'rit.1., cm t.ho ot.her 11ILIld, IICOlll8 Lo holcl the view Umt. rigorousllIct.hods of infereneo were developed prior to t.he firnt. syst.cmat.ie lLt.tempts Ilt. soeuring t.he foun
I'AIIT S.

T"'~

C:tINlt'rlllJ(:rlIIN III' MA1'IIICMAl'W!t

22!l

writ.ings; he quotes eX1Lml'I0I! from nULt.hcllll\t.ics to iIIusLmt.c his own idelLs, n.lld sometimes ho even t.ries to explll,ill the wn.y in which maLheIlluticilms reMon. Most significl\llt. of 11.11, however, is Lhe fuet. thn.t t.ho first. coherent discussion of the problemR connectecl with n.xiomati7.ing ml~t.hemILticsis to be foundin his Prior A71alylics. Adstotle, of course, iSIL 80urr" of some very vl\lun.ble informo.tion to the modern student of U,e history of mathematics. li'u rt.hermore , some questions n.bout the history of I\xionmtics simply cannot. he o.llfHVCl"Ccl without flt.udying his tcxLH. Y ct tho sLlLtcmentR whidl he nm\u'f\ ILhcmt. l~xionmt.iCH are orwn lI.rhit.rn.ry MId historiclLlly i",~cellrn.ttl; hClIlco they need to bo Lrol\L('cl wit.h Nlnne caution. }'rom t.he Jloint. vic'w of IIIILLhomat.ics, it. must. ho lL
A nalylica.83 A dist.inetion is fil'8t. dmwn hot.wcen ,?iC1/(:' ILllcI del(J)/lO. A lI&e.'li.'i, Aristotle SI\YS, is 1il(O ILII first. prineiplcR IInlll'ovl\ble; furt.hermore, it. need not. be Imown by one who wants to Icn.rn somcthing (about t.he scionce in quest.ion). Axiom8, on t.he ot.hor hand, do hn.ve to be known by thoRO who desire scientific understanding. He goes on to su.y tJIILt. 1\ thesis is eit.her 11. ,m&{hC1l~ or 11. C)llt(1"d~; a hypothuis stutes t.hn.t somnt.hing is (II' if! not, w\mt'C!lLH I~ Iwri,·mw.v (fldinit.iun) clo(,R nut. Dorinit.iCIIIf! (IiItO 1t!I1JO(Iu'.'jci.~) ILl'n lU!ltllllll't.iuIIH or HLipll);~t.iI)IIR; IInlilm l,.ypol"Cl1eis, howcver, t.hc.1y clo 1I0t. sLipllln.Lfl t.hll.t. sOlllet.hing is. (A derinit.iu/l 8t.ipuluu'.8, for oXI~lIIple, time. t.he unit is an indivisible qUIL/ltit.y, but. /lot t.hn.t the unit exist.H.) Thus ArisluUo's ge/lcrn.1 elMSifieat.ioll of principles is ILK followB:

thesis

I.

axioma

horismos Ii

vCln F'rit.7., 'Die ~xa( ill dllr griochiBclulU lIfllt.homIlLiI,', Archiv fUr BegriD6.

!le..rhir."ttJ 1 (1966), 26.

R1.AIU): TIIR IJlmlNNllfOR OY OIlKRK NATIIF.MATlt~

230

'1

LeL us now lIee whoLher we can roconcilo Aristot.le's elu.ssificaLion, o..~ oxpll~ined abovo, wiLh Euclid's. It can be slwwn conclusively t.haL, axiolllnlcl correspond to co,mnune.s animi conceptiones. Furt.hermore it. wnuld sccm t.hnt. AriRtoLlo's /,ypollU'.seis nro t.ho samo lUI Euclid's aite1II1I1a. H wo nccepL t.his laLter ident.ificat.ion, howover, wo co III 0 up ngaillRt. Home considemhle difficulties. In t.he first. placo it. implies L1mt. ArisLoLlo had n. n)lecinl collect.ive name for definitions and postulatc.Ii (yet, no oLher ancionL ILuthor, as far 88 wo Imow, used t.ho word thuei,., in thin sense), and in t.he second plo.co wo can eMily cstll.blish Llmt Al'istot.lo I\8cribcs tlllO dist.inct. meanings to the term aitema, neit.her of which hns mOI'C than tho most tenuous connection with Euclid's ailelllaia (or postull~tos).1U FilllLlly, it should be said t.ho.t even Lhe identification itself dOCR not stand up to critical scrutiny. l~lIclid'B postuhLLcs are not existentil~1 stat.ements of tho SILIIlO Idnd ItS Ariflt.oLle's "Yl)otltCUleis. Aristotle IIAACrt.s°5 t1ULt. mn.t.homILticilms 1L.'IIIumod tho exiswnco of the unit in ILriLiunet.ic ILIld of t.he line ".till l'oinL in geometry, and t.ho.t. they proved or showed the exiswnee of Ihe even n.nd odd in n.rit.hmet.ic and of the incommensurablo ali(I the tl'ilLllgle in geometry. However, them n.re no such exisLontiall'ostullLi.t's (\I he found in Euclid, and the proofs of existential st.atements ill tho F:lrll/tmls arc not at all of the sort ment.ioned by AristoUe. A ristotJn expects Lhe exis~JI(:o of tho oven n.nd udd 'to be proved', WhCl·Cll..'t J~udid de{inCUl even lionel odd numbers, hut. he does noL go on to prove I heir exifltenco. CICl~r1y 'exilswnee' meant something dilTurcllt to AristoLie than iL did tu l)I:~Lo, the J~lclLt.ics and nmthomn.ticiILII!4 ill J,:1·llt'ml. The ILrit.hmetical boolcs of Lhe mement'! (V II-IX) arc not precedcd by any existential llORt.uIILtos; indeed they nre not preceded I,y any postulat.cs ILl, all, but ollly hy definitions. SimillLrly the oxistenco III' II IIILII (.it.y i.v IwlllOshdnlcd ILL t.ho llIlginning uf tho .,IILJlimol.rit: IIoolcR, f·olll.mry 1,0 whaL AI'iRLoLlo Kooms to roquiro. It. is tru~ that ROIllO sort of 1:(Jlllpl\l'iHOn C:ILII 110 lIuule hct.w()en ]~uolid's post.ul,~tes on the olle hand ILIlfI Aris\'ut.lo's vnoOlo£u; 011 tho other, but. tho two ILIlI IL lung WILy from I,cing idenLiclLI. AriRi.t)Llo's viows about ILxioml~tif~ ILIIII

If

II

Ihic!., 1'. -12. POllterior Alla/!Itirll, 1.10.'1l1h;

li4-li.

cr.

K. von Ir rit.7., op. cit.. (n. 63 ahovc·), PI'.

I'AIIT 3. Till( C:ONIITIIUf:fION flY ~IATIJP.tolATICS

231

the terminology which goes 'LIOllg wit.h them simply emmol be nl'1,liecl to l~uclid, unless they arc first subjecLed to an ILrhiLrn.ry rcinterpretlLtion. The terminolugy employod lJy othor o.neienL mathemo.tici,Uls is cv~n fur~h~r removed from Aristotlo's. Indeed, if wo were to aclopt. AI·IHt.otle s IdulLS ',11(1 to J'CglLrd t.hem 11.8 somo leind of norm, wo would lJe forced to 'Lgroe wit.h von Fritz, that. when it. CILmo to naming Lho vILt'ious groups of dexat, tot.ILI confusion and vu.sueness prevailed ;ulIongst post-Al'istotcliu.1I mo.t.hcmn.ticin.ns. ii'ur this renson 1 do not WILli I, to phLce t.oo much reliallce UpOIl whlLt Aristotle hl\8 to so.y about. the terminology of o.xiomo.tics. As flLr lL8 this subject. is concerned, his views (or mUler his proposals) aro of moro interest to t.he st.udont of AriRtotlo than to tho historil~n of Greol( nllLthernntics. My own 'LtLolllpt to expl.Lin the origins ofaxionllLt.ie rnlLthenllLtic~s will be blL8ed on 0. historic.~1 analysis of the koy tel'rns involved. Deforo beginning it, however, I would Iiko to ment.ion some important. fl\.eta which have 0(/1110 to light lL8 I~ result. of previous research. TlLllnery Rooms to have boon the first, person to notice thlLt }t~\Ic1id's fClundl\t.iOlls, cspecially the definitions in Dook [ of t.he Elements, contain some interesting absunlit.ieR. The definit.ions of 'stmight. lino' ILnd '1'II\ne l\l'Co.' ([,4 and I. 7 rospeoLivoly). for eXl\lIlplo, are noL used in o.ny of Euclid's theorems. Furt.herlll()J·o, the tOl'minology int.roduced l,y t.ho definitions docs not always ILgI'Ce with t11lLt. founel in t.he propositions and proofs. Tho terlllR lU(!OII'lXCr:;, dOflflor:; ,~Jl(l eOJlflowUr; (rectangle, ~hombus and rhomhoid), fur example, nro defined ill 1.22 "ut. occur nO\vhOl.. ' else in tho Elements; Euclid Hpcnl(s instolLll of nu(!uJJ.'1).oy(!al lla. Another exnmple is furnished "y Definit.ion 1.1 !I, ' whidl is ine()IIIJ1I~tihln wit.h gudicl's f:llnvontiull fur IlILllling JlflIYJ(OIlS; fill' t.he definition leads 0110 t.o expo(:t thlLt t.hcse figUI'CR will be given III~"II!H whic:h "(,ff'l' to tho number of their sides (T(!(n).W(!rJ, no).vn).clI(!u), wh:"'C':I~ thuy (.111'11 uuL to ho clcmcwihf'll hy 1.11Il IIl1l11hol' of Umir (l1I!llc.~ (1"(!')'/I'I'Ol', n£vTeJY(()VOl', ow.).co Nnw tho L,(lJIdusion whic:h 'I'allnor.v drow frolll t.hiH is undoul.tedly t.ho ellrJ'ect UJ~o, 1II\lIIoly UlILt. t.ho f(IUIIIII~Lilllls ill HUllk I did not origilll~to

II P. 'fBIIIIC'ry, Itflllloires .')cienli/iqllc., 2. pp. 48-0.

232

I·

i'

.f ·1

II t

...

,if

8U80: TIIR III(OINNIN08 01" OILJ(ICK .IA1'IIK&IATICS

wit.h Euclid. a1 'fho author of tho Elements WI.L8 not tho first. mat.homo.tician to preface his work with 0. list of fundl~mento.l o.asumpLions. A greo.t mo.ny of his predecessors, especially those who thomsolves compiled books of 'clcmonts', did 80 as woll. Much of wlu~t Euclid Look over from thcso eo.rlier mo.thematicio.ns ho left unaltered, regnrdless of whether it was neoossn.ry for his worl, or whether the terminology used WU8 the same as his own. In foot the systemo.tio development of deductive mathematics began at IOBSt two centuries befom Euclid, and by the middle of tho fifth century Hippocmtes of Chios hnd compiled 0. book of 'elementa'.as Of courso this development simply coultl not have to.lten pln.ce without some mathematical principles being accepted M f1LluJamertlal. li'urthormore, thero is no doubt thnt Hippocro.tes' Elements begnn with II. list of fundamental (fssumptioM, even though today we do not Imve any idoo. of what these wore. In my opinion, these observations justify the belief that the systeIm~tic devclopment of mathemn.tics bu.se(l on fimt l)rillciples begt~n long before the time of Aristotlc.

I'AIIT B. Tille CONSTllU(''TION (JII MAl'lIKMATICS

233

Let. us now turn to the study of some mo.t.hematico.l torms. We sha.H begin with VnO{)lttElr;. As WIlS mentioned above, this word hM two (uU\Lhcmat.ica.l) meanillgs in t.he writings of PII~to and !)roclus; it. is used to denote tho 'foundations of mathematics' in geneml o.lId o.lso to refer only to 'definitions'. The corresponding verb, Vnod{)Ett{)al, mca.ns 'to defino' in both 0. mat.hemat.ical and 0. non-mo.t.hemo.tico.l sCllse; one can find examples of the former usage in Archimedes'" I~nd of tho latLer ill PII~tu.10 These meanings, of course, arc easy to expl;Lin from o.n etymologiClLI point of viow. 'Y"&{)Ett'r; from uno and T{{)Ett{)a, means that which is placed under, i.e. that which can servo

u.s II. foundation for something. But we n.re interested in more thn.n etymology here. We wn.nt to find out oxo.ctJy which uses of the word led to ita being to.ken over into mn.t.hellllLtics and to its bccoming IILter 11. precise nmthenllLtieal term. Let UR t.hcrcfore consider some eXI~lI1l'k'H which illustrate related senses of mUW£(1U;. It is n.dvisl~hle to look fimt 11.1. II. IIlUl8ILge from Pln.to's AIellO which den.ls with the question of whether or not virtue cn.n he tn.ught. l l Soemtes wanta to begin by investigating t.he nature of virtue; his complLllion, however, is too impnticnt.. As II. compromise, Socmtes proposes that in considering the fillestion he be n.llowed to malte IIS0 of n./'ypOt/U!8i8 (ttUYXcI'(!'1ttov Vno{)itt£clJr; aVld ttxonEitriJa" Men. 80e :I). 11e then goes on to i11l1stl'll.t.o ti,e /typothelical met1zod by men.ns of ILJI cxnmple from geometry: "t.() mlLke use of II. hypot.hesis - the 80rt. of t.hing, I melLll, f.!i'Lt gcomet.om often uso in t.heir inquirics" (U:yru IJl Td ;m'Il?lttErur; W/)E, wttnc(! ol )'Ud/lit(!Ul noAA&Kl; ttxonOVI1l'a'). These words are by themselves sufficient to show that. uno{lEtt'r; IUlcl ti,e "Yl)olllelicalmeilioillllllst hn.vo playetllL major rolo in t.he IImt.hcI11ILtics of Pll~to's dt~y. SOCI'II.IA~8 is t~ble l.o CILII on an exmnplo f!"Om gClOIllC'LI'Y becn.use, M he himsclf romarl(s, t.his SILllle method is frcqllcntly IIHm) t.horo. The example which Socmws discu8.'1cR bricfly is also vel'y infnrmlLt.ive, Ho ~ollRitlcm tho qllestion of whet.her n. cm·tlLin ILrClL Cn.n he illsCl·ibed in a given <:ircle, n.nd clt~illls thn.t IL geomet.or would most liltcly set It.bout l1.nswel'illg it in the following way: II I dOll't Itnow yet whether it fullills Lho '!ClluliLiulls, LuI. I t.hinlt llmve IL imJiJEtt,r; whieh will hell' us in the IIl1Ltter. It is t.his. If the n.reo. is such LhlLt ... ;72 thcn, I shouid say, olle result follows (I'llu T& ttvppalVEIv pOl ~oxEi): if not., t.hen the result is diffcrent." This re.'Ipollse is IIypnl/,eliml ill the sense that it is qualified. The geomcter bases his n.rgument on ILIl o.asumpLion; this twmml'tioll is his IIY1)olltesi8, and he goes Oil til S.LY what follows from it. If tho UllBumptillll (i.o. the Vnnl?El1lr;) worc cluLIIgml, then the concillsion would n.lso be different. We can scc UII~t Socmles

IJ Ibid., p. 66; cr. A. M. Fronldnn,[.e po3lulat chez Euclid el ches Ie" Motleme3, Pnria 1940, p. 14. • Ie l'mo11l8, 66. 7-8 (in 1;"'10I1I1IIn'8 IIIliLlon). .. Soo n. 48 ubovo. JD 8"1' II. 41i "bovfl.

J. cr. O. Bockor, 'Oio Archul in clor gril,dli8chon Mathonmtil,', Arc/tiv liir DegriOageachicllle " (1060), 21011. 'l'ho trn"RI"tion of t.ho plUlllllgn quot",1 frolll tho Meno 18 hy W. It. C. Oul.hrlo • JI Cf. P. Trmnory, 'JlUl/loirea ScicIIliliqrll'l" I, 3!l-46 an.1 2, 400-1i; "r. nllm 'r. L. Jlont.h, A HilIlon) 01 Grtlek /lfni/reflllllics, Oxford 1921, I, p. 300.

3.0

'J1YJ'o'rnRsICls'

Ie

Ie

23-1

IIZAIlI\; 1'IIK IIKUlHHIHU9 01' OIlIU
iH HI(otching a geomotrieal u.rgumont which is bll8ed on a Vn&'?Ecft~ (a.mt.fllplion or supposition).l'ho argument dopends upon this lLS8umption and is qualified by it. The actual wording of Plato's text reveals an interesting fact about nmthemlLticlI.I VnO{}AcfEtt;. Socrates ll880rts tlll!.t geomotem conduct thoir illvcstigrLtions on tho bMis of lL88umptions (l~ Vno{)£aE(J)~ aHonoiivTat). ~il1lilarly, Proclus73 maintains that geomotry is II. science whieh rests Oil o.ssumptions (Ie Vno{}la£(JJ~ Elva,). They both use the same expres»iUII, hut tho HitUILtioll1l which thoy dClu:riho ILre djO'oront. Honco tho word tJ1t&{}Eat' docs not have precisely the su.me RcnKO in tho two l'IL0;e8. Proelus means by it the tl&ndamental assumlltio,&8 of mathematics which have been laid down once and for all. (We saw above thu.t in Book I V of the Republic (510c-d) Plato o.)so refers to t.hese foundations, 1\lId in particular to the definitions among them,ll.S WroOlcftl'.) Socrates, 111\ the other hand, is not referring to the definitive fou71a.ali0n8 of malltelIIalics, but merely to an ad hoc assumption which is to servo us tho hlL'iis of /I, plll'ticulu.r mathematico.) argument. Furthermore, it seems dmLr t.1l1Lt ill nmthematics Vn&Otat, meant. an n.d hoc ll.II8umption before it ac<]uil'Od its other moaning. In my opinion, Socmt.ca usell two othor oXl'l'CK.'iionH which o.rc worthy or our ILttentioll. 0110 is tho phra.se with which he introduC('s tho 'hypothetical method'; he Il8ks to bo allowed to malto uso of II. hYl'othoHis (cf""XWQ7I(fOV IE tJ1toOlaECOI; aUTO cfHond(f{)at). 'fho other is his CICI'(:ription of tho cunRCl)lIenccs of II. tm:oDEa~ "thOll, I should BILY, ono rClmlt follows; if not, then the result is different (cillo n CIV/lpalvuv /l0' ')UHEi) ". It is Lypicu.1 of (:ontoxlssuch 118 the above (!\lono 86 3ff) thlLt t.he \.orm VnOOE(ftt; is found toget.her with t.ho words ctv)'XWOJ1"0V ILnd 'Tv/l{1alvE'v. Tho ono indicat.ca tho way in which II. 'hypothesis' C0Il108 into being am) tho other tho way in which it is used. Tho I'V/lOt/,elia" mclltcxl cu.n only bo used in I~ dio.loguo if both IJl~rtiCII tt(,ree to it, Thill ill why Soemtcs hM to a..,1e hiB companion for 1)crmi.orsion tu nmlte IIKO of it. Without such an agreement 0. 'hypothesi"' cannot J'Cal1y servo us t~ foulIl.IILlion for o.ny kind of joint inquiry. lIence in PhLLo's dialectic" /'Yllotllueis arc somotimes called o/loAo)"JItaTa thORO t.hing" which arc cunceded or on which 0.11 parties ILgroo.1& 73 PmllhlR (Friodldn's odiLion) '76. 6-14. 7. In LhiR book 'dialoct.ic' melUlB I'lat.o's cJw,uJetucq tiV'l' 7J RI'O, for OXRIllJlle, I'lnt.o's T"efletellllf 166n-b.

1

I I •I

I I

,I

I'AIIT II. TIIK ClONNTIlIIC:1·ION fII'

~1A'rIllUIATI~

Tho word 'hypoth08is' seoms thcl'Ofuro to Imvo heon 1\ dialecliml term lUI well lUll!. 1natlu~1nalical one. 'rho flwt that it camo from dialectic also tells UR something about the mathomlLticn.1 U8C8 of its cognat.o verh. Archimedes, for exo.mple, introduccs the prineiples of fluid mechn.nics in his worlc 0", Floating Bodie.., with tho phruso Vno"EiaOCd,'CI 'Let it. be Rupposed'; it is 118 if he wanted to SILY 'lot us allow oursel VCR to tnlto us IL foundation .. .'.77 Since tho wore) Vno{)Eat~ antI its synn· nymR78 c10arly derive from c1ialeclir., wo will h'Lve to invcstigl~to how (.imy wom lIKeel Lhoro if wo WILIIL to gnill I~ full IIndorsLILnlling of the'i!' 1'010 in nULthollllLtk'R. The verh ctv/4Io{vF.tV, which Socrates URea in connection with tho cmUleque,lCu of IL 'hypothesis', is equally intercsting. Plo.to'R v()cILbu\;LI',V (·ontn.ins a gren.t mn.ny synonyms for it. OneRuch is tho verb al'JltpWvEiJ,.70 'l'ho consoquen('.ea of ILn MSllmlltion nro o.lso (:alled Tci En&/lcva,80 TU UiiC;81, etc. An oxamination of PhLlo'R writings reveals t111Lt. t.heRO l'hrMCS ILro Romotinw8 uRed lUI predRo mlLthemlLticlLI lcrms82 ILIIII sometimes n_'I onlinn.ry dialectUoll1 oncs. Wo seo tlmt the technicn.1 term tJ1C,WF.l1tt; fOl'lllH plLrL of t~ wholo ellllll'lox of exprtmMiuIIs which 1\)lI'OILr logdhOl' in IICILh tli"lcr-lir. /LIlli lite language at "UltI,eI1Ultic.~.83 'l'hol'efol"O, wo will Imve to 11J'()U,dclI Cllil' inveatiglLtioll t.o t"I(O n.ccollnt of these a.'I well.

7IO'H!M OUlllin, 1',1. Hc.ibnrg, l.iJlflinn 1!1I0 Iii, II, p. :IIR. ,\rc:hillll',I"R IIH"II Lim nXJlmBlliOllll "molft:idO'1J .lIIll .;no"(Ii,,FO,. IIH "Lyli"Lic: Vlld,u.l" ur nil': IIl1nl.l ... r (lit'" II. 4ft). III Onmlt il. CBII btl Hui,1 flf n unuOulIC \.Imt it un,llfllru •• n l'!uclicl'R lisL bf Jlust.ulatt'ft l",ginR wiLh Lh" \\·(trllllrqIJOw ("I"" it. ho rr.quirr.d 11",1. .•• "); iI. iR rBqll;red or Lho oLh".· plll·,.icipIUlL in Lit" nrU"llnlnl. t1mt. ho l.. 1.1", Rt."t.... mnnl.ll whhlh rollow (Lh.. 1)IJ.~tlufllell) 'IR n "llRiR for LIm c1iHc'U!I8icm. ,. Nul. ollly dons Plnt.o Lmut. Lh" I!X,)I"1'f1l1illll!l rJ,,"AOy'1l'" RIlII .indOLIJI~ RA IIY"'"

lOII,,"

lIyll"', III, RUlnnLiltl,'11 UH41f1 Llro wunl ,iex'; ill " Hillliln,· """Rn. s..", for ,·xulII .. I.. , em'!lluli 43611. 70 Run, for ClXBITIJllu, 1'/'fIl'.Ilo IOOIL. 10 Crntylll6 4:11itI. 81 PlllIeJo 100c. AI l"or I'xnmpl .. , Lho oxpl'C8llioll td otJ/.(lal"ol·rrl in ftl eno 8'7n. n 'J'hn nxpn'rJRirllll' td Ind/ltJln, td l~'i', uLn. 110111,1, COIII·RIl, 111110 bt. tI"",1 lIN lI'I&LhuIIIBtical 1.01·11 lB. SUC! 1., JIIlu·lurvic, ''','R IlIBLlIIJlIlnl.icIIII'R d\llT. 1'lu\.on d A rlRwte', Bullr.lin 1,,'enan'iOlIlIl ,It! I' A NlfUrnie }' oll!l"slf111ft ,Ielf Sde,,· cel' el ,lell BeiliU: A rill, Clnsself des SeiellCtllI fllllt/IClllnti'1"e" el nfll"relltlJl, 32 (19:19) 1-21.

.,r

... ------_ ..

!l7.AII6: TIII( 1I1~IIINNINOR 01' OIUCKK IIIATIIICalATICS

3.7

·fUR '.ASSUMPTIONS' IN J)JAJ.J~C'l·IO

Tho ltYlJOlhuia in II. din.lecticn.l proceeding seems to represent n, starting point chosen n.lmost at random. Tho pn,rticipn.nts in I" discussion ngree upon some MSertion whioh is to servo u.s the boais of their subsc,!uent inquiry, and it then bccomes their tmO{)tc1C,." Some schollLl'88S have attempted to explain tho development of Aristutelinn logic from din.lectio nlong these lines. Aecording to them, UIIO of tho participlLnto in a dilLlecticn.1 dellate would select some statement and set out to persuado the other of its correctness. In order to tlo this, he had to fimi premisos which his opponent would I),Ccept ILS trlle (Lnd from which his originn.1 stn.tement followed 8S a matter of logieal necessity. These premises, chosen almost at random, werc tho II ypothe.seia. In f(Lct, PIlLto hlLd something IiIm this in mind when ho talked I"bout provisional assumptions in tho Republic. 8I Tho pn.ssn.go in question runs n.s follows: "we may proceed on tho lLS.'Iumption that what wo havo sn,id is right (vno{)lpEI'oc aI, TOUTOV OVT(J)' lXovro,), ILgreeilig thaI, (OlloAoYllc1aVTEt;) if 0.1, any timo wo lllLvo to chango OUI· opinion (lU" nOTE illn 'Pav"' Taih"a 11 TaUrn), all tho consequonces of our 1L.'I8l1mption slll\lI bo considored invalid (n&.VTa f}/iv Ta dnd TOVTOV ITII/I{JalvOVT" At:Avl'EVa llTtc1{}a,)." A ltypotltuia, howover, is very closely connected with what is illferred from it. Hence, far from being completely arbitrary, it must he chosen with thc greatest ctlore. This point is made in the CratylWl whoro wo reac1:87

.. 'I'ho illitinl proposit.ions or n dinlect.icnl debnto Wllro, in rnot., somoLilllC8 cllO!lCIl ill a coDlI,lelel!1 arhi'ranJ mannor; seo AristoLlo'B Topia (VIII. I.) 1;61129 nll,l V II f .:I.llilln:IIT). In tllII )lIIll1l1lK'1It in '11111111.11111, AI·illl.ol.lu refurn I.n t.I ...1I1I I'rtll'lIIdLiollA 1111 cl~uullllr". but. hll in cll!l&rly tiRing t.hin wurd DB n Bynonym ror IlnclOt:CJI' (I'r. ]{. von Fril.7., oJl. cit.. (n. 63 a"tIVo), II. 31). II Ie Villi If",I.7., 01'. I'it.. (II. 6:1 "Imvo), I'. 20. Suu "llIu I.hu "rt.lc!u 'SyllllgiHLill' lIy fl:. J{RJlJl in llIC (Puuly-WillllOwR, llealencydoplldill der l(l,IIJ8i.7chen Allu'ulll8· 'Ui8Ifen8Chall, I VAI 048, 1068-9) and E. ]{npp, Gruk Foundalion 0/ 7'rodilional /.Q!1U: (Colnmbl" UnivorniLy Prt'JlII, Now Yurl, 1042). II Republic IV.4:J7a. 'I'ho t.rannltatlon in hy A. D. J.lllllan.y. If Cralyllllf 436cl. Th" t.mnnlnt.ioll iR by Jowet.t..

-_._----------

l'All1' :I. 1'111( 1:f,Hlfl·ltuc:nIlN Ott at "Til 1·:&1 A'f1t:9

237

"If he did oogin in errur (tl YU(! TO newrov c1rpaAti~ 0 n{}lpf-vo,), ho nllLy havo forced tho rellllLindol' into ILgreoment with tho origilllLl orror nnd with himself (TcLUa ,j"'1 n(!ck TOUT' 1{J,&Cno "ai avrq; c1VptpwVEiv ,}v"y"dCtv); thoro would I,e nothing strn.nge in this, ILny more th",n in geomotric dilLCrn.ms, which ha.vo often 0. slight and invisible n:~w in the first part of t.ho pruCl'Sq (wl11tCe TWV d,aYeappar(J)v bioff. TOU neWTOV ITI""eou "ai ddJjAov V"1V"OV~ Yf-vopo'av), and aro conRiswntly mistalwll ill tho long c1culldiuns which fullow. AntI l,hiM is tho reaHun why ovory ma.n should cxpcnd his chief thought :LIld ILttention Lo tho considerution of his fil'st principles (l'Ei cl,j nE(!i Tli~ d(!X,j, navnk n(!&.ypaTo~ nal'Ti a""et TOV noAul' AOl'Ot' tT"a, "at T,jv noAA,)v IT"I""v) - are thoy or(Lre they not rightly In,id down (Err£de{)w" tlTE ,,,) tmO"t'Tac) 188 And whon he hlL.'I duly sifled them (l"tIVTI' .,' leETaaDtlClri' l)(avw,), all tho rest will fullow." Thero aro a numbel· of rou.sons why this pll.88l\ge is of inLercKt.. li'il'lit. of nil it. shows clearly thn.t Ph"tu aLLau:hed grent importlLllcc to tllll rolo uf foundlLtions in I~ system of illolLS. For, lL8 Sucmtos is IImllo to OXl,l!Lin, it is vittLl tltlLt the assumpLiolls 1.)0 chosen correctly sin<:c the smn.llest mistlLlm in !Lny ono of them can lead t.u completely flLlso cunsequencos. This is an iuea which rccurs often in Plato's dia"logucs. In tho Phaedo, for oxn.lIll'le, Sillllllin.'I decll\l'cs tlmt, oven though he cILnnot raiso any objections againsll tho a'·gument which Socrates hIlS just given, he still ho.rhours 80me doubts, l~nd Socrates rc8ponds in the following ml\llllor:Sg "Not ollly t.l1I~t, SiIlHUilL.'I, ... hut Ollr first lL8f1umptions (Tcl, vno{)lllfl&' Tut; "(!I:'TfIl;) oughL Lu he lIIore cn,rcfully eXl\mined. oven though they scom to you to he certlloin (El mITraL vl'i" tia,v). And if you analyse them complelely, you will, 1 thinlt, fulluw llond I~groo with the ILI'gulllent, , •. ". Theso words,-which imply that 'first a.'I9umptiolls' IlIwo to be ccrirLin, inovit-ILhly hring \.0 mille) tho n(!,illlu ,1(!xu1llf IIULI,hnllmt.il:H. AK fILl' lut wo Imow, Al"isLotlo WILS tho first. ILlILhor to discuss (lxpJiciUy Lho Imtll1"O mid function of first prinoiplca ill ml~tlmlJlatiCH.GII Yell SUI:mleH' rmlln.rks 1~lJuUt tho '01'141, lutHullll'tiunfJ' of l~ tlilLlllctiClLllLrgumcnt givo tho illlprcs"Tho vurb ,brdHaII'O' is lIisculllrod In n. '11l ubovo. It Plcnedo IO'1b •.'rho translation is by 11. N. Fowltll· (LOIlCloll 1963). .0 Cf. Chapter 3.6 above.

23R

i,

"",~

J ,I

,

""

11'

,"

l'AltT

HiUJl LIII~L I'Jato WIUi 1~ls() fully o.wlLro of tho problom of 'first IUiHumptinnH' in nu~themo.tics. This brings us to thc sccond interesting fCI~turo of our CJuotnticlII frulll the Cratylu.s. 1 am referring to the fnet. that it contains yet allot,her examplo from gc()metry. Socrates IloinLs out. t.hat. an iucoI'redly chosen "Yl'0thesu is like 0. geometrical diagru.m which contains Homo flaw; they both can lead to n. whole succession of false consoqll~nccs, ILIl of which aro consistent with tho init.ial OlTOI·. I t is sUI'Cly no accident tlmt l)lI~to's discussions of hytlolllc.sci8 fn!'1m'nUy contain 0. reference t.o geometry or to mn.thcmo.tics in genel1ll. lIo WI18 of the opinion that din.lectico.l l\rgumenLs should conform tu the high atlmonl'ds sot by Lhe methods of mn.thcnltLLics. In ordiJltLl·.l' (!ollvenu~tioll, all inconclusive o.rgument. whieh is at. best. plausible will often suffice t.o sotUo t.ho point. 1\1, iHSue. l\11~thomo.ticil~ll8, howevor, did noL ILccept such o.rgumenLs;" Lhoy wero concerned only wiLh convincing proof, Il.nd t.heir concern WIL8 reflected in t.heir ml\llller of choosing and handling /'Yl'0lhe.seu. Consider, for exo.mple, thc three intcresting OlloACJYf1paTa whieb o.re found in tbe Tlundeht8: n (1) "Nothing elm become greater Ol' loss, eithCl' in sizo Clr in number, so long ILS it remains equal t.o itself" (P'llJbr.on: P'lMv Ii,. /UcCOl' '''llJi: l}.anol' y~/u{)a, P7]n: l1r"fP P7]n: O(!,{}p{jI, lro~ Faai' d'1 aUto laVT{jI). (2) "A thing to which nothing is added and from which lIothing iH LI\ken away i8 neit.her increlLSed nor dimini8hed, but. always remains tho snmo in o.mount (q, p7jTt "(!OctT'{)OiTO, p'iTt Opat(!OiTO, ToiiTo ,,,in: Q1;~ovEa{)al "OTt, P~Tt rpOl"ElV, dd M Tool' Elva,). (:t) "A thing which WIL8 not. at nn earlier moment. cn.nnot. be at I~ I;~ter moment. without. becoming and being in process of becoming (0 Il~ "(!OTt(!OV ~v, VctTt(!o,. ilici Toii'ro Elva, TOU yulaOo, "ai ylyvto,?u, ac5uvaTov}." 'rheso fundnmental Q.88ertions are very remin.iscont. of ml~thcmaticlI.I nxioms.D' Yet., even though they are drawn up wit.h such precision ILIld care, Plo.t.o meroly calls t.hem opoAoy7jpaTo, "concessions which the pnrticipo.nts in 0. discuR8ion ho.vo agreed to malte". The no.mo c1l'lnly indicntcs tho diclleclical origins of the llind of 8tatement. to

:!I~ 'I'I ',I

II S"Cl T'leaclcll~ 162Cl. IIlhill. 16611-b. Tho LmnslRt.ion is by F. M. Cornford (London 1936). IJ ~. MRrkovi6, op. cit.. (n. 83 obove), p. 2.

Till<

CONRTllmn'lnN

III'

239

M"TIIR.IATJ(~

whioh it. refol'8. All thiR suggosLH tlll\t IImLhollllLLics ILIHI dialectic nolo only shared 0. common terminology hut. were ILI80 interconnected lliscil'linca. In fl\ct., it. 1001(8 118 if Plut.41 WILB writing I~t 1\ timo when mo.Lhomo.t.ics WILS jUBl a branch 01 dialectic. It bocomes even moro appu.rent tlmL dinlccticu\ Imd JIlILthcJIluticl\\ lIlethods were closely reh~ted, indeed thl~t they were identicnl, wh('n one examines tho way in which '&YIJolltesei8 Inc used in }>hLtO'8 nrguJIlonts. On nne occlLRion SOCrn.t«~8 dL'R(willl'R hiR IIlILllner (If thinldng P' in t.he fullowing Wo.y;DS "I U08sume in c/\Ch CMO some pl'inciple which 1 consider strongest. (Vxo{)lp~or; l"oClTort AOyov 0.' av ,,(!/vro 1(!(!CIIllll'ictTOTO" elva,)" [not.ice thlLt. he bll8es his iluluil'y on Rumo 'stl'OlIgest I'l'incil'le' in very much the sll.me way n..'! rtULtheml\ticin.lls u8cd to Lu.'!o theim on 10undaliQn.'1 which had IIcen I'Lid down once l\Iul for nil J, "lLItd wlmtever 8eems to 1110 to agree wiLh thiR, 1 rcgnrd ILa truc (li IllV dv POt dO"li Tounp ut'prproVtCV, T{{)qp, ([1< U.Al1{)tj ona) and wlmtever disagrees with it, ILS untruc (l1lJ' til'/uj, (r,,: all" dAl1{}JJ)". As we CI\n sce, in the ubove quot.n.tion consequcnccH nrc suid to agree (ClI'I'rp!JJI'dv) with Ute "1I1'oil!l!si8, o.nd not to lollow (""I,/1n{vEI" )VG from it. BoLh t.hctlo expressions belong to the or(lilllLry vu(!ILlrulnl'y of dilLlllCl ic mentioned in t.he precc(ling chapter. Of coumc l>hLlo ILlso IIR08 the verb d,apctlvdv, t.he opposite of uuprprovti." in contexts where something is said to di.,agree with 0. hypot/&eBi.s.D7 ] II fl\et., wo 1i11l\11 soon S(1e tlmt. the di8agrcemenl between sto.tements expressed with the hull' of thiR IILSt. word. is tho en.rli('8t form of wlmt we would cull I~ Ingicnl COlJtra-

diction. 3.8 now 'nyrOTnESBIS' WERB USED

«"tv

\11'

a,

Our next tiLBI, is to investigo.te the rolo which it!l1,otlte.'lci., plll.yed in dilLlect.ie. We sho.l1 eXlLmine in somo dot,\ilnn elLSily 1I.C('.cRBilJlo exo.mplo of dio.leetical argument. taken from the worl(s of Plut.o. It. should perhaps be ment.ioned that. Pinto's dirLlogllca contnin 0. great. mo.ny

"In tho Th«Jetdw (1891>-190) rloto dMl~rilw~ t/'inking vCl"BRtion which thCl BOlli hulcls with ItRnlf. fl IJ',acdo lOOn. '1'110 t.rnnnlnt.ion in hy 1,'0\\11",'. .. Tho T..atin word for t.his is con8C1J"i. t1

Pha«lo 10 I d.

08

n kind of con-

240

SZAIIC): TIIU 111<0I8N180S 01' OltJ'.HK .IATIIRlI.A1'ICII

examples of his method of argumentation, which incidentally wus often used in mo.themo.tics us weU. Furthermore, it reo.lIy mo.kes vory little difference which one of them we choose. However, there is 0. cllrLI\in o.rgumenL in the Tltco.clelus which suits our purposo po.rtimlll~rly well fur tho following rOMOUS: (1) Plato's own worda suggcst that it was patterncd after u. I"'~the­ IIlu.Lical argument. (Immediately before giving i!;, Socrates emphasizes Lha!; one should not be satisfied with reasoning which is merely ph1.usible ILlld con trusts such reasoning with the vigorous proofs OXllcctcd of mathematicians. [TI"co.elelus 162e]) (2) It can very easily be seen to have 0. counterpal'!; in early nmthelJIaLics. (3) It illustrates very well not only how the conaequ.ence8 of 0. hypotl,csis wore checJted, but also how the hypolhe8ia itself WILB subject.cd Lo scrutiny. The main points of the argument in question co.n be summarized IL8 follows. The iBBue to be debated is whether or not knowledge and sense perception are identical (1630.: d Ilea IlfTlJl bu
I'A Ill' :t, Tille (!ON!ITII U(:J'lflN (Ito' AI ATillmATlI:lI

2-tl

The sentence quuwd I~bove in buth Croci, I~nd English cuncludes tho tLrgulI\ont. It ~o"tu.ins 80mo intel·c.qting Lerms, ill JII1.rLieulILr the Greek wOl'd cl!5VJlaTOJl which is uscd to denote the 'imp0H.qihiliLy' l·t~.ItIILing fl'Om tho IIY110t/1C.'Ii,'I. Anyonu who hll..q rend ]~uclicl in Lho uriginlLI Imow8 thl~t ho ILlmost. ILhvlLyS cuncludes his indirect prouf.q with Lhis SILme wonl. Tho phro.se ho uscs is: 81lf.e IIfTLV dM.'arov.o o Plato's ILrgumellt is in fact no less of II. cOl'rect ·i.ndirect proof thu.1l the ones found in the Element.,. Socrates Wll.nts to provo that knowledge o.nd scnlle l'Cl'ccl'tion t\l'O not the same. ·Fm·thormore, ho feels Lhat hill proof must be patterned Il.ftcr 11. mlLtholllll.tical 0110 jf it is to bo convintling.'uo Ho therefol"O I1.dopts the indirect method ami sLII.rts out by o.ssuming the opposite of wlmL he wlLnts to prove. Then he shows that this o.ssumption leo.ds to IL olell.r contradiction (c)tatpfJ}J'Eij. He is thus u.ble to concludo tho.t his inititLIIl&qumption CILnnot possibly be correcL hecn.use n.n adynaton o.l'pcars ILlIIongst its consequences, u.nd heneo that its negation musL bo Lrue. This is cXILetly the samo kind of u.rgument as the PyLhu.goreal1s used to pl'OVC L1l1Lt the dilLgollal of a squaro is incommensurable with its side, for Lhey deduced the existenco of I~ number \Vhic)h was boLh evon 1~1\(1 odd from the ILRHlIllIl'tion thn.t Lhe two longths in question wero commellsurlLblc. Plato's d ill.log II 1'.8 , of COlll'SO, contu.in 11. great mlLny ILrguuumts which arc similar to this one from tho 1'lteaelelltB.'OI 'rho rCluler may (,y now hn.ve noticed Llmt P!t\Lo ILLt,~chcK sevcral diOercnl meaning.'l to the ditLlectical/maLhomn.ticILI I.eI'm ,;no{}uftl;. lrllrl.lwflllure, ht! mn.y ho woncllll'illg why 1 Iliwo nClL 1)(I1.llOn:cl (.0 distinguish between them. So, befol'e cmhl1.l·ldllg on IL more dettLiled investigation of tho argument outlined above, I woulcllilte to discus!:! thiK mu.tter briefly. De{initiona and the IOll,nt/alioll,s of scienco in geneml, which mathomaticimls talte to he self-Clviclont mill nul. ill need of ILIly fllrlihel' OXtLmifllLtioll, ,uz 11.1'0 clLlIed "!IlIO/Ite-sci., hy Pln.to, n..q ILre .. SOIno propositiona (for oXIlInplc. Elelllent.. LO IUIII TX.20, 30, 33, fUIII :14) coucllldu wit.h t.ho worda 8nte cbonol' or 8nf(} ollX Iinohtt:lTlU, Hol.h U"",o phrlUlr.a 01'0 nothing O1oro t.lmn atylisLiu vnriolltR of tho nlln IIl1'llLiollUd in thl! tox!., nnd !.hlly nro olao IIfWd by l'lnl.o ill his diaCIIRBiolls of clinlr.ct.ic, 100 In 1I10!.hernoticol cont.extIt, Pinto rc·Ii,", to 'COlldllSivl! proof' lUI dndcS£,~" Jtal (11'"),Jt'1: soo 2'lwaelelus 16211. 101 n"ClIUlr (Arc/liv fiir RC(1riUsf1csc"ic/lte 4 (I !lIiR), 212) hUll point.l·cI nlll. Umtn similnr leiml of nrglllllunl. is tn hI! fOllncl ill Pllllr.sin 10 I d 4-1i • 10111e1'1IMir. VJ.IiIOo-c1.

_.----

_ ------- --_. _.... - .. -- .. - •..

.- ... ..

'

242

--.~-

SUllO: Tim 111(01""1"08 Ott OIl1~ICK AfATIIK~IATIC8

thoso {irst principlea in a dialectical dobat.o which Socrates advises us to seloct on the blLSis of their strength. 103 On the other hand, Plato ILIHU uses Lhis term to donote any ll.BHortion whoso validity i" 10 be examined in a dialccticu.1 delmt.o. (Tho l\rgumonL which we havo just been considoring and its mathematiclLI euunt.orpart, tho ancient l'ythn,gOrlllLII proof of incommensurability, n,ro both built. around ItYl10f/,eaei8 of tho s~c~nd ~ind.) I do not wish to deny thl~t one can draw f~ legitimute (hslmctlOn along lhese lines, but I must eml'hu.sizo t.lmt it is not essontial for U8 to do 80. Clearly tho only (lifforence botween a 'fundamental' hypoll.eai8 and one which Iu~ merely been gmnted provisional acceptrLnce is t1l1lot the latter could IClIod to an adYlwlon (ill which ell8e it would hllovo to be rejected U8 fILIRe) wherclLH Lho fOl'mer does not (and hence can servo ILS a genuino 'foundation'). Tho statement thn.t tho 'OliO' iR indivisible, for oxn,mplo, ImClLtne ILccepted fLB a truo (fundalllonLfLI) /,ypotlteais of arithmetic lIeCl~uso ~IR ncgation, just like the falso hypot"eai8 that the diagonn.1 of n. squure IS commensurable with its side, WIL8 found to leud to n. contmdicLion. DiHeuBSion of this exumplo is 1'cserved for n. later chllopter. 1)laLo's argument is especially informative becauso it illustrates very well exactly what WIl8 involved in ml\king usc of a /'ypOI/ICSi8. li'rom whut hILS bcon said up 1.0 now it should 00 apparent thllot /'YI10 tI.eaei8 were laid down with a view 1.0 cxamining their COMeqrumcu. EXlLminations of this Idnd Imd ono of two purI'OlW.8. If the /'Yl ,olheais WILH 110 gonuine ono whoso t.ruth wlla not in doullt, its eonHo'luenccs woro eXlLminod in ordor to gn,in an undol'stunding of tho concluHiuns derivlLblo from it. ]f, 011 tho oLhor IULIld, tho /'ypot/,eai8 WILS a qUCHLiolluhle Il886rtion, then its consoquences were oxaminod to 8CO whether they containcd an adYM/O", i.c. to fICO whothor tho /,ypot/,eai., il.solf (:ould bo shown to ho untenablo. In Blly evcnt, a cruci,,1 rolo WI~ playod hy tho consoquences of tho /,ypothe.'lis. Tho qlU'Htioll whillh now lU'csonLH iLHulf iH htlw Hlloh C()f1HOIlllnlll:t!8 were olltninod in Plato's dialectic. Lot U8 attoml)!. to find out how thr. chlLin of ll.B8ortions oonnouLing n /'ypOI/£C8i" wit.h its cCJUsoquonccH WfLB c~OlUtLructod llolld what l'ul".8 govornod tho ohoico of tho illtli y idtml links in this chain.

lOa

PIKJalo 100.

t'A itT 3. TJIJo:

3.9

C.'ONlI1·IWt:nO" (II'

lIATIII(lI ATIC,'\

24:1

'IIYI'01'UKSKIS' ANI> '1'JII~ !II~I'II()() OF INJ>JJtRC:1' l'ROOF

If unc luultH ILl. !.IIOHc) PIL'iH;t.geR in l'!aL(,u wllCl\"O Lhe C\iILlocLicl\1 mothod iH UKed tu illVCHLigl\tu Lhe CIIIIHCljtlllllC'l!H of HOIII(! h!l1ml,.t','1i.9, then ono iH furred to cOIlC!ludt! tlmt KliCh invcHlig:LLionH Wl'l'e cuncerncd oxcluHiveIy with LIm (luCHtilln of whc!Lhcl' OJ' IIl1l. l he illl1ividu;\1 IL'tRCI'I illliR which wc'n' tak('11 to lie cClIIReclucllc"~,, (Tfi (JI'/lflnlt'fII'w)II'1 ngll'cII wit h olle ILllothCI' (f1"I"rCI)J·riv). In Pln.lu's t1iILIc!I'till, howc!Vc'r, LIm ollly way til deci(le LhiH qucstioll WI\8 by ILl'plying IL Itf'Un1iv('! c,.ilc~riolt. ThiH mc;LIlt thal hiH ILI'gulU(mIR, inHhmd of expllLilling t1il'ecUy why fllILt.cnuml.a wore in fLccortl wilh (Inc tLllothcl' (ff/IOA,,),fi,', (1V/lrpll)J'EiV, 01.1:.), invlLrilLhly (:llIlCclltml.c'c\ UpOIl l.hosc ILHSl'I'LionK w hiclt did lIof flf/rre ("mrprm'Fi,')' PlalH I'l·gardcd LhiH Idnd uf Illutu;d diH:~rcclII(,IIL IL.. IL Hure guiclc, 1\Iul hn alll·flY." tlRl'<1 iI. whcm hn RuL 1I1It. III (,XlLllIinn Lito c~lIlIfln'l""lwl'R (If I~ "YJloI'It!.'ii,~. Alul'Cl 1'l'Oci~dy: Lhc CIIIIHC!IJUI!I\C~CH IIf 1\ "!lpuil.t'.'I;s wero tulwlI to IIC LlllIse KLILLt'IIWllls which ngrccci with iL, wCl'e C()lIlplLtilJle with it or dicl not eonLmc\ict it;IOS howClv('r, if IL conLradicLion WlLK t1iHcovered ILIU(Hlgst Lhc80 conamlUtIllCCH, thiR illdiclI,Leu LhlLt Lho lIypollle.'I;.'i WILA flLIHO ur Llmt it led to nil (U/YlUllon. lIlO The L"OIIRC(lucnCCH of 1\ hypothesi.., were chcclwcl ollly for cUlmiHtency, ILIlc1 Lhl\ criterion UHl!d WILa hasiclLlly I~ IlI'g:Llivo UTIC. JL f"lIowH th('I'Cf,!l'o LlmL PI:Llo's dialectic WIt8 wholly elc'pcndent U1'011 the IItcth(ld of inclircl!t l'J"(Jllf. Thn cOI1Hiatem:y IIf IL sl.ILI~!lIIellt or of IL whllic flyslcm (If iclmL'i elLII IIIl clC·lltllllHtmlecl ill IL IIc'W~1 iv(\ Wl~y ollly lIy l'HLlLlJlishillg

1'04

/tJ CrlO R7". wil.hRtAI"'lIIlt'lIl~ ",hid. f"lIow",1 1.11'"111' 1'1"f'lIIi8l"H ",,,re lnll' or fIl18l·).

1t",'II" vllrh all/lp"la'fI" wnR 1180" in Cllllllnul,ioll IItlt~''RMrily frum 11011111 Jln'miRc'8 (wl",lhu,'

ThiH is t.lm

lIf'IIH1' ill ",hi,!h AI·iHLoU" IIHI'II LhiH "' ..... 1 (/'I,!",i("" 7.0.2:l9h30) wi ... " h" ,1i1lr.UMC8 7.(,IIU'1I U'·KIIIII.,lIt nlloll" thn rlying "rm",: .... r. .. ne·,-el,'/! Ihnl Ihn r.II'lf'llIlIiun ('1.1," f1yhlR nlTIIW ill nl, r"III,') f"II"\\'1I from till! I""'mill,' nf I."I! IIC'gll""·IIt., I,"t IUt/lflrl~ tllllt. tlllI "rI,ml,." iH flllllll. '0' It Rt!P.1I1R t~) 11111 t.JUIt. t.ho I'rillcil'lo uf "tllI·'·ollt.mtlid.iulI pluYII " lI",jllr rill" ill I'lnl.II'1I ,Iinlm:t.ill. Rtlc:I'IlI~'R iH nlwllyll l'III· ...... .\"'·cl II" t.'·yillg tn 11 ... 1 nllt. wl .. ·U ... I· 1.1111 IIlnl~'III"1I14l 11111 I Ill' ,1i1l"'UC,tillll III'" C'"l1l1iul'·III. wil.l, 1111111" Icy I ,"I ..,·,.ill ,,1,i,·I, 111111 nlc"'luly 1",.,/1 IlI!c·c· .. I.,cl 1111 tr"r.. If I·hn hYI'"I.IU'IIiA iA " 'AI milS' 011", nil ..14cl.·· 1I11'1I1~ which cnlll.nulir.t. it. RMI ... ,jr.cw,,1 1111 f.. lReI (III .. ' l'/IlJ~lo IOIIIY, ''''IN·r.inlly 1(1111). In ROlli" C·IIII,·H, Ic"",nvm', tic" hy .."U ..."h. ilH.·lr i,. " q ... ·III.i .. llIlhl" /IRR,'rt.imc whie'h hllll ollly h,·CtIl J(rnll"'"l ...·lltut.ivCl IU!""I'IIIIIC", IIn.1 it. IIII1Y 11I'l.lIlIlIy btl ""jl!r.I.(,t1, if it. ill f"mlCl to r.ollt.nulict. 8111110 "Uu·r 1I1.,,"',/lU·III..

111.,,"0: TIIK IIKOINNIN08 OF fllllCKK NATIIIUIATIC!I

tlmt. Lhe cuuLI'II.ry s\.l~LemenL or Lho oppolling sysLem conLILius a cunLmdidion, i.e. only by giving o.n indirect proof. ) L would have heen inconccivn.blo to exo.mine the consequences of ;~ /'!lllotlte8is wiLhouL using indirect arguments. Indeed, by its very IUlI""e llle ki,ld ofl'Y110tllC8is tuMc" we lmve bccJl. di.'lcus.'1i1lfl could not /IflVC 1,iaycd its part in Plaio's dialectic wil/wut the method 0/ indirect 1,ron/. I consider this fl\CL to be of prime importanco. As wo sho.)) soon

PIIl.to himsclf explicitly aclUlowledged it o.nd deliberaLely chose ttl oXl'rCR.'1 it through the chn.mcLer of PILrinenides. The observo.tion tho.t tho uso of hypol/'C8eis in dio.lectieo.l rensoning WIUI illseparn.ble from the method of indirect proof fits in very well wiLh the fn.ct that in the Thcaetelus PhLto describes his argument dealing with the relationship between knowledge and perception 11.8 one which is patterned afLer a mathematical proofj we know that indirect arguments figured very prominently in early Greek mo.themILtit'.8. Consider, for example, Book VII of the Element8 and in particular the first thirty-six propositions of this book. They are by 0.11 ILCCllunts of gren.t anLiquity,I07 o.nd no lOllS tho.n fifLcen of them have indirect proofK. Similu,rly six or eight of the sevenLccn propositions which make up the ancient theory of oven o.nd odd lO3 ILro cs\.l,bliHhed by means of indirect ILrguments. In a suhsequent Clll~l'tor 1 shu.1I atwmpt to show that even the central and fundlLmenLtLI theorem (If Pythagoreo.n arithmetic, which sLn.tcs that the one is indivisible, wu.s pmved in an indirect mo.nner. It seems therefore Lhat the use of indirect proof ~Ln be rego.rded 08 a eho.rl\Cteristie feature of nmthelluLties. No doubt Plo.to had this in mind when he compared his o.rgument. (TlIeaelel~ 162-4) to 0. mathematical one. /iN',

3.10

A QUES1'ION Olr l'RlORlTY

A number of intorcsting facts havo emerged from tho evidonce I"'csenLcd up to now. We Saw tho.t the mathematical term hypoll'esis and its synonyms, u.s well n.s a whole complex of rolated expressions, ILII ClLlIle originulJy from dialectic. Viewed u.s mathematical terms, they 10' n. I., Villi ,Inr \V"tlr.I"II, 'Uin AriLhl1lnt.il, tlnr l'yLhogunlflr', Mat/l . ..til", 120 (19(7-49), 127-63. 101 Reo n. 12.

I'A liT 3,

TJII~

C:ONIC"UU('"l'ION

ell'

.!A'I'II KMATICH

nrc in a son80 'loan words' which were IHLIWIl over from dilLleeLic, their ClLrliCHL field of al'plicntion. If we wmlLcd to illLerprot Lhisl\.'1 I~ reflection "f sOJlle rch~tionship hoLween tho two fields, we would hn.ve to suy tillLt IlllLthemtLtil:s WIUl n.t IClUlt in one l'cspect n. hmneh of clilLlectie. It scellls rl:IUlIJllable tu suppose that mn,themlLticiulls mll.de sudl extellsivc UKO or di:Llecti(,'tLI terms bCCILUBO lIIatl&ematir.'1 itself grew out 0/ tI,e more allcient 8uilject 0/ dialectic. It should ho mcnliuncd, h(lwcver, t1mt l his histol'iClLl conjectul'U ILPIK'IU'N In cOllflict with sOlllething elfl(' whid. we haLve lelmlL. 1'1,,10 hdcl up IIIntiwllllLticn.1 meLhmls ILB ILII e:mlltl'/e .. he wlLllLc'cl to illt,roelue·t: IlIIL\.henULLi~LI rigour into dilLlecLic. 'L'his suggcst.s tillLt it. WIUl nllLt.heIllILties which pTCceded ILnd influellccd clilLlecLic mUmr tlmn the other WILy ILround. The close connection between the uso of I&YJlot/.eseis in cli'Llect.ie ILnti the method of indirect lll'oof lends n.ddit.ional support to t.his lo.tLer view. Indirect tLl'gumenLH seem to havo been lL distinctive re~lLture of mathematics and, tLCcording to PI".to ILl. ICILBt, wero l'cculia~r til it. Yet, if wo I\CcepL Lhat mathellll~tie:H WILS thc olcler discipline, wo ILm flLl~ccl wiLh the proLlem of expIrLinillg how the IIULt.hemlLtielLI terms m(,llt.ioneel lI.hnve could h:LVe Leen ded vccI f mm dialeelic. TI", CtUCHl.ioll of priority is therefore tlr l"'IIei,,1 iml'() .. tl~nce, 1~'1l1 we will Imvc to LILI
"AliT 3. Till'. (~()NIITIIIJ(~rIllN III' "ATlIl-:alAl'lt~

nhout

II.

ItYllol/'C3Y of tho Idnd wo hn.vc bcon invClitiglLtillg. 'rho word" 1l'QCiiTOV n.ro cspocially Imggcstive; t.hoy bring to mind IIOt.

fi(lX'j nnd

only t.he l"I..'VIlLgCII from PI"t.<) diflculISed proviuuHlylln but. ILIHO t.ho f;(!XU{ of mlLt.hellllLt.iCN. ThiR interpretat.ion, however, is completely mist.u.ken. The cont.inu,,(.ioll of t.he o.hove (1IIot4Lt.iun J1l1Lltcs it pllLin t.ho.t. Eudmnlls is eoncerncd wit.h something altogct.hcr different.. The arclle or init.ilLI propoRit.ion (1f(]filTov) of Hippocru.tos' ILrgulllcnt. t.urns out. t.<) he an ordinll.ry genIIIdri(!nl t.heorem, which I'uns M fullows: "SimillLr segments of circ:ll'H h:L\'(' t.ho SILlIle mt.io to onc ILnother 1\8 tho squo.rcs on tht-ir hll.<ms Imve." It docs not. fit. Socru.Lcs' descriptinn of 0. hypothesis; i.e. it i8 not nil Il.R8ertion upon which Ilmthcmo.ticilLns bllJlo their invcst.iglLtions 1'1111 of which thoy givo no furt.her IWcollllt.lIl As 1~lIdemlls hilnllOlf goes on to SILY, Hippocmt.cs succeeded in l"oving hiR initilLI proposition (foum ,)[ l,}e{)(vlI£v l)( TOU )(d) tty showing tlmt it followed fmm ILlloLher one, no.moly "circles h,LYO tho an.me mtio M MIO S(IUlLrcS on their diameu!rs" .'12 SII l.hoRe words which Lhe (:118111'1 rOIl.tler of ElldolllllK' Iwcount. might. illlcJ"}II'et n.8 references t.o tho n(!,imll. c1(!Xal of nmthmun.Li(!R IlILvo in fact. IL mthcr different. RignifiC:lLIH:o. They ILrc uRed to denow the le"wUl.'1 which Ilippocmw.8, pmhlLhly right. 1Lt. the /'('ginning IIf hiR wurlt, found nec:csslLry for his arguJIlents. Nothing which EudemuR hM to Sll.y iR n~levlLllt to our present. inLereAt ill t.ho uso of hypotllC3cis. On tho othor Imllll, wo CIl.1l1l0t. determino whot.hor or not. HipPOCrAtes used tho 1I11~I.h"cl of indiroct proof without eXlLmining in dOt.ILil tho text. which c'unhLins glldomus' wordR. lIenco t.hiR (luCRt.ion must. remAin Un/LnHwcrcd hero. The fmgments of ea.dy Greelt mo.themo.t.ics which modern schollLrHhip 1111.<1 rccenl.ly heon ILhio to recovor ILlao Reom to indiCl\t.o thlLt. "!lJlol"c.~ei..; and tho met.hod of indirect proof woro used in t.ho fifth t:ent.ury. This evidence must. bo t.roo.ted wit.h CIl.ution, however, if nne is to avoid get.t.ing involved in 0. circulAr u.rgumont. Alt.hough it iR genomlly ngrccd nowl\(lu.ys tlmt most. of t.ho propnRit.ions in Book V Il of tho Elements daLe hack to tho fifth cent.ury D.C. and tlmt. t.ho 1l'(I(ome

O,.mylull 4:Jlicl

}'!.IU!,lo 10711. III /lcp"fJ/ic vr, 610r.-t1. In or. n. lI .. nlUII", A ,..:IJi'J Iii,. ll'lll"iO"IIMr.hilliltfl .. (I O/iR), 21 RIT. IIQ

Dncl

2-17

RO-CIl.J1cd thoory of ovcn ILIIII odel is liltOwise of great. ILntirjuity (I'rohILhly originlLt.ing in t.ho middle 01' ovell in t.ho first Imlf of the fifth (:
2·IR

llz ... n(): 'I'II~: nF.OINNINClS 01' UIIRI(K .....·I·llltAfATII:R

I· ... ILT 8. Till'. CON!!TlUJUI'IIIN 010'

exnt:Lly when ",nd under what circumst.ances /'ypol"cais nnd indirect I'I'00fo were first used. To est.ablish t.he claims which I have mn.do, nil Llmt neo,ls 1.0 be done is to let this evidence speak for it.8elf. 3.11 ZJ~NO,

1'nE INVENTOR OF DIALRC'l'JO

JII t.hc l'tlrmellide.s Plato nmkcs Zeno t.he EleaLic explain in his own "'ClnIH Ule nll~ill points of his wor/{.1IR From Lhis it. emerg(lS LllI\t ~cno'H t.rcntiliO WIl8 illLolldcd 1.0 ho a dcfcnco of Lhe t(!,~chings of PMIIIcllidcs ({lorjOmt ne; TfP IlU(lIl£1·{Jov Myr!,). Ptmnoniucs' docLrine of Lhe Olle Imd been ridiculecl by some on Lhe grounds thn.t iL led to conso'1ucnCC8 which eont.rn.dict.cd it (we; El tv [
Plnl.n, I'(lrlJlcnides 128.

117

RimJllic'hlR. COllmlClIlanJ

Oil tlu: I'ltysir.s oj Arutotle 134.2 (tho with l'hysics A3.1871l1). A ristoleiiN Prrrum. (Ild. V. ROllO), r.ipsinn 188n, frnglll""t. nli.

COIIC~"I'II<,d II'

1"181I1lg"

240

MA'I'III~M""rW!(

here. EleaLic dialectic WILS ImilL on I.hmic Lwo thillg8, amI PhLto's dialectic, 38 I have remlnltcd Oil nmny provious occlt..'IioIlS,lI9 is nothing but IL In.tor and perhaps more highly doveloped form of Elon.Lic dilLlectic. So wo IULYe been led to conclude tlmt Lho olnliCliL IUlown applications of "ypotliuois and the method of indirect proof were nmde Ly memLers of tho ElelLtic school. This in it.8clf would enablo us 1.0 o.nswer our earlier question about the reln.Live (l.ges of dialectic n.nd mathemll.LiM, were it not lor a 1UJlewortTty ob.'1ervalim, TUM a/len bee,~ 7Imdc word Ze,w's dialectic. 1 havo in minel Lho conjecLure t.lULL inclirect proof, or reductio ad absurdum, lVll... cJuvclopcd by Lho carliclit GI1.·ek nHLLhemn.ticilms 0.1)(1 thnt Zono ICILl'lu'cl ILbout it from theh' \Vol'lc. H this were su, it would mell,n that indirc(:t ILrgulllont WM origilULlly nmthemaLical or geomeLrical in clmru.clel'.l2n IL is apparent tlmt tho (IUcstion of whether or lIot dialccLic ILntedl~t.es mathemn.ties, which cropped up in connect.ion wiLh OUl' study of Plato's argtunents, cannot be answcn!d in a sn,tisfactol'Y mannel' simply by referring back to the EleaLicA. Tho c()nje(l~uro thn,t Zono tool( over tho method of indirect, proof fwm nmthemlLLieH is not without some plausibility. After all, Plato el\l}lhlt.flizca that whcn he uses "YIJotllP'.'Iei.'I and indirecL proofs he is following the cXILlIl}lle of ItllLthcmlLticians; so perhaps Zeno did the SILmo thing. I myself lUl.Ve stressed thnt Zeno's dialectic WM 0. forerunner of Plat.o's; flO perhaps tho Eleatics were also forerunners of Plo.to in t1mt Lhey IClLrncd fnml mathematics. All this incJicates tllII.t we arc fnced with a problem which requires serious and cl'iticlLI examination. Tho first. thing.u) be IWliL ill mind is t.lmL wo Imvc 110 dil'Ct;L ICllflwlcclgc of GI'eek mathematics in Lho ago of ZOllO. Wlul.t liLLIe we do know about the mathematics of this period is bll.8cd entirely upon mouern o.ttompla to roconstruct it. This fnoCL should in it.8olf make U8 a Iitt.le wary of the cl"im that indirect proof WM used in mo.thomll.ties be/ore tl,e time 0/ Zeno. There is absolutely no evidence to support. it. insist that Zello Icn.rnetl this technique fmlll 1lI11.t.lI('Illll.t.ics is t.o explll.ill what is Imown by something which js ulllw()WIl. 011 Lho oLhor Imlld. iL is VOl'y elUiY u, 111(;,,1.0 Lho sourco frulll which Zcno obtained Lhe method of indircct proof jf 0110 docs 1I0t diHcount

1O"ic"

'ro

iH

II.S llO my I'IlJlOr ·Ele.llt.i(~Il' rl'fcrr(,11 1·(1 ill 110

A. noy, I.,a JCllnMlle (Ie 1<1

NcumcCl

II. 33 nhovn. !lre,·qlU!. I'IlI'in I !l3:1.

I"~

202.

250

I'AIIT :I. '1'111< t:fINlO'lIu<:nnN III' M'\TIII':M.\TI(~

S7.AII(): TIIR ImOiNNINIIS 01' OImmc arATIII~aIATICS

('(llIll'lelely I.ho ancient trn.clition. Zello was II. pupil of Pn.rmenideR, ILIIII his gl'ClLtcst debt. undoubtedly WI\8 to his tolLcher. Thero WI\8 no 11('1.'11 for him to leul'll about indirect proof from some 'ancient mat.heIImlicilLII'; this method is to bo found fully dovoloped in the didactio )1111'11\ of PILrinenidcs. It WI\.~ PILrinonidcs who proved his thC8Cl! hy wruLing Lhoir ncglLt.iolls. l21 Tho discovery of indireot proof wa.'i perII/LPS his grelLt.cst and most lusting contribution to philosophy. wo n.ro ('\'('11 ill IL position to hc morc Jlreciso n.bout. the circumsLn.nccs of tho dilicuvl~".v. JI. prublLbly came about fI.~ II, rcsult of Plmnenidcs' crit.iquo or J\liI('siILll cosmogony or, to ho mol'O exn.ct, of Anaximencs' cosmo~­ oily. In AnaxilllenM held that tho world cn.me into being from a prillllLry suhstancc by means of 'rluefaction' and 'condensation'.lu 1"01' exn.mplo, ho mn.intained t.hat tvaler could be reduced to air,' wa/er, bofore it. came into being (i.e. before it was condonsed), WILB 'ail" or 'non-willer'. On the othor hand, this Iino of relLBoning could liCIt ho applicd to tho av. As l)armenidcs relLIiscd, it would ho sclfI'IJ11ll'luliclory to slLY of the av thlLt, lmforc it eamo int.o being, it WILB IL lui 01'; this would ho incollccivlLhlo ILnd henco impoHRible. What i,9 IIl1lst nlwlLys IlILvc exisLcd; it could 110t havo come into boing from ",!:al i,~ nolo (Using tho terminology of Zono nnd l)lnLo, we could dl'st!ribe the ait.untiun us follows: the Ity},olhe8is thlLt whn.t is 'cnmo illlo Imillg' lon.ds to 1\ cont.rlUliction or adynalon, for it implies that ",lUll is WIIS onc:o ILOli-existent.) 'rhus the notion of II. logical corttradirtion WILS fonnultLt.cd by Parmenides in the course of his criticislJl of l\lilc!iilLn cosmogony, nnd t.his in turn led him to discover tho met.hod of indirect proof. 3,12 l'I.A'rO ANU 'l'lIE ELEA1'ICS

We ha.ve now shown t.hat bot.h the usc of ',ypoll,esei8 and the method of indirect proof ClLIl be tl'll,cctl hn.cl( to tho Elc/Ltic schoul. Our next tlL..'ilc is Lu seo what light. this Imowledge throws on tho origins of

n.xiomn.tic ml\themll,t.ics. }I~ir8t, howcvel', leL 11K t.1I.\(0 1\I1othol' lool[ ILt PllLto'H mothod of I'CILRoning n.nd pid[ out. KIIIIIO fCILture8 of iL whkh rellLto tAl lIiR use of hypotheaeis. III so tilling. wo will alHII ICl\I'n ROlllcthillg ILhuut elLrl), Greek IJlILt.hcmILtics. Thero iR no douht tlmt 1)111.1.0 tlKlULJly dlllOKCS to (,xl'rc8.~ his own views through t.he dllLl'II.ctor (If SocmLcs. It iH wcll Imuwn, howcver, thlLL SOI:I'II,W.s is portl"lLyC(l in tho dilLlogues 1L..'4 heing extremely distr.ustfill (If Honso l'ercel'Lion~ i.e., ho iR Rusl'iciollK of ILlI kn()wledgo oiJLtLlfled I hy nU'lLIlR of the sonKeR. ]" tho I'ItaCllo, fur eXILlllple, wo rel\d: ' . "When t.hosoul nmlw8useofLhe hody fur ILIly ilUluiry, either t.hruugh seeing (lr hearing or ILny other of Lhe SCIIKC.~ - for inq~li? Lhrough the hody means inquiry t.hrougb the SOnKCS - Lhen It IH drngge~ l.y Lho body to things wMch never rellUlil~ tlte same (oV(~br~Te "~TCJ Tmha lxona ), o.nd it wanders ILbout. ILlld is confuscd ILlld dl7.zy hl[O n, clrunlwn man ),ecnuRe it lays hold on Hlleh things. Hilt when tho Roul inquires aluno hy itself, it deplLrts into the rel\hn uf the 1'lIre, the evor\:LBtillg ((~£i 0'0), the illllllOl·LILI nnd the , ., l XOl•. )" t:lmngelCR8 ( a'{) uvaTCJV "aL,waaVf(/)~ III t.ho ILbove qUULILtiull t.ho two ldnds (If Imowledge 1\1"0 contrl~Lcd wiLh uno n.nother. Pl:Lto'R dist.inction bet.ween knowledgo obtlLllled wit-h t.he help of tho senRCS n.nd purc, nOli-sensory lmowledgo, 1\8 well u.s hiR preference for Lhe t:Ltt.cr surt, iR flf course plLrt. of t.he legacy. uf Ell~ILtie l'hilosophy,lZ5 It. WIL.~ PILrfHCllitleH whu gavo tho followlIlg wn.rning to his disciplo:12G "You must. dehlLI' your t.houghL from thiR WILy of sOlLrch, nor let onlilllLry experience in its vo.riot.y fon'e you "long t.his wn.y, the eye, sightless as it. is, nnd the en.r, full of sound, (Lnt! the tongue,

.:4

S(W UUI pllpUIlI of mille liaw.1 in II. 33. Cr. K. lloinhllrdl" l'armel&ide~ mm die GesrhirJlte rlcr griet:/list:/um Phi/o:rOl.!.i/J 1111.1 Illy JlIIJlI.r '1.11111 V"''BLiimlniA .Inr 1'~I .. "f~'II' (Hnll n. 33 "hovtI), IU llipIN.lytllR, llof. 1,7. 3 (Dics .ullI J{rlUlz, Frogmonte ('.13 AUllXhnl'flNl '11

lit

,,7).

2!i1

1'1'''1'4" 790-11. '1'1111 LrIlIlRI,,!.in/l iR hy I'owl,,,·,

In "'hin iR 1101. !.hn pinon 1.41 cliAI:U1IA wh"t.lmr it. in 1'"Mit.l" In .1I·lllIir.. 1U1'~\\"I"'I~" wiLhout I.bo Ilid of 1.1.., Ron81'11, hut. 800 Illy Jlllrtlr '1.ur Gt:8chi('ht.n tlur DIUI"ltt,llt

eI,," I),,"I(I'IIR' (rofnrrn,1 1.0 in n. 3:1 "hovn). • . It' Dids and ICrl1II1., 1'·I·II~lIInnl.<., l.2M l'arllll'lIid,,.., .,7; t.l1I' !.ruIIHI"I,nm .n by IC"I,hl!'nn Freomllll (A ncilln ro IIIIJ I'rc·Sor.raIit: "/Ii/o·'''l,/u,r.• , Ox fon) 194R). Stln Ilhm t.l1n fr~mn/lt. flf M... liRIUlR, nj,'IA " ... 1 I{ .... m:. l~mJ(III!1nlA •• 1.30hR.

252

87.AIII': TIII~ 1l1(UIHHIHUS 01" OIlI~~:K MATlll.:AlA1·IC8

l'AltT 3. Till'. C!IJNII1'IUJ(:\'IOH

to rulo; hilt judgo by means of tho ROlLsun (My'p) t.ho much-conI{'~lcd proof which i~ expounded by me." Thu~ Lhe Idnd of ILrgulllent. from Itypollll'.8r.M which is t.ho .mltjcct. "I' OUI' prcsent. inquiry IIl'l'SUPPOSC8 t.he LoI.ILI rcject.ion of sense )l1~1'­ cepl.ion. This l'enll\l'lt is t.l'Uenol. only for t.ho]~leatics (Zeno, for oXll.mplo, I:ould lIot Illwo nmilltlLinml L1mt his 'plLmdoxos' woro vlLlid wit.hout ILl. Lhe SILllle l.imo donying tho truthfulnc88 of sonso percept.ion) but nlso fur PIILtO. The crilcrion of consislcncy, by which ILrgumcnts f!'llm 101Iu'.8eis were judged, elLn only ho ILpplied in the domain ollmre t"oliU M. Ohjects l,creeived through t.ho Ronscs IUO u.hvays clllLllging; 1.I",y 1I00'Olllll tl':Lnsfol'm(l(l intu thoir opl'ositcH ILnd, honce, Ino cOlllrutli,;lory. Only such idcILI ontiLil'S u.s thc concopts of 'IdcnLiLy' (TIl ram') ami 'Lho BelLuliful' (nl HaM .. ),27 are unclmuging ILnd novor Lurn into t.hdl' 0PJlo!lite~. l~or the I~I"ILticlI, therefore, ILnd for nil who elLlIlI! aner (.hOIll, argumcnL from "!l1lOllteseis l\Jul Lho rcjccLiun uf KCII!lCpCI'I:cpt.iull fonncd n. singlo orglLnic wholl!. (It is worLh roclLlling hero LhlLL Lhe llIeLhod of indirect proof mlLdo its first appearrLnco in Orcolt llllLLhonllLLicK ILL Lho sll.mo Limo 118 maLhomaLicians begn.n to till'll ILWILY from visUlLI evidoneo ILnd Lo udollL an ILnt.i-empirical viewJloint. A full discuB.'lion of this Illat.tor is to be found ill Clmpter 3.3 ILIJClvo.) PI:Lto himself provides us wit.h incontrovertible evidenco for Lhe vicw HIILL he wlLa following t.ho exaJllplo of Lhe EleaLi(:8 in rejccLing HellNe-pcrcepLion and ILrguing illclirocLly f!'OJIl lIypotltc.'Ieis. Pal'lIIcnidcs, in Lho dilLloguo of Lhe alLmo IULmo, lI.ddl'C88CH Lhe following words of I'milm to Sucmlcs: l28 "Thoro was one Lhing you slLiti to him which impreRRClI lIle very much: you would noL ILllow the survey to be confillcd to visible things or Lo range only over Llmt field; it W88 to extend to t.hoso objects whieh ILI'O sJlceilLlly n.pprchended by ronson ... ". Thcso worda, whieh PhLtO puL into PlLrmenides' mouth, 1101'0 consoIlILnl. wit.h Pn.rmenides' own WOI'dB quolcd above and wiLh PIrLlo's dialcct.icalmet.hod. Their prcllenco in Lhe dialogue indicates LhaL PIt,Lo

"!I1

cr.

to

II' /'''nctlo 7St.!: aurd T.) laoI', allrd TO xaAdl', allTd [xaaros> 6 [aTlS> 61' II~ 710ft: lura(loA,p. XIII ~l'rl"oi1l' i,odixuIII • • • dd a"twI' [,,"aTOl' 8 [aTl • • • CillJllliTWC x"rcl IntlT,} EXtl '1.. 1 Otl.5brOII! otlda/l;; otIJrrJlw, dUolwal" oUdI!Jllas> i"Mxnal. I:. PIll to, l'(lr"'Clliclc., 1350. 'rho t.rul1SIRl.ioll is adapted from 0110 by I". III. Curnfurtl (""(1 II. 134 hulow).

fW

MA'rIlI~MATI~

had IlO wish to cuncou.l his dobL to Lhe EI(~u.Lics. The cont.illlllLLion (If Lhis I'IUlRn.ge is also of considenl.blo inLcl'e~t.. Parmenides gol'.A on to discuflS how uno should set I,bout. arguing frolll l,ypoillesei.'i. lie ll.'Viel'ls LhILt. in II. giVtlll Kit.1I:LLifln iL is nul. llJIIJIIgh Lo considcr Kome fixeel II!I1JOllw8i.'1 mul iLH cOIlSC'lumwCH; On(l IIlUKt. 1,1110 puL forwlI.rd t.he OPPOKing II!Jpo/lteaM nueI subjoclt. it.R (!onseCluenCI'S 10 elLroful exalllillation. 1211 He du's ZOllO 11.8 ILIl examplo of Bomoonn whoKo invcRLiglLLiullR ILlwlLYR (!onfonn to Lhis plLLlcrn. The I'CIL8011 why Parmenicle8 insists LlmL IIOt.h "YIJolhesei:J IlIwc to 110 considCl'Cd should ho oIJviouK. Sinco ILII Kcnse perccption, l\Iul wit.h it I'll phYKiclL1 experiencc, must on principle be rejected U..'I 1(lls('. or tmreliflbie, lUI mnpil'ielLI t.<:Rt ClLII he llcviKI'cJ to choclt wheLlml' or IIClL ILII llo88ol'tiun iR correcL. Tho vlLlidit.y of IL KtlLlcmcnt clLn he demonstralcel in only ono way. The conscquonc:es of both Lhe stu.tomcnt. in fJucsl.inll nlUl it./i ncJ.tll.t.ion must !.II examineel t'(l fiCO which of t.hcm cont.ILinK IL (:Clllt.ra.dict.ion. Om:o LhiK hlL'I "(~em dlllcrmincd, Lhe KLILLcllIont. musL bo rcjelClcd OJ' ILCf~U"Lcd, elllpl'neling II1'0n whct.hCl· it. or il.H l1l'glLLion loads 1.0 Lhc illconRiALcllcy. l~vel'y illllil'ect proof. wlwu clLI'riccl ouL in filII, uLilizeR Lhis meLhocl of invoHt.igILt.ing IL pILiI' of I,ypoillt.,('.i.,,furLhcnllnre, Lhc whole of PllLlo's dilLlel:Lie: iK IIII.qed upon it.. 'I'hero iH no dOllht. UmL hy t.he fifth l'cnLury II.C. thc prOf~e!lllrc dc...'Scribcd ahove WI1..'1 already heing wieldy uKccl in lLrit.hmoLic. The propoAiLiun Llmt Lhe One iH illClivisihlo, fnl' oXlLml'le, coulcl noL IULvo bcon oHt.ILhlished wiLhouL Khowing Llml. il./i ncglLt.ioll led t.o IL cClnt.m-· didion, i.(I, iL could IlOt. IlILvo I,een r"st.:LhliHilcd wiLhClut uKing IL IIILil' of hypotlu~.'ieM in. Lho IIlILllllCl' pI'CH(:rihcd hy PlLrmonide.'1.

3.13

'U V1'O'l'U Esms' ANn 'I'nm l~OUNI)A'I'IONS OJ!' MATlII~MA'J'JCS

Sinco wo have now McorLlLillecl LhlLL "!I1IOIIte.~(!is ILIld the moLhod of indirocL proof Cll.me from ElelLt.ie phil(}!l()l'hy, we ILre finally in IL pORit.ion to dllcide whcther dilLloet.ic HI' UlILt.hollllLLics is the older ReiOlle:(!. Quito ohviClURly it WUR dilLlect.ic which CILmo firsL. The mere fiLet. l hILt. I:D Ihill. 1:lliu-I:IIi: X(I.) ,~~ xrrl "I,~t ,,, n(lc}, r",iTeI' :no'Fi,., ,,,) "d,·".. tl {,rrll' [xaafOl' '~TroflOil'FI'O" ".,conti" nl all/,{I"I"ol'rn ix rric ,i7lo0iaEwt;, dlAd xal tl II~ [an Id atlrd ToiJm ,J"IJrIO,n8fJ1,' d {I",iAn ,(fUM" )·../"·rrnOij......

25·1

!l7.An(): TIIY. n.:CUNNINOS 01' IIIlRKK MA'rIlKlIAl'ICS

"II Ihose lerms w/.iC/, "elale to tile 10"tu1ations 01 Jlwl/u!11wtic.'$ a,.e 01 dialcclic.ltl origin should hn,vo led us to Lhis conclusion. Din.lccLic did IIC1L Lnrrow any (If iLR voclLhullLry from mILLhellln,Li~; inHwad, pcrft~I:L1y onlinn.ry cxprcliflionll from dhLleeLic were LmnHfurmeli inLo \.eoillliclLI IImLhcnllLLiclLI terms. Heneo elLrIy Creele IllILLhellmLiCfi, ILL 11'11.'11. whon iL is viewed 11.'1 ILn ellLlIomtely coJ1st.ruot.cd system of Imowlcd~e, elm properly "c (:a.llcd n. lImneh of dialecLic. In \' icw of I.his, Lho f:wL Lha.L PIaLtu held up t.ho mll.LhclnlLLielLI lIIeLhod lUI all cxulIlple to lie illlituted is of (~unsidcmLlo hisLuriclL1 inten.'8t.. It. illtljcat(~s LlmL by his t.imc I,ypotlle..~eis ILnd t.he mot.hod of indirect proof had Lecn completely incorpomted into IIlILLhomlLtics ILnd weI''' IIC. lunger ILNSociut.cd only wiLh ilialccLic. In flwL iL scelns Lhat Lho Iwrim) IIcIIlLmting PILrJIIOllidcs ILlld Zcno from l>llLto SILW thol~lolLLic (C'lLching nllOuL cOlltmdictioll ILIld (''Ollsistency dovolo}! in t.wo diffcrent. dirl'cLiOIlR. On Lho 0110 Imnd iL dcgollemLed into sophisL1'y, 130 ILlld on (he uther iL LloS8ollled illto tho lIIat.hellllLtics of tho fifth centUl·Y. 't'hiH laLier sulJjecL, of which we IUL\'e ILL "cst. IL fmsment.l\ry knowledge, glLvn hirth 1.0 l~uclid's Itlcmelll.9. Thero WI\.S, of courso, no (JuosLion in Plntu'll mind ILbouL which of Lheso t.wo developments ho should ILI'I'I:UUI. We have not. yet cxhausted t.ho inwn~Rt. of t.ho fn.et. t.hlLt l'IrLto mfors \.\1 mn.thollULLics in connecLion wit.h his uso uf Itypotlteseill. Nul. only tlcll~S iL ullow us to conclude LhlLt I'YIlOlll.C8eill plnyed I'n import.:Lnt rolc in mnthemlLlics ILL LlmL t.imo, hut iL also ermlJlcs U8 to maim quito IL plnusihlo conjeeLUI'o aLouL tho kind uf /,ypoll,esis which WlLR tiRed in 1'1'0- PItLtonie InILLhom:Ltics. Tho qucstion n,L iHSuo is whoLher Vno{)£UtU; in Clnly Creel, maLheIImtics wcro for tho most }tILrL j1lst ad Iwe assumptions liko tho gconwl.ricl~I/'!lpol"esi,~ mcntiuned in t.ho .M e"o (8603), or, t.o put iL ILIluLhcI' WI~y, wlUlUwr :my ILLtmnpLR woro mu.do in pro-PIILLollio Limt'H 10 IlLy IIlIwn IL YCIIl'm[ loundation for IIl1LthellllLl.ics. III l1flsw('ring I hill qUl'slion \vo must !.alte a(~('O\mt. of t.he following fad,H: (1) At~('ording to l'J'OcluR, UJO first perROn to compile a Look uf 'EI(~I1ICnlli' WILR Hippocrates of Chios, IL fifth cenLury nULLlmmlLl.icilLll. 131 Ill)' J"'JlI,r 'l'.lIr G'·Imhi.,hto .J,·r gd""hi~dll'lI DinldILiI,' (ror.,rn'" til ill :1:1 /llmv.,). IJI I'mdllH (,·,1. J,·,·j."II .. ill), IjG.7r.

no & .. , II.

l'AltT:I. TIIK GONln'ltllCTION fIV

MATIII~AIATH;!\

255

Jt is difijcult. to heliove Lhl~t IL worl, of !.hiH Idllll wOllld noL IlILve been prefaced hy sumo list of princil'lCH. . (2) We have nh'CILtly uillrerved (ill Clmpll'r :I.fi ILLuvo) tlmL IL lIumllel' of l~udid's geollleLl'ical definiLiuns ILJ'n n'dlllllirLnt; either Lhey }lilLy III. I'I~rL in his I'l'Uufs, 01' tho UlLmes which LImy scrvo 1.0 inLJ'Udlll'(l ILre never tlscd. FurLhermore, it CILn lie shown (lly (lxlLmining PhLlu'H voclLhullLry, for oXlLlnplo) LlmL Lhese al'o of pm-guclideu.n ol'igin. '1'hili 1lIC1~n!l t.Il1LL in ILII prohlLllilit.y Lhey ILI'O tho l'olllnlLnLa of Home lisL of )IILRic rnathenULLit~al principles whi"h was (~lImpil('d hefore the (.jlllll of l~uclid, (a) PlaLo nWlIliOIlS IL ftnv mlLLhllnuLtimLI hm,0Ilte.,ei., (concerning 'UIIl odd ILIld t.ho oven', for cxn.lllpln, ILII(I 'tho Lhrco Idnds of l~ngle')Il:! which turn up in Eudiu'H IiHt. uf uefinitions. Ho ILl so rellllLrlts that. IIlILthenllLLicilLJls "will givo lUI fUI'Lher lLt:t:uunt of Lhem (i.e. t.hcSI! "!lPolltcsei.'1) oit her to UlenHlcl Vl!H 01' to others". ThiH 81lggcsLR LlmL in pre-PI:Llollic timcs fJllLthemlLUdlLllS wero wont. to pref:\ce Lheir dillCIfR.'1ioI\R wi~h (t sl,ec:ial li,~t 01 ,~ltd, IlYl/(}l/u·.'1ei", just 11.'1 AutolYCliH of IJitlLJltl, l£llclill ILntl ILII tho 01.1101' IIlILLheIllILt.iciILIlH of II. hLt.UI' 01'1\ dill. Ollly hy doillJ; Utili (;ullitl Lhey Imvo indi(~;LLed UmL theRo prul'oliit.jonH, unlilUl 1\11 tho mllu\inillg OileR, would not be IJrOl>etI. So t.Imre iH 110 duubL LlmL ovell Loful'O J?llLLu'H Lilllo elTtu'LR W(,I'O JIlILtl" to l:oIlOt:L togdl\('r IL group of funtilLmentlLl IlllLLhenmticlLI principles. Since l>lrLt.u's dialecLic was ImtLerned ILfter ml\LhemlLticlLl rOlum"ill~, W() (lI\ll infor from wlmt. SocmLes sll.ys in t.ho Plwedo (WOIL) Lhll.L ;nIlLhellllLt.icirLJlS LeglLtI clLeh of t.heir in\'(~LigILt.i()n8 by nS8ulllillJ; Home prilU:il'lo which Lhey juclged to Lho strongesL. Whell it WILR realized t.hILt. by ILnd IILrgc t.ho RILmo principles wore nceded fm' eVN,V IIlIL(.honmLh~1L1 invesLigaLt.iun, tho idolL uf I~Olll!':Lillg thorn t.ngothcl· ILL the h"J;illlling of 1\ wol'l< clIuld ClLRily IULVO sllggCHled itself. I helieve LlmL wo CILIl go OVOIl further Umn t.hiH ILIUI (~"njecl urll LhaL th" ('ILrliest/Oluu/alicm..'1 woro (:()JnpIIHml only of cil'fini/ioll.." AfLm' ILII, wo Iwow Llmt Lho WOl'd {}1f(Uhl1tt; (n..'1 IL lIIaUlI!IImLimLI tel'lIl) hlLIl t.wu melLningH; iL coul,1 tienul.o eit.her a 11I1U[amelllal prim:i]Jle or jUl,t IL definition. 'this ill itsolf Reems til iJl(liCl~t(! I.Imt IlIllIia lIIelttflllJrillt:ilJle.'i were I~t one t.imo idenLified wiLh dfl/inilio" ..I. JI·u .. Lhennol'(~, iL iR obviouH UIILt. (.\10 very first l&YIJotll('.'Iei.'1 Oil whidl I.ho 1'lLr(.nmll in IL ,liILlf'ctil!al

"0

•

2liG

RZAII6: TIIK JI.:UINNINOS 01' OIlKBK &IATIIK&lATIC8

debate hn.ve to agroo take the form of definitions. One needs to establish 1.III~t whn.tover ill being discussed t'xisl8 .~nd thll,1. it C.Ln ahvays bo c1iHI.inguished Irom wllat it 18 not,' it. cannol. bo both itself ",nd its ClJlJloRiLe ILl. the SI~lIle t.ime, The following qUOt.ILtioll from 1)ItLto's 1'~uLhYJ'hJ'O iIIust.mtes this III.st puint:':!3 "ls not the holy (Tf) uatov) ILlwlLyS one lLnd tho sa.mo thing in overy action, o.nd, ILglLin, is not. tho uuholy (TO ChOcfHW) n.lwlLys opposit.o to tho holy, l'Iut Iiko itsolft" The Grcolt word fm· 'to defino' (Oe/ClaUae) ncl.ul~lIy IIIClLUslo JlUlrk oU. 1\ Ilefinition WILq intendcd to mark off Lhe Form or I~ido8 of o.n object fWIll thn.t which iL was not o.nd in this way secure tho consistoncy of the Form in question. The e'....liest definitions, therofore, were designed tu distinguish o.n object frdm ita opposil.c. Tho first l,orson to mako IIlIe of ono WII..'i Purmonides, when ho drow a sho.rp distinction bet.weon L\w 0.' (Wlmt is) and tho /I') OJ' (What is noLl. As far lUI t.ho l~leu.t.ies were conccrncd, there could have bmm no dh~loctic without t.his pro1~('IIIlI'O of murking olT or defining. Similarly, Plato USlIOrts thaI. it is impossible to englLgo in 0. (lialectico.l debo.te without first. la.ying down tll'fini Lions :13. .. Jf ... IL man refufles to ILdmit thn,t Eidc of things exist or to distinguish n definiLo Eido8 in every CllSe, he will ho.ve noLhing on which tu fix his thought, so long IlS ho will not o.lIow lIUll each thing IUl.'i a r/Iflrtlc/er which 1,9 always tile same " I~nd in so doing he will completely dest.roy the significance of all wlLlect.ien.l debn.l.c."

Platu's monning iR e1C1.... : if uno dueR nol. h~y down definitions ILl. tho UIILIII)t to cnsure tlmt Lho subject of one's discourso must "'\WIl.yS relllain t.he HlLme in clULflLclor (i.o. to ensure t.hat. it mn nover be(:omo its "I'posilo), it wiJI turn out to bo contradictory (i.e. to lJO bot.h 'itsclf' I~nd 'not itself'); bu~ one ClLllnot sellsibly o.pply tho wo.lectico.l method 1.0 ILllythillg which is not consisl.cnt..

au EIlIIIY1,/aro 611; cr. I'IIuIllIo 102u. l'rmllenidu 136b-c: d ,.i '" cSlj ••• aJ Ilq laae, d"" flUp 6pTaw tIro, ••• '''1M TI delf:ira. d.loc I~ belfafl"', oMt! IIno. T(!/ya. l'qp .,.dPOllIp l~"" "q iliip lMap 1.;11' .i,·m", I.Ir.il1fnu TI)' ntlTl)" dd d~IU, HIIloIITOI' T')I' ToiJ t'S.aAi,.r:oOa. "WII/IIV nlJl'Tlmao, l}m".o,(!Ei. - Tho trnll81"tion iR tnltl1n (with 80mn minor o.ltorno.tions) from tho nil" I.y F. I\f. r.orllfunl (/'Iato mad /'flrmenide~. London 1939).

"AILT 3.

TII'~

CON!n'ILUUrlON 01" MATIIJUIATJ()S

257

Since eo.rly Greek mo.Lhemo.tics ClLn ho regn.rded us 0. LnLnch of din.lectic, t.ho first 8WP in ,I. mnthellllLt.icIL\ investiWLtion must ",Iso have bcen to Io.y down drfi.ltition8. Such definitions, in flLct, provided II. Luf,lLlly new lotuulaliolt for (I"re Ilmthonmt.ics (ILUd especin.lly for ILrithmetic). Thoy mo.de if, possible to guamnlce conRistcncy in II. !lew ",net ullt'xpcctm\ wny. This cll\im will he suLRt:,,"LilLtcu in the next cluLl'kr', which iR"concel'lled with "YI101'tC3eis in m:"thonmtic8.

3.14

'l'lIm ))JWINl'l'ION OF 'UNI'!"

13y II. 8Lrolm of good ludt it. is relatively CIlflY to invcstiga.te the definitions I"t. tho Leginning of Elements Doolt VII. As we InlOW, tho so·called theory of cve,~ n.nd odd, which is to be found at tho end of Doole IX (IX.21-3G), represents the mLrliest. known eXl\rnplo of d(~elucLivo nmthOillatics; it. WII.H duuhLleR.'i includcd in t.he Elemellts only on o.ccount of its hist.oric...\ inlcres1,. Decker .rewscovered t.his theory in 1930 13& ILnd ('.sLirnal.cd it.s dlLto to 110 I.he middle or sOllle timo in tho first. IULlf of the fifth century.l30 Now une needs to know who.t it means for 0. number to bo 'ovon' or 'odd' before ono co.n const.ruct such 0. theory. In fo.ct, ono needs to Iw()w more UIILn this, ILnd it is ineonceivl\ble t.llILt tho ILrchiteeta of the f,heory were ignortLnt of Definitions Vn.G-9 llomt 12. As nct:1wr himsclf reILlized,l:!7 however, this implies tho.t the definition8 mentioned IIlUSt. ",Iso have been formultLtcd no lal.cr than tho middle of tho fifth century. ] n my opinidn thia 1".ltL roml\l'lt 1\1'plieH to sOl11e oLher definitions lUI well. It is sClLrcely possiblo to distinguish betwcon even ILnd odd numbers without giving somo thought to tho conccpt of number. lIenco tho definition of 'numbor' (VlI.2) is o.lso presupposed by the theory. Numbors, howover, o.re dofined as multitudes composed of f""i18;I38 80 Definition VII.1, which introduces tho concept of 'unit', gocs together with VII.2. As 1\ mat.ter of fiLet. the concept of 'unit.' also '''Pl'eo.rs in the definition of 'oelll number' ILnd in nULuy of t.he propositiolls and III'oofs which go to mlL\tO up t.he theory of oven and

UI

I» RI'" n. 12 .. bnvu. 81m n. ) 1:1 "bovu. m Ibitl. na 'A(l"}"~ in"p ,d be "ol'IMwp nI7Hd,,,VOP nAij{loc.

III

25R

I'AIIT 3.

I17.AII6: TIIP. IIROINNINflS OF OIlREK MATIIKllATICS

odd. Thcreforo my conjecture is th"t Definitions V 11.1 and 2 are "n

inl.cgral part of this theory and UULt they wore known n.t the time when it WIl8 first developed. It could bo argued that tho Euclidean dofinitions of 'numher' ancI 'unit' might have be on formul"ted Il8 a result of somo philosophiCl~1 lI.ctivity which did not take plilce until long nfOOr the middlo of tho fift·h (:cntury. Most of tho propositions included in tho thcory of even Imll (ldcl aro very trivil~1 mId Rimple; they must. h:LVll been ILcccptcd wil hout proof lIy anyollo whu 111"1 tho slighlcst. IUlowlcdgo of arithmc'til:, :md they do not Reem to roqUil'o t.hat tho concopts of 'number' and 'unit' I", given eXlLct definitions. Cont.rary tu what ono might. thinl', howcver, the preceding ILrgument clOCK moro to con{inn my conjocture than to discredit it.. It hlLK heen ohflcrved t.hnt t.ho proofs in the theory I~re carried out in painful detail.ll' This indical.cs that whoever compiled t.hem wll8less interested ill ('ontent. UULIl in form, His intent.ion must havo been to show huw I'Ulllplc(.e proufs could bo given for theRe simplo propositions. Without !'he definitionB of 'unit' and 'numbor', howovor, tho theory of even ILlld oeld ill not coml'lcLo. Moreover, t.hero Ilro oUlor mlLKons for thin\{ing t.hat t.hcso two definitions aro of great ILnt.i()uity. Boolt VII of tho Elemenls begins with tho following definition. lIo A n unit. is Umt by virtuo of which Clwh of the things that exist is elLllcd CIne," This is such a succinct statement t.hat at first glanco it ia difficult. 10 comprehend what lics behind it. Plato tlLlks on ono occasion abollt t.he lIo,clLlled 'theory of tho ono',m We would do well, therefore, to try Itll(l cli!!ct)vCl' ILK much 11.'1 wo call about. this n.noiont 'theory'. In t.\10 /lrll1lbiir. PIILt.O tells UH tho following about tho unit: J4Z II

•• You Imow how it ia with sltilled mathematioians t If any ono in nn ILrglllllent ILttempf.B to dissect tho one, thoy laugh at him and will not nllow it. If you cut it up, they mUltiply it, taking good Clno t.hnt the olle sha.1l novor appelLr not Il8 one but B8 many parl.,. ..'

n, I..

130

RIl('

110

Mo~.l, /11111'.

dnr WlLlIrdflll, !ll,rtl,. A"", ]20 (1047/.10), 130. HuiJ"11' lHoaTOP TWP IIrrwl' It> Aiyua,.

Villi

~

,.';0"",,.

III

Ur.public V 11.626:

1ft

Url",I,li., V 1I.1i21i.1- 620,1, 'l1.n t.rnn"I"LiClII ill hy A.

neel

Tel

I.

n.

J.ill.l"ny.

TIII~

250

('ONflTIIUCfION OF MATIIF.MATIC'!I

If you were to SILY t.o them, "My wc)l1Clel"ful friClul, wi liLt. Flort. of lIumberB nro you discu8.'Jing in whidl the 'olle' allswers YOUl' dl\imH, 143 in which en.ch 'ono' is equal to every otllt'r 'one' without t.he smallest. difT(~rellce and contlLins within it.Rolf no purlR 1" How do you thilllt they would reply 1 I f!Lncy by sayillg that t.hey are rolltillg ILhout thoso lIumhel'S whidl CILII he apprehended hy the undcrst.lLlUling ILlone, hut in 110 01111'1" WILY."

So, lU'c'OI'(ling to the nhov(', t.lm 'ono' is illlliviNihlc!. Of (:ourSl~ ILny ullit. ('ILII ho divided in pmcti(:o, ILI\(I fl"OlII tillle imlllelllol"in.1 Gr('clt tl"l"lcsm('1\ ILml engineers used fnwtions in t.heir elLleulations. The mwil.·nt scicnce of arithmetic, howevor. admit.ted nOlle of t h('se things, It Lool, 110 lLecount of fmctions unt.il t.he tillle of Arehimedcs. As Plato SlLY8, illlltel"l of dividing the unit, mnthematicians multiplied it. Theon of Smyrna oXl'lains how theso word!! I\re t.o he undol"st.ood:1U "When t.ho unit is divided in tho dumlLin of viHihle things, it ifl ccrtainly reduced 118 a hody and divided int.o p:Lrts which ILre Hrnallor LImn tho "oily il.tlclf, hut it is inc:relLKed in IIllmherM, hm:ILusc IImny things tlLlw LIm pll\co nf one." This CIUut.ILtinn not only iIIurniJmlcs tho IIlclLlling £If Platu's remarlt, but. ILltm helps ue t.o understund the signific:lLnco of Euclid's (h·finit.ion of a unit 11.8 "tl,at by virtue of UI"icT, eacT, of 'he t"ings '''at exi.9l is cal/rtf one". This flLct in itself malteR it a lot. cl1.8i('r for us to ILseert:Lill tho age of t.he definit.ion. Plato nlld Euclid shUr£l IL common conceptiun Clf whfLt it moans to· bo a 'unit'; hence the EuclideAn dofinit.ion of this not.ion II1I1St. hn.vo been form II hLtml in l'ro-Plntonio tinws, Jnclecd, I beliove that it dlLt.es lJlwk to t.he vcry beginning of dmluct.ivo nmthemalics. My rClumns for thinking 80 ILre given helow. Plnt·u's cxpllLnation of why Ule unit in nrit.hmot.ic WI~ rcgn.rdod /l8 indivisible is very instnlClivc. Ho ILKSerts LlIILt. mfLt.hcmat.iciunB wanted t.o exclude tho l'os!libilit.y of t.ho one being sonwthing other LImn itself, i,e. of it. ovor fLPllearing to Le mlLny (ro;tU{Jmjl'£l'O' lUI noTt pavn Ttl Iv I',) tv IDei noUd I'Ofl1a), If t.he one were di viHihle, it would no longer .u . , • olm' "~,6w wllI'lI

'~/'ri, ,1~loilrl: WI! '~()1II1!

•..

,V"

"h .. 11 luwn " .. "-,, I.n

filly

1111,,"1. LIm vn,.h

In cli"cURR UI/I ,""/I"/lUU:S (/11;",i ronccl"il/l.tl~.

"'I'. 18 ill lIi11or'" ,·,IiUon (J.iJlllilu, 111711); 1.1", II/UlllII!;" ill 'l\lolA',1 hy vall \\'/U.,....... ill Sri,,"rr. II ""'/("11;"". 1'- II Ii.

t.I,.,.

2liO

81.AII6: TJII~ JIImlNNINOIi 01' olmme MA1'UIo:MATWII

he one, hut. many, Henco the concept of 'unit', would be both itself (Lnli its opposit.o at tho same time. Tho ideo. that the unit is indivisible hecamo o.ccopt.cd onco it wo.s realized t.hat tho opposing idoa (bocauso it involved tho notion of 0. wvisiblo unit) woos solf-contrn.djctory; thus (.11(' stu.wmont t.hlLt t.he unit. if! divisible had to bo mjected ILa flLlso, which meant. t.hat. its llego.Lion WILB Lu.lwlI Lo be truo, The procewng intorpretlLtion of PIo.to's words suggesta thu.t the fullowing conclusions CI\n bo drawn from them: (I) It is now c1eu.r what lies behind the Euclideo.1l definition of 'ulliL'. It, is definitely not the vn.cuous descriptivo stat.cment which iL n.ppelLrs to bo at first sight. Hiddell in this definition is the theoretical j IIHLificlLtion for excluwng fmctions from the ancicnt Greek science of ILriLhmetic. It I"Cpresents the conclusion of a carefully considered ILrgument and iml'liciLly determines wl,eJlter or not tlte one i.s divisible. (2) We o.re also in a position to say how the Grocks arrived ILt Lhe idclL Lhat the onc is inwvisible lI.nd u.t the EuclidelLn dcfiniLion of 'uni!". The stlLtcrnent Lhat Lhe ono is diviHiblo was found to conluin an i71cO/l,.si8le1lcy, u.nd this discovery was tlLltell to mOlLn tlmL its nogl1.Lioll had to be true. So Lhe l'mposition that the ono is indivisible WIIS obtlLinecl ILS the conclusion of nn indirect argument, \-Vo Imow thn.t indirect proofs in eu.rly Greek nmthematies alwn.ys tool( the su.me form: the inconsistency of somo' assertion WIIo8 demonstrated in the course of an argument; this ILSsertion had then to Le abandoned. and iLs negation was considored to hn.ve bocn osLablished, As we can seo, Lhis u.lso describes Lhe way in which tho Euclidean definition of 'unit' WILl! worltCd out, for it, 1,00, wu.s built upon an indirect argumont, In view of this fact, it 8cems highly unliltCly tho.t rigorous methods III' proof were dovoloped in Gmok mathematics be/ore a start WBS nmde 011 IILying the axiomll.tic foundations. us Our example suggests rathm' tlmL Lhese two developments must have taken plrwc at about the s:~mo timo. A close exu.minlLLion of Plato's description of the 'one' in arithmetic leuds us to a furthor conolusion, He states that the one is indivisiblo and cxists only in the mind since "each 'one' is equal to overy other 'one' without the sma.llest differonce and contains within itself no 1'ILrLs". All this, espeeiu.lly if Lho lo.st phmse is rond in the originnl ... ROll 1111,

2

IUIII

01 nbovu.

l'Aln' 3. TII« CON!I'fllll(,'l'WN (W

IIIATIIJo!~'A1·It:S

2Gl

Greek (((fOV TE t"actrov nliv navTl "a1 oV~lITJlt,,(!oV ~,atpl(!ov, JlO(!u$v T£ lxov lv tavTq; ovMv). inevitably brings Parmenides to mind. 'rhe ElelLtics ILscribed exactly these s/l.me clIlLract.cristie~ to 'Deing'. The following pu.ssa.ge, selecLed almost ILt random from Parmenides' poem, nerves to iIIustmt.e this lloint: 140 "Nor if! Being divisible, since it is ILI1 n.JiltC. Nor is there ILnyLhing there which could prevenL it from holding together, 1I0r nny Icsser thing, but 1\11 is full of Being. Therefore it. if! ILIt.ogcthf'I' (~onLinllOUf!, for Deing is dllse to Being." Jt is no accident.. that the lmcatics ofLnn RpoltC n.~ if lJeing (TCJ lJv) and lite Onc (TCJ ll') were intcl'clmngeahle ('oncepts, H is fn.ir to RILY, therefore, that the J~t1clidcnn definition of 'unit' is lloLhing but n. concise summn.ry of Lim RlelLLic doctdno of 'Being'. 'fhe definition woos obtained by the 61LII10 Idnd of indirect rea.soning ILS PILt'fIlcnide8 tlscd to devclop his thcury of 'TIeing'.147 This is wlmt P);Lto had in mind whcn ho mentioned in pn.'lsing the "Lheory of the Ono" (7) m:(!; n} ;,. l'U{}71f1't;),IUI 3,15 ARl'rJlAm'l'IO AN]) 'rIlE 1'EACIIING OF 'rIlE EJ.EA'I'ICS

OUI' investiglLtion of Lho dofinition of 'unit' hILS led us stmight to philosophy, This definiLion, which tmdiLionll.lly lul.8 heen atLributed to Pythn.goras,lCo turns out to be, in a Rcnse, a t'ectLpitnllLtion of Elcu.tie doctl'ine, HowevCl', wo Imow t1l1Lt there are ,"n.lIY other fClLtUl'cs of ClLrly Orecle nmthcnllLties which undoubtedly reveal the innuclH:o of the Elol,ti('~, Hellcl) it is IIILL\1ml tu specui;Lw tlmt the ell.dicHt Hyswm of Gl'cek li.dthmoLic mll.y hn.vo been nothing moro tlmn u.n oxtenHion (If tho wlLching of tho ]~Ic:,LiCfl, 1 hope to show below LlULt this cOlljeet)J1'O is indeed cut'recL, Beforo doing HO, however, l~letLLic

,•• Dinls Ilnd I{r(Ul7., Fragme7l/e 1.28 p"rllll'ui.I'·A n rrngmf'lIt R,22IT, Thl! LrumduLion is by KnLhlulln Frenumn, u71·UI'IIICnidIlA I\IAO pl'Ovocl his LhuAiR (Lhnt, t.l1II 6,. Ilxiat,(J(J) hy rnrut.illC ita III'g'lLioll, Sce nlso 0, Oigoll, Der Urs]Jrtmg ,'er grice/,iselien P/,ilosop"ie. O",n,1 1946, p, 260, U~

Ilepublie V n,"26,

n. cr, ROXt.UA Empil'icuA, A,lvt'rsus fIIat/,. X .2(jO-I: II 111101lVdflll, dflXl}1' 1'1''1''£1' dl'uc TWI' 61'((111' n}1' "ol'lf,lu, Il!: HatU ,,,,rox,},, {,mnrm· TWI' 6rr(lll' '" Aiy£TIII, 'flm conclmling wordA or t.hi!, (ItloLut.ion nrt! "hnt'RI. "hn 8nmo liS 1t;uclill's d"finiLinll or 'unit',

2G2

51-Alit); TIm IIIf,tl/NNINOS 01' OltlmK .,A1·U,'.NATJC:S

I will Imvo t.o - discu8.11 I~n orronoous view which was quite popular among el~rlier scholars. In much of Lho Hcicnt.ific Iiterat.ure t.ho relationship boLweCJi Lho Elcalics ILnd t.he Pytlmgoreo.ns is described in Lerms which differ Higllifh:lmUy from t.hoso used ahovo. It. is claimed LhaL Lho Elel~t.k'8 alltl Pyt.lmgClrcl~nl:l wcro opposed to ono I~not.her. This view, whieh J.II.~il·ally goes hlwl( t.o 'l'anncr'y, 1$0 is u8Ul~lIy nH8oeia~tcd wiLh t.ho following threc Lhcscs: (1) Tho ten.ehing of PILrlllenides, despito Lhe fl~eL t.haL it. resulted from IL close sLudy of Pythagorean doctrines, was directed agninsL the PyLIIILgorcmlS' plrtrali
' 'ey

uo /'our l'''istoire ,Ie I" .~cie"cc /161l~"C, Purill I 887, pp. 249-60: "Panllcllill,. II\·/lil. lim'iL lion pll"lIIlI dllllll 1111 lIlilinll Oil, COlllfUlI IHlllllnlll'll, 1,,11 pyU",sorici"IIH ,"'lilA ,:Lnit,"Loll laonll"'I"; i1lwllit rnprodlliLphlAOllllloins nxud.c'III'·/lL IIIlIr cnlll·ig/l'!. II"'II~ ,'xlI"'iriqll" .•. miliA l'lI t.cJII~ maR, iI uVllil. niti III vuriM ""I"lIr t.J1t~n "uulill"". - LI'II nLLII'111t'fJ ,!ontro Ron pOt)rnn ,ll1rlllll. dono v"nir Rurt.clllt tIn pyLlll1sorit:iuIIR, .. I. '''''II~ "IIX qun 7,6noll prit. " pllI·Lin." nnd pp. 248-9: "Zdnoll n'a nllllolnunt. nid I,· IIIOUVI'IIll'nt ••• iI a 9Ulllomont nffirmo ROn incolnIJat.ibilit.cS avoc In croY'lnl''' ,\ lu I'llImlil.ci." .. I Fur ror"rOnC08, BUO C. Timor, 'Ant.iko MuLhonmLik 1900-1930', in nurlliarJII .111/m'-,berir./al4 iU,er ,lie F'orhclari/U ,ler Id4-'8. Allerlum8wi"SlJ1I8c/,all 28:1 (1943). .... n. Io'"r 1111 oppooillg viow or Zono'R argumonlll, ROO n. f •. van dor WIU'rd"n. M"t" . .... nu. 117 (1!l40), 143ft. 161 n. VIIIAI.oa, in laiR rovinw (ana/liP" I Dfi:l) or" huol, by .J. Ii:. Itllvon (/'!I""" !I'''-'''''''' 111 ..1 1':1,."';,'11 . .... II "I tilt, iu't!rtldi .. 11 t11t11"~' "1'[Hllletl "r./..",,-, ,1"riIl9 '''e liltl. and earl!! I,mrl" cenluriu ]l.a., Cumhridgo 1948), qllol.c'fl U, .. Ic,lIl1wilig r"lIulrJc mndn by lV. A. Hni,lol (,Tho PythogornOllR ami arook l\inLh.,. mnl.il'II·. Am . .T. }'I,. 61 (1940) 21): "I.hurn IR 1I1lt., RO far 08 I Imow, a Ringlu hillt ill ollr BOllr".. !ft t.hllL tho arenl,s worn nwaro t.h" (lurflOllo or 7,eno t.o crit.il1i7.11 tll n rllllllmm'ntul ducLrilll'-8 or tho l ' yl.llllgoronlls".

,,1"''''''''

I,"'"reo"

or

"AILT :1.1'/11( CI)Nln'IIUI:l'ION 01' AlATJlHAIAT1CS

263

lakiJlU seriotUJly. NonoLhelC88, Lho Lasic idel!. Lehind t.hem deserves 0. closer examination t.o seo exactly what kind 0/ 'opposition' there was bel ween Ihe Eleatics and PylllUuoreans (i.e. arithmeticians). As we know, the EIOl~tics maintained t.hat. the on.ly Lhing which (>xistcd was 'tho Ono', 'What. is'; Lhey denied that. t.hero was I\ny plumliLy 1~lId olu.iml'd t.o Lc lI.hlo La 1'I'OVO t.hl~L Lhe VCl'y noLioll of 1'IumliLy WI~ solr.coilLmdictory.'s3 '1'0 dmlY 1)luralily, however, is to 110 ILway wit.h arit.hmetic. Henco n sLudent or arit.hmeLic, o.lLhough he would havo Imd no t.l'Uuble in n.ccopLillg the not.ion of 'uniL', could noL Illwe followed t.ho J~lef~tiC8 in Lheir rejecLion of fIl tLlIi 1)licity. lie would SUllwho\·. have had t.o rett~in Lhis laLter concepL. In fl\.CL, J~uclid'~ :lUcond nrit.hnwti<:ILI definition (Vl1.2) is deHigned eXl'reK.qly to savo Lhe concopL of plumliLy. For iL st.aLes: "A number (Oe,O/lOr;) is a multitude composed 01 unils (TO Ix /llU'cMwv (fvyxelpt"ov nAipJot;}." This definit.iun illdiclLLcs that. Lhere WIUl in fiLet. an essen LilLI differlllll'o lteLwel~n t.ho J~ICI~Lic 1~Jl(1 Py Lhll.gurel~1I t."CI~t.ment.s of 'plum.IiLy'. Nonetholl'lIli, iL Wlluld bo wrnllg to SILY LlIILL Lhey wero opposcd to ono n.noLhcr. 'J.'ho PyUlILgorcl~ns did not. disputo tho Jmcatie docLrillc of the 'Olle'; all Lhcy did WIUl to develoJ) it I"rtlter. According t.o Ph~L(), 154 Lhoy JIlulLiplied Lhe 'One'. My roll.IIOnS for claiming Lhll.t. this repro· sonts n.n oxtension of ]~le"Lie Loaching ILre gi ven helow. (1) ArithmoticilmH Lrel~wd 'numhers' in eXII.cLly Lhe SI~nlC wlI.y Lhn.L Lho Imel~tiC8 LrenLcd 'J3eing'. Wo know from Plato'55 L1Il\L numhers were rcglLl'ded nul, ILEt I'hysiOlLI o!JjecLs which could bo 1~I')ll'Ohelldcd hy Lhe BenSCH, !JuL I~ u.bRLru.cL oncs which could !Jo ILpprOl\.Chod unly hy wl~y of pure .LhoughL. !roo SimihLrly, PlLrmenidcs wl~rncd his dist:il'lc lI.gaillsL Lrying La grMj) 'Beillg' by IIlCl\.ns or Lhe sClIses 1~lld illsiswll t1I1LL it. could only !Je comprehcnded by pure rel~on"57 Since 'numbers' WCl'C construed us mere ab.dractions, thoy ropresenLec.1 illStances of a now Idnd of 'l)lufILlit.y' whoso exisLencc was less easy Lo challenge. The ElelLt.i<'8 were ahlo to show L1m.t., bcelLUSC 1)iumlily OIL a l)racticltl level

to'R..n \V. (~IlI".I1 .. , /Jill I''''·.... /,· ....'i/,·'·r. I...il,,-ig I !1:IIi,I'I'_ 11:11T: 'Z""""M nnw,·jlCl' g"CI'1I ,Ii" Alllmhll'" ele.r Vinlh"il. Ilur Dillgn'. 114 l'lut.c), flo//IIIllie vrr 1i21i... III Ihicl., fi2/i,I. IlII Ihill., 1i21i. 117 Su" n. 126 nlll/v" ..

sunt): TOR

264

IIEOINNINOS OF OIlRY.K MATUIWATICS

self-contmdictory notion, there could not be 0. 'multiplicity of things'. Their arguments, however, did not apply to the concept of 'number' which was obtained by multiplying the abstraot 'Ono'. (2) It becomes even more apparent that the Euclidean definition of 'number' did in fact develop out of the Eleatic doctrino of the 'Ono' (or '130ing'), if we keep in mind the reason why arithmetioians found iL necessary to multiply the 'One'. Plato explicitly tells us that this W/Ui done to lI.void dividing the unit. ~"l'ILction8 were replaced by "umerical ralios in Pythagorean arithmetic. 1M Benco the new conccpt of 'number', which was introduced by Definition Vll.2, made it pmlsible for the Eleatic doct.rine of indivisibility to be retained. We are perfectly justified ill speaking here of a new concept 0/ number, for it is not the naivo and t.raditional notion which is defined by VII.2. Tho Pyt.hugorean's 'numbors' diffcred from t.he usual kind not only LCCILIiSC thoy did away with fmetionR, but also OOcn.URO t.hoy did not includo tho u"it. Counting begins wit.h OM, and, according to the usual WILy of thinldng, one is also a numbor; but this is incompatible with Dcfinit·ions VII.l and 2. Numbers, if thoy are mult.itudes composed of units, can also bo decomposed into units. Tho unit itsolf, on the other ham1, cannot bo decomposed ILt all. Hence it. had to bo excluded from the domain of numbers. This in tUnl led to the use of such cUlllbersome I'hrn.se8 118 'If a is 0. number or one •• .'.15" ! It is int.erosting to noto how difficult tho earliest Greelt mathematicia.ns found it to trent 'one' consistently as a non-number. Let us (,(Hl8ider just. one example of t.his, which is provided by a plLir of prol'osiLitlll1l in Boolt VII of t.he /iJ/emenis. Tho first of these propositioliR runs as follows: VlI.fI: "If a number bo part of a number, and anothor ho the Rame l"Lrt. of ILnothol', alternaLoly also, whatever part. or IIILrts tho first is of t.he third, t.he slLmo part, or t.he Ramo pnrts, will the second also he of Lho fourLh." So t.he t.heorem st.aLes t.lmt if a : b = c : d, where a, b, c and dare wholo numLers, then a : c = b : d. Tho fact that t.he unit. was not considored t.o bo IL numbor meant. that the preceding theorem lmd

WIIS

III

lit

II.

D. L. VBII dor Wuordon, Erwachende WWen8c1J1J/I, p. 189. (bitl., p. 180, n. I.

I'AIIT 8. TUH

L'QHSTltUl:flOH

OF MATIiRllATICR

205

Illso t.o Le formulated for t.he CILAe in which a = 1. This is accomplished by Proposition VII.15, which rell(ls: VII.1S: "If an unit mOlLRurcs any n\lln[,or, n.nd anolher number mc/UiUl'CS ILlly other numher t.ho sILme num),or of times, alternllLoly also, the unit will mon.sure the third number the slLme numbor of times thll.t. tho second meusurcs tho /011 rtlt. " There is no doubt t.hlLt this hLtt.er t.heol'Cm differs frolll Lhe fOI'mer in just ono respeot; it t.allls abuut. the 'uniL' insLoILd of n 'first number'. 'l'his i8 the only rOILAon why it received IL sOl'aro.lo Rt.awment ILIIlI pl'Oof. One should notice, however, thlLt. t.he terms 'another number', 'third number' and '/ourll, (number), are inappml'riaLo. If the unit is relLlly not a num.ber, thon the formula 1 : b - c : d cunt.ILill8 only three numbers, allli it is nonsonso to speak of IL '/ourtl,'. By his unfort.unate choice of words, the ILuthor of V!l.15 reveals tho fiLet tho.t, ILlthough ho only formulated this proposit.ion bcclLuso t.ho unit Wll.'! not a number (acconling t.o Definit.ion VJI.2), he L'Quld not riel himself of the ovcrydlLY \'iew t.hat it. WI18. We CILn summlLrize t.ho results of this Illst. chn.pLor by slLying LIm\' t.ho first. t.wo definit.ions in Euclid's ILriUlIlIet.ical boolts dCllLrly b('t.I·ILY the influence of tho Eleat.ica. Tho problem of divisilJility (or t.he ElelLtic doct.rine of indiviRiLilit.y) IIILlI IL decisivo (!ffcct upon the form which t.hcso 'fumllLmenwLI principles of aritlllnctic' toult. This c()Jlclusion is furt.her confirmed by tho contents of tho next. chlLpler.

3.10

'l'1If~ UlVlsrnn.l'l'Y ()Io' NUAfnRn.~

''Ie havo secn t.ha.t t.he EIOILtio t.heory of t.he indivisible 'One' cnuRed t.he Pyt.hngoren.ns to int.roduco numbCl'S ILA Illultiples of a unit. They were t.hon ILbio to get rid of fro.ct.ions in favour of ",,,,,erical ralios. More WI18 required, howover, if they wore t.o solvo complotely and in a conRisLont. manner the }lrolJlol11 }losed hy l~IC:Ltic philosophy. Alt.hough it could now ue mainw~incd t.Imt t.ho Ono itself was indivisible, tho old problem of divisiuilit.y D.Cfl'lircd IL neW mCILning when applied t·o multiplos of tho Ono. Tho difficult.y lLrOHO f!"Om tho fact that 1L number 'composeli of units' would ILlso III~ve to 110 decomposable. Arit.hmetieilLllS, thorofore, found t.hemselves confronted wit.h a neW prohlem. If they ho.d· merely hoon fl\ced wit.h a plurality 0/ thi"g." i".

2GG

I17.AIIO: l·IIJ~ IIKUlNNINOS OF OIlY.Y.K NATU)QIATJC8

fhe n'(Liworid, LlICY might porlllLps havo been ablo La deny that it. wns

divisiltlo. Tho Eleat.ics, ILfLor all, had discrediLed tllia noLion of diviHiltilit.y by showing L1mt it WlLa solf-contradictory. (Henco in Eleatio philosophy 'Beillg' wns ono Ilond indivisiblo; thore WILa no l,lurality lIor Imy diviHibility.) Tho situat.ion wus comploLoly difforcnt, howovor, flllW L1mt. t.ho Eleatic One, which existed only in the mind, hud booll mult.iplied to form a now and abstr.a.ct 'plurality'. This made it Ileccsfmry t.u tnlw a fresh loolt ILt the problem of divi8ibility. It appears L1ULt tho investigation of Lhis pmblem becamo the contrILl and fundamentlLI concern of Greek arithmot.ic in ita earliest stn.ge of development. The so· called theory of even ILnd odd, which is the oorliest pie(:e of III~dllctivo IIl1Lthomatics Imown to US,IGO sooms La havo arisen from tho 1'1"1/1110111 of division, or from the problom of Imlving. Tho simplest killd of divi8ion, of COUl'80, is division into two p,nts. 'l'he Greek word 1~lIltl!r.iI', which in ILritiunoLie moant 'La halvo', WILS usod in the ancionL philOlml'hiclLllcrminology of tho Elealics to exp1'C88 any kind of decomI'osit.ioll or division. 1'lLrmeniues, for OXlLlIlplo, writes in his poom,IOI 1II11~F. I~/IIleETOV idnv (NOI' is Heing uivi8ible), by which ho meall8 I.h;Ltlh·ing CILnnot bo split up in any way. There is Iittlo doubt that. aL firsL ~tal(ltiv simply meant. "0 divide' or '10 deCOmlJose' .. t.ho SpOCilLl lIIelLllilig which iL aC(luired in ILriLhmetic (to Iwlve) devoloped from IhiH earlier ILIld moro geneml one. It. is Hignificant. thlLt Euclid, when Ill! UHeH t.he vorb ()LUI(!£iv in his definit.ions (VII.6 and 7), only finds it. lIel~(,SHI~ry Lo I~dd tho 'llllLlifying ILdVOI'b MXa (twofold). On Lite oLher hand. J'hLLo (Laws 8060), who wroto 80Illewhat ell.rlior, dofines an I~\'ell Illllllhm· liS neno!; den?"fl!; ff dLfJleOV/IEJ'Ol; til; ida ,)vo /lleTJ (IL Jlumber di\'iliihll! illt.., t.wo equill pl\rla). Tho latter's ehoice of words inuielll.cR I IlILL t5'flle£iv, when it fimt camo La he uscd inlLrithmetic, was lL880cilLted with il.R l~leILt.ic meaning and hn.d to bo (1lIlLlifiod in (I. preci80 Wily. Lat.I'" 1111, hllwover, Llao lLI'it.hmotical IIS0 of this verb WILa confined t.o t.ho lIwllry or ovon {LOU odd; ILa I~ result. of this it no longer needed such I'rcciHe qlllllificILt.ion, ILnd t.he 1"lverb ("Xu WILR t.hought to bo 8uffioif·nL. Tho flt.ymological col1sidomtions in tho proceding pamgnLph clearly imlil'lLlo t.hILL it. t.ook SolllO t.imo hoforo t.ho ILrithmot.icnl problom of

I'AllT 3. TUB

COHH"rllU(;TIOH

01' )'ATIIY.hlATI~

267

which numbers cnn be halved WILS sOlllLruLcd frolll tho geneml probl(,111 (If 'uiviRibilit.y', which the ElelLtics had rormulaLed for Being (the Ono). Wo Imow from DefinitioJls VJI.6 ILlld 7 tlmt Lho PYtlULgOrCILn L1wury or CVCJI. "lid odd parLitiullCfl I1l1mlt"r8 inLo t.wo groups: t.huHe whid. cUIII" he Imlvcd (evon III1I11IJCrs) nlld t.hose which (:ould 11111, (mid OIl(,.H). II iHturit:ILlly RIIf'ILltillg, thili IlllLrlwd Lho first Htep of ILIt investigat.ion into tho divisibilit.y pl'operLiClt or numbers (i.o. mult.iples or ILn indivisiblo unit.). Tho distinct.ion between oven and odd numbers ult.inliLtcly led to ,\ theory of t.holll. It is clen.r that t.ho problem of divisibility also suggested t.he (liRtinction het.wccn 1,rime nnd composite nllmlltlrs. MaLhematicialls mllst. (1lIicldy hlLve renlized t.hat. ove~ry numLel' c:ould he divided by t.ho IInit.; thi8 WI'-'! gUILrnnt.ccd hy t.he dofiniLion of number. They wcre thon Icd to eliBt.ingllish bet.ween !.hosn nllmbers which wcro divisihlo mtly lIy t.Iw unit ILnd t.hoso whieh woro nllt. (i.o. t.hose which were c1i"iKihlc Ity mIllie other nUlllhcr IL'i wc\II). The furmer were cILlleci 'prime' (Delillitiull Vll,12), ILIIII th" liLU.cr we ..e clLllcd '"on-primc' 01' 't'"mpusilo' (Dofinition VJ1.1.1). The prohlem of divisihilit.y ILIKO liCIt hehilld HIICh definitiun8 u.s: Dr/initiorL VII.3: "A numbet· iH a 1'1Lrt. or II. number (,"eOl;) , t.ho leM of LIto grelLLor, whcn it men.surCK t.ho grtlILt.4lI·"; a"d lJe/initiOl& VIlA: "hilt pinta ("leTJ) when it. c1ell'R n"t IIWlUillre it.". Let me mellt.ion ill l'U8Sillg t.Il1Lt. I'Hed.. , lIw mU8t. import.lLnt. exprtlHHiun in L1WAo definition8 (V11.12, 14, :1 ILIIII
•

I'. SCI!

The J;L'It. t.hreo clml'Lors wero intetlllcd tu Hhllw L1ULt. tho fOllneIILtions or ILI·it.hlllllt.ic which aro "Lid clown in Book V11 of t.ho 1~le"umI8 cll.n bo regarded as lion exLonAion of Eleatic l'(lILching. \Ve now wanL to tllrn 0111' (Lt.tention La another grollp of 1~IIc1id'A mllothollllLtical prin-

III

ciples.

n. 12 "hove. Dil·IR I1l1el I{mll?. FraUlJlclI'c, 1.28b, rrugment. 8.22.

20M

3.17

'rUE I'nODLEll1 OF 'I'JlE 'A 1'1'RlIIA'l'A ,

o lie of our major concerns in the present invcsl:.igu,t.ion ia 1,0 discover (~xlL(lLly

\

l'A1tT 3. 1'UIC

117.A1II"I: TII~ IlImlNNJNOS OF OIUmK lIATIII~lIATIC9

what signi6clLllce should bo attached to tho fn.ct that tho nmthcnmliclLI principles at the beginning of Ble11lenla Book I are grouped under throe headings, o.nd to give II. historical explaLnAtion of huw this threofold division co.mo into being. Up to now wo have only bccn ILble to throw 80me inciden .....1 light on theso questions. WI! alrcluly Imow something About how the tmO{)/C1Ut; or dexat, tho foullIIILLiulls of nmthell1atics, wore discovcred. Furthermore, wo Imve slIlTcclicd in cllLrifying to Romo extent the problems connected with t.11I! (mrlil~HL mlLthmlllLticlLI elofinitions (lleOl or simply tmOD/C1W;), ('sl'ecilLJly the arithmeticlLI ones. We hAVO ILB yet, however, hardly tlluched upon tho much more dimcult problom of I~uclid's alnJpaTa (pnstlliala) and J(ou'al lVVO'Rl (comnUtne8 animi conceptione8). Let UR firHL Lo.ko u l' the mll.tter of tho so-called postulo.tes (alT~paTa). At the moment all we know is that the wonl afT'7pa, ILB 0. mo.themAti(':d t.(:rlll, ILPPOArs to ho.vo heen merely II. synonym for tmd{)£C1't;. (This J'('lIl1Lrl( docR not Apply to li!uclid's USILgO, which I propose to ignore fIJI' Lim time heing.) I hl1.Ve ILlready mentioned thAt in ono of his writillgs Archimcdes 1II0rn LImn once uses the wurd vnoTl{)EC10ru. (in tho flll'lIl {monOl,u()u) to inLl'Otiuco defiuitiom;; IGZ in another of his wori(S, tllll principles of fluid mechanics 11.1'0 introduced by the expression l~nm,cfC10(l/ (let it· bo (.",kcn ILB 0. btulis),IG3 On the other hand, similar prilll:il'lcs of mcclllLnics aro introduced in a third worl( by tho word uiTll1i,IEI?(f.IG4 Proclus comments on this I1LRt plL8lJR.ge that j\rchimedes WlluldlllL v(~ du lie lIcUor to speak oC' axioms' , 165 ILl though A rell.mcdes himSdr,lLftcr Iillting the principlc8 in qucstion, refers to them ILB ToVT(O" Uno"""lh(O)·,1GII We cl~n infOi' frolll this thAt Archimedcs did not distinguish hdWC(!1I Lho m:LthcmlLtical terms imd{)£C1,t;, tmoJ(E1flE"O" And a'iTf/pa. Tho u'nn aiTf//lfl (II. 'request' or 'dmnl\fHl'; in l'(Ltin 1'oslulal-unl), just Ijlll! ilK sYlionyms hypnlluu;i8 ILud "ypokcimclIo", is obviously of di:~-

CONlITllUl."fION

OV at.\TIlt-:MA'I''':9

IccticA! origin. It comes from tho verb alTlw ('to Mil, to requesL or to domand). Wo Imow tho.t one of the po.rticiplI,lIta in a diAlectical deb:Lte W88 obliged to start out by InILldug II, vory iml'ul'tlmt request. SoerILtcs, in the 1'888o.g0 from the Jleno discu88ed previously (86e3), boo to request thAt he be o.llowed to U80 the hypothtltiCl~1 method of relLRoning. Tho words which ho U8C8 110m "allow file ((ftI)'XJJ(!'1dOV), in considering whether or not it OAn be taught, to mAlto use of II. "Yl'0lltl'.8i8". (A joint investiglLtion could not he hU.8ed on an IL'l.~umption or l,ypolhe8i., unlcss botl, 1'ILI·ticil'".nts ILgreed to it. llellco olle (If them IlII.d tu a,'1k for t.ho Ilgrcement of l hn other.) Dialectic:~1 (lelllLtcs lLl'e full of such exprCII.'\ioIlR 11.8 'a,'ik', 'rlernmul', 'reqrrl'.'1t' I~IIC I evoll '/(1 kc' (Aallflu,·F.CI') on Um IIIlC siele, ILnt! 'give', 'allollJ' lLml 'disaliol/J' nil the OLlIO .... G7 An alnlpa al'l'elLl'K to IIIWO lmen SUllie ltiml uf initilLI 1'1'II1'osition which (lnf\ pl~rtieiplLl1f, in 0. diaLIecticlLI ILrgulIlent dcmrmrled that the othcr Iloilo\\,. 11'01' this rcason, such propositioJls were clLllet! dcnuzmb (po.91ulflta). If the demand W88 conceded, i.e. if the second plI.~ticipILnl:. ILcl.'Cpted UtlLt tho sLutemcnt being put forwa ..d WII.8 corn: ct , It ~clLllt tIlILt both pArtners wore in Agreement !LIIOUt the stlLrtlllg pOInt of thoir dobn.Le. In RllCh CMCS tho stll.tcment upon which they ILgrtlcd could ILlso bo clLlled a ollOM'Yf/pa or II. v1UiOEC1'C;. This explaLinR why Art:himodlm WILR allill tu URO ainll,a (or the verb alrovpE{)a in his l1\:LthematiclL! writings o.s a synonym for MO{)tC111; or {moJ(EIC1{)ro.) It mAy ovon h",,'o been the CILBO thAt in WILlectic the wo ..d urn/pa WILR used for the most part to denote An 888umption. Yet the mere flLCI:. that Euclid chose t.o CI~lI It. Rl'o(:iAI group of ul1proved principl(!R alT~pa~a leuds one to RIISl'Cllt tllII.t this nn.me, ILL Ihnat originally, W88 intended to pick out pl'inciples which had SOJll~ di.91in.gu.i,'1I,inf/ fealure. A complLrison hetween tho JlILmCS o/,n).o)"Ipam (Lhose things upon which there is ILgrecment) lionel aln/f,aTu (dentlLnds) immediately reveAls II. difference bctween them; 01,oM"'1/lf1 seolUs to strC88 LllII.t tho two 1'ILrtners IIILYU rOlu:hCll It true (fyreemellt, .., ('.ulIsicl.. r, ror nXlUllplCl, tho IIlUlorlinC!CI \VunlH ill ..lin rnllCl\ving

.., 'nn .:clIIui.liIIl1S .... IIpJlluu·oitlilms', OpBrfl Olllnia (otl. lloiborg, Lipairto 1010-lu) I, pp. 2481\nc\ 262. 113 'Do 1~C)rJloribus f1uil.nnt.iblls', Opera (ac!O II. 162 above), II 318. '61 'n" ..1"IIClrnlll ""'IIImtJriis', O]lt!m (81'(\ II. 162 utJovo), II 124. 'u I'roehm, (Ji'riCllllnin, J.ipsin" 187:1), 181.171T. '" (II""'" (R"" II. 11i2 "Ionvn), 11.1211.

2(iO

1'"I11UIJ;C"';

l'/uudo IOOb6: dnoOipnol; d,aI n HuAdv odrd HuO'ourd ••• a d 1'0' (1ldwc ~E Hal rrlJ)'xcb(!e,c elva, Taura • • ,'AAAd "",_ bp,} 6 la{l,}I;. w, (11('6vro, 00' odx c'lv .,,00MI, nrealrwr, nnd ArillLol.ln, P11!l1J1ca ZIl.231lb30 (Um Ruhjne!. \lllIlnr cliRulIlVlinn is 0110 or ZOllo'8 urgUIIIOIlt.e): 6n " 01C1T~ ~/,l.r'l 101ll"t:F· oU/,lIalrf;' di no"d Td AII/'lIdl'clP rdl' xedvoll oll)'JtdoOol be nuv vi/II. /'~ ",,'0/11.1'011 yd(! fOuroll. od" lara, d auA.tu),la/,d,.

270

wllercll.'1 air'll'a indicates merely that something hu.s Leen demanded Ity Clnc of them. Therefore it seems possihle tho.t the word a l'OJpa , wllcn it was first usod Q8 a dialectical term, wn.s intended to signify 1\ t/(!IIltt.lul mnde by one of the pn.rtncrs mther tlllLn ILn agreement (opolo} " " I(1 ) I'CILchl'11 by hoth (If tllem. In flLCL, this is exactly thc diffcrcnce which lL})pears to exist, undcr 1'l'rLain circunlst;"nccs at least, betwecn the meaning of alT1]pa 011 till! olle huml ILnd that of v1f(Uh;uu; or OltoMY1]lla on the oLher, when t.Il1'RI! \\'(JI'(Ja n.re considered 1\8 dinlectical terms. J>roclus, for exo.mple, ~i\'cll Lhe following o.ccount of Aristotle's opinion lC8 ILLout this 8uh1GO jet:L: "Whenever, on the other I""nd, the RLtLtement is unlmown ILlld IllwcrthdCM is taken M Lrue withOld lice slude,r.l's conceding it, then IIl~ (i.e. Aristotle) &LY8, we call it a postu/flle (alT'lpa): for example, I haL nil righL nngles al'e equnl." It. hILS correctly Leen rcmn.t·I(cd tJmt A l'iHloUc ia not LILllting n.Lout 1IUlllumwlicnl 11ostullltC.Y in the pl\8liago 10 whieh ProcluB (slIJncwhat i"ar.curalcly) refcrsYO ProduR iR wrong (0 !'ite I~uclid's fourLh postullLLe n.s l\n example of what Aristotle hlLtt in lIIind. The IILtter intended to define afT'1Ila n.s a dialecticlLI te..m I\llel not 1l.H IL mlLthonmLicll.l one. 'l'his need not concern UR ILL Lhe moment, however, since our present gOILI is simply to obtILin IL geneml underHI.ILIIlling uf wlm.t. afT'lpa meant in diILlectic. It'rom Proc:lus' words ~t ILppears tho.t Al'istotlo would uphold the following vicw: In the terminology of diILlectic an aft1]pa could, under l'l~rtain cin~urnsLtLIlCC8, refer to an I\88Ortion which ono partner in a dl!lllLlo WILli Lcd to lI.ssume (whoso ILccel'tanco ho dcmmulcd), LuI. which (hc other partner refuHed to concede. It is Lrue thaI. J>rocluH spel\ks of IL 'Htudcnt' and II. 'telLcher', but this fact is of Iittlo significance. Tho I'dlLt.ionship between student and tcn.cher is obviously tho so.me 1\8 the one which holds betwccn two persons engage(l in a dilLlecticlLI dehlLtc. Tilt' f)lI(m(.ion which now prcsente itself is whother tho meaning of lIi"lIlU ill dialectic custs any light on tho hiRtoricIL) prohlem of l~uclid'H ''''I'h"

I'AIIT B. TIIR CONH'fILUt-TIfIN 01' .IA"III~MA1·JCS

R7.AII(\: Till>, m\fJINNINI/R Ol' OIlImK t.IATlllmATI(:H

11I18l1llgu

I. 10.7Gb 27-:14.

which )'roclus hod in mind

WIIB

ubviuuuly l'ollterior AnalytiCll

!Co I'rudlJlI (l'cI. Fril'llIoin), 76, 17ft. 'fhll t.rnllnlut.iun ill hy Olulln It. )\(orro", ("rilll·"I,OIl 11I7().

170 IC vOIl}<'riL7., 01'. cit.. (n. 2 abov,,), p. 42 ",llIlro ho reft\I'B to p. 640 of Arilltotle'll l ,r ;(Jr mid 1){)8/er;flr A'IfII!ltiCll by D. 1t0fl8 (Oxford 111411).

271

alT~J.laTa.

Perhaps theso postulatcs should lJll illLerprcLc(l as mILthemo.tical 'demands' which somo plLrticipants ill a scientific discussion were unwilling to accept immediately nnd wit.hout reservation. We will only be aLle to answer this question by taldng a c1osor look at the ain/PflTa t.hCfIIRCIVeR. Fimt, howcver, we IHust considtlr two uther mo.tlcrs; one is the treatment which Euclid's postulates have received from previous historiILns, and the other is the question of how they arc to be dl\tcd. Theae will be dealt with ill the next two clmptcra, after which \~e will return to our original (IUt'flLion.

Euclid's five ainipflTa relU:) n.s follows:

j

"Let Lhe ftlllowing lIo posLuln.Lcd: 1. 'fo drn.w II. straight line from any point to ILlly point. 2. To produce II, fhfite straight Iino continuuusly in II. straighL line. 3. '1.'0 describe II. cil'cle with I\ny conter !LIlll distl~IlCO. 4. That nil right ILnglea aro equnl to one ILIlother. 5. TImt, if a stmight line flLlling Oil two RLraight lilies ml"l(e the interior anglcs 011 the sl~me side Icss tlmll two right I\ngles, tho two strnight linea, if produced indefinitely, meet on t1mL side on which arc tho anglC',s less t1llLn tho t.wo righL ILIl~lefl." Sinco t.IlO apl'C'ILr;~nco (If Zcuthon'8 pioneering worlc, there IU\8 been littlo diso.greemcnt I\mongst scholt"rs "bout thc rnClLning anti significance of tl\(,'so pnst.ulates. 17l DecltCr, for eXILmplc, describes them in the following wn.y:l72 ,/ 171 H. G ..Zouthen, 'Dill 8l'OllllltriRCiao ]{onRt.rukt.ion ula Rxillt.cn7.hmvlliR', lolat/,. A,m. ~7(i861l), PI'. 222-8. Tn hill Gl'.'l·/li4'J.Je I/er MallumUJliJ: jill Altertl,,,. MiUelalur (J{0(lflnhu8un 1806), 7.lIlIl.hon ",rot" (Oil p. 120) t.hat, sinno tho prohlol1ll1 flf t.ho uncionUt wllrn in l'AAUIl(:n JlrllJlOIIiUons aoollt oxiAt.cncn and ~"ir soiuLiollH wero proofs of UxiRt.cIll~n, t.ho ,t08t.lllut,,'8 weru fUlIertioru abollt the e:rilllence 0/ certain /orm", whoso notmptnnco WIUI dcmAnded wi~h~)Ut. proof.

,,"41

ViIlW, 1100 A. Fmjt·nn, 'Rllr I" Rif,:lIilil:nt.iom a11'H pllftLlllnl./t I',mli· di~ilB'.' ',Arc},iv.;6 Ital6rtiiltiClluJ/ell d'IU.,'oire ,1M Sdl!llrr..• (1961) I'P. :IR2-112, " ... 1 La maletnntim tUl tnondo IIntVo, Ronlll 1I1IH, Jlp. 02-7. .71 O. Hodmr, DII" ltfIlI/ICf;lalilfrJ.e J)llrIken flu AtatiA-1! «(lnt.l.inRI'1I 1967), ('. 19.

For u dlftnnlnt.

272

51-Alit): TIII( JlY.OlNNINOS 01.1 OIml~K MATIIJ(MA1'ICS

"The PoStuullC8 serve Lo gUinanteo the existence of certain basic forms, nl!om~ly~'!!:!g!!J.l.iJ!-,,-~. circles and thyir ,Points of intersoct~on. Ollt of which tbe other figures are built up in a constructive manner. Thc famou,s.Ji(tii post~late, fo~ e~ample, guarantees tho existenco . I of n. poink~~ whi~ .tf.vo convcrgent Iin~ will mecL." Howcver,. tl~el'e i~, 1I10re disagreement about tho dating of theso postulates/ n.:~nner:y!)for example, started out by observing that Aristotle scems not to have known about mathematical postulates alld to havc I'egarded ail'1pa exclusively 1\8 a dialecticll.1 term. lIe concluded from this that Euclid should be given credit for the first three pOHtullLLes at leo.st.173 nut when it came to the fourt~ and fifth ones 'I'ILllIlcry conjectured that they did not originate with Euclid and argaed somewhat surprisingly that they were unworthyofthe author of tho Elemellt". m Heath, on tho other hand, was of II. different opinion: "I t.hilllt that tho great postulate 6 is due to Euclid himself, and it80ems prolllLble thaL Postula~ 4 is also his, if not Postuln.tes 1-3 as well". 176 LILt.cr schoilLl'S, although they may di8tLgree with Tannery on some points of deLILit, seom hy and large to accept his contention that Euclid should be given credit for at least some of the postuln.tes. lIOn'lIllLllII is nn exception: in hi8 history of mlLthelllatica he rClmLrlts Lhll.t the postuln.t.es may be regn.rded as an important methodologiCiLI cOIII.rilmtion mude by Euclid himself.I7O It. is only ill very recent times that noticeable efforts have hcel1 mado to Lrace theso postulates bacle to the pro-Euclidean period. Von 1~I'iLy., fol' example, hu.a dl,imed on tho basis of a pn.ssn.go in Proclus (I" 179 in Fricdlein's edition) that the constructive postUlates (2-3) weru illLl'Oduccd by Spausippus, IL pupil of Plato. 177 I do not o.ccopt Lhis clIlljeetUI'O, nor oan I find anything in Pl'oclus to support. it;n8 IIcvnrLhelcss I think it noteworthy that such claims havo been made. m 1', 'J'ullnnry, Mimoire8 scicnti/iquc8 2, 48-63, A. lrronldon, op. eiL" p. 16 (n.· 67 nbovo), p. 16; BUO alBo C. 1"mor, J>ie /t:le"umle von Euclid, p. 22 1'1,£, L. JIeat.h, A Ili8tonJ o/Greek MatllfJJllalicB (Oxford 1921), Vol. I, p, 376. Ill.r. ]1}. HolTmon, 'Oeuohichto dor Mat.homuLik, Part I', Sammlung Goscllcn, (Bol'lin) %26 (1963), 32. m K. von Fritz, op. cit.. (n. 2 abovo), p. 97. 178 cr, 0, Boclwr'B papor in Arcltiv fur BeuriOBU88clticllle 4 (1969), 213.

.,. cr.

I'AIIT :to 'l'UH (:ONII'l'IIIJ(;TlCIN 01"

MATIII'!AIATI~

273

'1'horo havo also bcon other attempts t.o cln.l'ify tho prehist.ol·y of Euclid's postulates. necker, for examplo, has succeeded in reconstructing what could very well be an early form of tho fifth postuln.Le (and hence uf th~ fourth 118 well).178 Not even the most plausible attempt of Lhis 8ort, however, is sufficient to establish that postUltLtes were \lscd in pl'e-J!;uclidelLn timcs. '1'his must. be tn.lcen for granted, i.e. one Illust at the very least n.ssume that tho kind of mathemu.ticu.l principle which Euclid calls II. postuln.te WIl8 known in pre-Euclidean times, if such reconstructions arc to lmve any hist.oricn.l interest. 1 would now 1iI(C to show that such n.n I\8sumption can he justified. In what follows I shall attempt to explain how postulates originated. Euclid's fourth u.nd fifth ainlpa will not be discussed at all, since they form 11. separate tOllic by themselves. Instead I wn.nt t.o concentrate on his first three, the postulates which den.l with const.ructions, and address myself to the question of their age. /

~i'l ~ TUB CONS'I'RUCTIONS OF OENOPlJlES /

\

In my opinion, an answer to the question mentioned 11.1, the end of the last clmpter hn.II been ILVlLilable for quito 80me timo. Since it, ha..'i gone almost unnoticed, however, the facts l'ertlLining to it. noed perhaps t.o be stated in a 0101'0 porspicuou8 and precise mn.nner. It will be recnllcd that Uecker wrot.o tlmt Euclid's l'ostuln.t.ca were dOHigncd to gUlLl'ILnLee tho oxistence of certain bn.'iic forms froll1 which the other figureR could bo constructod. 'rho plL.'lSII.ge in which ho mlLde this statement continues ILa follows: 1BO '''fhe constructivo viow of mu.tlioll1lLticILI oxistenco goes hacle to \ Ounopidt'-8 in tho fifth cent.ury. He WIl.li t.he firat. }lomon to carry out clmncntlLry constrl\ctions, like thlLf, of the porpendicular to 11. line, using only n. ruler and COIllPn.ssCS. It ll.ppOILI'H that. in the sohool of 1'1ILto the uso of any other insLrumolltH WILS avoided whenever l'u88iblo."

In

180

18

Ibill., Jlp. 212-8 •. 0, Bllci(or, J)(18 1Il(1t11p.m,rti.,r./I/l Szab6

/)clIl.:ml tlf!r

A .llike, pJI. 'lilT,

I,

274

I

I'

S7.An(): Til I': III'.OINNIN08 01' Olll(l~K MATIIIUlATICS

Tho prehistory of l~uclid's post.uIILt.cs cannot. bo discu88ed sILt.isflLct.orily wit.hout. making Bome referenco to Oonopides. His name is mont.ioned t.wice in Proclus' comment.ILry 011 ]~uclid. Tho t.wo plL88ngcs in which it OCCUI'8 deal with ccrtlLin conRt.ruct.ions ILnd aro of Btlch importanco t.hlLt. t.hey ILre hoth quoted below. One of Euclid's problems I"OI"IH: Elcmcn/.9 1.12: """0 I" given infinite straight. line, from IL given point. which is not. on it, t.o draw a perpendicular st.raight. line." Proclus mlLkes tho fullowing remarlt about. this proposit.ion:1 81 "This problem WlLB fil'8t. invest.igaterl by Oenopides, who t.hought. it \U~cful in astronomy. In archaic fashion, howevor, he calls the perpendiculn.r a Iino drown 'Ullomol1wisc' because LlIO gnomon also is at right. ILnglCA lo lho horizon." (rho pointer on n. sundilLI was clLlletl :L (/ IWII/Oll. )

The other plL8,~I\ge whioh COflcernH Oonol'ides is IL commenL on

Propositioll 1.23: "On IL given sLraight. lino ILnd at. n. point on it. t.o I:

.ij \\

Ii

"I I!

eOllstruet a recLilinel1l anglo eqlllLl t.o IL gi veu l"Octilinenl IUlgle." It rUlIs lL'I fo11owS: I82 "~t'hi8 too is 0. problom, and tho credit. for its diRcovery belongs mther t.o Oonopides, 1\8 Eudomus tells us." Buth thcso cOllstruct.ions ""'0 RO simplo that. iL is difficult ILl. fil'8t to understand why Proclus bothered to mention the name of their dilicoverer. Geometry WI\R no longer in its inflLucy at t.he timo of Oenopides (n.round440 D.O.); it. can sClucely be claimed tltlLt t.hese simple const.ructions woro unknown unt.il thon. l83 Oenopides, after ILII, WIIS no more than IL fow yen.1'8 older than Hippocrates of Chios, n.nd Hippocrates' geomotry WILa n. highly dovcloped scicnce, remnrl""blo not only for t.he number of theorems which it containcd but also for the rigour and precision with which t.hese were proved. lIt Therefore it is c10lLr Umt Proclus' words reqllire a difforent interpret.n.tion. There is one to be found in t.ho worlts of Hcu.th, whose opinion ILbout thili 1I11Lttcr elLn (,0 summed lip in thrco flcntencca. Alt.hough eac:h ono

.1.

I'rodllll (",I. FriCllluill) 2ft:l. 7R. 'I'lfI trnllHlnl.ion iH 1Iy Olt'rut n. Murrow (Princeton I !l70). m Ihicl., :1:1:1, Ii-Il . •u ,\. n. RI.... I", 'Ob"r ,Ii.. Ile,lIn Villi l.il'l,nl 111111 l.hlHnl ill ,I!lr griUl!hiHdu," MuLhcIIIILW,'. Quelle" mul 81udien •.. n, 3 (10:16), 287R., 304. II. n. r•. Villi "(lr \\Trum.lnll, Erlllrrr./Ilmlle lI'iltltClI8Clllllt, pro 21 :1-4.

I'AIIT 3.

TIIJ~

c:ONII1·UllC.'1·JUN Ol' MATJlKMATJC:S

275

of Lhem 'IILYS milch tho SILI1IO thing ILR the otherR, all t.hree are worth quoLing: 11IS (rt) "Tho point. mlLy ho thlLt. he (Oen0l'idcs) WILli t.ho first. t.o solvo Lhe problem by UlfflJl.9 o/Ille rider (mil COlnpIM.'ic.'l only." {b} "Oenol'iclcs mlLy IlILvc been (.110 first. t.o givc Lho Umorctical, mthl:1' LImn t.lIC l'rlLcLil::LI, eonfltructilln fol' UIC problem." ((') "It. mlLy thcrefol'll be t.hlLt. Oellupidt's' significlLIlcc IlLy ill illll'rllvcllIenlH (If met.hod fl'Qm t.he point of vicw of t.heory." ] finll t.hCHC conjcc:tm'cH vOl'y l'hLllllihlc. The only commcnL which J wOlllel Iilw \.0 lIuLIClI CUllC:ClrnS t.he phmso "by m.ealM 0/ ruler aJul com!m.',ses (lilly". Pro<:lus ducs nut mcmlilUl these instl·ument.s in t.ho plLR8agl'.8 eluou'li ILlmvc. It. might therefut'e he thought Llmt (a) is mUlcr IL flLllciful l:onjccture. III my opinion, huwever, t.his is not t.ho I!MIl; 1Ienl.h'lI II..'UltlrLillll iii IJllito jUHlifi(lcl, 1L1\.IlIIlIgh Llw mlLnllcr' in whit:h it iii cXI'I'l'8,QctI I'IllJuires ROIllO eXI'IILlu~liuJl. Thcm is 1111 Iiou hI. II\ltt the: I'rohlmllH (I;:Ip.menl.~ 1.12 ILnd 2:1) whic:h OI:IIol'idcfI ill 14II1'1'0smi to Imv., Holv('11 nro intere~ting rntl!! from rt. lheort!tir.tlllJOillt 0/ view. Pmctir.al solutions U) t.hem wero ({1I0WIl frolll Lime immelllorial. 1i'lIrt.llf~rlllore, rulm'H I~nd compns.~cs were used hy (:mrtllllll:n 1lIIl,(Ll1fol'O tho t.illlil (If Olmlll'i(les. Tho pllrposo of l~uclicl'H two propositions WIUI theoretical, noL J11'11.cticILI, ill othel' words, t.hey wero inLcncled to demollstmlc how the pmblOlns in question could be solvcd OIt the busi.'l o/llte firsi/Itree pmlllliu/e.'l. POIlt.uhLtCH 1-:) nre clJuivILlcnL in t.hcmy to ILllowing tho ulle of ruler ILIIII (!OIllPIIAIIt·M. 18lI A rlller ('/lILlJlclS Olll! 'UI elmw u. straight Iino from ILlly point. Lu .ILlly llUillL' ILIIII 'to l'T11lh/{:e IL finite stmighL Iinc conLinuously in IL lIt.raight lino'. SimillLrly it is eMY 'to describe IL circlo wiLh nny cent.ro lLIul disl""nc<:' by using a plLir of compn&.'IC8. Thlls to HlLy thnt. OI'llIIpid('R 8olvl'cI the prohlcm 'by menns of lho I'uh'r and COJllP"SSOS' is just. to say Llmt. hc soh'ed it by making conscious WIll 0/ Euclid'[t {irat tllree 11O,'1tultrlr.'l. 11111('1'11, he rnlLy ovon havo Illmn t.IlO originator of these }losLuhLtc:S, if he renll y WIL8 t.he fil'8L person t.o give t.hcoreLical const.ructions for PrupolliLions 1.12 nnd 1.23. This is

'.1,..

r •. n""I.h, A lIilll"r!1 "I Greek JII"tI"lIItrr.tir.1l (Odo ..,1 1021) VIII I, .,. 171i; .. r. 1"nlj"H"'N JI"I'"'' ill Arr/ai",r.d" n (1!1Ii7), 1'1'. 2RIi-O·1. I " cr. J. R. HUrnll"'", '(l''fIuhid,LIl ,I,· .. JlfIL"'"mllltik'. ParI. I, 8rrllllllll1ll9

008r./'CII (nurlin) 22G (101i:1). :12.

276

I'I1.AJlO: TilE 1180lNNIH08 01" OllKll,K MATIJRMATICS

t.he only sellsible interpretat.ion of Heath's conjecture that "Oenopidcs' significlUlce IlLy in improvements of method from t.he point. of view of t.hoory". A nyono who disagrees wiLh t.he conclusions roached above ILud hcli(w(!fI LllILt. Post.uln,\;cs 1-3 ILre of more recent origin must rojoct. cVl'I'yLhing which Proclus tells us about. Oenopidcs as mistaken. I L IULrlUy scems pOB.'Iible to find a satisfactory explanat.ion of Proclus' words which is not along the Iincs suggested by HCILth.

3.20

1'1I~ FIn.Q'I' 'I'URRR P081'ULATRS IN TUR 'RLEAIEN1'S'

In view of wlmt. we learncd in the preceding clULp\;cr, it. 8eOlilS rca.'1C1nlLhle t.o conject.ure t.hat. Euclid's first. t.hree post.ulates originated ill Lhc fifth ccnLury and t.hll.t their author mlLy have been Oenopidcs. The l!1~rline88 of this dILLe should not surprise ua. We shall seo in I~ subsc(}uent clmp\;cr Lhat Euclid's so·called "otvai lVJ1O,a, (cornmrme.~ /II~; 11/ i c:onccl'tiones) l~ro no le88 ancient.. LeL \IS now t.urn to t.ho qucsLion of why it WIUI Lhought neCOBHILI'y 10 Illy down postulaLC8 of uny kind. It scorns to me that t.he WILy ill which t.his qUO-'It.ion IlILs been tacklod in t.he past is not altogoUlO1' slLtisflLctory. Wo n.ro told quito correctly t.hat in ancient geomotry t.he only figures which (.'ould be snid to exi8l were t.hose obtained in lL l·on81.nlcLivo nmnllor, ILl III Lhn,t t.ho posLlllat.es woro intended t.1I gllamnLeo Lhe existence of curLain bu.'4ic forms (sLmight. linus, cil'c1us al\(I points of intersect.ion). Howevor, these facts do not. explain how Lhifl eOllcopt.ion of mll,LhemaLiclL1 exiatence arose, nor do they hell' us to understand why tho unproved statemonts which guaranteod (!xilltmlcu were called alnlllaTa. Wo know that tho torm alT1l1"a 011.1110 frum dilLlecLic where it WlLR used to denote a 'domand' about which thn second pn.rttlcr in a dialogue had reaervaliofUJ. LeL us see whother thoro iH I~ny connection botween this early moaning of the word and Nlldid'H P()fItUhLW.A. At. first gllLllCO, Postul"w.8 1-3 appear to bo Bueh simple, solf-evidonL ILnd C1LE1i1y fulfilled 'domandH' that one is tcmpted to disregard t.ho 1iI.'ml IIl1!lmillg (lr t.hnil· IIILIIIU, 'I'hnl'O iH Ill) HII1I80 ill illHiHt.ing thILt. t.lm l'oHtulll.lt!8 I~ro 'domandH' or 'Iuvmmpt.ions' of tho kind dcscribed ILboVlI jIlRI. hlll!ILIiHO I.hc~ir 1I1~1II0 WllA 1I1I1Ic1 to (~CJllvoy t.Ilifl mcn,ning IIlIclnl'

!'AIIT :I. TIIK (!()HHTIlU(!TIOH lit' MATIIRalATICR

277

ccrt.ain cireumstanccs in tlialectie. Nonet.helcss Lhero are, in my opin· ion, good rOll80nlj for thinlting tlmt the postula\;cs wero alTtl/laTa in thoorigirml dilLloctiefLl sense of the word. In order to he ILhlo 1.0 jUHLify Lhis chl.im, 1 musL bogin by quoting a plUJ.'1ILge from ProclulI' (:orn IIIcntm'y CIII ]~udid: "The dmwing of II. line from any lJOint to ILny point (ollowfl frolll tho concept.ion of tho line as the {lowifl(/ of a point and of the straight lillo ILfI its uniform and undevitLting {lowing. Fur if we Lhillit of Um point ILfI moving uniformly over tho I:Ihor\.cst. JlILLh, wo almll come to Lho oLluw poinL n.nel 1:10 Hlmll havo goL Lho first posLuh~tc wiLhnuL I~ny complicn,Leti l'rocCSH uf thuughL. AIIlI if we tnlle 0. sLrlLighL Iitle ILEI limited by n. poillt. and simil:nly imagine its extr~mity I~ mOlJing IInifurmly over tho short.cst ruuLe, the 6Ccolld post.ullLte will Imve been estlLblished by IL simplo and flwile reOect.ion. And if we thillk of n, finite line lUI hUoVing one ext.remil.y statiorULry ILllli the ot.her oxtremity 1rwlJing "bout this stlLLionlLry point, we slmll have pro· duced tho third postulat.c."I87 (Tho tmnshLtion uf the nbove is by Glellll H.. Morrow, but t.he itlLlics mille.) I do nut WILnt. to rnaintnin thl~t 1'J'()c\us, in this sltor·t. exphLllaLiull of l)UIlt.UI:Lt.cs 1-3, mOlLnt t.o define 0. 'lino' 1\,,'1 'the flowi.ng 0/ a point' nnd l~ 'sLmight Iino' I~ 'ita uniform (l.1Icl 1LIldcIJialing {lowing'; Lhere lLt"ll (lel'l.:Linly no fUlCh dofiniLions to 1m founci in Euclid, At. Lho moment, I um muoh mord intcrcst.cd in tho flLCt thnt ho u8cd cerLain simple leindH of 1110vr.llumt to eXl'll~in Eucli(l's fil'at. threo lIOStuIlLL(~fI, As n. nmt.tor of fuet, t.ho 'demands' uuulo by t.Ilf'-8C postuln.Lefl cl~nnoL he BlLt.isfied witllOul1novemeul.lt is qU(,Bt.iunt~hle, however, whether' mot.ion' iLself is really as simplo and unproblemILI.i(l /1.'1 it ILppelLrS t.o be from Proclus' words. Our commenLaror WILfI dmLrly ;L\YlLre uf this diffieult.y, for ho went. on t.o SI~y:l88

1\1'0

"1f someone should ilHluire how we (!ILn inLroduco fIlo/io".s into immovablo geomot.l'iclLl ubject.s and 1//(I/7C thingH tlmt nre without •., ProduR (,.,1. Irri",Udll), I RIi.llfT. lOll

(hi,l., IR7.2lilT.

27R

111.1.1141: 1'111" IIJ~OINNINClII 01' OIlImK MA1'III~alA"'If~

purls - operati01l8 tllat are alloucl/'er impossible - wo shull ILIIlt t.ImL he be noL ILlmoyed ... nul. let us thinlt of this moticn not /1S hodily, but U8 imaginary [in the Greek text: "("la,e; 9'OVTaO'T''''i), :mel admit 1101. tlmt things withouL PlLl'ts movo with lWllily motions, !'ut rather thlLt they nre suhject to tho wn.ys of tho imugilltLtion. l"or "1I0US', though partlc8.'t, is moved, Lut. flat. spILt.iully; l~nd imagination hM its own motion corrcslxlIIdillg to ita OWII pn.rtlC8Hn(,8H. J1\ at.lending to bodily motiolls, we lose sight. of t.he Illations Llmt exist. ILmong Lhings wit.hout extemledness, etc." Luckily We do 1I0t IlILve to wOl'ry overmuch nLout interpreting Proclus' n.ttempt to I·ClI.'lRUre hiM relLdel'S. II. is suRioient for our purposeR merely to note the fu.ct that the above quotaLion, in 0. Bonsr., mn bo said to reproduce a dialectical debate. The deLate is of interest. IlcelLlIso one of (.he participants in iL ILCts lUI IL Rl'0ltesman for thuHo who oppose Lho ideo. thlLt the ltind of molion required is 'cOllceiVlLhlo'. So we soc that the unproved propositions whioh Euclid clLlled 'l'UHtuInt('s', nnd which ho regllrded U8 nCCC88ILrY fur Lhe foulldlLt.ions of geomeLry, am in Lhe eYCH of these 01'}l0I1Ont8 lUcre dema71d,'l (alnipuTa), whid. Clmn()t he accepted wi\.llOuL rl!H{WVILtion. l'rlll~hlH RCClns to Imvo Lhuught (.IIILL Lho 0PllOnonl.:t of 1II.olion in geollloLry oJJjccted only to Lhe ide:L of II, moving puint (i.e. to Lhe idClL of tllILL which hM no PILl"lS, ILnd honce no hodily existence, being in III11Lioll). lIe readily conccdes, therefore, tlmt the bodily rnoLion uf fltfll IIIltiell Ita.~ 710 body iR ineoJl(:ei vlLhle mul HUggCHts U8 an n.lternlLLi ve IIi:LI. Jloinl~ IIIUVll in ILIt 'imn.giIIlLI'Y' MLt.lUlJ' t.lllLnlL 'hullity' WILy. Of mllllllll Lhis is little moro than n.bstruse speeult~Lion on his }lILrt. and is noL to lIe Lalwn too seriously. However, wo lLre lefL wondering, whether he ha.'{ ell'nlt wit.h ILII of tho ohjcctions which were miscd in dilLlol:Lieal (Ielmlo againsL t.he idea of movement in geometry. . AN IL IImLt.eI· '~f flLct, wo know t.lmL in tho yen.rs which preceded LIm 1.11111' of OCllul'ulcs, tho put.n.tive lLut.hor of l)ost.uhLtcs 1-3 there ~IOIlriHh{:tl Il very inll.uentilLl group of lIlen who succeeded hriililLntly III Nhcnvlllg tlmL llloLwn of ILlly IlItmdiotol'y and, hOllce, inr.mu:eitYlble. Of course I o.m referring to tho Elcat.ics and, in pn.rticul:Lr, 1.0 Z(~1I0, who glwe t.he mosL incisive formulll.t.ion ofPI,rrnenides' idoIL8. 1111 I" H"" 11. :1:1 !Lllllvn.

l'AItT S, '1'111'. C:ON81'IlUUl'IIIN 01'

MA'J'lIImAl'JL~

270

H wo LUlU t.hiH in minci, it iH elLSY to llIulcl1itlmd why Buclid's first threo 1'0sLulILWs hnd t.o ho IILid down. 'I'ho ollly wn.y to mn.lto geometriclLl const.ruct.ions theoretically possible is to admit. n.t. lCMt t.hoso Idfl(ls of movement which ILre illdi8ponsn.hlo to t.he product.ion of the very simplesL gcolIUltril:ILI forllls (namely stmight lines, cirdes ILIld t.hcir points of illtcmcctioll). 1n flLct, POSt.UI:Lt.eS 1-3 glllLrantco Lhe t'.:ti.yteltce of ccrt.ILin htL.'tic geoJ)letriclLI forms; they ensure t.hn.t. it. is l'ossihlo to carry ouL the simplest cOllstruct.ions or, in oth~r wonlH, LImy require those leinds of movement which nrc ncce.'!Sary fol' t.his purpoKO. '.r!wy l'clLlly ILm del/muds (ulnipara) n.nd not agreemenls (opuloy~para); fur they postulate mol ion, n.nd anyone who n.dhered consistently t.o Elen,tic t.(!(Lchillg would not hn.ve beon o.ble to I\.ccr.pt. sLILtoment./t of Lhis kind M II. IJlLsis for furLher diSCU88ioll. Therein liM 1.110 difference hetween arnll,a Oil the one hand, l~nd V1tOOEC1,e; o.nd dpoMYWta on Lho othol'.

• I hO(lo thILL in t.ho lu..'tL four cho.l'tem 1 hllve 8ucceeded in est.lLLlishing tho following impol'LlLllL f'lcLs: (1 )l~uclid'H l'UHtultLum or ailelm,", h:wn Lho alLIllO OI'igillK ILII \.lit! gellcml foundlLtiolls uf IIl1LthellllLtics (hYIJoll,t'.~cis or archai) ILnd IIllLtheInILLicn.1 definit.ions (ltypotlte.seis or lIoroi), which is 1.0 ao.y thILL LImy CILflle from diILI(lcLic. (2) The nlLIllO (tier-malt' rof"I1i to (.ho fILet LhlLt t.hey are IlAAllmptions which W(lrO lIuL .ILm:0l'(.cd wit.llOut r(,.fl'~rvlLt.ion (lly 0110 of Lhe pn.rticipILIIL't in lL dialogue). ]1'urLhe... noro, wo HILW LhlLL when nllLLhenllLLicin.n8 in pre·EuclidelLIl Limes IILid ,lown Lh(,.8o nlnJ"IJTa Lhoy extended t.ho tolwhing of Ute ElclLt.ica no Ic8.'t than whell LImy inLI'Oducod the bMie ILrithmetieal dofinitions (uf 'unit', numhcr', 'ovon Jltllnbol", 'odd numhr.r', 'l'l'ime lIulllhcr', "eulllpusite 1111 111 Lo ..' , ctc.) (liHCUKIfOd em'lier. Tu lILy thu fOllnwLtion8 of deduetivu nULthellllLLiCH, it. WILlI nccl088ILry to do moro LhlLII jllHL ILCluJlL Lho illolLII (If 1.110 J~ltIILI.kH (Lito lIIeLhod of illdinmL proof, o.nt.i-empiricislll, et.e.); honce Lho clLrliest. rnll.t.hemo.t.ieiILIl8, when Lhey heglLn (.his tlLIIli, W(ll'O ohligeel Romcl.illlcH to depart from ElelLtie tolLOhing. ,£ho definiLions in al'ithmotie nm uno IIl1Lnifest.n.Lion of this, lUI nre Lhu poStUItLt.cS (alllongsL uthor Lhings) in geometry.

2RO

3.21

'l'JIH 'l(OINAI ENNOIAl'

Wo luwo rcnmrlmd moro than once that the fundamental principles whidl are staLed without proof at the beginning of J~lemenl8 Boolt I ILre liated in tltree separate (Jroups. Two of theso, the definitions ILnd t.he l}oslultlles, II/love already been discussed, So lot us now turn our a.ttcntion to tho third group, which Euclid calls "otvai motat. '.I'his IlILllIe is uSlllLlly tmJlslnted into Latin by the phmse 'com.mU11-e8 animi couccptiouc.s' . Let me begin by pointing out an interesLing ft~ct which it is imp0l'tant to lwep in mind. Proclus' excellent commentary is full of quotat.ions from Euclid's text; these quotations are very accurate and, as fILl' a.~ 1 lmow, havo invariaLly Lecn found to reproduce Euclid's words eXlLct.1y. It is all the more surprising, therefore, that Proclus' names for (.)10 three IdlUls of nmthematical principle ILrO not ahvays the SILme a.~ ]~uclid's. The terms which thoy use are listed below for purpmms of (!Ornparison: Rudiel

Proclus

])llfiniLion8

8eOL

Wco{)If1EU; or Den'

p()st\lh~tC8

alnjlla-ra

al-r~lla-ra

Communes a71imi conception.ea

,",tval mOta,

dELffJ"aTa

As we cn.n seo, it is only in the case of the postulaie3 that Proclus cOJlsistently follows Euclid's terminology. He refol'S to the definition.., HOllleLimc8 M l£ypotlte3cis lOfl and sometimes 1\.8 horoi.'Ol In my opinion, however, this fact is of no great significance. Wo know thll.t in early t.il\leR hypothesis meant. definitio1& (as well as 'fundamontal matho!IuLl-iclLl principlo'); fUl·thcrmore, the word lwros OCCUl"B with t.his llIelming in t.he writings of Plato. It is apparent that both these names worn in Ulm evon boforo tho timo of Euclid. As flLr ILB wo ILre concerned, the fact tlmt Proolus tLlways refol's to tho "OU'ot b'.'otat M dEtlb"a-ra is of much groator imllortance.JD2 If Lhis Wl!I't' I~ dolilJlLml.o dOp"'l·tl1l'O from Euolid's t.oxt, I fool S\lI'O t.hat Jll'oclus '001';",., IiII' ""l\lIIpln,

1'1'1101118 (ml, li'rioclluill) 10.4fT, Iln.1 118.1 fT, I hid., 81.26. Itl IIJ icl., 10.0; 17,1; 178,201; p088im.

101

l'AnT 8,

81.A1I0: TIIR JIJ(OINNIN08 01' OmUCK 'IA1'IIKMATIC8

TlIJ~

CONR"I'llUUl'ION 01' MA'l'III(IoIA'I'WII

281

would hu.vo given his nm.sol1s fur mn.ldng iL. Yet ho US08 tho Lcrm dE/rolla u.s nu.tunLlly u.s if ho IULd wLlcell itsLru.ighL from Euclid, He eve,! tlissocin.tes himself, so to spcu.I(, frum thoso who use the term "aLva, lvvotaL, su.ying t/ULt "n.ccording Lo Lhem, axiom !l.utl 'common notion: mean t.116 sn.mo thing" (TavTov yae lf1nv "a"r«i TOUt'OU, dElwlla "aL lwora "otVJ]j.I Q 3It 8cems tu me thaL this fILl:L ClLIl only be expln.illC(1 in tho following wI\y. The ltintl of mathemlLLiClLI principlo which is ClLlled II. "am) lJ'vota in our Lcxt of Euclid obviously bore tho nl\lno dE/mila in pre-Euclitlean timcs. Aristotle refors more LImn onco to nmthonllLticlLl axioI1UlIn.;lo, furthermore, he frequenLly uses Lhis term Lo denot..o the unproved proposiLion whioh we know as Euclid's third "OLv1j lvvoLa (whon cqun.ls ILre tn.lmn from cqulLls the romn.inders are equn.l).lV5 No doubt the name axiomala II.I so appelnecl in Proelus' copy of Euclid, ILml Lhis expl;Lins why it is used so consistent.1y in his commentary (illsten.d of "otval lvvotaL); on the other hantl, it wu.s replaced by Lhe nower expression "oLval lvvotaL in the texts on which only modern editions of Euclid are bu,sed. This conjecture presents us with two problems: Lhe first is to expln.in the origiluil melLning of dE/wlla, ILnd t.he second is to discover why this term is ILbsont. from t.ho texts (If ]!:lIclidIUlOWIl to us, n iH perlmps worthwhile to reewl aL Lllis point how tho problem of tho "oLval mOLaL hM been tren.ted by hisLorialls in t.he PMt. 'l'U.IlIlCl'y observed thll.t tho terot "Otv,) lvvOLa docs not occur witl, ils leclmiclll meant1i(j in the works of Plato or Aristot.Io and thn.t Lhe Stoics I~ppetu' Lo Imvo been t.ho fil'St to usc it. in this Hcnse. I10 eoncludClI therefore that tho group. of pl'inciplc.'J which nro ClLllcd by this IULIllO in the Elemcnts hn.d Lo be of pOOl-Euclidean origin. lUG Now, if he hn,d merely ch~illlod that Lheil' lII~mo WI\.8 posL-Euclidmm, I would hlLYO no qlULrrlll with him. Tn.nnery, howover, WILa not so motlest; he nmintainccl t1I1Lt the principles t.~lomselves could not IlILve been gn.thered together Soo l'roclus (od, li'rimllnill), PI" 19:1-4 (.mI'UI:i"lIy 194. 8-9), Soo, for oxomplo, .M elal,/ty.,jl'.8 1':1.1 ()OI),,2(). l"lt.id., J{4,IOOIt.20; CllIlIIl'unt t.hiA II/UlIIIlJ;" wit.h 1':1. IOOlio19fT. 1"1). '!'Ollllory, lUdmoirc8 8cicllti/iqU6 2, 611: "C" t.orlllO d'lI'polu n'llAt nllllu· ilion I; do 13 IUl1guo I'hiloAOphilJllo .111 l'opoqllCl; lin In chol'd",roit vuilllllnClIlt. flYOO uno signifi('.utillll t.uclhlliCJ"n .IUIlA l'mllvrCl ,I" 1'1"t.lln 1111 ,IRIIA .,.. 110 d'AriRt.()I~'; i1llppurt.iollt. fLUX sl,oluioll8 dUll" 1'.jcmlCl COIIII1lC11I\,uiI. 81l1l1"IIIt'II1. "II 1.01111'8 d' gllcliclo 01; donI. il no pOllv"i~ aubir l'illnUllllcn t\ AI.'Xflluh·in." In 10'

.

1 2R2

S1.AI14): TIIK IIKOINNINUS Oil C1I1KKK alATlllutATI~

unLil ILfLer tho timo of Apollonius of Porga."7 It is romlLrkablo MULt IIIl ignored completely 0. picce of evidence which contradicts this view. I!Lm referring to tho fo.ct (which was corLo.inly known toTILllIlery) Lh;LL Aristotlo was obviously o.cquILinted with the xotva, mOtaL, only IIl/(ia ltIwllier 7lame"" (fn.nnery WILS novel' tho mOllt consilltent of Hc:ho!:Lrli.) Jt is oven moro surprising to find him oxplnining why tho 1I!Lllle ,If{colla hud to bo chnnged to XOtV~ ivvota; o.ccording to him, tho Sluirs were responsiblo for giving the word delCl)lta nn ontirely new IIl1mning.lllD (This lIuggcsLR Llmt tho inRuoneu of tho StoiCR only hmughL ILbollt (& r./U"'(li! of lImne. Yet the pll.per in which 'l'ILllIlUry offored this (~xl'llul!Ltion (I,lso contlLins his lI.88erLion thILt tho whole list of llrincipies Iwown ILA "alva, lVVrltaL could not hILve been compilcd boforo tho timo of Apollonius.) I must confess Lhu.t I would bo moro sympnthot.ic tOWILI'(ls Tn.nnery's conclusions u.nd think thom of greater worth, if \.Iwy hnd been propel'ly thought out ILnd wero consistent with one allot-her. They ILrc, howover, mutu!Llly contnLdietory, despito being Imsed on oiJservlLtions whieh nre nt least in pu.rt correct. 3.22 1'JII~

WOIt!)

de1ml'a

We have conjectured that the nllLthematicn.l principles which go by 1.110 IIILIIIO of "otva, lVJtoLOt in Euclid's text (ILA wo Imow it) wero .n Ihid, 2,60: "QUIlIlt. IlUX notio,~ COmfflllJU8, olles 1It1 IIOruillnt. pas do lui 11':lIcli,l" I: iJ It'lt uUnlit. olnllloy6MJ commo .1I1Ilnt. do sui ou commo SUPpOIlO('S pnr I.." tltHilliLiunB; I'uttontion no IIOruit. porWo sur cot.to qU08Lioll qu'A I'opoqllo d' ,\ PUIlOlliIlA, 'lui cl8llfl.yn (10 dtimollt.ror OeB propooiLiolls oL rooonnut. lour liuiaoll 11\,111: In e1cillnit.ioll ell! l'uguliLo ot. des opOrnt.iollB do l'oocJiLioll 01; do Illsoust.raot.ion gc:"lIlt;t.riIJCWB. Lua tiditoul'8 suoocssifs d'Euolido Burulonl; pris cJnpuls 101'8 I'habi· 1.11,10 ,1'i!IRurur UII r(!cuoil plus ou moins complot. do coo Ilot.iono suivIlnt. 10 point. tin VII" nlll1uol i1a atl plu~uionl; maia In t.radit.ion aorait. rcat.Co 10llgtomps 0880Z nUU.lllll.o A cmt. agllrd." .va I hid. 2,62: "Pour 10 mot. axWmc, Arist.oto I'mnpluio A pou prc\8 danB 10 11,1\11\0 SOIlA qllo noUB, ot. iI nOlls approncl qu'iJ et.ait. en usagn chcz les mllt.llo· IlIlItide'lIB, lIIaiA pnrLiclllit}romonl; pour dtSsignor lea nolio~ COmmUn4!8, II cit.o 1I1t'lIIn ':01111110 tlXUlIIJll1I qllO, Bi on rot.rllncho dOlI chosclII tignles do llh08tl8 ugnlOll, II'A f('Sl"A BOIlt. eigllux." 1001hi,I" 2, 62-3: "II (lilt. A fOllIllrqllor, quo lea st.()ICliClnO clmllge}rollt. com· ph)!.,.. ", .. !. I., 1It'"11 du 11101. delwpa, ot. IlJlJltlll)rollt. do au nom uno proJl09il.iulI fllll'I''''"CJ"II, vmiu (JU rllllKllt); c'cst. IA CJlI'iJ fllllt. chorchor 10. raison cJo l'adopLion ,1'11111' "lItrn ,Mfti~IIIlLion dRIIO I)(m t.oxlA'fI t1'F.llditln."

I

l'AllT :I. -rIm C:HN!ITllIIC~ICll'l Ot· MATIII·:M.\TI(:!I

origim~lly clLllcd ci~Lw#aTa. Let till now IL'i.'JtlmO t.IULt. this WI\8 indeed their original namo u.mI turn nur ILU..cnlioll lo ILn invest.iglLtion of

its melLning. The uriginILI me!Lning of the term ci~lwllCl, I\ILhnugh it. is UILAicILlly no mom complimLted than Llmt of aiUlllU, cml no longer 1.0 CHtlLblishml flO casily. l~ven in_ o.nt.iquit.y this exprcsKiun ILnd its significance ILA a mlLthenmticlLl tel'lIl wore explained erroncously. !i'III·t.hermore, this 'l'Rl!luloox)lI'Ln!Lt.ion' is still current. in t.he literaturo t.odu.y and hiLA nm.llI ii, ILllIUlHt ill1llClHHiltlo to glLill II. (:u ....(!c:l. IIl1Clul"Kt.ILnding of thl) IIl1Lt.ttll', Wo flhlLII IlILvu t.o hcgin hy cxalllining tho flourc() of thiH misWLlte. l'rocluR, ILft.er (luoting tho fivo EuclidcILn axioJIIs which he t.ILlws to be ILlithenl.ic, commonces his expll\lULtinn of t.hem wiLh t.ho fullowing wurd8:%OD TOUT' lefT, nl XaTa nciVTa~ avanOtlEt"Ta "alovlLEva d~tw­

l,aTa, "aOoClov Wit; nOVTCI)V oiJT~ lXEtv d~tovJ'Tat, "ai ~WI''1'tCl{J'1TEi "al neo~ TaVTIJ ov~t:l~. Mormw's t.mnRIILtion of l.hiM HCnlonco runK ILl' fnl·

lowfl:20\ "These !Lre wlmt !Lro generally culled illliemonstmbio axiol/f,'] , ilUUilllIlCh ILS they ILI'O decmed IJy everyb\ldy t.u Il(l I.ruo ILlld no uno ever disputes thcm." Although this transitLtiun cunveys the flubsllLlwe of tho Urt'ck fum· tence quite o.cmlmt....lly, it obscures tho fILet thlLt 1'l"Ochls (lxpllLinH the noun M{CI)l'a in terms of the vel''' ciEtofuat. We CILIlIIOt. l'elLlly reproach hilll for doing 80, since ciEICl)lla i8 in fo.ct derivcd from U~tow. IL seelllR retumnaIJle, thorefore, t.o investigate t.he exu.ct meaning of this notln hy l(Jolting at tho vcrll which is ita ron I.. In my opinion, howover, this iK not WIIlLt.·!lrochIH docs. Instead of conducting !Ln olJjective 80tLI'ch for tho menning of ci~(wpa ho sto.l'tS out wit.h IL lJrejudice whinh ho t.ilen o.ttompt.s to justify. This is IIhown by t.he fl~ct. t.hat. he mcntions Oftly one of lite possible meanings of ci~t(1C1). The prejudice to which I am reforring is hifl belicf tlmt the wurd fl~{(I)/ta WILA ulled (in EuclilielLll mll,thcmlLUcs ILt Icn..'Jt) to denot.e II. stl\t,oment w/to.'le trulll collid not be doubted. As wo slmll sco Inter, AriRt,oLio seellls to hlLvo heen hLrgely real'ul1siblo for t.ho widt'sl'rolLd ll.(:CCl'tILIICll which thill viow glLincti ill ILllti'luit.y. PrudU8 himHulf, UII ILIIOt.lwl". occIL'4ion, 1~t.tributcH t.o Aristotlo the opinion that "oVl!ryono woulcl

"0

IDO I'rocltm (,'(I. Jrl"il"n"ill), 19:1, 16-7, 1011'. 1r.2 in o. It. Murrow'lI t.mllllinlinll (R"" II. Ii "hftv .. ,.

2R,.

SZAIII): TIIJ·; llJo:mNNINUR 01" nlwlm 'IA1'Ult.MA')'WS

diHpulled to ILCCCl't it [tho truth of an axiolll] ellen tlwug/, some might

cli,'1pulc it for tho sn.ko of argument".2M Arcordingly there could he no real dispute ahout tho correctl1e88

Ill' Euclid's axioms, and in flLot no ono in nntiquity (or at lonst no ono who lived after the timo of Aristotle) seems to have seriously enterL:tined the iden. Llmt t.heir validity could be qucstioned. Proelus WI\.'J amollgst those who agreed with Aristotle aLout tho nature of n.xiOIllS, :Uld ~,e nt.~mpt.ed to Lolsl.cr up his views Ly appealing to etymologic:tl ('Ollllldcra.t.JOl1s. It WRS fort.unaLo for him, t.herefore, LlULt t.he vedl Il~'liw, when followed Ly an infiniLivo, had MOllO of its moaningH '10 thinl< fit'.203 Vnn Fritz has recently given ILn eXl'lnrllLLion of tho word delenpa whiell follows the t.rn.ditional one very elosely.~. Tho only differenoo hetween the two is that in tho modern explanation an effort is made 10 derive tho mathematical meaning of clE{wpa from ILn oven eal'licr IIIclming (lo valuc, to thinl[ worthy) of t.ho vorL de,ow. Von Ji'1'it.z howover, just IiIm Proclus bofore him, tlLcitly lL88ullled Llmt tho matheIIlltlielLl mClLning of this t.crm (dE{wpu) WItS unlLmLiguolIs and woll eHilLhl ished, I now wlmt t.o Iiat what I consider to ho tho welLIUle880s of this posi tioll: ( I) delmpa ent.ered ~athematicn.1 tel'minology Irom dialectic. It is Lhel'Cfore wrong to investigate tho otymology of this wOl'd with tho ailll of finding out only how it aequil'ed its supposed mall,emaJical IImmillY· Ono cannot gn.in n. preeiso understlLllding or wlmt it signified ill 1II1L1.lu'lIIaLi('.8, without. first llOing dtllLr Itilimt tho 1'010 which it played in dialectic. It is oLvious that dE/mila, oven M a mathomatical lel'III, WII.'J originally used in it.a dialectical ROIlSO. (2) li'url.hormoro, Lho IUlIgungo of mlLthenuLtics (:ontnills elelLf inw(::LtioIlS that the cn.rliest mathematicn.1 uRngo of de/enpR WM ignored hy Pl'OcluH, Tho mcn.ning of thiR torm Roems to havo becn di.,'or'rti ill }lost-Aristotelian timcs, whon it came to dellote a stnLoment which was U3 a 11Ialler 01 course 'thought fit' (to Lo accepted). 101

PnmlllH (cd. l"ricdlcin), 182, l7ff; LIm Lrun81nLion iu hy Morrow (800 n, 6

I'AIIT 3. '1'lIle

CONRl'll\l(:}'IClN

III' MA'I'lIlm ...... II~'1

In flLCt, it is vel'y OMY to ootlLlJliHh tho dilliectit-Al meaning of LlIO word u~{(J)lla. Von li'rit7., for oXILmple, lutS writton:205 "Evon when ho is not. don.ling with (lirueetic or logic, Aristotlo uses tho word dUenpa simply to mean II. sltpposilior&, OP'UiOll, doctrine, etc.; this usnge is a natum) outgrowth of tho pre-Aristotelian mClLning of Lhe verL ~'&(J)." He ILlso IIllLdo the fullowing pertinent rem:LI'k, which t.hrows some more light on the subjed:!08 "When diHcusl,ing in tho 'l'o])ics tho tli'Llecticnl technique (If quesliun and answcr, Aristotle fl'cquenLly used tho word ue,oiil' to lnlk n.bout the stat.cment which tho qucstioner hopcs tlmt the rcspondent will aceept. This ILCCeptlLnoo is thcn described IL8 nO/val (SCO, for , exa.mple, 155b30ff, 150a14fr ILnd many other 1'1L88ILgCS). If de,Oiil' is followed by nObal, the dilLlcoticl\1 argument eo.n proceed."

if we now wILIIL Ln find utlL wlllLt ,Jewii,· IlICILnL in tho context. do· 8criIJed ILbovo, Ulcro is no neml for UR t.() go Imcl, t.o tho original moaning of this vel'b (to thinl( worthy). I I, is much simpler to luult in a dictionary (PILI"" fill' OXILllIl'ln, or Liddd ILnci ScoU.). whero lihe verh dEnIm is listed 1\.8 hn.villg 't.o donllLlld' IIr 'to rCCJ"ia'o' amongst it.s mCILnings. This WIL8 such a common Imo.go ill ILntiquit.y tlmt it selLrcoly needs to be iIlustmLcd by oxamples; [ will c:ontont mysclf with citing only two, both tn.ltcn from tho writings ur Pln.to. Tho first one is naea TOV laTeoii pa(!pRXOl' decoul' (/ll!lJ/tiJlic 11 [ 4011 - t.c, ask tho I'hysicil\ll lor medicillo), lIoIld tho Recond iH (l~u$(jJ VllIic; ,ilA,jAovc; ,SuM,IXt'v (AJwlogy Illtl - [ (l.'Ik ymt t.tI infc,..u' UIIO ILllutlu:r). Of course T do not wlml. to deny UULt this IIlClLning of tho verb ('1.0 demand' or '/0 require') nllLy very well IUWI! ,Ieveloped from the carlier OliO, '10 [I,ink ",arll,y' or '10 '''i"k fil'. Nov m't.1 IClIOR.'t, ( think Llmt it is misletLding t.c) coneentmlo on this el\rliol' sOlllm in IL diseussioll of hnw sllhH~fJUent mOlLningR evolvecl, It RUru;MLK th:Lt ,U'O(" WIl.R used in r.nllncction with "mnamlH whidl wel'o cc!rtl\inly justifiecl Itml 'eleenIt'CI worthy' (t.cl bo ILCeepLed), whol'Oll.R this vel'h WRS in f,wt used just :tB nften t.o CXPJ'(>.AA a 'Ialse dellllLllfl' (IJ' n. 'lal,vl! supposition' .707

IlIJOVI'),

• f h or U' liB mClUung 0 L 0 word dEulw, 80n Um ruforuncC8 (Xcflophon, Cyr. 2.2.17 IUIlI An, 1i,6.0; Pindur, Nelli, 10,39, ute,) quoted in I'UPIl'8 diBLioOlLry, '''' I{, Villi Fritz, op. cit., (II. 2 ahove), pp. 211fT. 10, I~

2M!)

Ihi,l" 1" 31i, Ihill., p. 32, II. 32, 101 For Clxu"'JlI .. , I l"rlH lotuft G,H7; !,Inl.ct. J'II """rr.,.".~ 2:I!l,', .,I.e. lOS

lot

2RU

R1.AIl(); TIII~ 1\I':IIINNINClII 0 ... OIlV-JCK .IATIIF.aIATII'S

CorrcHpollclingly tho 1I0un delwllu, wholl usod us a clil~loctiel\l ull'm, wa"" nrigilll~l1y synonymous wit.h all'1lla nnd just melLllt. I~ 'domand' or 'requmIL'. Somet.imes it. had t.his mCILning ouLsitlo clilLleoLic lUI wC'II; Hophoc:lc!H, fur CXILlI1plu, IIHml it to dOlloto tho deuce (or tl('",mu/) of LIlli (:mIH. 2t18 JIIcil!l!d, dUw/fa or detCJIl1t1; WILBoven uRed on occtUliunH 10 rnfc'l' I~I lL '11f'li/ion' OI'IL 'writtell refltlc.'1I' .:OU Of cuunit' this word nlso ILl:'1 uia'('cl IL VILriet.y of lllighlly different m('anings in di"lectie, but t.h('so ILro now CILBY 10 cxpllLin. '.IU{wpa, Iileo arn/pa, lIigllifi('d nl\ osserli01l which ono of t.hc participo.nts in 0. dialectical dohlLLc demanded that tho oU,or gmllt; IIt'llt(: it. "Iso CILme to mClm 'UR8orlioll', 'supposit.ion', 'doct.rino', c(.c, J IIhould elllplmsh<:o horo UIILt. t.ho word detwpa, when it. WILS used ILB I' dialcctical Lcrm in IJl'C-ArisLotclinn t.imes, docs not. seem to hnvo «'''llIIuLed n 'dellllLlld' whid, WILS ClUlily satisfied or n. rclwily ILCcel't!Llllo 'a~('rtioll'; all ILximllIL did 1101. IIILVO 1.0 ho IL 'platulible assumpt.ion'. JII fad uno gcLH t.ho ill1pl"CJ'II!ioll Umt. Lho rcvcJ"lolC Was true; t.ho 'IIMcrI iOIll~' which were called axio"m(a (or uilemula) in dilLlect.ic ILppl'nr 1.0 Im\'e heen OIlCH which wcre not. ILcc(!pu~11 wit.hout re.'icrvatioll, ThiK !,oinl, is iIIust.mted by the following 1"\8Sago from Plu.to's llcll1L blic (V J1.526), which WILB quoted curlici' in IL different context.: .. YOII Imow how it. is wit.h sleillcd mlLthomnticilLl1s t If ILny ono in nn argument. ntLcmpts to di88ect t.ho one, t.hey laugb n,t him, and will noL "lIow iI.. If yoil cut. iL up t.hey multiply it, t.n.lcing good care I hat. t.ho one slmll never ILppcnr 118 not one but us nULny plLrtH .. , If )'IIU wenl 1.0 lilLy to them, "My wonderful friolld, wlmt KorL (If IIlImhcrs ILro you diKclJssing in which LlIO 'one' a.nswers yuur dn.illlll (m:(!i no{wJ' u(!dJ/uiiv ~,aU'Y£a{h:, Iv ol~ tv olav vp£'i~ dE,oiin II1TU') , ill whieh clLch 'one' is e'lulLI to overy other 'ono' without tho BlIllLllcst. difrerClneo ILnd clIntlLins within il.llelf no plLrtsl" How do you t.hink tllllY would I'oply 1 1 flLncy by "'Lying thlLt. they are t.allcing ILUOUt l.hoRo IlUIII),crs whic:h CILII 1m ILJlprolulIIclcd hy t.he undel"Btnrulillg alulII!, IJllt. ill no oUwr WILY."

I'AIIT 8. 1'11)1, CONRTIIUl:J'ION 01'

MA"III~"ATlr:ll

2R7

delOiiv in hill description of t.hoir claim.) It. KhouM be noticed, howe vcr, tllI\t. this WILR not a claim whieh tho oth('r sirle in IL dilLlcct.icn.1 delJILlo (i.e, non-nmthenllLt.icitLns) could Imvo lu:col'Locl very rclLllily, The IlIILI.h«lIllILLic:ilLll'S 'UIlO' WILt! HuL IL I"LI'L (If Lhe IUlllsihle wurltl; hellco Lim ncm-nmt.lummLiciall mllHt hlLYtl 1111.41 l'f'sl·rvILI.ionll a),ollt. ILllowing il.!f (1X iKh~lIt·t!.

We (,ILn sce t Imt fi~{(011U ILIlIl u;nlllR wl'm interclULllgf'ILble cXl'reR.'1iIlIlH ill dilLloct.ic; t.heso two tenns WCI'O c:mrl ,'1!1'IOIIyms, as were Lhe vl'r!Js cl~lnm Imd alTiw, 'J.'hiK IIUit rcmlLrle ILpplies to IIlILI.henmt.ics lUI well, if wo igllore liu" t.hl! momont Euclicl's wodeK ILlld tho liu!rlLturc rohLting to thllm. We need only look through their writ.ingH lo RCC tlmt I1lnLhemalicilLllH l'I!glLrdc!d ae(C/JllR Imd aiT7l1lU ILB synonYIllII. (ThiK is cspccin.lly truo of Arl'lliulI'dt's, for whom {t;ltUJEI1", VnflXF.{/,tt'()J', Uil'1PU, de/CII/la, ).a/f{lUl·()o /1£'fOJ', etc. wero all eCJuivlLlellt. eXl'l'e88ionR.21U) li'urthermore, ProdllR, ILn~r not.ing tllILt Archimc!dl's intrmluccs wiLh t.he word alT/co IL principII! whidl might. rlLthel' 1m cnllc!d ILIl ILXiulll. gOIlS 011 t.o SILY ILI)()ut. firKt principles in geneml:211 "OthOl'" designn.to ILII of them axiom.~, IL8 thoy CILIIII. (/t('ore-m everything tlmt requires IL delllonst.ru,t.ion." Pl'(lc!us WILS fully ILWlLro of t.ho flLct tlmt tho terminology of Grocle nmthcllmtica WIL8 neit.hCl' uniform nor consiHLont. IL8 fILl" IL8 the Imming of prillc:ipl('s WIIS (:olwcrned. No doubt hI' RCarched Lhrough Lho worltll of ol.hl'r mlLthcmat.icinlls, hoping to find ILl. lelLSt. t.he trlL('CH of IL dill' tilwLion (~Ump:LrlLIJle to tim ono dm.wn by Euclid Letween axiO"'''(tl !Lilli "ilr.rmlltl, lie lIluHI. hlLVI: 8(!ILI'dll't1 ill vlLin, hllw(wer. Hillce tI",St! I.c~rll1H W(lI'(1 Ullt'd Kimply lUi KyllUIlYIlIK hy IIII1RL IIml.hcIllILLiciILns.

Our t'L"I, now is to find Ollt. why ]~t1dicl'H a:r:iom,'! wcro givf'n IL RIII'I:illl IIILIII". WU huvo conjocLul't!tI tllILL Lho 1.1'1'111 «i~If;,/utm WII." flri~illlLlly IIK(!(I iIlHI'(!lLd uf xtlCvul f'f1'Om, I" (luIlClu) UliK ~1'II111' of I'riJlI'i "ll:K, Hn I hn

'\""orcling I" LhiH '1IIClI.,LI.ioll. IIIILI.hl!lIIlLl.i«:imllt daimetl ur r('lflljl'rtl

tim oxisLoncC) of

IL

very IIpecilLI leind of 'one'. (Plu.to USC8 tho verb

t080itl. (Jul. 141i1. H,·,· t.J", "IIL"~' ,,,,,I,,,· '.llill wore I ill I'IIJlII'H Ilid.illllnry.

tn.

cr. ]{.

(II. 2 nIHJvl,). 1'. 1i7 'UI,I ... ·('ulIlI"ry. JI'o!l/It)i,.,.~ 2, 79. IIIl'molUB (od. Ji"riNlluill), IRt.lllfT.: wIIJ., (II I"" nth'HI ulr'!/fflr, • ••. 0/ M

110

von )i"riL7., op. dt..

.~drlltifiqttCIJ

nUl'ffI fl!II;',lRffl n~oall)'o~wouall'.

2RR

fl7.A lie): l'Uy' IIIUIi HHI HOS 01' (JUVoKK "fATIIKl'ATI~

'1Ill'HI,ion n.t iSHuo is whother ]~uelid, liIto Arehimedes, considored axioto he 0. synonym of /,ypot/,esei.'i or whether tho filet thn.t certain principles in tho Elemelll$ were cn.lIed by this nn.mo tells us something ILhuut t.he nn.ture and historicn.l development. of u.xiomn.ties. There 'Lrc nine n.xioms t.o bo found in t.ho modern text. of Euclid, allel, wilh t.ho oxccpt.ion of t.ho last., clwh ono of t.hom l\88erl.8 somo ImRic propert.y of equality. Becu.usc t.ho first. eight form such II. homo!!l'nl'OUS group, Axiom U (Two st.raight. lines do nut. onclose 0. spn.co) iK u9utLlly thought. to be a IILtor addit.ion. 2Iz These principles may t.hcrofore he dt'8cribed n.s n.xioms for equalit.y. (Even Axiom 8, whioh states I hnt. tho wholo is grou.t.or LImn the pn.rt., is a Itind of equality axiom, albeit 0. negative ono, for it denies thu.t a 'part.' cu.n over be equal to I IIII 'wh(lle'. We shall hn.ve moro to say u.bout. this lu.t.or.) We now want to cxplain why theso stu.tements were cn.lled clEu.oI'ClTO. Tho significn.nce of t.heir namo hna alroady been discussed; axioma wna illlclUlcu to donote an n.ssortion which, although required at. tho beginning of an n.rgument, wn.s not accepted wit.hout reservation. Let. us in\'('stigate (,)10 cxtent to which this meaning is applicable to Euclid's axioJIIS. It iK wort.hwhile t.o begin by comparing his axioms with two 'c1uasiIIlnt.hcm'Ltical' 888CrLions about equality which are ma.do in the 'l'/U·(If'lelus. 213 The ass~rtions in quest.ion arc: (I) "Nothing CI\n become great.or or Il~, either in sizo or in number, flO long II.'! it. remains equal to il.'1elf." Imel (2) "A t.hing to which nothing is addm.l n.nd from which not.hing is InlH'1I ILWILY is noither increased nor diminished, but o.lwo.ys rema.ins ('qual to ilself." On closer inspect.ion, the n.bove st.at.omonts turn out. to be a kind of double definit.ion of whn.t it means for somet.hing to be 'eqUlLl to itself'. The fimt. one says tho.t. a thing which remains 'equn.l to itself' cannot. "ccome bigger or smn.ller in lI.ny wn.y. 'fhe second is tho converse of

1//al(,

III /':uclidca: Elelllenln (od, J. I•. Hcioorg, Liplliao 1883-1916) LID, and 0, n.,dwr, Grllntlltlflcn lier !I(nJllClllnlik in Ue,9r.llichlliclaer Entwicklunu (Fn,ihurg~11I1Ii.'h), p. lin. IU "Iutn, 'l'I,eflr.lctlllJ llilia. 'rho t.mIlR11Lt.ion iR tlll(OIl (with 0110 minor ohungo) r..... " I". TIl. Curllrllrcl, 1'lllla'lI 'l'''rllr!1 ,,/I(unt111f!11!l1I (Loneloll, 1936).

I'AIIT S. TIIK COHRTILU(''TJON UI" MATIlItMATICS

28D

thc firsL o.nd M801'lS that. IL thing which docs not. heeolllo higger or smaller romains 'oqual to itself'. It, ill instructivo to noLo t.hn.t. theso two assertions n.bout. equality n.re not. CI\lIed dEu.o/4a-ra; l)IILt.o rofers t.o t.hem 1\8 O/IOAoy,ipol'O. We already Iw()w thILt"omologcflu, IIonu/'YlJOthesi.'i were inlcrclmngelLblo in dilLlect.icnl terminology; 0. 0PoAoY'II,a WII.'! a st.tLtemont upon which bot.h po.rt.icipl\nl.8 in 0. debat.o were iJt agreement. It. seems rel\8olll\ble to conject.ul'O t.IULt PhLto'S n.ssert.ions ILbout. eCJUlLIit.y wore clLlle.1 Iwmologt-main II(\CI\U80 th(\y wore unc()lItrovomit\I, WhOl'CI\8 Euelid's were cn.llml axiomrr/" II"cauRo t.lley cuuld not. 110 ILccel'wd wit.hout reservations. In wlm\. li,ll(J\vs wo ~llILlI C'lulClLVour tu verify {.his conjecture. It itt, u.s 0. meLLor of fl\ct, hn.nlly conceivahlo UtILI. n.nyone could ItlLVo objecte(1 to PIILto's IWlIlolOfJr."lllla n.bout. 'equl\lit.y' (or, mOl'o precisely, about. tho prop~rty of 'being equlLI to it.~clf'), for hot.h these st.I\Lomenla ILre purely formal. They merely eX1'lI\in in ot.her words the meo.ning of t.he .,h1'l\90 'eqUiLI to it.~elf'; hem:e t.hey eonst.itut.o 1\ corrocL fOrllllLI definition. Such formal slaleflum/,~ could cert.l\inly IlI\vo been regarded na tho 'st.rongest.' (in Socrutes' senso of the word - ef. Plmedo lOOa) by Lhe ILncient. dialeet.icians, Thoy wore not obtained by I\bstmeting from everyulLY exporionce, hut by 0. procC88 of pure rcfloct.ion. Henoo they must Illwo boon considomd no less consistent LImn t.ho prototype (If all lI.88orLions of t.his Idnd, Plmnenidcs' fn.mous thesis -ra 0)1 iCf'r{ (What. is, is). Euclid's equl\lity axioms, on tho ut.her Imnd, n.re st.n.wmonl.8 of 11. rat.her different. ltim!. 'rhoy nll\cl lUI f"lIowli: 1. '1'hings which ILre equal to t.ho Sl\mo t.hing are also e(lual to ono

lI.nother. 2. If equa.ls be n.dded to equals, the wholes are equal. 3. If equals be subtrncted from equals, the remaindel's o.re equal. [4. ] f ClIUn.ls be added to uncquals, Lhe wholes n.re unequal. 5. Things which n.re double of tho sll.me thing n.ro equal to ono anot.her. G. 'l'hings which are ImlvCK of tho 81LIll0 t.hing are equal to one another.] 7. ~rhil1gH whiCllI (:oint!iclu wil.h (11\(' ILIIIlLhm' ILrO (!(lllILI t(l uno n.noU\l!r. 8. Tho wh"lu iR gren.(.er tlmn \.hn JlILI'I.. J 11

flI,lIM

290

Not. ono of thcse st.ILtcmonts CtLIl be tlcscriiJed II.'J merely formlLI. 011 tho conLrary, t.hey nil lU!8ert. properties of a rehLtion (equalit.y) whidl IIIURt hn.vo becn rcgn.rdcd (by the 1meatiCII at Icnst) 1l..'J 'solfc·olll,l"I\(lidoI'Y'. It is by no mCILnH evident LllILt two cliHtilH:t thingR (LI!. t.WII things whieh nrc not t.ho sumo) can over ),e 'equal to one (moiller' ; fUl't.IWJ'lIlOl'l!, it. llIakes nu Ronso to speak of two things unle!18 they ('1m 1,(, disl,inguiRhed f!'OllI ono another in SOIllO WILy. The 1mclLtics wero willill~ In concedo only t.Imt IL thing coulcl eqtllli ilsrll, 1l0t. Umt. it c'",ald r'lllfli CLlwtll"r II,il/Y. l~udicl'H nxiolllH aro lLA8CI·tioIlS whic:h ILI'C jUHt.ifiecl lIy I'l'ILcl,iciLI I'xpHl'iclll:Il ILnd, in 801110 ClUiCS, direnUy hy Rcnse-porccption. Axiom 7, Jill' eXlLmple, stlLt.cS Umt. "t.hings whieh coincido with one another ""'11 cII"nl to onc another". It. can IiLomlly bo seen UULt. l'hme figures whic~h c'oim!idc! ILrc actually eqlllLl; henco t.his ILxiom is vorified by sellsory I~XI"'l'ic'IIC'C" Similarly. tho Himplest. wILy to convinco nncHolf of tim 1.l'ul,h ofAxiolll 8 ("The wholo is gren.ter t.han tho pint") is by IIlClLIIR of tim RellseR. One need only loole ILt. n, geollletric figurc 214 to soo Umt t.he wholc is grclLt.er tllILn I"ny PILI't.. Am!ol'c1illg to Eleatic cloctl'i 110 , Icnowlcdgo uf UIO tl'llth could lIul, be ILc!IJllil'cd through sonso-porception. 211i It is apparent., therefo/'t~, I hat 1,lm IItlLwmonts which wern originlLlIy clLlIod Clxiomalct ill tho N/PlllclIls coulcl never IlILvo pllLyctl tho rolo of lto1l/ologcnlClltl. or hYll11tltrs/';s ill IL clilLlcctical n,l'gllmcnt. Theil' n.ccepttLnco miLy woll hn,vCl heell dClllflIuiecl ILl, tho heginning of 0. dolmto, but. t.he othor party woulcl !lot agree to thorn just hecfLuso thoir truth W:Ui gUlulLlltecl1 by HCIlHC-PO/'ccption, Honc:o thoso u.8SertinllH, instolLcl of heing ClLII('cl ,1/l/I).0Y"I,am, were given tho no.mo c}Eu;'llaTa. AR IL IIl1Ltlor of flLCt, it iH well known thlLt. tho E1oat.ics also discus.o;cIl I Ill! I'l'lIhlolll uf eqrlCllily,' hut thoir' treatment of it. was not tho samll as l~uclid's. Amongst thoso who have written "bout this subject is BedlC'l', frulll whose worlt tho following qllotlLtion is ta)een: 2IG ZII I L WIUI ol'ighllllly thought. t.hat. J~uclid '8 Bxiomn wcrn vnlitl only for g"oml'I,ry. S,·., O. B ..c1cl.r. op. cit.. (n. 212 Bhov.. ). p. 90. 116 S,'O n. 120 above. III O. JlI.d",r, lIfaIIJell/nlisr./le E:r.islenz, Unler.",c/lIlIIger zlIr Logik tlnd Onlolor/if! ""Ilhclllalisclll:r 1'1u'inomcnc (1111110 1927), p. 14.4. DoclcI!r rofun thoro to Tanllury (Re.,..c l'hilosophiqlle 20 (11486), 381) and RllMol (Till! Principlu 01 Matl.c· ,,,,,tir., (C"lIIhrillg" I !l0:1), Vol. r §:1:11, p. :1fiO nllll §§:140-1, I'r. :1fiR-OO).

/'''111' :\.

T"'~

CIINRTlI1I(:rHlN III'

~I.\TIII·:M"TJ(~

2f11

"ZCIlU'H 1'ILrtl.C)UX of Ac:hilleH :Lnd lhc lortoisfl c:nn he given IL S(~t­ theoretical interpretation. Whcn rcfu/'/IIIlIILtccl in this way, the I'I~l'ILdox runs IloR follows: If AchillcH iH In ovel'l,ILlw UIC tortoiHc, who IIlLR III'CII ~ivlm IL 1(~n.c1, tlll!/'O IHUSt III~ I~ olle 1.0 C11ll~ C!OI'/'('Hl'"IIC)CIII'O hl'twel'll the disl:LIlco whic'h he ImH to c~ovcr (vicw(!d IL'I I~ Iml. of I'OilltH) ILnd the djst.lmce covcI'Cd lIy thu tOl'toise; flll'Lhel'lllorc tho c'OI'l'('Hl'onclcncc Illust bc Ii\l('h that Hlo tillle which it Lnltcs Achillcs 10 1'(,lwh any I'uint ill tim HalllC as (1m t.ime whic!h the l"I'LoiRe LILI(cH (." n~lwh tho (~()l'rcRI'(JII(lin~ point. 'l'hiH ill jllRL 1I/l1l way of C!xl'/'(!Rflillg Ihn w('1) Iwnwll Hc~t-(,hcOl'(\I.ic1L1 flLllt tlml, Iwo infinitc "iri.'1 /lilt!! l/flve! lite .~(Wle! ('.(trdilWlily even Ilu>ugll Olle o/Ihcm i .., a l,ro1,er .'Wb.'Ipl ollhe oilier. III tho pl~Rcnt caso t.he distlUlcc tmvollecl hy tho tortoise is a '1'1'CI1'1l1' subsot' of the IliHLI~nco l.mvcllecl hy Achilll~s." Thi:; ""Mel'vILtion of Bcdwl"H, ill Illy ol'inioll. iH '1uiLo c(ll'r('(~t. ILIlcl it /:ILII bl! l/lied to cst'Lhlish tlmt pl'C-l~udicl'~ILIl IlIILLhcllllLtic!inllH pmlml'ly hLicl clown ftxioma/r' for ellUILlity ill "nll'I' t.o ILvoid UICl I"Ll'lulox('s "f XCIIO. This is wlmt I hope 1.0 show ill the ncx\' dmpter.

3.2·1 "'I'lm WIIO/.K IS ORl'.A'I'ICIt TIIAN 'I'IIJ1! l'AltT" l~udicl'fI eighth ILxiom HI'ILl.c>s tlmt "tllll wholu iR gl'eILtc/· tlmn the JlIL/·t".m Tho filllt thing to notice nllUut thiR I'rincil'lu is tlmt. it is I'ILl'cly IISl'cl in t.ho J~lcmrn/•.,. (An (IXILlllpln of IIno of I.ho fow pIILc!C~:i wlUlf(} it (I(!ellt'R is tho proof ur Pl'Oposit,joll 1.11i.) Although it iH '1uutllll vCI'Y flcidolll, it is mther silllill\r to a formuln. which appC'ILl'ft at the ond of c)tlite IL nU/llbel' of Eudid's indil'Cct ILl'gUIll/lIlla. 'fhe proof of PmpUHi(.j1l1l LIl, fur (1XILIIlI'I!!, c'cllldlldeH in t.Im fullmving wl"Y: t.J1O fl~IRo sUl'posil,ion which is to bo rcful.c>cl is HhoWII to imply Hmt "_ will be eq1wl In _, tile le.lis 10 Ille yrm/er: whir" i.Of ab.mrd". 'rho I'hl'll.'JO quoted ILllOYCl (in itl~Ii(:R) WILlI IIH1Ht l'rnlmhly uRcd ill geometrical argument.s lung Lefore I,ho timo of Euclid. It is (Illite COlli ilion nut only in hill worltll ),ut ILIHO ill Ullmo of Autolyclls flf Pitl~nll,

117 Elcmell18, Bonlt J, COlllmlllll'ft Rnillli I" it.SI' Inul'.

l!cnw"I,lillllC'H

VII r: fO 6Aol' toil "I(!OII'

2!l2

l'AllT :I. TU" CONRTllU(,7ION 01' 'oIA1'IIKMATlCS

81.AII6: TIIB IIKflINNIN08 OF OltRKK NATIlKIoIATIC!!

who lived somowhn.t en.rlior. 218 HeiuCI'g, in his Latin tmusln.tion of tho J~fr/l/(,l/.t.9, 1~lwn.ys referred hael( to Axiom 8 when ho came to tho formula in IJllCstion. uo This is n.n n.lJpropriaLto l'Cfcrencc to nmlte, of coumo, Hilll:C IIIILh sLatcmenU! HILy very much Lim HILmo thin,;. NoverthdCHH, il. is worthwhilo to invCHtiRILto tho rellLtionRhip hetweml t.hem. Talllwry WII.8 tho first to point out UII~t tho axiom and the furlllllhL are nol identiClLl, hut he offered nn rCI~1 expllLllation of thiR flUlt. He HlWII18 to Imve thought Llmt the fonnor WII.8, somewhat llufol'tunlLtcly, ILbsLmctcd from tho Ia Ltc I' at IL rell\lively late date. 220 Apart from this, however, he hnd nothing much to RI\'y I~UOUt tho nmttcr. The first tn.sl( which fn.ces UR is to tl'y lo decide whether Axiolll 8 WliS clerived from the formula, n.s Tannery believcd, or whether it WII.8 t.ho ILxiom which cl~mo first. In my opinion, this question can Lo 1~lIsw(lred with IL fnir degree of certlLinty. If hoth p08.'tihiliLieR ILre exall1ined soriously, it SOOI1 becomcs n.PI"\.rtmt lhn.t ono of them is Illuc:h moro likely thn.n the other. TWII notions, 'tho wholc' Imd 'tho 1,ar", nrc cOlltmst.cd in the axiom (tlUw wholo is grClLLer UUl.Il the PILl't"). The fOJ'lnulrL, on tho othct· Imllli, involvcs Lwo entirely different eoneepLs; 'tho wholc' is replaced by the greater mul 'Lhe part' uy the Ius. It. is e:l.'1y to uccount for UICRO fluhstilutions on Lhe MHumption tlmL the formulll. WILlI obt.."Lined fmm Lhe !Lxiom, for it is ILlwl\'ys the CIUIC in practice UtlLt. t.ho wholo of IL thillg iB the grelLtcr n.nd its pu.rt the leRS. Bence it. is quite possiblo l·haL tho formula which wo find in Euclid developed from nn Clulier Cllle which denlt, with the narrower conccpts of 'wholo' and 'pn.rt'. So t"ere arc good reasona /01' tllinking tltal the axiom could I,ave !liven ri.~1'. 10 lite formula. If, on the othor hand, ono starts out uy ll88uming that tho formula ill 8UII\e unspecified wny Wlla wscovel'ed first, it is extremely difficult to cxpllLin how tho u,xiom might havo arisen subsequently. l'ho '~rt'ILIt!r' iB nnl. ILlways IL 'wholo', nOl' is tho '1C88' ahvays II. 'PILI"t.'. 'I'hcm BCllms to he aLsolutely no ren.son why t.hese more gonoral eOIlt'c'pl~ Rhould ItlLvo hoon TOJlln.cod hy nlLrrow('!r nncs. In my opinion, m Ii'CII' uxum"ln, in J)c Bp/lflertl'lltfle lIIol'Clur liller, (oli. ) 11I11.8ull), LiJlsiru) 1881i, l'n'IH'Hit.inn 3 nlluls: fa'i dell inrl.. ,) All (Il lino lIegment.) til AZ (IIDot.hor litul ... ·glll.. IIL). ,) /A,lan",!' tli l,tlCorl. 8n,(! inri .. d,1.i... no ... III SlIn, for nxnmplo, Vol. J, p. 26 ot. JllUlllim. 110 JlUlllo;re$ Ifcienli/i'lllcif 2, 64.

2D3

t.his i"diclLLes quiLc clearly Umt tho uxiolll preceded Lhe furmuhL. The areclul coulll ell8i1y h".vo tmnsfOl'med "Lho whole is groater than the III\.rt" into "t.he grClLLcr CILnllot cqual t.ho 11~R.'t", whcrell.8 the reverse 1'1"CI1'eduro iH ILlmoHt illcol"'t.~iv:L"lo. We "ow WILllt.tu find ollL wlllLt.lell to 1-110 fO"'"I1IILLionllfAxiolll 8 alltl when t.his t.oolc pllLce. The stlLt.cuumt tlmt "tho whulll is gl"ClLl.I~I· Limn tho P'LI'L" is at fin;t. sight KO ohvious tllfLt one CILI1I10t hcll' wondering why such n, LILI1ILllrut11 WILlI (:ollntcd 11.8 an UXi01lllt (i.e. 1L.'t lUI IL.~'tertion which could not be IL(:I'('plocl without l·oscrvILLion). Furlhorlllol'C, ono's flllrprise is (Inly illcrl~lIscd by tho disl~ovory LllILt Ellclill nml(c.'1 Vl't·y little use of lhis ILxiom; instclLd, he ofLen uses a prindplu which nllLy very well hlLVe bcen RuggeRLed by tho ILxiom hut is not idcntical wiLh it.. Euclid, it scoms, would IlILve dutil, ueLLm' to give t1efillil.ionR of 'grcater' ILnd '1('88'; fOI' Limy would have Leen more useful to him tlmll Axiom 8. The fact thlLt he IIcglectcd to do so showR how closely he followed hiH ori,;illlLl or, 1I10l'C IiItely, 1\11 earlier tm
III

TO) ..

,1/IIIJIII".

2!H

II1.Anc): 1'1I1( IIIWINNINGS 01' (llUmK ~IATllnaJATIa-

I'A 1l1' 3. TIII( (:ONlrfIlIlUl'ION (II'

MATIII(~IA"'WS

205

lIwl'll ILI'e threo groups of hodicH in a SLluJiulll (I,rranged u.s in Fig. 10) ;LllcI UmL one gwup (A A A A) is ILL rest, while tho other two (BBBlJ and CCCa) ILI"O moving ILl. Lho HILIIlO specu but in opposito directionH, I)III'"IIHO, furthorlllore, thaI. the B's and G'a begin to move at the sallie tilllll f'mlll t.lw )losiLiunR showlI ill tho figure !Lnt! tlmL t.lIOY continlle ill lIlotion until Lhe first B l'elLches tho 111.'11. A (011 Lho flLl' right) ILlId Lhl! find. C reaches I.he fil'Ht A (011 tho fILl' left). Sinco tho /J's and C'H 1L1'Il1,mvclling at Lhe samo speed, they will elelu'ly l'elLch their rcspecLivll J.(lIltlH ILt Lhe samo time. Now, if wo WILllt to melLSuro the distancc (!ov('n'" Ity Lhl'HO LWII gl'UUpH of bOllies 01', wlULt ILUlUUllts t.u I.he H/LIlIO thillg, UIIl time which iL Loole them to cover Lhis distullce, tho mngniLutin

in CJ ucation CI~II he I'cprescnted by A A, for in both elLllCA the first hody hILS pussed exactly two A's. InstCl~d of rnen.~uring this period of time relative to the bodics at rest, howevCl', wo could just ns wcll IIWI~ure it rell~tive to thoso in motion, It. would then bo rCl'rcHcnlcd by JJJJBD (or GCCC) because tho first C haa pnssed four B's (and \,jl~O vcnm), Thus wo arc faced with a pltradox; IL fixed period of time elm he memHll"cd either hy two or by foul' lel.ters. Honee we IniLy cOlldudo that "tho half-time is equal to its double". Before dismissing the {~rgument in mom detl\il, I would lilw to point ouL olle notlLhlo fCILtUI"O of it which 11U.~ belln ignored hy previous C(JllllllentILloI"R. 1 I~m referring to t.he flLd that, ILl though tho letters A, JJ lLIul G ill l~ig. 10 arc SUPIJOsed to dcnote bodies which ILro either Itt I'l~KL 01' in motion, IlILlf wu.y thl'Ough tho lI.rgumcnt they n.re used to denoto certlLin /f'lIyllts u.s well, It is clelLl' L1l1Lt, if tho Limo which the firRt, IJ (or C) -t:Llms to pll88 two A's is represented hy A A, thcse I wo lctters (A A) arc being uscd to denote not only two hodics ILL I'm;l. hut 1~ls() !L cortlLill le/lytlt, JlILmdy the distlLnco tmvollcd uy (or G) in }JlLIIsing two such bodies. 011(\ of the rea.'101IS why t.his flLCt is ill1(1ol"tlLIII. iH UtlLt it t'
m cr. ,I. I.nd..,li.,(', Hr.IIIIC ,1,1 Jlfc!lllJl"!lsiqIlC (!t tie .1ITomla 18 (10 III), (1(1. !I
IU Al'chywlJI (ill H(leLiufI, /).! i'lstitutj'",e "",..;m, .,,1. (l. Frin
0111' pl'evious observaLiolls ILlrclLdy mlLlm it seem rct~ollable to COI\jccLuro thll.t Axiom 8 WI~ lILid down to Iwoid pILl'Rdoxico.l conclusions lilw the nne mcntioned ILlJove. Tho plLro.dox, however, is concerned wil.h I,he IULtUI'O of timc, whereas Axiom 8 is a purely mall,cmalical (HI' geometrical) principle. AILhough Zello's argumcnt undoulJtedly Imll SlIlIIC influence on mathelllat.ieio.ns, wo o.ro not yet ill II. positioll 10 SIL,Y wheth~r it was tho direct Co.uso of their decision to fOl'lllUJ:Lto this I~xiom. To answcr this ItI.'It qucstion, we will need to eXILmine tho det.ldls of tho ILI"gllmenL. 'l.'hl~L plLrt of Aristotlo's text which tlmLls wiLh Zeno's ILrgumollt p""Helll.a ma.ny philologiclLI difliculLics, some of which luwe sLill 1I0L Iwen solvcd saLiHrlLctol'ily.223 Novel'thele8.'I, we CUll quite el~ily l'eCOIIslmcf. ZOllO'S tmin of thought. Our tu.sle ia facilitated Ly a certain diagram Wig, 10 helow) which is to bo fOllnd ill Simplicius' commentary 1111 Aristotle. This diagram (which Simpliciufl borrowcd from 1m carlier ('lIlIIlIIonLI,Lor, Alexander of Aphrodisill.H) SCI'VCS to iIIustmte Lhe ILrgu111('111.,224

AAAA IlIJBB - .. ~-aocc l~ig.

10

A e(:ol'(ling to Al'istoLl(!, Zello I'ClI.fI(JIlcd \ I~ follows: Stl ppose thaI.

n

"III"

2!l1l

!l1.An6: TIII~ IJROINNINOS 01' OmUtK ~IATlllmATICS

lineH). In view of this, iii becomes evon more likely IiblLt Euclid's eight.h axiom was formulaLcd in response to the para.dox. Lct us now examine the 'fallacy' in Zeno's argument. AristoUo WII.'! firmly convinced that tho enLiro "rgument was incorrect, ILnd hiH pupils tried even more zealously to ..efuLe it. Eudemus, the first hilltorian of mathematics, was perhaps the harshost. critic of Zeno's rCll.'loning. According to Simplicius,22o "Eudemus asserts that. Zeno's ILrgument is totally absurd because U conta.ins a blatant.ly obvious fi~lIncy." It Rcems tha.t Zeno ha.d littlo lucl( with hiB plLrndox in ILnLi'luiLy. Even Loday, many scholars ocho tho judgment of ArisLoLlo 1\1111 his school. It is wort.hwhilo, therefore, to take a oloser look ILt thiH 'fallacy' as wcllas at the ingenious idea which lies behind it. Zono WI\S right in thinldng that, when bodies move at. a constant speed (c), Lho distunco which Lhey cover (8) [md tho time which iL LILltes Lhem to cover this distanco (I) C'LIl be represented by a single ICllyllt, whosc plLrLa will correlll'ond to mn.tching distan('.cs 'Lnd timcH. However, his mistalw WI18 to ignol'o tho fact that theso bodies will l'O\'cr twice tho distlmco (2s) in the samo time (I), if Lhoir specd is douhled. It is clca,r that if tho B's and C's are moving in opposiLo dil'CcLions at a constant speod c, thoir spced rollLtive to one another will be 2c. (They ILro only Ll'II.velling at speed c relativo to bodies I\t rcst.) Consequently, the distance which they cover in a fixed period of tillle will also double' (28 = 2c . t). This means that, oven though Uw 'half-length' AA represents hoth distanco and time, the 'wholo lellgth' BBRR (or CCCC) can only represent distance. It R(l(lIllS Llmt Zeno cOlJlluil.8 I~ twofold error in his rOILlloning. 011 t.he ol1e Imlld, he flLils (,0 montion that IL spced whieh is supposod Lo I'Illllaill 'co"sLILI1L' suddenly doubles, ILIl<.l 011 tho other ho cllLims LhiLt a ('ort:Lin longth 1'01'rtmllllt.s both distalll~o 1\11(1 timo, whon in relLIiI.y iL is only a faithful representation of distanco. This is tho 'fa)JILey' which Aristotle discovcred in Zello's argument. In my opinion, however, tho IJreceding account does not do filII justice to tho argument. It is misIea.ding to tallt about a 'speed' c which is 'doubled' I\nd bocolllcs 2c. For this concept is nowa.dILYs defined as tho ratio of distanco travelled to required travelling timo or, in O(,JWI' words, ILl! Lho distlLn('.Q (lovemd in a fixed amount of timo. no ~illlplil'illll 1111 AI'iIlLoU,,'s

PIIY"icR, 1019.:12fT.

l'AIlT 3. TIII~ CONSTllUCfION 01' M'\T"V.~IATII:S

2D7

Such a definition presupposes tlmL boLh dil!LILnce ILI1d Lime CILn he broken up into units and tlIILt some measure of length or dumtion can be lL88igned to these units. ZOllO, howe vcr, held n.n entirely (lifferent viow. He is criticizl,d by Al'istoUo for llIaintaining thaI, timo is composed of instants (viiv, literally 'nowfl').227 Similarly, he roglLrlled space as a eolloction of dime'l.8ionle88 1)oinl8.Z28 11oth tho 'half' and tho 'wholo length' woro considered by Zeno to ho infi·nile se/ll o/poi.lfls. Henco, when ho lL88ertcd thlLt the Imlr-Lime WI\8 eqlllLl Lo its duuble, ho was I~otually eqlllLt.ing two infinite sets. If a modern mathe\lIlLtician wauLcd to ostablish this cOlldusion, ho would probably ILrgue 1\8 folluws: Supposo that AD is twice lLa long 1\8 CD ILnd that boLh theso 10ngLhs consist of infinitely many points. Wo say tlmt AD is 'equal' to CD if thore is a one to one correspondence betwoen tho points on An ILml thoso on OD. The llxiHI.cncc of such IL correspondence CILn be domonsLraLcd by d\'ILwing Lhe lines AC and JJD n.nd oxLending thelll llllt.il t.lmy inLcmect ILL al'oinL E (sen 11'ig. 1 t). Now, let x he an ll.I"ilit.rn.ry point on A B; wo c()rrcllLLe wiLlI x Lho point ~ aL which tho lino Ji)x inLcl'seC!l.8 CD. SimillLrly, wit.h em:h point I' on CD we COlTcliLte Lho point 11t obtained hy extclHling Lhe lino E" until it. inLcrseets A JJ. In this WlLY, evory poinL on C]) elm bo ILssign('(1 to IL uniquo poin!i on An ano vice versa. Considcrillg Lho flLeL tim!. CJ) is shorter tlmn AB, this sitlllLtion ILppcars to be somewhat Illm1.(luximLI. It is dcscribed in the Inngun.go of Bet LhcOl'Y by saying !ihn.t An lLlld CD hll,vo tho same c.urditUllily even Lhough ono (CD) is IL propcl' suhseL of Lho oLher (All).

FiJ.:. II n7 Arist.otlo, rl.ysics, Z2:J!lh:lO. U~ If wo 1l0ll1Jlllro J'ltysil'.JI y.!I.2:m IUlII 2.22:\,,21, it ill "pp"r!!"L \.Illlt Z"/I0 Lhought not. only of Limo lUI mmlpo81,d of infinit.nly many 'nowA' (mllmnnl.a WiUIOllt c!urntionR) hilt IlIAO of "1'"1:(1 lUI 1l01ll1)(IR... 1 of infillic..,ly rlllmy l'oill11l.

2f1R

l'Alrr 3.1·111<

1I1.AII6: Till< III
l'ONlITIIUGflON

01' MATIII-:&lATJ(:S

2m.

So iL IU'llms Lhat Zeno wns righL; if Lime is mll.llo up of infinitely

point..'J ILntllinos: ho MScru!cl (Dcfinition J.3) UmL "tho oxtremities of IL

m:\ny illsLILOta (nows) and 'equality' is intcrpl"Cted ill IL sot theoretic liense (i.e. I.L8 C!lLrdinal equivalence), thon tho "ho.lf-t.imo is equal to its ,lauhl,,". No doubL, his main purposo in devising Lhis plLrlwoxicnl ;l"~UIIIIIIIL WIUi to show tho inconceivability (or inconsistency) of 'Illotiou', 'spaco' ILIIlI ·Limo'. llut ho o.lso 8uc(:ecdcd ill demonstrtLting I.h:Lt Olll' illtuilive cOllcept of 'equaliLy' elLn unly be applied Lo finite seta. AristoLlo's account suggests Llmt Zello WI.L8 con corned exclusivcly wiLh tho IIn.tuI'O of timo. In my opinion, however, the Eleatics must Illwe gellemlized tho above lLrgumont to cover the ClLaO of geomctrico.l II'II9'''S ILR well, for it is clenr that tho letLers used to denote hodies He ..... e ILl. Lho same time IL8 a measure of distallce. Zeno's argulllent c10l'l~ll(ls upon showing thll.t tho o.mount of time tlLlmn 1.0 travol 1\ (:(, .. laill fixed disLlLnce is '1IILIf of 0. longth' or 'twice that IULlf' according 10 wheLhur the disLILOce is measured by bodies 0.1. rest or bodies in 1lI0tion. Nevertheless, the conclusion which ho dmw fl'Om iL is valid ro .. Rl'llell lUi well us for timo beco.uso tho structure of Lhe one is simiitLr til t hat. of the other. (Points in SplLCO corrospond to iJ\stlLnLa of timo.) III otlllll' wurds, Lho nrgumont nlllO esLahlishes Llmt tho ho.lf-lcll!JI!, (IIr diHLalH:c) is 'oqUlLI' to ita douhle. There nrc Lhree fn.cta nhouL tho foundations of Euclidean geometry which inllit:nto LllILt tho blLRic principles of Lhis sciellco wero laid clown l'XIII'Cs,'ily to nvoid (.ho ·pn.mdox sLnLed ILlmve: (I) Euclid's filllt geometrical dofiniLion relLlls "A point is that whi,~h hll.'i no part." 'fhiR mOiLns LlmL 1Ioilll.'i ill gculIlct.ry wero OXILCtly :umlogllllfl Itl Zl~nn'8 inslmus (vVv) uf Limo. (2) We ctltuLocl earlier n. plLSS."LgO from ProcluS= in which he defines "the lilw 11... Iltc flowing 0/ a point untl ... tho straighL lino n.s its 1uti/orm IIl1d II Ilt/CIJ;fltil1(/ flowiIlY". '1'h080 dofiniLimlK, howovor, ILI'O noL to ue fuulld ill Lho ElemcIlls. Thoy give tho iml,rcssion tlmt n, 'lino' is coml'oHc·d o[ IlOi"ts just WI IL 'uu/IIIJOr' iR cOlllposed of unit." nnd ZOllO'S I"Lmclllx ILpl'lil's Lv such Iinm. Itl' lungUls (i.o. to 'r&fi."ile BCtS 0/ lwi".t.'i). 1~lIdid, who WILIlt.ctl Lu IIS0 LlIO cOIII:4'I'L "f 'illfinity' lUi IiLLIo 1\8 possiblo ill gf'IIIIWI.I·,Y ,:!2oa WW(I n. c1iIT"I'l,"L clofillit.ioll "f Lhe mlnt.illllHhil' hl!LwmlJl

Ii Ill' nro points". (3) Axiom 8 stlLtes 0.11 eml'irielLl flLct which WILR obtaincd by I\bstl'acting from cx}Jcrir.llcc lVitl~ finitc .w~18. Othcl'H of muclitl's ILxiomR ILlso scom to 1m CtlIII!(~rncd wiLh hLying down proPCI't.il'H of 'Cqll11lity' which hold ill gmlcml ollly [UI' fi,~ite sr.I.."

::0 :-;•• " II. IR1 "Iouvl'. :.... I L iM WlII·tI, """ullillg Irrujl'l\l"" ill\.t,rl'rt!l.nl.iull "r l'/li/cbu., 26,,-h (,I'IU\.o1l1l ,. In III1ILc·lIInl·i,," II..! 1111111110 nlltillu'): "Si pllh ""gllirn, "Imollo dn 1111 illlpor\.ulI\.o

3.25

A COlltl'I.EX Olr AXlOlItS

In tho I'I'OViOllfl chapter iL wu.s ILrguccl LlmL Lhe formulation of Axiom 8 should lJe rcgtmlecl 1\,'1 a responsc t" Zcno's I"Ll'aclox ILlJOllt 'C~(II\lLliLy'. Wo now wILuL I." C,xlLIHino IL li,w 11101'0 of E\lcI~cl's n.xi~)JlIR. HcilJCrg, lUi iK well known, rejm:lecl Lhrce ,,[ Lho fimt elghL ILXIOIIlS (I he fourth, fifth ILlllt sixt.h) lUi Sl'UriIlIlH. II is ILuthoriLy for rt'jccting Axilll"H !i ILIIII (j W:Ui ]'I'OdUH,73D who l:oIlHiclC\'('el I.hesc l'rillc:ipl('s lo he SUl'ernUlIlIH.231 11l,lecd, lldllcrg even UIIlIIghL \.ImL ho IlILd djR('overed illl(~rl'uI:Lliuns ill I.:n nml (.:18 heclLuRe tlll'R(' I'r()pO!~iLioIlR COllt.ain ILII nlJllll/it lituml reHIILt.olllenL o[ Axiom G. I'rucluK ILml llcihcrg IL ..e l~erLtLinly right ill Illaintainillg LlmL tho ILXiolllR which they rejecL nrc nnL requircd lIy Ji!ucliu; the whole o[ his gClI/IIctry 1~ILn he huilt Ill' wiLhnut II.'i8tllllillg Lhose I'rilll'il'lcR. H.ILthN' UIILII diRCUKHillg LlIC qucstion ufLhcir' ILUUWIlt.icil.y, however, let. us firsL IL\.I.(:IIlI'L to fincl uuL what, Icd til their inc~orl'0ml.i(}n inul the JUr./IIclIls. TILllltel'Y hdilwml t.iIlLL 1\11 of l~uclicl's I'rillc:il'lc:s WC!J'e l\hsll'lLc:LI~II fl'o/ll Lhe '1.1~xt of Lho gl(!)"r."'.~ II-'I It ki'/,ll 01 nlll!,.t/wllyltt ;7'J2 huL hiH VillW HCOIllS tu he' c~OI~ll'let.dy unu'I\ILhle. l\t.wh o[ wlmL wo find ILIlIOIlg1lt. I he f(lul1,IILLions of t.he J~I"lIil!Il/,<; is Lite hoglLl'y of n.l1 c'ILl'lier (.m
:Jj-

P"II\.II viH\.n, III "villi!,,") ,Inllll IIIIII.t-II\1l\.j,·II ~rn"" ,'IIII11l il MIIIlI!.· ... ·llIi eli ··I,iK•."li .Ii 1IIIIIIul.I.II ""111..." l·i .. n"iLc •. 111'1'11'.. 1.1, P"RII" 1'1,,1.'11 .. i.. fT .. r"", d",ln 11",1 .. ·.. ",111''' ,\ il dOllliniu 11,,1 liniln." nn I !Iii. 26fT, (ill J<'l'imll .. ill'R ,·.!iLiOIl). :JI ~." 1':lIdi.I.'ft! 1':1",,,,,,,/,, (.·,1. I. I •. 1I.·iI .. l.iI'Ri .... IRR:I 1!IIIt), \,,,1. I,

·..,!.

1'. Ill.

,n 1', "'" .. II"..Y,

,11 .'"",ireN .'I·;lm';fr'J/lr.~ 2, 1i:1.

:JOO

I'ART 3. TIIR CONRTRU(''IION 01' MA1'1I1':~IATICS

SZADO: TIIY. lIROINNINOS OF OIlEF.K MATIIIU1ATICS

AI! flLr 1\8 1 know, tho scientific literature of antiquity mentions only (1110 thing which could p08Kibly have prompted mathematicianH to JI~y flown theRo o.xioms. I ILm refol'ring, of COU1'80, \'0 Zeno's pn,mdoxiclLI d;~iJll LhlLL "Lho lIal/-timo is equo.l to its double". Wo SILW in tho IILBt chapter that this nsserLion WM ilIustm,ted by moans of len(Jlll8 (viewed u.s infiniLo sets of points); hence Zeno's argumont o.lso established thll.t :L Iml/·length could be 'equal' to its double. The 1l1l1thematicians of antiquity must surely ho.ve ILttempt.ed to refute Zeno's claim. Eleatic doctrino, however, prevent.ed them from appel1ling to everyday experience or to the evidence of the senSCR. l~uI'Llwl'mOl'o, thero is noLhing wrong wiLh Zeno's argument, so they ('oultl noL ho.vo presented a counter-argument to show that his rCIUlollillg was flLlllLcious. A 'refutation' such ILB Aristotle and his llUpils ga.ve. which simply disregarded Zeno's fum)u,menLn.1 insight into tho IIILLure of ' ill finite seLs', would not even IULve been considered by anyone who had the slightest undm·Rt.lmding of ElelLt.ie difLlectie. Therefnrt" it HCefJIS, that the pamdox could only JIILVO beon avoided by laying fl(lWII ILxioms designed Lo bloelt Zeno's ILrgumenL. Of course this is ttt 1"'('H(mt. nothing moro LlULn IL conjecLuro on my IIlLI·t; noverLheloHlf, it. iH worth wLking seriously hcclLuso it Rheds now light 011 at ICMt /ollr of ]~uclid'R axiomH. Tho principleH in queHtion Ino listed below; if ~lHlO'R plLmdox is l(Cpt in mind, they seem to form a coherent group or RI.ILlemcnts whosc conlmon purpose is to dony tha.t a IwI! can ever bo ('IJIIILI \.0 it.a double. Axiom Ii: "Things whit:h ILro double of Lho SILmo thing arc equal Lo CIllO ILllOthCl.. " Axiolll 6: "Things which aro /,alves of Lhe same thing are equal to one IUlothcr." Axiolll 7: "Things which coincide with ono another aro equfLI to OliO ILlloUler." (The lengths involved in Zeno's argument did not coineido; novorl.hl'lcss, t1wy wero said to bo 'equal',) Axiolll 8: "The whole is grenter Lhan the llart," Of COUrH!', Axiom 8 wit.hout Lhe other three suffices for tho develop1I1!'1l1, IJfll~uclidlllm gCllllluLI'Y. JIILI'llly ILny UHO is made ofAxium 7; JU!IH'C it could be rejected 011 much the same grounds Il8 Axioms 5 allli n,n3 My opillion iH Umt. JI~udiel includ(~cl Umtlu f.. ur axioms bOCILIIHIl

it WM traditional to do so; thiB also oxplains why he gave definition!J of 'straighL Ii no' and 'plano surflLCo' ILllIl illLl'od~ccd such terms. IL.'J £U(!OP'1XEt;, eop{Jot;, e0l'{Joe,Mt;, T(!/n)'W(]U ILlld no),t",;),F.V(!O (all of winch, lLro HupernuouB MfaI' M hiswOl'lt is CtJlleel'lIed).~3.

• Our st.udy of J£uclid's axioms hM led UH to the following conclusions: The n.xiomB for oqualiLy which appoal' ill the Elemellt., are Cll1pil'il'al ILSHCl'tiOIlS bu..'Jcd upon expcricnce with finite s('l.'1. Their vlLlitlity is glllLrlLllLecd ollly by sensory evidenco; hcnen they could noL IIILVO heen ILccopted hy tho J~IOILt.il'R. who rC(l'lirccl that all Iwowlcdgo be u\'tlLinetl by purely intellectlULI me:LIls ILllel without ILppel\li~lg ~o t.he HCIISCR. Thesc principles wero origirllLlly clLlIed dl'1mmds (.crElwllOrn) IlI'clLUSO Lhe other party in a clin,lccticlLI clelllLte IlILd rescrvlLtlons about ILCC('pting thom 1\8 IL blLBis for further illquiry (Jr, ~Il, oth?r won~, iJeciLuRC Lheir ILCCOl~I.ILIlee could only be demmlded. (Euehd HILXIOIllB WCl e no I(~KH incuml'lLt.ihlo with 1!!l(,ILt.ic LtliLching t.Il1Ln th~Rn llOHLuhLtes \y(lI·o.) Afler J>In.to's timo, howevOI', tho cMOlltilLIH of EIClLLio dilLlecLic \Yore lin longer very wuH ulldersloml; hOllco the ILllCiont. term (lEiwllo "CflUil'etl IL now mooning. Since it Imd always heen used to refer to IL grou~) of principles which, from tho viewpoint of cummon sr.nse, wero eVIdently valid, it OILmo now to denote thORO RLlLtemelltB whose. t.rI~Lh WIlli " accCl)ted as a maller 0/ cOII,rse." (J~f1'ol'ts wore even mmlo to JlIsLlfy tho new menlling on otymological groundH.) Aristotle's dismiMILI of Zeno's paradoxC8 M mere sophistries undouhtedly helped to make this change possible. As 0. II1lLtter of fact, Aristotle seems to hlw~ hc('n the ancient world's leading proponent of the idelL tlULt IIULLhematlC8 h;\.~ t.o be blLSed on "evidently true, indispuLable and simple foundILtions". Wo CILn now Reo why tho old name "Elf/'llaTa wu..~ roplaced in tex~ of Lhe Elements by the 1'lll'l\llO XOlvallv,·otat. TlLIlIlel'Y helievcd t1I1LL It. l"'I:ILmo nm:ellllal'Y to IlllLlta this BuhHtitlll,ion when LlIO StoiCR heWLn 2J liNing Lho wortl dEiml'o to melLII ILny dechLI'ILLi YO Ht.ILtoIllOIlL. & ~ 1\ lilY opinion, however, ho failed to mention tho IlllLjor l'CILBon why aXlOmala

cr. P. ']'unnory, lIU",. /i(:i611I, 2, li40-4 IIlIel 4R-fl:t. u. '''i.I., 2,62-:1. R"uuIRu Chupl."r 3.21 "" .. v ...

lSI

rn CT Ie

VUII J~ril.7:, 01'.

,·il..

(II,

2 "I.nvu), pp. 76ft.

301

--_._---_.... _----- - . -302

II1.A"~: TIIK IIJ':CIINNINIII! 01' OIl.:I{1\. AlATIII!AlATIL'S

I'AIIT 8. TIIR (!ONlfl'IlUt,"'UIN 01'

gl\ve wny to koiJUli e1tnoiai. ThellLtLer Lcrm wo.s preferred to tho former 11I0sIIy heclLusc it betLer described Ute new (Aristotelio.n) conception IIf'ltxiOIll'; one would be evclllesB inclined to c.loubt the 'evident tl'uth' of ILxiol1ls if they were clLllcc.l common 1lOtions (xol"all"Jlolat), 1.'hus, in th(' l'ourRe of time, thc dialecLiclLI origins of hoth tho term d!{wpa ILIIII !.I\I! Idnd of principle which it hnd been used to denote wero nlmoHt COlli piett'ly forgotten.

:1.2G '('1m 1)lJ"I"Jo:lmNCI~ m~'J'\\,lmN ]'OS'J"Ul,A'mS ANI> AXIOMS Olle (If t.ho factH which hll.H emerged fmlll our disCUBSioJl in Lho 11I.'1t few dmptl'rll is (.JlILt ailenmlrl nnd Clxiomala were synonymous. Furtherilion', I.ho mallller in whieh they wero used by evel'y ILllcient maLlleIlIat.il!iall ('xcept l~uclid HUgg('st-H LltlLt Uume t<'1'1II8 were illl.(:rclllLngealllc wil h I~ lIulllller of other words Imving to do with the foulldations of III1LLlmlllll.ticlI. In the latcl' liLcmt.ure, however, ILtteml't.a were mlLllo 10 c1iHLillgllish lIeLwoen tho variolls sorts of unproved l'rillciplCH upon whidl JlIILI.hellHLtics WIlS LILIIl·d. For eXlLntplo, noole I (If Archimedcs' On I/'I! S11/'cre ruul CyliJU/er begins with two lists of slIch principlCH. Tho 1ls."('I'I.ioIlS ill the first list. am called d~ltnll((TU ILnt! nre obviouHly arfini/ioll,", wherells those in tho second lLrc called ).ap{lav&ll£va ILlIli :<1"'111 to Le rclnted to Euclid's postulates. (For example, Archimedes' IirllL ).ap{l(Jvo,'E"O" reaus: "Of ILlllines which iULVO tho 81Lme extremities (.Jill sl-might line is Lho 10ILIIL." ).ap{lrJ"w lJ~ nu,' TIl miTu nleaTa fI1l O',u,t J'(!III I UilV lAax{unJl' Elmt nil' £lindav.) ' Of ':OllrIlC, no distinction of tho Idnd dCHI!J'illcd ILbove WILlI eV(l!' ut.Hl'l'vl'd ('ollllisLeIlLly in ILntiquity. Even J~uclid's thl'C('fold divisioll of I'rindpll~8 in!.u defiIJ.itioIJ..s, postIL/ates ILlld flxioms WILlI not 1\.(I0l'L<~d Ity ILlly other ILncient mathemo.tician. It is truo tluLt AristoLlo dovised It R)'lIlem of arc/lal which embodied his own views ILhout wlllLt distillglliHhml din·crent. ltiml" of blLllic l'l'incil'Jol!. In my opinion, hOWOVOl', 1.1", hist.orian of n.llciont miLthellllLtics hlLBIiLl.Jo L<) 10lLrJl fmlll Aristotle's t.1"I!ILLIIWllt of this IIllLtLer.

'X

Produs mnde soveml ILLLempts to oXl,I;Lin tho differenco betweon ILl II I axiOlllallt, buL IIUIIO of thcso is olltircly 8lLtisfu.cLory f!'(J1II IL hililuril!lLI point. of view. Without going into the dotnils of his CXpIILlIal illllK, I WUIII,1 lilUl 1II01l1.ioll two of I.holll hOi'll.

"i/l'/11I1I"

I.,

MA'I'III'.MATJ(~

303

(1) 0110 of Proehl II , remn.rks which BcelllS Lo belLr on Euclid's clU88ificntioll of first ilrincil,les is the following: .. , .. postulo.tes are peculilLr to geometry, while ILxioms ILro common to 0.11 sciences thn.t delLI with qUlLntity ILlld mngnitude" .2311 1~\H'lid'8 110stuiules (ailemala) nrc indeed purely g('omctl'icnl naserLiolls, J),'sl'ito whll.t l>rochlll SIl.yS, however, the axiom,., which nre found in the Jt:/cmellis do lIot ILII IL}Jply to ILny Bciellce dealillg with qtumtil.y lLIlIl IIl1Lgnitude. Thero is ILL lell.'1t OIlC (If Lhl'lIl which ollly mn,ltcs flellse within t.he context of (1'I:Llle) gcollwl.I",Y.:!37 I lun reflllTing to Axiol1l 7, "'I'hi II~S whil~h clli lI(:idc with ono ILJluLher ILI"e equn.1 to one n.notlwr." So, if tho ILhove distinctioJl is tl~lwn Roriuusly, somo of Euclid's axiOllllllft n.re not rCl~lIy n.xiolllR tLt ILlI. But this lIlon,ns tlmt Proclus' words dcscrihe I~ mol'O 01' les9 11.(!eidentnl f(mt.ure of the principlcs found in the ElemcIIls .. t.hey do not explain Euclid's syste-Ill of cln&~ificlLtion. Perhn.ps ]~lIc1id did not ulle t.ho t.crlllR a ilrlllflla nnd a:rim"ala in I.he WILy tlmt I'I'0clllR' "CIlIIL..t1 BuggeRts, ILnd it Wll.~ not until Ill,ter Llmt Lhe equlLIiLy ILXiUlll1t (or n,t IOILIIL 1II0st of Umm) woro scon t.o be II. foundll.tion fol' ILrithmet.ic M woll.:1lR Of COtll'llll, nllwl\.(.);LYS, I~l'ithnwl.ie is blL'ied upon IL mOl"e comprehollsive colled,ion of such ILxioms. (l~0I' exnmpl(" 1·~II('Ii,1 Ilocs not SLILLo !.Imt erttllLlity IlILs to bo reflexive lLnd tl·u.nsitiv(" alLhough 11101 Ie I'll IIl1\themlLLiciu.ns usually do RO.) Tlaifl, hO\yev(:r, is IIl1t to filLy Llmt 1~IIc1id'R equlLliLy ILxioms pllLyed ILny rolo ouLHido geomol-ry when tllf'Y wcre first formullLted. As n mn.ttcr of fact, I clLn find no tmees of Lhe conscious usc of these principlos in elLrly PyLlIILgorclI.1l ILrithmetic (flool' V 11 of the lClemeJlI,'f, fUl' OXILllIl'lo). Berol'Cl "olll'huliJlg this (1iS'~UHHi(l1l ufl1rudus' remlLrlt, I would lilm to mention fllIO further l'ruhl(!l1I whidl it miHes. Tho rOllllLrlt suggCIILs tltlLt Uucli,1 limy Imvo cllI.Rllified hiH prillcil'lca ILCcordiJlg to their degreo of gmlt!mlity. 111\.(1 ho dono flO, howuycr, ho would undouht.cIUy hn.vo pl,"~d I.IUllllllre gellcml axiol/wla before tho purcly geomotriCiLI ailcmala .. yot in t.ltO Elemenl" this ()n.h~r is revelllcd. (2) AIIOLh(lr cxpltl.llII.tio/l "Irorod hy Pl'w'hlH iR LhaL tho ,lilrol'Clwe betwconaxiomaltt ILnd (IileJlltUa corrc.fljlOlulR t.o t.ho one hetween theorems

l~rit',II .. ill), I R2.61T; RIll' nl,lft fiR. '1 IT . It. mBy bn t.Imt. Axiom D, whid. iK Vllli,l nnly fnr W'(lllll't.ry, ill "puri",." IHThi8 WII8 nlcogni:wd Blre'rufy by AriRtoUII (/'fJ .•rr.riflr Alla/ytir.1I I.IO,'1611371T): ur, It. V,," Fril.7., (II'_ !lit.. (n_ 2 Ilhnvtl), p. 21i.

UI I'roduR (.111.

III

304

I'AIIT 3. 'l'IIK

fl1.Au6: TII/~ IIKCIINNINOS 01' OIll':KK )IATIIKbIATICS

(,?t:W(!'l/ UlTU ) and l)roblellls (nl!of1A~llaTa). lie enlarges upon this idea in tho following pn..'I8Il.go:230

"A postulate prescrihes UIILt we construct or provide some 8imple or ClL.-.ily grlLBped object for the exhibition of 0. eharacter, while ILIl axiom IIMerL'! some inherent o.ttributo that is known at OI1CO to OIlC'H ILUditors - such ILB that fire i8 hot, or some other quite evident truth ahout which we Bn.y that they who are in doubt. lack Bellse orglLlls or must be prodded to usc Ulem." 'l'heRo words nre instructive because they reveal tho kind of distillction LllILt WII.S drawn betwcon aitemala al1d axionmla nfter Euclid's Lime. However. they alsu revelLI UlILt at this late daLe the true origins of thclle two groups of pl'il1eiplcs had beon forgotten. Fl'Um our previous diBcu88ion we seo that Euclid's postulates differ agaillst tho ElelLtic doctrine of the impo88ibilit.y of motion, lLnd thoy Hervell 1.0 guamJllce the existence (i.e. tho (,-Onstructibility by means uf l'lIlm' 1~lId eompasscs) of certain goometrical fOl·IIIE1. The axioms, Oil tho other hand, were designed to ILvoid Zello's pnradoxCB, and they laid clown empirical f'LOLa ILbuut 'cqunlity' which bad been lourned fmlll ('xl'lll'ionce with finite BCts. Most ofEuclid's furulamenLuI principles wei'll JlI'OIJlLhly to.lcon over unchanged from his I'reccdo88ol'8 (i.e. fl'om UIOSlJ el~l'licr nllLt.!lcmlLtieilLIIB who had themselves compiled boolts of 'l~lcmenLa'). So, acting in ILccordo.nce with tradition, he retained tho (liHt.il1t:lion hetween axiomnla and ai/cmata even t.huugh it had 10Rt 1Il1I('" lIf its originlLl importo.nco hy bis time. 3.27 AlU'rUlllln'lO ANI> OKOltlWfRY

WllILt we ho.vo lell.l'Iled 8U fiLr in Purt:1 of this boolc CILn be summnftlllow8. ElelLlic philosopby hnd 0. decisive influence on the development of dcd\l(:tive mo.themnLies. It WILB tho teu.ching of PILrmenides and ZOllO which (ill Lho fil'st IlILlf of the fifth cclltury, most l'J'(Ibl~bly) illlluccd ri~('(1 ILK

01' MA'I'IIY.&IATICS

305

mathem:LticilLIlIt to turn ILWILY from empiriciHIII ILIlt! to adopt the view UIILL IlllLthonmLical ohjccLa exist only in Ihe mind, Tho 1II0thot! of indirect proof, 80 typictl.1 of mathonmticlLl rensmning, o.lId the theoretienl /ollmirnion.'f of ma.thellllLLics 11I\d their origins in Eleatio dialectic. li'urt.hcrmorc, ILII tho term8 which were used to denote fundn.mentlLI nllLLheumticlLI principlos camo from thel'o tl.'! well. Indeed, ILB we have H(~ell, Euclid'lt very first arill""elical definitioll (VII. 1) pre8upposCs the 'theory of tho Ono' (llel1ublic VII.525), which ilt in 0. sense identical with Parmenidcs' doctrine of 'Deing'. 'l'ho definition of 'number' (VII.2) ILB "IL multitude composed of units" marked IL depo.rturo from Eleo.tic lclLChing. It is true that ILriUuneticians treated 'numhers' in muoh the so.mo way that the EleatillS treated 'Being'; they empllluiizcd thll.t numbers were ideal entitics which did not IlILve visihle UI' tlLllgiIJlo bodiclt Imel could ollly be apprchended by tho understanding. Arithmeli(:, howovCl', rcquired the existencc of a 'plurality' (or, ILt lelLHt, of ILII 'it!cal plumlity'), whcreas ]1~lcILtic philosophy admitlcd only thc exiHLonco of tho 'One'. In an ClLl'lior chapter wo Haw tllILt the JmelLtic II1'oblolll of 'eli visibility' toole 011 II. new menning when numhers came to ho rogart!ed 1\8 multiples (If the 'Ono'. This HCCmlt to hn.vo given rise to one of the most important allli hasie problems of prc-EuclidClLll arithmeti(l, na/llely tho problcm of divisibilil.y for numbers. When tho foundntiolls of geometry wero IrLid, tho giLl' betwecn mlLthcnULtiCR alld Eleatic philosophy widclled. The gcomelcrs of preEuclitlelLIl times IlILd to forllllllrLl.o 1'0slu/alt'., which (:olltmdicted the ]~IClLtic docLrino (If tho 'illlllOHHilJility of mot.ion' ILJlCI UXiOIlUI whidt dmracterized Lhe rellLLiun of equality hetw(~(!\1 fillite ReLa (i.e. which l\Voided Zello's 1'ILrlwoxca). J r thoy hlLd "ut dUIIO HU, it would hlLvo 1"~I'n iml'ORSihlll to givo n. HysU!mlLtie devdl/l'''J(!lIt IIf gC(lIlUltry. So thero is 1m imllOrtlmt difference betwecnlLl'iLhmctic ILIld geomeLry. JI'I'OIl1 all ]~lelLlic poinL of view, Lim theoretic-ILl formullLtioJ1s of tho funllcr a.m much simpler tllILlt those (If lire IrLUer. This is the rClLflon why tho ancionts thought tltlLt IlriLlllnetic toul, prcccdcnce over gcolllutl'y, As P"IIIlIIIS Wl'Ut.fl ;'411 "'I'IIILL goumetry I,. IL I'ILrL of gl!lluml 1l1lt.Lhnl'",duH (ml. l"ric"lIuill) 4R.!HT: 6u I'tl' 001" ,) ,."IIIIUelll ujc; BA'le; /'nl,IllO'III<1U' ,'wr/{!'... 1l.1I r';~I" ",-,,1 n) .. "e,O/I'1uH'i.. • • • de'lr", Tol, naAllloic; Kill au nulloiJ ucira. Adyuu neUe; to nlled". ('1'1111 LrRlUduLion is by Morrow, l'rillcut.on 1070.) 110

"'l'mduR (c'(\. Jfricdluin), ISl.6fTi Bell uloo 17S.12fT. (Tho LrlUlslnt.ion is by Morrow, l'rincet.on 1!l70.) cr. A. Frenkinn, IAl postubtt che% Euclide et chez les 1IIotie.rnf!1I (Puris I !lola),!,. 10.

C()NSTJl\JI~JON

H'~ 1"lJlJ( Hoi OU

20

!lui....

:Ulli

R7.AIU): TIm IIRtlUININOS 01' OJlI"I~K &fU'URa\ATfCS

nmLics mId occupies I), plo.co second to ILrit.hmet.ic .•. hl\8 been MSert.cd IIy LlIll ILllcionts ILml needs 110 lengthy argument hero." It. is rehLtivoly ensy t.o IIl1LintlLin L1mt numbers are idelLI entities whh:h Imvo no lJOdies. Any given lIumhel' (23, for exalllple) if! reooily Ewell (0 he 1\11 ILbst.mcLiun ollce it. hilS been (list.inguishcd from the IIbje'ds which it counts. A higher level of allRLmcLiun is re:Lehetl when I'ropert.ies (01' sots) of nUIllIJOI'8 IJ.l'O considered instead uf individulLI oneil. Whell it. comes to ueometrical fiullrcs, howevCI', t.llCl·O is IL case lu he lIlade for sl\ying that they are Icssabstmct. t.han numbCl'8.1'roclus I!CelllS to luwe been fully aWILre of t.his, 1\8 the following remlLrk of hill imliCtLLt>.s:2tl "That numhCl'8 are pUl'Cr and more immlLtcriai than lIIagniLudes Imd L1mt the stu,rting point (If numbers is simpler t.lmn Ih:Lt. of magnit.udes al'C clOlLr to everyono." There ill ILIl ohvious difference between IL number ILnd the leUtlr, IIl1l1wml or KLmighL lillc whieh iR used to represont it. JL is leRR clel\f, hO\\'(l\'CI', LIULt Lhe 'idealslJlllLro' differs from lL figure dmwII to iIIuRt.rntc it. Plnlo l'ml'lm.'1izcd, of CO U1'8" , th'Lt. ge!ollloLors:!C2 "uso t.ho visible! squIUc.'J nlld figureR, It.nd IIllLke t.heir ILl'gumenLR ILhout. t.hom, though t.hoy are lIoL 1.llinldllg ILllout t.hem, IJUt ILhout those things of which tho visiblo MI' imng('s. Theil' nrgulllentR concern tho rcnlsqU:LI'O u,nd u, rOILI diugonnl, nuL Lhe (lillgonnl which Lhoy draw, I\nd so wit.h everything". NoverI hdl'IIII, it. WlUi not so eMY to distinguish Lho 'relLl' fmm Lho 'visible' ill J,(c·lIl1lcl,ry. PIILto even found it neccfllmry to reprolLch geomoters for (!lIl1fm;illg I.he two. Ho wl'ol,o:2t:l "(,JIlIY LILllt in Illost ridi(mlolls lLIul hC·j.(g'Lrly fa.c!lliun; for they sponk Jilm 1II01l of IHlSilleR.~, ILnd 1\9 though all I.heir demolllltmtiolls Imvo IL pmctieal ILim, wit.h their tlLlle of

on

III Ihi.!., !I1i.2:1fT: /lip Oil .. dC!cO/lOi TtlII' /lfyiOOIl' ullAduC!OI xui l((10ll(}IOU(!OI, )(11; ;;" Tri,.. ,1(lIO/lIill' '1 riel') Tli~ tliil' /ltyiO(l1l' 1l1t1J, dnAolll1Ti(}u, 1wpti xurutp
:'11

,,,j;

I !Ifill.) tU JlJi.!., 621: .tirol/al nOIl/IIUa ytAolw, n: XIII drayxal",,· W, yd(} nl'drrovt/, ,; Itll; lI(!I/eCW, la'INa nuvra, tau, A&,'oll, 1l0l011/,,,''O1 Alyol/tJlv n:t(!llywl'lCtll' n: Itnl :IIIl",nil"'" Itlll nC!orJraOI"(II Itlll 1IIII'rll oOreu rlJ,yyd/'''''III. td d'lnn nlliJ mil' tol y",ill""" ' ..fit.. brn'/~tlld/j[I'OI'. (,fitII trullHluLiulI iR by A. 1>. J.iJlcl8llY,

"i,

1"iIl,,,,,,

1':v"'.I·ClIIIII'1I I,illl'liry IIl/iO.)

"AllT !I. 'fll1<

COHSTUII(:nOH

111'

MATIII(MA"'J(~

307

RqUiLring (tt:Tl!aycovICw') ILIld lLPl'lying n.ncl ILClcling, lLIul so on. BuL Hurely t.ho wholo study is clLrl'iml on for t.J1(I "ILlto of kllowlc(lgo." W(~ ClLI1 Heo t1mt t.ho foundlLtiol1s of geolllc!tl'Y II1llst IlILvo seemed Illllch more I'roblmRlLLic t1lnn t.hoRO uf ILrithmetic til the nmthenmlicimls (If unt.iqui Ly. In Lhe llClXt c1uLI'Lor, thcrefore, we slmll (!XILIII ino t.ho (lfful'(~ which were IIlILlle tu J'(!culldlo ge(lIllC'I.l·Y with t.he t.elLehing of tho 1~1l"LticH.

3.28

"'IIJ~ SCII'!NC": 01·' SI'ACI':

ILl'plyill~ Lhe bn,'Ji(! telletR of l'hiloRophy to goollwtl'y seOIllR non·(lXiRtcnt., 'I'ho 1~lclLtiCR IIlLt.ly Ilcni(!cl t.lml. Lhl'l'" waR Rlwh IL Lhing n.~ .911flf'.r.. 2H ZCIIO, in I"LI'Licllltlof,
011 tim fIL!:f) of it., tho l'oSHihility of

]~IClLt.iC

'l'/'cnC'.tetu.. I RO ... m cr. W. ell III,lIc' , J>ie l'orsllknrtik"r, L"il'zig I!I:IIi, "I', 112r. UI l'Ulllllil'hllH, I'j". "!I"m!}"ri .." H!I. RII" nill" Illy 1'"1''''' 'nl'ilc".\'lIIi, IIln til

1II1I1.iR,·h,·r'I'"rlllinIiA [jir 1,.'w"iKI'''' ( ... ,r"I'I·'·.! I .. in 1'''ln'r in lIt,,;n, N. R II (I !III!I), 2.c:J Ii.

II.

111111.1 ... •

H "I,uv.. ) 111101 1'·... ·uld",,·A

:wx

!\7.AIII): Tim IIICOINNINCIS Ott UIIIU'.1\ ~IATUICMA'I'll"X

bm"H) ILlld clLllcd them 110I7,i,laTa.217 The word l(1'foel'1, ~n the other halld, wmi reserved for empirical Imowledgo which had been acquired by o\JscrvILtion.2C8 We cnn conclude from what Io.mblichus says thn.t gl'"nUlLry WILS o.t fil"8t considered to Lo II. pro.cticlLI o.nd experimenLal s.,il'lH:C (llfTllel'l rather tlmn 0. true malllema). ProcluH, in the pn.'i!ln.go quoLcd o.bnve,2CII o.lso 8eOl1lS to be RILying t1u.~t. geometry did not. he(:ome part of general Illo.thcmn.tics unt.il 1'l·latively IILt.e, and that even then it took second pltLlle to o.rithmot.ic. Plato ILgreed wit.h him ILhout the rolo.tive posit.ions of geometry ILIIlI tLJ'il.hmeLic. 2so In flLeL, Plato's works ono.b\e us to show conclusively that iL WILK t.he ElclLt.ica who first made this o.ssessmonL of geomeLry arlll pOlled t.he problelllH uf 'spn.ce' ami a 'science of splLce'. To 8eo LhiH wo will need Lo rcCILI1 lllrLtO'S elu.'1sifiml.t.ioll of t.ho vILl'iouB ltint's IIf I{ nl)w\cdgo. 251

• I t. is w(!11 Imown LlIILt. Pln.to distinguished bot.wcon t.he wOl'ltl uf I,ecomillg or t.he percepLible world (oeUTOV) nnd t.ho wOl'ld of boillg (I·m/Tov). (II Lhe Re-IJublic t.his distinction is used to dcvelop an int.crcHt· illg I hcury of kJlolI'ledf}r, i!l'wrance, ILIllI belief, which Plo.to eXI'"Lills in t.ho fullowing WILy.%S2 lie who IULsIU1owl(l(l~e IUIOWR somet.hing (0 ytyVWc1"WI' ytYVWU"E' n),' furl.llIll·nllll·e, he IwowH "Holllet.hing which ill (Oil), ]rur how could UII"!, whidl is 11I)t.I,,, IWCI\vn (milt; )'Uu UV In; r,I' yt n YV(lJc10El'1)? So kn.ollllr.,ly(! iN 11.'I.'1l11li'LI.cd wit.h what. is (brII'EV nii ovn YWOc1tt; tpo), ILnd ig1wrmu:c with what. is no!' (dyvwc1la d'l~ av&y,,'1~ lni Ill) ovn}. 1·'or t.hat which lie!; l'l~Lwcon wlmt. iH ILnd wlllLt. is II0t. (i.o. for t.1mt. which inhabits t.ho m B. I.. VOII cJur WUllf.lulI, Mat/I, A,m, 120 (11147/40),127. Aristotlo (1I[01."l'hys, i"A Ali) IIxl'liciUy II\.aLe'S thnl. tho Pythngoft'IUIB worn till! Cantt peoplo to COIIUI1I'1I th.'mRdvl'II with IlfJiJril"UII, cr. ,,1110 1(. HtlicJulIluisLer, Das e:mkltJ ])er~ tier Orie· ('''1''', lIullllml"g 19411. p. 62, 118 n. SlInll, Die AIl.!drlicke/ur den IJeuriO lies lI'issen6 in Iler vorplatoni.RrI/I-" /'''ilo''0l'lIio, BUI"IiIl 1024, I'p. 60-71; A. ]rmldlUl, Hevuo Ilell Jttl/lJes r"doeur·ol,tI,m. ne." nllcul'ust.-Puris 103R, pp. 468-74. In S,," II, 240 nbovu. n. I'I"IA" El,inoJII~ 9!JOo-d. lA' cr. my Jlllpur 'Eluuticu' (1Illtl II. :13 uhov,,), 1'1). 981T. UI Ilnll11l,/ir., V 4760-47711,

I'AIIT :to TIIK

C()NlI1'llU!:J"I0N

III' M.\TIIICAlATIt:li

30!)

world of becomi"g and 'pas,fting aWlly) t.here IlIl.s lo be wscovcrcd II. COl"' resl'OJuling int.crmcwlLw bet.ween lmowledgo and ignom.nco (lni {)E Tfr IlETa~v Tovnp IIETa~v n "ai CllTlITiOll uyvolat; TE "ai 1mc1Ttlll'lt;}. 'rhis inlermedilLw is clLlle(1 belie/ ((jJ~a); it. ill "dlLrlwr UlILn knowledge ILIlt} urighh~r t.hall igllorlmce". I n my I'pillion, pilLlo's dist.inet.iun bet.WI'I~1I knowledgr., ignorance ILIUI belir/ cort.ninly hILS itR rools in ]~lclLtie philosophy. PlmncniticR 811.y8 the following Il.uout. his 'sccond WII.y of illlluil'y':2S3 '''rhe ulh('r, t.hlLt it i.'1 not, I\IUi t.1mt it is uuund uot 10 be: t.hiH I tell )'OU iR IL IIILt.h tlllLt cn.nnot he ex }Ilorctl, fur yOlt multi neither rrCOQnize t"al tl'hir.lt is not (olin: )'Iie «'iv yvot'l!; 11' ,'1: I't) 101'), nor eXl'rcs8 it. 254 "

I f, us i>lLrllwnillcs cll\imA, wlmt. is 1101. I':lll 1,1' lIl'ithcr ... ·cognizctl lilli' (!xl'rC8l;(·tI, it. obviously CI\IIIIOt. he I\n III'jl·c\. of ku()wledyr.; ollly what. i.'1 CILn 110 \mown tLnd recognized. Whell PIII.t.o connects klwwledye with Being o.lId ignorance wit.h Not.·Bcing, ho is dru.wing on 0110 of }larmcnidcs' idens. It. ClLn ILlso be shown t.hl\!, I1lato's ILCCOUIlt. of belie/ (do~a) derives frum ]'arl1lmlidca. Tho II,U{lr disculIRt.'f! his 'I·hird wn.y of iJl(luiry' (i.e. the 11(';/;1'/.'1 of lUor\.tLls - fI(!duIJI' d&~at) ill Um folluwing t.orm8: 255 "But. Jlnxl, J dehlLr you from Lhn.t. wl~y alollg whidl wl\Jldcr lIlorL,LiH IwowiJlg nothing, t.wu-hlllL(lc(l, fur' Imrl'l(!xit.y in t.hcir "OSOIllS sLcm1i t.hcir int.clligcmcc o.struy,·ILIld they ILrc carriccllLlong lUI deaf n8 t.hey ILrc blind, ILIna.zed, uncriticu.l hordes, hy whom '1'0 He Imd No!' Tu He (TO ntAw' TE "al ov" elva,) Il.rc regarded ILB tho HILIUO ILUd nut. t.ho HI\me." ] have argued clscwherc2SG LhILt. l'armeni(\l'!; J"(ljcclcd do~u (1LI'I'cMlLnc(', illusory Imowlctlge. helief) lmCILuse ho cllIltlilh'red it. \.t, 110 incoJlsistellt.. 112 Ruu Illy JI"l'ur 'Zur CI'·lml.iuhLo <1"" Di"I,·III.ilt <1,," ))"nlu"If~', PI" 6·1 IT. rcrurrccl to in n, 33 above). IN DiclR Bnll I{mll7., Fragmen/n. 1.2Rh, rniAIlIf'"L 2.Ii-R. 'I'ho LmllAI"l.ion iA hy U:nthltlcn }rrcc'mnn, 1M Frngmnnt. 0,4-.9, ,.1 R"o '7.11111 VumUin,lnill .Iur 1<:1",,14,"'. 1'1'. 21.2 r. (n·r"rr..,1 tu ill II. :1:1 "!.nv'·),

310

Tim cunt.mdicLion which he sa.w in t.he concept. of clox£iJ1 involvcd t.he not.iuns of Being and Not.-Doing {dva, xal otlx dva,}. Plato associllotes belief wit.h thoso t.hings which lio betwcen what is und what. is not.257 (in t.he world of becoming ILnd passing away): accordingly, ho pllLCCS il. IJCt.wcen knowledge mid ignorance. I t. seems flLir to SILY t.hat Plato, in the l,M8ILge discussed lI.bove, is merely recapit.uIILt.ing PILI'Inenides' view of knowledge, iglwrance and befit'/.

* l\Iy 1"(!!l.'IOIl for giving thiB "rief IL(!(:ount of some fClLturCft of .I'IIL(,U'H ('l'istemology is thlLt thCKO havo to be Iccpt in mind, if we uro to make HOIiRe of two import.lLnt l'"ssuges in which he t.ull'R ILIJOUt. ycolllt>lr!l. From WIULt. hILS been slLid ILhove, iL Rhould bo applLrent. tlmL PIiLto rcllLted knowledue n.nd belip./ to the world of Being {VO'lTOV} rmd the pot'(,(\ptiblc world (rSeuntv) I'cRpectivcly. Belie/, Mea, WIL'II&HSocitLtcd with HenHe perception ILIlli tho wodd of becoming /LIId 11U,ssiJlg aWfl!l, wherc/LA t.he dOllltLin of true knowledge was sOlllothing iltvisible rLlld aC(:eRHi .. le oilly te) reason. The I:LUCI' WIIS often called lld'1!I hy boLh PILrnwllides lLud Plate). Nut. only were t.hese t.wo I'hilos0l'lmrs in hl&Hie agrc(!mcnL nllout tho IULture of Imowlcdgc, but. thcy lLlso Rlmred IL conHIlOII terminology. To subsllLntilLLc this chLim, I would need t.o quoLc some 10llg 1'1\8.'IlLgcs from the 1'i",IICUS in the original Groele. Insten.d of doing I.his, howcver, I slutll content myself with quoting t.ho following (,rlLnSIILI.illn (If /L smnll }IILrt (If 1'1lLto's tcXt.:258 "Thcl'I' is one Idml of lining which is always the Ra.me, uncrclLtctl IUIII illdcstructilJle {rlytV"'1TOV xal o.vw)'dJ(!ov}, nover receivillg ILllyt.hillg intu itself from wit.hout, noritself going out. to /Lny othor {0I1TE ric InuT,1 elrJ,1£x,$IW'''V tiAAo{)£v, oi1n; a1~TO £l~ ti).).o nm Mv}, inviHiJ,lo and iml'crccptilJlo by ILny sense {aoeaTov Ili xal tf.t).w~ rlva{f1lJr/TOl'}, ILnd (If which tho (!unwml'lnt.ion is gmnLctl to inLclligClwo only (Ii .l,) 1',hir1l~ rlA'lx,£J·lnulxoneil'}' AmI thoro ill ILIIUUIIJI' lIut.uro of tho /{e/JUblir., V 478,1 5-0. 7'illlaew 62a-b; Lhn tmlllllnLion ill hy Bonjwllin Jowutt. (TIIIJ Din/ogllu oJ I'/alo, 3rcl Olin, Oxford 18112). In rAil

I'AIIT S. Tim CONl\TItULTION 01' AlATIIRaiATICS

!lun\): TIIK IIHUINNINOS 01.' OIlRKK MATIIKaiATICS

311

same name with iL, ILnd liko to it., perceived by sense, crelLtcd nlwn.ys in moLion, becoming in pl;Lco ILnd aglLin vanishing out. of place, which is apprehended by eldEn ILnd Renall (Men p£T'alf1{)'irl£CJ)~ nee').'1nTov}."

'1'ILlccn out of context., thia plUlSage seems to cont.ain nothing more than a rt'.statement of the differenccs bllt.wccn the objccts of k,wwleiJge ILnd t.hose of belief. When it is read toget.her with its cont.inuation, however, it provides us with sOllle vnllmhle ncw informat.ion. In Lho Timaeus Plat.o is concerned not. only with tho l,erc(!l'lible ("oralon) ILud ideal (nnelon) wol'1lls hut ILlso with a third world,25t1 which ho calla 'SP:LCC'. l~il(Q t.he world IIf Heing (,wt>.lon), S(l/LCO is f'LcrIULIlLlul indestruct.iblc, nevCl·t.hel(!811 moLiIHl, gcncrat.ionlmd dest .... ct.iou tlLl(C I'IILL'O in it..zeD Unlilto tho other two worlds, however, SP"'CO is apl'rchcnsilJle hy a Idud of 'hl\8tlm.l reasoning' {IIET' riva'f1{}']f1{a~ Mrov ).0YlfTpq; nv, J10{}rp}201 and not lJy pure illtelJigcnce or 8enso perception. Other plL88ILgCS in PJILto's worl" eR)leciltlly two which Itl'O to be found in tho Republic,2G2 atteRt to tho flLct UIILL he identifies 'bn.lltltnl reMoning' ILbout SplLCO with geomctry. 'fhc L\Yu plL88llges in CItlCfttion contlLin l'Ofcrrnces to tho faculty employed hy ~1~olllcLcrs; this iR clLllcd Illth'OlO, /LIId it is said t.o lie sOlllcwhere IlOtWI!I'1I lIIere dOl~a (Le. l'"rc('l'lioll hy mellns oCthe senses) ILnd the )lurely inteJlectlULI power of ReMon {vovc}. I believc t.hat. Plut.o's ideM about geometrical 'reMoning' ILI'OSC us rL l'C8ult of the following historical circtlll1st.unces. The ElclLtiCS fllLtly denied the oxifJl.c.lIlc!e of Sl}Qce. They showed tlmt. nil I'henOtnOlllL 1101(lllgin~ t.o I.hn I'fll'l!C!I'f.illlfl worlel, CRI"JciILlly tllIIBO which involvcd such conccpt..'1 IL'I 'motion', 'c1l1LllgC', 'genem.Lion' ILlld 'dcRtruction', wero contradictory. 'l'his led t.hom to tho relLlization Llmt 81JUce Imel time WOI'O ILl So incolIsiHtonL, for LlICRO t.wo COnC(ll't.a ILrtl dORely rdlLLed t.o tho uthl'l'R. (In IIILrl.ic:uIILI·, their argulIlonts ILglLinRt 'mol.iou' forc(,el them to conr:hulo 1.lmt tho notion of '.'1pace' WUS I " "illlOC1l6 600-11: I" ,"Oil" TIp ""(I,il''' Xl"} )'/1'" r),o"o,,o;;..o, '(lIlt II. Tel /,/;" ),.),,'d/lel'ar, rd ~'I" 'T' yiyrHIII, r,l,)',11" .. ,1'1""",.. 9'';£1111 Tel y,)'I'd/II·I'Ol'. 110 Fnmkinn (rAl l'oillu/Ill cI.u Ew/i,lc cI c/',,% les mOOerne." Paris I !l40 p. 26) diacU8IICB Lho nXl'n!88iol1l1 uSIIII to dnscrihn ""pucu" and rerur& to Zollor's Die

"l,.rI·O,.

P/li/ollol,hit! der ariccl&el& 11.1 {4th ("1. IRR!!), "lOt" 722. II. 7'i,naeu.., li2b. III Republic, VI Ii~ IcJ-fl nnd VB 1i:l:Jn-Ii:I·tu.

:H2

lIZAII/): Tille 1I~:U1NNIIUJlI 01' OUlmK MA1'UKMATJ(:!I

im'uhcrcnt.,) As far 1\8 t.ho Elcu.t.ics wcrc conccrncd, huwevcr, ILnything ,'untmdictory wu.s inconceivable and could not be; hem:e the very (~xiat.enco of 'spaco' WlLS called in question. 1'1I~to's conception of space evolved from a later and more refincd \'('I'liion lIf tho train of thought sketched in the preceding parngmph. At. his Lillie, spaco wus no lunger regarded u.s something intermcdilLLo heLwcen Being I\nd Not-Deing, a mere phenomenon which could bo ~(Hll!I'nLed nnd destroyed. Although iL WILH described n... tho 'l'ecol'tl\clc' III' 'JlllrsC' (If genemLion,2G3 it. WILli nlso c:onsidcred to bc nil 1(!H.q indcsLructiltlu (rpOor!(h, oV 1l(!OutlF.IOI'tl'ovj%tU Umn Lhe 'idclLl' (l'O,,'fov). Thllll 'SJlI~CO' wn.q t.hought. to Imvo IL kind of dlml n;LLure; on t.ho IIJlIl hand iL was eternal ILnd indcsLructible, ILIld on tho oLher it WIUI il\l!xt.ril:lLbly bound up with the phenomenn. of the pcrcepLible world. III \' iew ()f Lhis, we should not be sUl'prised 11.1. the fn.ct tlmt. the opithet 'Ims\.nnl' WII.q applied to geometry, 'the science of Spl\CC'. SUJllllthing ahout whn.t I'lato means by 'bnstard rel\sonillg' CILII bo "'I~l'IIed frolll a plLaso.ge in the Republic2G$ whore ho points out. thlLt LIm IIILt.III'O of geomctry is in (J,Lt contmciiction to the ll\ngtmge used by W~ollllll(ll's, Pln.to hM in mind such locutiolls M squarinf/, f7.1JplyinU Ilond midi,,!!, which, ILCCOI'Wng to him, givo tho en'Oneoua illll're88ioll \.IULt. t.lw W'lIl11oter is engnged in rL pra.otical I\CtiviLy. Ho goes on to sn.y I hILI. t.ho aim of geometry is to gain Itllowledge of eLornal l'Cn.lity: it. ill not. tu en'cct the construction of ephemeral figures.

3.29 TIlE FOUNDATIONS OF UEOMETRY III Lhe previou8 chapter we discu88ed brieRy some of Plato's reml~rl,s nl 11111 t. SplLCO ILnd the science of spaco. I believe that wlllLt ho hILA to SIL)' 1~IJ(}ut these subjects helps us to understand tho proc088 which II~d lo t.he formulation of theoreticu.l foundations for geometry. TIm IiI'8L step in this proCCII.q neCC88l~rily involved a revisioll IIf 1~lcl~Lic views about 'space'. Thore could be no science of spn.co ns IOIl~ as the very existence of BplLCe was 'donied'. (The E1ootiea, it will 110 I'lICILllcd, rejected 8pl~CC, tugethor with nl11,honoll10lllL Lelongillg

:
,ill'''\''X') "II'I"(I~ ur

"'''CII'',

II. 'I'j t·' R....

II,

1i2/L, 24:1 /Lh''''It,

"O,J"'I 111,

,'cp/ruw,:

11110 II.

200 nlluvo.

"AllT 8, TilE OON!ITllIIUI'ION 01'

MA"'III~~IA'I'ICll

313

to the percoptiblo world, M cOlltrmlictor,Y.) Geometry wns at first nothing more than a ltind of lU'fo(!I,,; i.e. it WiLli a body of empiricl\l Imowledge obtained by mea.ns of tho scnRes and tl/LAed prinl'ipo.lly on visual evidence. MILthenlfLtieians must soon have rCl~lizml, however, LI lILt , nlLhough o.WILrencss of sImce is invl\rill.bly ll.B8ocill.t.cti wiUI awarellcss of those things which are loca.ted in 61)lLCe (ILnd which movo, change, nre genorl~t­ ecll~lIc1 d(',sLroyed in it), ono ClLIl rLhatmct. frolll olle's (lxperienccs and thinlt of ,'llflce i\.self as indepondent of Lll('se phenomenA, COllsi,lel'ed in t.his purely nbstmct WILY, it appel\rs to 1m somewlmt similrLr to the 'idlll~I' (Jl.oelon). In flL(:t, t.he attempt to develop n.n cxlm.9cI"'inry cOl1ception of fll',~ce 8cems t.o hrLve marked the beginning of thcol'eticlLl geometry. As we hrLVO suen, Plato says that SIIlLCO is npprulmllllible not by sellllO percep' tion hut by a Idnd of 'blLBt.ard rensoning',2/1a Aft.cr sLI~rtillg out ILIi n Idllli of [d'fOel" whose chief tool WM the SOllA" of siUlit, geomet.ry bCl:l\me tho science of space itself. Le88 omplllLHill WILli phLced on tho role of senso pm'ception in this science, lIineo flplLC:O WM now I'cgll.l'decl lUi somot.1ling uncluLnging, irlllcfltl'uctiulo allcl l'lll'oly IloiJlIll'l1.ct., which WM disLiJl(:t f!'Om tho objects locl\wd in it.. Thl' n:nuncin.tion of selllle percoptioll II.... II. 1II00LI1,1 uf Iu:cluiring geu, mctricll.l Imowlcdge mii.nifested i\.self in I~ IHllllber of illlcrl'flLing Wll.ya, For examplc, overy effort WiLli nutde t.o l'xc'ludo tho concept of motim~ from the definitions in Boolt I of the Elemelll.'I. 207 We know from Proclufl that a. line was sometimes defined ILa 'the rtowing (If II. point.' an.l n. stmight line M 'ita ulliform rLnd undoviJ\Lillg rtowing', lIowovl'r, L1leRIl, I\re not' tho definitiolls given hy Euclid; hia nrc forlltuh .. ted ill lIuch IL way u.s to avoid any ment.ion of mot.icm. All II. nllLt.Lor of fact, tho RO' called 'geomet.ry of mot.ion' doC's not 8eelll to IULVO ILchievcd respect.ahility until after Plato's time.2GS It 11M oven heen argued t1mt effurLH worll Jmtdo to banish refcrcm.'{'.fI tu visible pl'Opcrt.ics from g(~oll1ctry. Heath, for example, helieved t1mt Euclid's definition of a strn.ight Iino wns ILn attempt to explain wil/lOut (Ill," ('l'petll 10 sight whllt. it mCl~llt to Lo 8t.mighL.2IIO 1CC'l'imatlUlJ, 62b. H' J, N, IIl1fllll1llll, Ot!8c:1li<:"'n JII"lIlc/II,lIiJ,'. VIII. I (B,.dill I !lIi:l) , p. :12. 1111 Cf, A. n, RIAI("", Up" roil•. (II. 1R3 /Lhnv,,), p. :m!l. H',)', .. , Huut", If lIilfCaru n/Or,.,./c «hefonl 1921), VIII. I, p. :17:1.

.'"r

Ma''',."",,;., .•

I\1.AIIO: TIIJ~ JII~OINNIN(lR 01' (ll!JmK )fA'l'IIJo:MA'fJCS

Inovitl~hly, howovCl', n.1t tho l~tLelllpta which were nmde to treat 'apacc' a.'i HOlllcthing c1oso to Lho 'iden.I' (noclon) ended in failuro. Thero We~l'Cl cOI'Lain Heomingly conLmdictory facts about space which Rimply I'IIl1lel not bo IIILIldied wiLhin Lhe frtLmeworlt of Eleatic philosophy. TIIIl merc flLCt Lhat lilJace presenLs iLself to Lhe human mind as somothing infiltitely divisible posed n. considerable problem. 270 It led Lho (: I'eulul to Lhink LhaL thore were no 'lolI.st magnitudc.s' in geometry; /!\'('I")' 'numbor' could bo builL up out of indecomposable 1tni(,'l, hut I.he~ Halllil could noL bo BI~id of aplLee. In oLher words, they CILllle to 1.lu~ e'OIwlusion Mmt Lhere wus nothing in geometry which corresponded III LIII' u.nit in ILrithmetic. As rroclus wroLc:271 "in geometry IL lClUll "'''!/II;/IIr1f'. lULl! 1111 pllLe:n nt. nil. J'nmllilLr Lo goolllotry ILro Lho I'fOl'osiliollR ... al.lOut irmLiOJmls (for the irmtiOJULI (TO a).oyov) hM a pl:Lce Ollly whem infinilc divisibility is p08.~ible)." Al'il.hmeLie was based on numbers and the D,te. These were ideal anti eOllIplotely ILbsLmct forlns; henoe they were immn.terilLI I\.nd eont:~illedno conLI"I1.dicLions. Thero was, however, no such simple stluting)luinl. lill' gconmtry. Proelus descrihed the situ;Ltion in the following way:2n "'fhll.t IIUIllbel'8 are purcr and more immatcril\.l LImn mlLgllitll(ll'll ILIHI tJmt the sLI\.l'ting-point of numbers is simpler than that of lIIagllitlldcs ILI"O dcnl" to everyolle." 'l.'hmm elifficulties cffeetively pl'OvenLod l~uclid fmm rormulo.ting tho II II Iii I. IJlIsie; geollletrical defiJlitioJls in 0. wholly consitltcnt mo.nner. Ile~ :tHIlPl'w(1 1.111\1, "A l}oint is lltal w/,ich /ms no lJflrl" and "A line is IlIr(lflIMr.~,'1 lellyl"". An ]~lelLtic philosopher, however, would lmvo n·g:mlc·el UII'Hn R(,ILUmwnL.. IL." grounds for donying the existenco or H)la'~I! alit I, hCIICU, ror dOubLillg LIm possibility of a true scienco lIr ~1'"J1I1'try . Tho infinite !Ii v iHihil ity of SplLCO was by 110 melLnK the only diffieulLy whie:h HLoocl in Lhe WILy of those who wanted to lay tho theoreticlLl rllllnel:LliolUl of gcometry. However ILbstmct the concept of 8pll.CO nmy havc I.colI, it WIL8 insopll.raLlo from the inconsistenL cOllcept of molion. Ch·olllcLricl\.1 'points' or 'least mo.gnitudes' could not be obtu.incd without 'infinit.o division', a procedure cho.ro.cterized as in'ational (l1).I}),(w). On tho other hand, any attempt to obto.in 'lines' from 'points'

I'A itT :I. Till·: e~ClNS'I'1I U(JI'IClN ell' M AT II e·:M ATWIl

(in the SILmo WILy Llmt 'numbers' \Vern ohtaiuml fl'oll\ tho 'Ono' hy Illllitiplying il.) nOl!eH"~ILrily invCllve~d 'Illolioll', l~tlrI.hcnll()re, Ln sluely R)lILCO iL WILS nceessary to consielor pmcl.icILI (1IIcHt.ions (Kueh IL.'I whetlwr ()\' 1101, two finite seLs were eqlml) which ('oulll hc ILllllweretl only wiLh tht' hell' of tho SCIlSCS. The ImmLLiCR, howevcr. rCPIICIi:LI.(~d Lhe irlt'a t1mt Lrue IUlowlcdge could he hlL.~ecl on Hen!-lC l'rm:cpl.ioll. In Illy opinion, the Grecl(l~ limy very well Imvo been led tu devise a:dnmlltic /Ofl1uilltiolls (illiLilLlIy just f(JI" ~eollloLry) n.~ I\. response to Lim climeultil's doanillcd ILhllVl'. They JlllIsl. hlLve felL the need to IlLy down eol"lliin elllpii"icnl faels whidl wew illeliHl'ellslLhle to tho eonsLrueLion of IL Bdcnco of fll'lLce, ovon though lltl'I\O flLe:ts clid nut saLiHfy LIm 1~ll:IL(.ie: J"(:lluirCIlUl/It t.hlLl. 1(I\CIwll'(I~" III: ILeIJuire'el ill IL pllre·ly intcllcctlULI w:\.y. Their fin;t l.IL.'JI( W:LS to IImlto it denr LllfLt the formH IIf gl!llllwtry (lincs, l'oinls of illUlI'liCcl.ioll, ILllglcR, figul'CK, elc.) wrrc flot lit flll tile .~ame as lIwl,e perceived by the senses ;273 tl'UO geometriclLI flll"lllR WUI'C idelll cllLitiIJII (Iilw IlIl1n!Jcm) whwm viHihle counlerp:Lrl.ll flcrve,,1 ollly to rOl're.8cnL Lhem. Tho dcfiniLionll were inlcncled to clilllililLIe IL.~ mlLny scnsiblo fClLturcs 1'-'1 l'o8.<;illle frolll geomctry, :LIlt! Lhe axioms ILnd 1JC1.'1lltlatt'4'1 tliRcufl...ed cal'lil~r wl're ILIHU !Limed ILt nmldllg the foundations of this scionce purcly ILhHLmct. It. ill nowwml.hy, ho\Yev{~r. tlULt in lLlIl.i'l"ity Ruclicl's ycomclrim/ dcfiniLions, l'ostull\.tes ILIlll ILXiolllS \Ycm nover considered to he :L com plcLoly BILtisfnctOl'y r"ulldILtion. Plat.1I "hHen'uti th:Lt YCOlllrlrr.'1 did lIut hothm' to give ILlly rlu·t.hor ILC('III1I1L or the 11.H.'IlIlIIl't.ioIlH on which thllir Rcicncc WI'-'I bILl!m1. 274 ]~urthcl"Ill()rc, \.Itey were evcn ILCCIISI'II of IIILHing thc:il' illvcstiglLtions Oil Itll,~f'. HIII'POllit.ioIlH. AI'iHl"Ue. who ILLLemplctl Lu clcfend gtmnwtl'rH "WLiIlHI. I.hi» clllLrgc', 1.l'olLll!d iL II." IL SllllHt!l(!HS ILnll uninteresting uhjlle:tion. Ile~ WI'CI\.(!:275 "Nor ILI'O the gcometer'H hypotheses flLhlll, 1l.H BOllin hn.ve hdd. \lrgin~ LlIILt, uno III \HIt not elllpluy flLIRllhol)l1 I\IHI that the gcomdcr is UL\.(II'illg flLIRohollll ill sLILLing LlmL Llw linn whi(lh lin dmwR is IL rool. long or stmight, whon it is ILctUlLlly ndther. Tho truth is tlll,L t.ho geolllotel' docR not dmw ILlly conclusioll fl'lIm tho heing of the plLrIn

nu Run A. A. Fl'itlllwl, A bstract Set 'l'hcory (AlIllltol'dulll 196:1), p. 9-1 :i.

e1. Fl'imllllill). 00.11 cr, I hid., 91i.231T,

171 1'1'01l1t11l ( ..

m

:J \Ii

tfl

176

llllTmblill, vr lilO,1. Ihlt!. l'oRtllrior .Ii nl1/!/ticR LlOj loll" I.rlllllduLiull ill IIy n. n. n. l\f III'''

11IliY"rnil.y

1' ... ·/111

11I2R).

(Ox ron I

:11 Ii

I'AIIT :I. Till'. CIINln'IlUI7ntlN III'

I il'ul:LI' linc of which ho SllelLlts, but. from wlmt. his t1i;Lgmms symJ. .. li;w," IIt!Ml!rihcd by ArisloLlt" t.ho objccLion r~i8ed nglLillsL Lho met.huds of ~I'omdry WM Imrdly worLh refuLing. So Lhero is Borne reason t.o Lhilll, I,IULL 1m limy Imvo miRulltiersUlod it. This somna evon mOl'll lil,dy III hlLvo bcen the C/I.HC, when we consider lho fILet. t.Jmt. his refuI al iOlls "f Zello's tLrgumenls ignored ('ompll'I.cly 1,110 ingeniolls illl'l~ \\laidl IlLy behind them. Wlml S(!(lIllS to be Lhe sll.me objcct.ion is l'I1.ised in 11, different form lay 1.111: 1I11o-PIILlonisL philosopher 1'1'oelus, who bolioved Llmt gcomcll'y had to 1111 Inll "oUl 01 CalYl)So'S arms, so to spetLl(, 1.0 moro perfecL illtl'II11dllai illlligh\,". Ho wonL su fm' 11... to !:ILIi upon IIl1LthellmticilLIIS III 1I11c1l'rlalto t.hiH t.11.'ilc in the future. 17o On lhe blLais of tho plUlSlLge ill whic'h LhiM \. iew is cxpresscd, HI~rt.rn!\llll aJ'gucd tlml Proelus should Iw •· ... ·dilcd wiLh nlll.icipatillg t.hc diflclJ\'ery of 1~1I1~lytic gCOInel.ry (lLIl C·\·1'1I1. which dill 1101, llLlw pllLce unLit LlIC timo of D
• I wlJultllilw to concludc lhis chapter by recalling HOIIII: of Lhe points whidl wpm ,mule earlier. AI. Lho h(1~innillg of ParI, 3, I sl..'1.wd the cOJljecture LllILt. new LochlIilllll'II0f pl'Oof, which Ilol'pclLled Lo sUJJlct.hing o Lhtll' Llmll viltihlo evidClwe, \\,I'I'I~ CICH'ld0l'ml IL.. IL I'UIIIIIL of IL geometrical diacovllry. I ILHllcrwtl I hal. <:1'1·.,11 Hdflneo IIll1lerwonL IL c1ecisivfl dllLllgo whit'h WILR vory dUlld.V I~"""e(!lcd wiLh Lho discovery of lillc/Lt' illctlJIIlJlensumbility. Thill dmllJ,t1l IIIlLllifeHI.ctl it.self in Lwo WIL,\'H; 011 Lho ono Imnd, p"rely II".",.,., i,." I 1III'IIu I{/.~ (1/ }'1'fI1I1 wero ill t.l'mlll t:ed ill t.u IlmLh('III1Lth:8, ILlld, (III 1.1 .. , 01.111'1', cmpirici.·rm atld llll! viSltalwcrc rcjccl".d. When Gmelc mat.hoilia I i.·illllH firM!. 1~ILIII(\ ILI'I'IIHH g(\(lIII(II.ril~nl lellgLhH fill' whidl !.lilly c!lIl1lcl lillli 1111 1:0 III JIll III IIl1:lLHUrO, LImy WI!I'C coml'ollml ulsc"lt lIew tC(:hniljllcs III' I'l'IIlIf. Although LlIO discovory of such 10llgLhs UlldllUbtClUy propared 1.1 ... WILy fill' Lho discllvllry uf illl:ulIllllollsumbiliLy, t.ho cxi.-llc1lcC uf ". 1'1'1,,·1 ...1

( •••

1I·.. i .• "

:117

illcommollfllu'ubilit.y could 1I0t. be concluHi\'cly proved Ity pmd,i"al or mnpiricnl moLhll(lR. Henco IL complex of prohlems n.!ulociated wiLh illconllnellsumhiliLy mllde it. neccEI.'mry to adopL EIClLLic techniqucs () f proof ill geUluo Lry . The ILppliclLLi(l1l of J~ICILLic IIIcLlIOllfl Lo 1':LI'Lit!ullLt· geollloLl'il'ILI 1'1'11 It1c~IIIH IIIILy well hlLvtl slIgg('HLcd Lim idmL l.hILI. lUi ]~lelLLic 1I'1'l'l'Oaoh clluld llO LILlwn Lo Lho whole of geomoLry. II, Lurnctl ouL, IlOwe\'!!r, !.ImL thl' U~ILChillgs of Lho 1~leILLiC8 wcre IIOt. nlLogeLhcr compaLible with tlais hmlwh or nmLlulIlmLicH allll, n.~ 1\ rClmlt., it Wll.'! fOUlld 1Il!l'I'lIl1lLr,r III provide guclidClm geomoLry wiLh t.heol'cl.iCl\1 foundat.iUllA. My ellLim is LlmL tho constl'ueLillll of nmUlemlLtics ILa IL deducLivn system 1:ILmo ILbouL htlCILUSfl of ccrLain I'roblollls cllcountered in Ueolltelry. I \, is tl'llll (.JlILt I~IC:Lt.ic doctrine clm 1,1I ILl'pliod more elLaity III ILrit.llll1otic tlmn U, ueomelry alld tim\' Lho Gmells therefore I'cglmlcil nriLhlllctiu ILa tho fluperiOl' scicnce; hllwevm', Lhis mnltillg WIt... only IL Lhrnrrlical ()JII!. guclid's IlIlLLhollmt.i1:8 is I'l'ctiolllinlmtly geollleLrieal in dmrnl:lur; IlVI'1I hiH ILt'iLhnwlic LaiwH IL gC·IIlIlI'l.ril:lLl flll'm. ThiH lllIO"lcl 1101. SIII'priHC UK ill view of Lhe flLcL l.hlLl. I· he prohlcllls whidl f:ILUHCll llmt.llemn.LichLIlII 1·0 hrOl~k with EllmLic philO!mphy CI~tn(l prilll'iplLlly fmlll f}f'olllelrYILlltl Lhe outcollle of Lhis Lrt!ILk WILt; IL Lheurclical foun(11~tiC)n for geomclry.

3.:10

A 11I~(!IINHlIlIm,\'I'IClN IW SIIII"': 1·1l1J1I1.I~JltS In:I.A'I·1 Nil '1'0

I~AIU.\'

mur.lr.IC JltA'I'IIJo:IIIA'I'U:S

Pal't. :1 of LlaiH hoolt hlLH J"mil dcvou!d III all IIX 1'1 all ltf.io II of Lho tll'igi 1111 ill whidl CIC1(luct.ivo IIIILLlmllllLl.il'lI I~:LII\I! illl.u Iwilllt. 'I'hl' t:lllldllHilJlIH whidl WI: II/LVO 1'(!lL(:lwcl IIllLy \'('ry well IULVO some IWILring on ol,hol' histuric;LI pmhlems (lIlIlCl'l'IIillf( cnrly (:I'I'nl( II mLhl"'lILl.it:s. I fill lIul. wall I. t.1l J,tu inl"l ddlLil/l ILhClut ILIly IIf t.J1I'1I1! hili'll, hut. 1 \Vuultllilw til dilll:lIH.'i I.WIIIII' LllIlm IIl'il~lIy.

or gudhll!lLll ILxillllllLLit:8 ILIIII t.Il1l WILy

t. II·ri.·.lte·in), PI'. Ii·t-li.

,a N. /I II rLIII...... ,

M",/,,.mfltile

~1.\l'III':MA1·IC~

(GiI'II.'mli

I) .." J'rolcills IJi"rI""/1II11 l,"i/"'."I,"i4cJltI .A 1I/'IJIy,'yriiJlJtI e/r.r I !lU!l), p. 44. S.·o ulao A. SI",iHtlr, /Jie IJIal/ltllJlali8cJ,e lJenJ:.

(lI ...wl-RLuLLgllrL 19/;2),

pr.

1i4f.

An illlport:LIIL IJIII!liLitlll ill the Ililollory of ILIll'il'IIL IIlILLlIC'JIllll.il:8 is how nllIll,emllti('.(J1 cxi.9lenr.r. C:~llle UI he IL pmllll'lII. AR fILl' IL.. I ('IL" Rf~O,

:11 Ii

11:-''\111'1:

'l'1I1(

JlI':ClINNINIlII {II'

Cllllml{

I'AII·I':I.

M.\l'lI1(MAl'l(:l'

t hem is no ntlC(JIII~to fOl'lIIUIlLtiulI of thill (IIIclltion in tho Ii t.cmt.uro.

Many HehollLrB Imve luldl'llSf!Ctl tholl1HclvCH to it, IIlLliillg their rC/l(lI~l'ch 011 IIhHl·l'val.iolls which ILI'O at l(llL'IL I"Ll't.i:dly corl'~}Ct" hut their cfl'ol'LR hlLVc~ Ill'l'Vetl II1Il.inly to obscure Lho tmc I1ILtUI'O of t.ho prohlem. II. iH W!Iwmlly ad, lIowlcclgcd Umt Lho lIoled Dllonish scholl~r, H, O. ZCIlI hl'n. waH the fil'st JlllI'HOIl t.o c1ILrify (.)IC princil'lca uscd hy Lhe Imcicnt.<: ill 1.III'il' 1'mo/.~ of r.ri.~/rllli(l1 (LS,'i('r'i(m..~,278 ncclwr, forexl1.mplc, wrote:2 '11 .. Accorliing t.o his LZcuthen's] reselLrch, the one nlu.l only met.\lUd IIf I'l'Uvillg (lxislonce wu.s constructionl furLhermo,rc, Rince ancicnt lIIallwlIl:LLil's cOllsisted only of gcomeLry (boLh nriLhmcLic mill algeltm wcre 1)J·(,Rcnt.cd in I~ geomctrical f01'll1), this mcant construe!I.ioll of fi!l/lre.~. All sueh t:ollstruetiollH Wl'rc buscd on two fundu.melltl~1 1I111'H, tho (Imwillg of I~ stmight linn hetwl!Cn two given poillt.s lI.nd of lL drdc wil h given mdillR mill ccntre. Tlmt thesc cunstructions am 11(1·~,~iMt! or, wlml. lLIlIOUIII.s til lim NlLllIC Lhing, Llmt tho fi,.~UrI'H .y ieltlf'tI by I.hem rxi,'1t is pl'ceiHdy WIIl~I, is u."'·mrtcd Ly )~u~lid's fil'l-l(. lwo I'0Sllll:~I'(!H ... " The illuivitillal sLatcllumla in this qtlotlLtioll may sccm at first !lighl. 10 ho hoLh dCILl' alld convincing, LuI; thcro is 1m importMlt wl'alwc'!ls in the position whic:h Lhoy 1'(~pI'C8cnt. Although UIO Orcells ILI'I'I!ar to IULYe pilL 1II0Ht of their IImthcml~tics (including the u.rith111('1 ic foulld ill 1.110 R(e1llC1L[s) in gcomctriCl~1 form, it is ext.rcmcly dOIlI,I,flll whet.hel' I.he 0110 ILml only meLhod of pl'flving cxist.cnco in ariUlllwlic WI\''I rOll,~/,.uc[io1/., H wo IlOep in mind that Zeul.hCJ~ mClloIlt, It,\' '('ollsLl'uctiotl' l,he'(lollsl.l·ueLion of figurcs', tho wOlLlmcs6 of hiR poSit.illll l)(lcolll~s 'even moro ILpplJ,i~ont. In III)' opinion, it is CIuito CICILf that Zeuthen's view cannot possihly I,,: ,·IIC t:ol'l'e(~t OliO, C()l1sidcr, tor (lxIL11l1Jle, Lho following two JlI'OpO!Ii(illIlH (If 1'~lIdi(I(~ILn ILI'it.lllllotic. \ 1?/('.1/It!lIt,~ VII.:n: "Any composit,n l1umber is mCl18urcd by 801110 I'l'illll\ lIulHbur," gl(~I1/(·,,'.~ IX,20: "l'dlllo 1I11J1I1Illl'II ILI'tl 111"1"0 thlLnl~IIY ILliHigllt.'d multilucic! of prime: nUIII1II!l'lI." II. ~ L • ./:'·III.lItI", lIIa(II . .111m. 47 (IH!lti), 1'1'. ~:.!2-H. (I"dmr; Jlfntlli:illldiscile JtJ:r.islcnz, (/ntcrsllr./lIllIgcn zur [,(lyil.: VJI(l Ollio/"!I;C /IItl,I",,,,,,'; .•r.llcr ]'I"/",,,"c"o (IInllu 11127), ". 1:lU. I;.

110 ().

~

'1'1111,

(:ON!lTIIII{:rIfIN ()It MATIII':MA'I'U:!I

:119

The proofs of lmth tl!cf!o }ll'opusitions IU:~ undouhtedly) ~xis[entifll,: Tho first onc f!hOWH Mmt, givcll lIoIIY ellIllJlIIRil.c nUl11bCl' (/, lhcre e.,:i.~/s a primc lIumber which dividcH iI.. Simi!!lorly, (.0 IlI'oVC the RCClIlll1 t.heorem, it ImH t.o bo showll tlmt no n.i'liglll'11 (i,I!, fiuiw) multitudc (If primc IlIllllhcl'R (!ILII (~()ntlLin nil of them; ill oLher WOl'tis, given /lny (filliLc) lIot of I'l'imt'Ii, Lhcrc c;d,'1I.~ :Lt lI'n..'II. IIlIe primc 1I1111ll'CI' whic;\J is 110t II. memher of it. There is 1I0Lhing which evml ':O-'IeIllIJlc8 II. gC(}lncll'.i(;{ICCOllslrllcli(}n in Euclid's l)roufs of UtcsO two prnl'oHit.iunH. (I n f'LcL, it is tlcbllotlLhle whet.her t.ho pl'oofs shoul(l he described IL~ (!ollsl.l'ucLiolls of mIl' Idl\(l.) lf wo fLrc to lIIa.intain the \' iow t.J1I~t roll!i/nll':/ioll WIL'I thc ullly method tlReU by the Oreel(s (.0 provo cxislmU'e, Ute IInl.ion of construction (:Im- , lIot h(l lillliu'u to W'lIml'L"y; it Jlllllit Ill! gi ven :1. lIlueh 'l!ro~Luen inlcrpl'ct.l~l.illn dt·Rpit.t: whlLl. is l\oRRt!I'!.I'(l in till' l'a.~lIag"· lJuoled Ilot,uve, . li'mjtlsO hlL'! poinLed uut uthcl' wcu,IUlcSlics in Zcuthen's position,WI Ono of hiH ol,jt:t:lioIlH, whid, tll'H(lI'\'eE,l kpc~t:iILI IIIc'Jllion, ill LIlli foIlClwill~: Thcre iH gClltlral agrcmnent 1~IIIC1ngHt hoth ILIICi(!II(HVII Imel 1lI0dC"IISVil on t.he fl~t:t t.Ju~L l~udid WIIoS 11•• PJILtnnist', Bul. the idcllo that gconwt'l'icl~1 consll'lwtiOits cl~n HCI'VO IIoS III'oofR of cxiHt.('nco WII.,~ curlll'lcll'ly f!lr('ign Lo PIILlo.283 III (.ho finnl mmlyliiH, 8uch COIIHI.nwlionR nrc I:ILITicd mit in the domain of 8cnsil,Ie Ohjl·cllI, 1~1l(1 PIILI" tho light t.Jmt '(lxislenm' wn..~ to ho foulld elllcwhcl'C. (At:t~ording t.o hiR vicw, what exi,~[cd (Trio,·) wn..~ 1I0t in the wol'ld of thc 1101"{/[on bul. ill tlml. of tho norton,) - There is 110 dOllht thlLt thesc I'OlIIlLrJUI of li'mjc80's nm very much to the point. III the light of nllr inV('HtiwLtiol1s, the hilll.oJ'i<:1101 Jll'ohlmn of 1I11111te1l1fltic:ttl {';l'isl('1/C:e elm Im'·1,1'C1IL(.ml in ILII I·nl.il'dy new WILY, fill' WI\ know how tho Elcl~LicH I'(~WLJ'(letl oxistcllcO I~ntl, of 1'01l1'RC, how 1,lwia' I'hiloRophy illflucllc('d J'II~LI)'s vicw of it.. ]rllrUll'l'lIInI'C, wo I\/Lve /lcen th:lot dcdllct.ive nmthmmLlic8 wotlill 1I0\'(H' Imvo (~Olll(l illt,o being, had it not heen for tho IGllm~ic8. Hence it Hemns mlLllonlLlllo to I~LLmnp~ 11011 CXI'IIIoIUIoI.ioll of I.ho l;rohlt!lll of IIIILI.lumml,il!1Io1 exisl,em:" whit:h RLILl'ls out frolll t·hc problem of 'cxiHI;cnc()' ill J~I('al,ie philosophy. ,... A. 1"/'lIjl'm', 1", ",,,,,,,,u"in. ",,{ ", ..",1 .. I/"';t'" (J( IIlI'n 1!Ir,I), ". !I2, PI'llr:luA (1111. ,,'l'incll.,in), IlR.201T.· tAl A. Fmjf!/ltl (op, nit, (II, 2HtI "IoIIVi,), ". !lIi) ""'III.ill"R .111 ")("1111'10'/1 If. VII"

:III

\Vill,,,.,.,,yil.;r.·J\1u,,lIu,,,lul'lT (I'llI/ull :1 ... 1 ,,,III, 1I ... lill 1!12!1, 1'. 7ti.1) 11",1 K /-1"dlll (IJir. /i;II/ 1,lrrloni8cllcn lOir1,e,', n""li" I !1I7. p. I ti!I). tn A. Fmjc'Bo, 01', !lit. (II. 2RO "I'flVI'), ". !Iii.

:J20

~U"II'l: TIm IIJo:OINNIN()S 01' OIlImlt MA1'III~MATIL'9

Thc ElcnLics never dll"IIu~d t.Il1LL /leill!! (TC; lir), ur tile One (1'1) ;,.), I'xillll!d, IIuL Lhey did noL csLILhliKh Lhil! flLcL directly. JlIsLlllLd, i>lLl'lneniIII's glLVC ILn indirccL proof of iL. 110 stlLrted ouL by supposing LlmL Uri "!! did not exi.5t !Lnd Lhen wenL 011 to show LltlLL this suppositiun ('Ollld nul. l'o8Hibly be Ll'ue hectLtllio iL WILa inconHiflLont.. Thus n. proof II)' l'onlrailicLion was thought. sumcicnt to csllLblish Lhe existence of

liei,,!!.

- I ,1~xistcntilLl

proofs in ILfit.hmeLic were simillLr lu the argument sketched ILbove except tha.t they in\'ol vcdan abstmct noLion of ",allY (i,I!. number) which WlLa ohta.ined by 'multiplying' (in ILn idoalsonsc) 111l! l~lea.tic Olle. The Eloatics hnd origilllLlly denied the oxistence of l,[umlily ILnd hnd thus ruled out the pussibility of arithmotic. Thernforo, iL WILa found neCCBBILI'y to hMO arithmetic on tho following definitioll (Elements VII.2) of number: "A number is IL multitudo composed of \Il1its." This definition is really l1.n unproved propositiun concerning flU' (';l'istence 01 numbers; it WtLa intended 1.0 provide a firm foundlLtion fill' the whole of al'iLhmetic, and, in addition, it. ClULLled arithmeticians 10 1111\1\0 uso of PlLrmonidcs' indirect. mothod in LImit· proofs of exislence. Olle of I.he Loclmi'lues which Euclid employed to establish the oxist.1'111'0 uf ariLhmetical furms (Lc. numbcra) WlLB proof hy contmdictiun; hn did nul. IIRC COllfltl'Uetiollfl. Pmpoflit,ioll V 11.:n I for oXlLnll'lo, ILRRCI'I,fI I hal, lwcry compuund number IIILa n, primo diviRur ILnd is pruved by showillg tlllLt IL conLmdicLion fullows from its ncglLtiun. (Any compuund 11I1I1IIt('1' which JIIL.~ no primo diviRur mllst lmvn 1m infinite dOcl'l!lI.Ring SI!C(III!IlCe of divisurs, and this is inll'o8.~iblo in Um dOIlULin of nllllllmrH 1II'('ILII!i(' iL contradicts Definitioll VJI.2.) 'fllUs, given any cOl1ll'o\lllll Ilulllher, the existence of a prime number which divides it is cstlLhlished illllil'l~cUy. (Sollie Illodern intuiti()ni8t..~ would dony Umt. a prouf of I·hilt Idncl esLabliflhcs tho exisLonco of a nUlllber,2111 hut Lheir viewlJ need IwL concel'll us here.) Proposit.ion IX.20 is proved in u. simillLr mlLnncr; Mivl'1I a filliLo set of primos, t.he cxisLonco of IL prime number whil:h lil'l! outside it, is esLablished lJy shuwillg that, if no such number oxists, IL 1·IIllt.mdicLion follows. ... ", Hnilin8{ln (in 1'roblclIIJII in the "I.illl.qn/,l,y 0/ JllnllUlmalics .,,\. by l. J..nks· AII.sLurlilllll 1!l1i1, p, II) huspointm! ouL Lllllt. "A cont.ollll'ol'ury int.uit.iollillt. wunl.1 rl'jr.I:t. t.hill proor a..., it. Rl.unclR, RillCU it. t.ri<'8 1.0 cllLablioh tbo proor or " IHlflil ivf'. I·xi" ... ·nt.iul "t.ut.olll'·IIL," I.~ ••

I'AIIT 3, Tlllt r:ONIITllllm'ION I)J' MATllltMATII:ll

321

Jt. ill :Ll'plLrlmt, t.hcrefum, L1mL cml.~lr/lc/i(m \Va... lIut the Olle :Llld Ollly meLhud used in Greek IIlILLhemlLLics tu PI'OV" existonce. AriLhmeLiclLI I'l'Uofs of exislence, even wlwn Lhey had IL ~comcLril!al form, were not. l'elLlly conRLructiunslLt ILII; UIllY IIop(,lIIlc
R1.AIlc): 1'1I1~ ImOi NNI NilS 01' I1ll1mK MA1'III~MA1'I(:R

:122

1~leILLi<:/l'IILLonie

philosophy, no visiblo or porccptilJlo ohjoct {lolLst of whieh WII.8 ohtlLined (LS tho result of a construction, a ltillcl of ",o/ion) <:olllcl ho reglLrded (LS consiutcnt; such olJjocts woro IL8IIigllod IL l'oHil,ioll 'hoLweon Boillg and NoL.Doing' simillLf to Lho ono lJoLweoll I.·II011l/rclyc Imd ignorance whic:h Wlut occlllPiccl hy 'r/ox,,'. All theorcLiclLI gcolllctry heclLlllc moro allll nlUro indol'cJl
nil

0110

II I woulel now lilto tc) tum t.u ILllOther int.orOHLing cluosLion which ill worLh rcc:ulIsidoring ill tho light of wlmt wo havo ImLrnod in t.hiH ),lIl1le It. COlt<:CI'IlR thn I'ClIlLtionshil' IJdW(l(lII tho philosophy of PI"I", ~..

O. Bockllr, /Jalllllllthcmatillcl,e Denice" tier Antiltl! (Oott.iUgl'lI 196'7), p. 20.

I'AIIT :I. Tim CONR'I'UllCrflON 0\>

MATIII~MA"'II:S

323

Lho ono Imnd ILlld ueducLi VO IIIILLhellllLties 011 the other, Ll't 1110 hegin by outlilling tho Idllli of tl'CILtlllcnt which lhis qUC3Lioll hn,.'1 IIcen giv(!n ill Lhll Iit.tlfILturo {ILllcI, ILS II. IIlILLLcr of flU:t, whic:h it st.ill rt!ceivC8 from sorno inouei'll HchohLrs}. I II 1!H i~~Li.~~'I 1'")'liHllC'cl lUi iml'ortallL 1'1'}IC!r IIncier the titlo '~ul' loa c..,(III~r;~;~ gcomclriqllcs cleH Crt·CfI n,vlLllL la re/orme plaloni· 1/ cicrm~/280 All LhiLt illt.crl.~ts. us hem is this tiLle, [01' it suggCflts tlmt ILL HOllie BtILgC Gr~cl( geomc(.ry \llIcll:rW(:III, IL flo-called' P\ILtonic reform'. (\ Jn flu:L, Zeuthen WILS of the opillion t.Imt I>I:Llo himself {if noL c1ireelly, thtm ILL Il:ILSt through his pupils} exerciRl'l1 IL very important influenco 011 gc'ollll,t.ry, which 1'f'.8ulkci in IL Lmllllful'/IlILtioll of the 8uhjt'cL. 'I'wl.'l\'o yelLI"S ILfler tho 1L1'ponrance of Zeuthcll's l'l~per, Tuplilz grwo the following account of how mOKt historilUis eXl'll~incd the rdlLtionshil' l'otweml PIILto ILnd InILthellmtiCfl. 287 Oil

:i\

.

...

"1'IILlo, of courso, did nul. IIlILlte any nmthenllLticlLl discovories; the tradit.ion which ILLtrillUt.c1l Lhe diRCOVCl'y of t.he uodecahcdl'On to him is noL to he tlLlwn scl'iuU.'dy'. llc did, howcver, influence Ilic 'fle.ltcml (/i,.rcliolt 01 muIIIClIInlic.'1. '1'111'. axiomalic siructurc 01 IIII'. , 'Blemc7l/,y', tilt? rt'quircmeltl IIwl only rufa (Uul C07nllas.'1('.'I be u.srrllor r:cnl.~/rucfi.tnul, IIII'. ClIwlytimllllellloti, IIIr.'1c arc (Ill Plalo',., work. ']'IIc,,('.· Icfu8 alld Eudoxtt..'1, tl,e grclrie...t 1IWlhel/utlicicm.s in Iti.s eirdt', ercalt'ri \ 1~IIr1id('(m 1IIft/hemalic... uruler I,i.'! influcllce."

\\'e clLn Hee frulll Llac wonls ill itlLlil:li thaI. tho 'Platonic rcfllrm' whi(:h ~euLhell n.glLl'lled IL'i IL lIIilc.'iLoIIO ill tho history of ILllcicnt sciencc~ WILEI t,huught t.o l\lLvo givon rille Lo LIm wholo hody of sysLcmatic anll cicdlllltive nmLhclIlILLiCfl I'l'IIclucecl hy LI\t~ nnmks hefore ILllcl (lurin~ tho tilllo of I';uclicl. In I !I27, B(!c1wr, l·xl'rc....'iC'tl hill agrcelllcnt wit h 7..f!uUum'a vi()\vH IUIII IL'iIIC:rl.c~11 !,JIILL Plalo'... 1l.I.'hillvc'III1!1ltS IlIlLr\wd U.., ht,,~illllill'~ or n, IIt'W e'l"wlt ill UIt' Itililw'.\' tor lIu~I.III'lIml.it'H, II .. wrtll.,:%1iII "Bnlll,clly BIIC'ILltin~, lIIal.lmllllL(.ie~ Imftlm I'IILI .. , c:lm 1m RlLiel to IIIWI! c:Clllt:t'r!lC'cl il-Kdfwit.J1 VilllllLlllhlLlwH (~""I.h,'" c\ell(:ril .. ,t1 it l'-'i IL 'gt·tllII IlII Olter~iut til" A:!lI. 1hm .•k I' i,/~n.d"tlbl!rlle" Scl."'~lIb.9 f'nrltnllllliny"r, Nil. Il, PI'. 4:11-·1:1, lA' 'l'Oplit7., 'Mtlth"IIIt1til, 111111 Antiltu' Dill A .. like I (11)25), 175-203, IU llltllltl!"",ti.,e/te Ezistcn%, p. 250 (BIlt! II. 2'79 Rhnvc).

21°

I!II :1,

:12<1

dry h:uleci 011 percepLioll·). Thc propcrLiCH IIf various HLrildll~ fi~lIn~M (KYllllllclriclLI ones ILncl Lho IiIw) were invc.'iLigILLcd in IL dcslIllory IIIILllller. l~urLhormUl"O UIOro wero 110 sLrieL rules governillg ·(·oIlHt.nwLiolls'. ('Vcrging' 1\.11(1 1\.11 Idllels of IdllllllllLtiC CUl1sLrtwt.iollS wern !,ormiLLed; Hippiu..'i of l~liR. for CXILllIl'lo. IImdt) URO IIf NIWh lechniques t.o c()lIsLruc:L tho fluILdmLrix). Plato c;arrif'.ll oul a tlwru'UJII!/oiny reform 0/ malltemnlic.'1. wl&icl, gave U8 lite (Ixiolll.alic melllOti amI

the definilion

"I'

0/

11lalhr.IIuzlical exi.'1llmce in terms

0/

cmullruclibility."

III Lho III.8L few dectL(lcs, ZeuLllOll's Lheory ILhouL IL SUl'poHetl 'PIILll/llil: rdorm' of IImLhonmLit's IlILs glLineu sllch wide lu:copLILIICC LlmL \',~ry rIm' diAAellLillg voiceH IlILvo heun hUlL... 1. Tho slighL mmlificlLLiun of iL whid. WILK pmposClI hy ·l'iiplil.." iii Llwl'IlfuJ"O very Hignifi(!(LnL. Tiil'liLz ILrgued lImL .ZouLhcu·s Lhe()J'Y ILllowcd Lho dcvelopllwnL uf (:J'('e1( geolllcL.·y t.o he di'vi(iecii',~Lc; Lwo HLlLgeR, ILll empirical (p ..e-rdllrm) 141011.(1' Imd IL 1.lworeLil:ld (l'0KI.-l'Ilfm·m) olin. Jle I.hen wcnL (III 1·0 RlLy:211U .. NIl\'(!rLhd(·R.~, wo CILII inH(~ .. L I" Lhinl KLltgO t.cLwC(·n Lhe oLhel' Lwo; ILL thiN inLcnlleclilLlt~ sLago pl'oofH WOI'O alre!Ldy in UKe. huL LIm 'llUmlillll of how lII!Lny UlIl'ruVILhlo ILxilll1l8 wem nlfluil"Otl til enNuru thei .. \,lLlitliLy IlILd noL yeL becn SySt.cIlIlLLicu.lly invCHLiga.Lcti. The WILy in which umLhclIlILLiCR is oft.on Laugh I, to children inclicaLcs UmL Lhero eould vOI'Y well hlLvo boen a SLllgO of this Jdnd."

or l~OIlI'NO, Lho Iu.ldition of 1I'lI:h 0.1l int.crrnodilLI.c stu.go leMelis l:()I1Hiclm'ILbly Lho significanco of I'IILOO'S ILllegod 'reform'. This hrings 118 1/1 t.lm Nc.(:onclllllrL or LIm 1'1'01'"80" mOllificlLtion: 200 "It, ill l'oHBiLJlc tlmL tho greILt nmLllOrnl"LioilLns, illcluding thoso who \\'1'1'l' ",c,mlll'l'll (If Lho Ac::ulcIIIY. ,"ere l,~tl to Ulldertake t/,i.... task Iwt II!! (I !iIl!lye...lion 0/ Plalo'S bUl by coIl.'lideTCIlioll8 which UJcre e.9sCltliall!1 1I11,'''rlllalirul in lut/ure .. 11('.r/""", illl't'.~ Plaia ",ho learlll/ro", l"-e", a"d ;1II:orI'OI'UIClL Iltc l"i,tciple.'1 01 tltci,. 1IIcllwcl inlo a !lelleral t"em'!! 0/ ~·""IIIICllge."

Beclwr glLvn Tiil'liLz's itlea..'i rat.hc~r a c:ool 1'I~l!epLiClII. I\.q frolll lim followillg 1'ILHRILgtl ill which 1111 diHCIlAA(!!i Lhcm :%01

,.. Ill'. dL. (n. 281 IlhuvlI). ". 2111. 01'. l'il,. (II. l!R'l "h"vn). 1'1'. 201-2.

WI!

ea.1I see

'''l'hiH hYI'UUU'NiR C~I\Il he nl'illJllI' c'lIlIfil'llll'cl 11I1I' l'eflllt~(1 1111 Lhe IIIL~iR of Uw fmv ... ~c!ords whh:h IIILVo C!III1l(! (IOWII to 11K frolll LlmL Lillie. We (~ILII ILL lell..'iL ho Rllnl of 011(' Lhing. huwc\'er: rlaio 1(VlS tile (tTl,l l)er.~onl(Joblrlin a

cil'ftrwuier.'1lmu/i)lg 0/ IIIe.,tridfy mellwtiicallerlllliqllc

0/ elclIumltlT!I

r.tm....frucliOlt (Old. in .~(J ,foil/f/' "e t:(mhiimtrcl gre",ly III I/'e devr.llllulltmi o/l'fJ... ililJe I//(/I/'rl/IlII;,.,,1 I'I's('(,rr/'."

We Nhon"l /luI, be u..'iLullhd,ml lo find Bl·da· ... ill 11)27. I:LJllCllLing Lhe RI~lLrciLy of records frolll l're·PIILlollie t.illleR and writing 11..'i if LhiA Helu·t!iLy preHlllllc·d ILn inNtIl·II\(lunt.altl,. oh:IIIU·I" \.0 It'ILrnillg mow abouL Lhe lIIat.lltmml,i(!H of \.1m periud. I I. WILH 11111. ,,"til IL fl!w yelLni IILI... r LlmL ho N"e(~I,,,,lml in J'('('OIlHLl'\wl,ill~ Ihc! lII'igillal ""''llion of Lhe lIu·mdlctl LlH'ury of (wc'n ILIIII ml,1. (ThiH \.heory iH \.III! olllesL pioee of tledlldi\'o IIIILI.hc'III1LI,il'H Iwuwn Lu IIH~U'l ILIIII lIIay WI'II dat." IIIL"', Lo t.lm fil'HL Imlf (If \.1m firLh cOIILIII'Y )l.c.) WlmL is SIII'Pl'iHillg, however, is LhuL lleclwr H('OIllH IWVnl' 1,0 Imvo allllllliolled ('lIl11l'lc'l~lly his helief in ZotlLlum's Lh"ory ILiuml.,,, 'PI'LLullic I't,forlll' of lIIat.lll'lImLit~'i. III lIl!i1 ho I'cHplJlIllccl ill Lhl~ fc.llowillg WILy 10 vall d'~r WIL(~rcl,,"'H c:nlljecLuro Lh:LL BOIlIt V II uf LIm 1~·lellte"J.<1 origill'Lt.ctl ill \.1m nft.h l:cIIILury 11.0. "Before IJcillg puL in ilN fhULI form. Book I'll muy /lcrhupS "ave been revised b!ltlle lIuzll,rllwlir.ums 0/1I1c Accu/clI/y. "~03 .BeclH'l· himHelf musL Imve ro"lh~ed· 1.llILL Lhis WILH ILIl illn.IIN/IHLlo wl'ly. fo .. III~ ILdded: " I I, NllUllltl ho mOIlI.iUlWd. hIlWU\'C·I·. Llml. 1.110 ILllthm' (VILlI der \VILeI" clc'n) miK('H ILII illl}lort'LnL aI·gIlIlU,"I. ILWLiwil, I.hili; he pllilll.~ 0111, LluLI, il. iR illlPIlAAil,lu Lu difiLill~uilih I,dwl!l!ll Llw "ollltmL lLIld f"rm of lI.sLril~t.ly 10gieiLI ill\'OKI,igILLiuII lIf Uw Idlld 1I1111(!I'Lalwll ill Boult VH." Till' NlIggC'NI,i(lll \.IIILI. "Lhc! IlIILI,III'IIlILt.il'iILIIH (If I,h" AI:luluIllY" limy hIL\'{l revised Book V II serV(,8 olily III Hhuw how relucLanL Bcelwr WILH 10 ILIULIlClulI 7.c~II\.1l1m·R I'"Hil.iulI. M"t/:"lIIflliacilc 'ExistcllZ (Hillin 1!l27), p. 21i1l. 1101.0 2. QIlt!lIclI 'tnll Stlltliefl ..• :I (1!I:lfi), r.:I:Ir.:1. (~r. niHil O. JI,.dwr. /Jin Or",,,I1,,!!,,,, ./er ",,,II,,m,,,li!.: ill !I'!.••·/,id,tli.·lw,· I·:",,,,i.·k/""!I. I'·/'I·jl'/lrg -\\IUllio-lt I!lliol. 1'1'. :lIm. H'

101

,DO

32!i

l'AIll' 3. TIIP. eONSTIIIW\"IUN 01' MATllt:AlAl'IClI

lI7.AJIl~: TIIK \l1~OINNINOS Oil' fIIIV.I~K .IATIIY.MATH~

10' 0110111011 2:1 (11161), 2!1!1.

n.

:121i

K....... IU): T"I~ "1~IIINNINml 01'

ilium"

l'All1' :to TIII(

MATlilcaIATIC:1

I f(!cl 8111'0 LhlLL Illy ILLLiLudo LOWILrd" PIrLLu'H 'rofi)rlll' uf mlLLlllllIllLLic:H C'IUI 110 inferred from Lho oLhol' pnrLA of Lhis boolt, HO I will not cliscuSli iL ill nlly 1I10l'e doLnil hero. InsLcl.L(l, I would IiIw to giv~ IL brief ILCCOUlIt of how iL camo ahout LhaL l>IILLo WILa Lhought to hlLVO had such IL clc'eiRivll influence on II1nthomn.Lics. "liLLo's clwiC connecLion wiLh Lho IIULLhumn.Lics of hin Limo ill mell· tilllll'ci repenLedly by the ILuthOrB of antiquity. Proclus, for eXILlI1plo, wrole Lhe following ILhout iL.2OI

.. Plalo, who ILpl'l'ILrNIILftcr Lhem, grulLLly n.ttVILIICC·cl nULthenULLics ill J,\"l1eml alld geomcLl'y in I"LrtieulrLr Locmlso of his zelLI for LhcHO Hlu(lies. It iR well 1
1IIalic•., willlluL find tho iclclL.of IL PIILt.cllli(l 'refurm' of IIl1LLhellIlLLit:H

I.' l'rtHlltiB

(ml. Irriudluill), GIi.HIT; tllU Lmlllll"LiulI ia by Ol""n n. Morrow.,

V(!"),

2D5

I'IILUHihlo. 'rhey "1'0 mnro Iilwl), to ngreu with Iteidomcistclr LhlLL tho mothod of indiroet proof, U\O mOHI. im porLILIlL Lool of PhLlonic dilLlecLie, WILEi lmrrowcd frum mILLhoIlIlLLicH. 2VG Wo ILro now flLc:cd wiLh Lho LILRk of explaining what led scholars to I.hillit LlmL PhLL() inHtiglLlcd IL 'refurm' of mn.thomlLtics whon thoro WII..'i 1It1 hlLrd ovidonco Lo support Lhis view. (Proclus' refel'Cnccs to Lhe gl'l!lLt zeal which PIILLo showed for the study of mathemn.Lics and the grent ILIlvlLIll:eS whil:h ho IIIade in Lhis sllhjcet. IULVC, of C()llrHO, SOlllO hel~ring on Llle III1LLlelr, hut LImy I~I'() far from J)(ling HuOicient leJ jmlLify tho c'onduHion Llmt Pln.to reformed lIlaLhemtLLics.) I Lelieve tImL Lhis curiollH modern idca nr080 in part n.s IL result of \.Ito fnd Llmt cn.rHer schularH (fur no good rcnson) misLl'Ust.cd II ipl'ocmlcH of Chios, nil uuLHI.ILnclillg III1LLhollULLieilLll of Lho fifth cellLur'y II.C. ThiH iH noL tho pllLl:o to discuRS Hippocmlcs' fllUulmLures oflullcs, ItuL I would lilw leI mention some of LIlli things which have been said abouL Lhis ILdliovllmcnt III' hiH. Th" ronllLl'ltS in qucHlion are worth recallillg heCILUHO I IIf~y Iwlp us tu ullc1cmtlLnd why n 'PIrLlunie refurm' wn.s thoughL to IlILv" talwn 1'11\1;0 in LIm hiHtol'), of Orc'clt IIULthenmt.ics. Ld me lwgill hy montioning tlmt Hipl'ocmt.cs \VILa not credited wiLh uHinJ,\ inclirol!\. nwthmlH of prnof hy parlier schIlIILI1I. 2U7 ThiH opinion altuliL J lipp()cmLc~R' Itnowll:c1gl! ILIld cILl'lLhiJiLicR led lo certain le!xLtlal ellUlndlLLiolls hoing made. PILRSILgeH in Simplicius which contninecl mfnrcmc(,R leI ill(liroct proofH wero freely exciRcd on Lhe grollnds \.IULt Lhoy could not l'oRSihly hn.vo been Lho work of Hippocmtes nnd mURL h,LVO Leen luldod IILle,r by J~UdOIl\UH.2V8 Not conLent wiLh Lhis, scholam Pl'CIf:I'(,llml leI gu flLr Im),,,,,,1 thn ommull'll Lext in thoir inle:rprcLILI.ionH mul made lIil')locmLcs out leI ho IL Lruo sophiHt. For l:XILIIIl'lcl, Siml'lilliuR le:IIH IIH Llmt JJil'l'oC!rnl(!H provod the 1'01I"wing Llu:orclII: "Gird"K havo Lho HILIlIU I'tLI.iu n.'i lhll 8(IIIILn'S 011 tho dilLllIOlcrs", but ho failH to RILy how this result WU.H csLrLhlishcd. Con-

"' IC noitlt'lIIlliRLur, DRS ,..:mr.lll /Jell,"", Iler r:riec/Il'n, HILlIlhurg 1949, pp. 44 Rill I

lit;.

n. I.. \'ullllnr \V"unl"II, ErwRclielllJe lI'i.•.,ensc/lII/t, .,. 247. "' Re·" II. C:. l.tHILh,,", 'Rur I,,,, .mlllllliRHIIIII·'·R W;UIlII;tri'I,"m dlla Crol'-'I', 1'. 44 R (IL rull rnrnrulI.m in givull in n. 2R6 "lIIIYO). 111. Ibitl., 1'. 41;2. III

I'riltt:l'Lon 1970.

327

(:ON~TltIIl:I'I()N ()It AI"'T"I~AI""I'IC!I

1l1.A1II): TIII~ ImClINNINOR 01' 01l1'.1(1{ A1A'l'IIICMA'I"I(JS

HI''IIII'nt.\y, Home comment.l~lOI'R WOI'O led lo Iwl{ whoLhor HippocrnwR wa,'1 ILbio lo givo II. convincing proof of iL. \viLh ILmo.zing solf-confiucnce, Lhey argucd Lhat ho coulu noL IULvo l
not tile only one in wltich such gUllS occurred. Simi/a,r one.' were 101m

I'A ItT :1. '1"111': C()NMTllIICrI'IIIN III' M ATClI·:M ATII:M

:12!)

"1,'IIe SOllbisliCCll JI1!Mkl'.", on Llw OIl(! hiLmi, \\'hClI~XC~I'III('d ('legILIlt.ly t.IlC myslci'iuuslclLp into Lhe infiniu), 1Llllltlte Acm/eJ1ticia.JIo'; on U\(~ oLhcl', who exposed all fllLWS in fl~ILSonillg lIy lllelLllS of tltcir irr.yeniou., IItctlto-

dology, Itc!'L t.he fmmewor). of fillite geollll'lry inlIL(~LI~llu lIu"l" overything follow more gr.mllf'trico fmlll IL singhl IImv ILX iom uiscovcrccl lIy ElidoXIiR. to

It. is npl'lLrcnt from t.hese la.'1L LwCl (1IIot.lLl.ieJllR UmL we ILre uelLling wiLh n.lw%ld /,isloricul comltrue/;on. All inlcrc'Ht.ing hody uf'sophisti('al IIHLLhcllllLLics' ncc(h~ lo 1m cClnH(.I·tld'(~II!lU~ in 111'11(,1' UI ('HLILhIiHh 1.lmf PItLto's 'reform' of mu.LhellllLtil~ ILt:LUlLlly toull l'!:LCC; Lhe 'reform' it.,c;elt can Lhen be int.erpreted ILS II. gl'olLL ILIl vallce 0\'('1' this clLrlicr ILppro:Lch to science. I need hardly sn.y t.Jmt neither plll't of this hist.ol'ical construcLioll will sll\lld up to mlll'h critic:,,1 HcrulillY. There iH 1~lloLhcr (IUcsl.ion whidl iR HOIlU'LiIlU'N ILSIH'cl in Lhe (!ontoxL ur PllLlo's ILllegcd 'reform of nmUI(lIJIILI.icH' ILIIlI whic:h CILII, ill Illy opinion, now Ill! anHwcrcd with somo (:cI'La.iIlLy. TiJl'lil.z, ILL t.he clld uf his 1'IL[lCI', fOl'lllUllLtes iL in Lho following WILy: "WILS LI)('I'U eVeI'lL t.irno whelll'hilosophy pl;~yed fL dncisivo I'Olu ill IlIILt.Jumml.icN ILIIII helped lo fm'm tho uisLincl.ive chamclcr of Lhis 8uhjceL, 01' diel iL (i.e, IllILLhcllml.icR) evol VB entirely Oil iLs own 1" \V c Imow frum our om'li
/Oitlul in doz(m,.'J of otl,er pro]lOo'Jitio118 (1) wlLielt !tad l,ce", formulaled by 'he SOJlhi.'lls." lie thclll g;LVO II. "riof ILCCOll nL of Lho J~ud()xilLn axiom which is required in !.III! proof of Jlipl,m:mlcR' (.heOl'mll;300 Litis eno.bled him to mlLlto ILIl nlrecLivC) comparison between tho Sophists and t.ho 'true Platonist fwhul:Ll'R' :301 :'11 /)it! A ntike I (I 92/i), 182-:1. '''. :-;.'" II. 112 nllUvn. 30'

lJie ,b.like I (I !J25), I !J2.

lu·cr.

B. L. vIln c1"r

W.""·.,,, .. , Mil'''.

war-lunule IVissenscllll/t, 1'.214.

A,III. 1211 (1!I-I7/,I!I) I:I!I-·I() /l1U1 Nt··

1"):\'I'~')ltll'r

POSTSUlt] Pl'

After rcnding this hoolt, the nOll-mathematician might welcome somo remarks Bbout the ways in which Euclid's axiom system differs from modern axiomatizlLtions of mathematics. He might also like to see how tho various Bncient scientific concepts which we have discussed, 11.1'0 Irc.Llnu in conLomporlLry nmthenmtics. For example, it would be instructive to compare Eudoxus' theory of incommensurables with lll'.leldnd's theory of real numbers. AR IL IlllLtLor of fact, I thought seriously about tho possibility of at.Lumptillg such a compn.rison; finally however, I a.bandoned the idea. II. heclLme clelLr to me tlllLt Greok mathematics could not be placed in its )lroper historical porspective until .dot more research had been done. 1",1, 1110 iIIURtru.t.o this point with n.n example. 11. is usulLl nowlLdays 1.0 djsLinguish between rational and irrational 111IIIlhers in the following wn.y. Rational numbers can be written (l.8 /m!'lions or ratios (mIn), whero(l.8 irrational ones on.nnot. (This is why tho lal.Lor BI'e slI.id 1,0 be irrational.) Historians have been of tho opinion t.hILL Lht' Grcolcs must n.lso luLYO made Lhis disLincLion. A ratio in Grcel{ IlllLLhclllaLics wus culled B A&yo~ (a : b) ILnd so they ren.8oned, the wOl'd (j).(/yo~ in Book X of Lho Elements must hn.ve had the sn.me meaning as our lluLLholllatieal term irrational. I do not want to comment upon this view here, but I do wish to point out that ehe problem 0/ mathem.atical c'j).o)'m' 1I(1s not evcn bccn {ouellcd on in ellis book. I have ignored it complot.ely sinco it docs not bear on my main Lheme, the discovery oflineo.r illcomlllclII8urn.bility, (In tho quotation from Proclu8 011 p. 153, the Will'll IiAU}'/I1' IIlLA nil onLil'Oly clifforcllL JIlolLlling, of coumo.) Anothor topic which we did noL rcally discuss (ILlLhough it W(l.8 mentioned in I'ml!~illg), iH R"doxus' t.hoUl'Y of [lmport.ioIlR. As I havo ILlrOlLdy 1"0IImriwIl ill tho illLroducLion, wo 1Ll'O sLiIIlL long WIl.y from being able 1.0 I-(i V(I II. dcfiniLi ve new pictur~ of the early poriod of Greek science. There-

fore, 1 ILIll ohliged Lo (:ollCludo t.his lIuol, wit,holiL l'rl'Rcnting IL l'OIl\1'ILriROIl of ILncicnL lLIul modern IllILthOlllnLiCiLI concepts. There is a very infonllBLivo boolt, "'"at is /IIctlltcma.tic.'I hy CoumnL ILnd Rohbills (Oxford UnivemiLy !'I'CR8 I 1l6n), which cn.n be rccommend· ed to Lho nun-llllLLhomlLticilLIl who is lIolleLhelcss interested in thcRO nmLlcl's, Clmpter 2 of UULL work (,Tho nUIII her sysLom of mlLLholnlLLics') is eS}lccilLlly I'OIOVlLflt (l.8 fiLl' lI.q ru.tiollILland irmtiOlllLlnumhers are concornell. Also, tho sections un tho Euclidean Blgorithm, axioms ILlld ILxionHLtic~ ILI'O woll worth rending,

AI'I'I~NIlI

X

iR conool'llml wiLh LllI! cOIIRt.ructioll of LWII /lLI'I~i~hL !iliuM whoRO .m.m (/.1/11

IlOW

THI~ P\''J'IlAGOIU~ANS DISCOVI~ltJ~D

PltOPOHITION IJ.5 Oli'

I

'J'1·1I~

EMallIEN'l'S·

'1'JII~ l'n~VAII,INO VI~W

N()WlUIILYH lIIo!!L schollLl'H woultl ILgreo UIILL ProposiLion 11.6 forll1H pari. or rllIC:iclll PytlUlgorl'.lm 11Iat/tClIUllic,'1 01', more precisoly, \.lULL iL iR IL I hl'Ol't'1I\ of Lhe so-clLlled 'y(!omclrical algcbra o/tlle PytllOyorCllns'. Silwc it iH 11811a.lly disctlRScII LogeLhllr wiLh II. 6, 101. me begin by quoLing boLh Iff t hplie propositions: 1'l'Ol'llRil,ioll II. 5: "If IL sLmighL line be eu(, int.o cqUlLI,LIlU ulloqulLl 1'''I~lImllls. Lhtl I'ectallglo contlLillcti hy Lho llllCIIIII~1 HCgnlllllls of Lho whole, lClJ{elhcl' wiLh Lhe KlltllLl'C CIII LIlt) KLmighL lintl heLwclllI Lho l'0illl~ of Iwdillll is cqul~1 Lo lhe RIllIILro on Lhe Imlf." Prupm;ilion lI. 6: "If 1\ sLmighL line IJO hisected ILllt! II. sLmighL lino btl lHlelc'd 10 iL inlL sLmighL lille, Lho l'eeLlLnglo conllLillod hy Lho wholo with IIII! aeldl:11 HlmighL lintl ILIlII lho ndllcd sLraighL lino togdher wiLh Lho 1''1"an~ Oil lhe half is (~CJulLl U) Um RI)tllLro on LIm stl'l\.ighL line IIlILde up III" IIII' hall' 1~lIcl LlII! lultll~tl KlmighL lilltl." AI'I'ol'dillg lo Lho stlLndl\l'u hisLoriclLll\ccounL of Gl'cel( mn.LhotnILLics, II. !i 1':\11 lIe l'{'l(ardcd IL'J a gcumoLric CIlui vl\lcnL of Lho ILlgchmie fortntll;~ (,2 _

1,2 = (a - b) . (fl.

+ b)

• ThiH I'''p,.,. \Vila ",I'iLIA'1I (nn... r Lilli r"14L of t.h" bool, hocl blinn cnrllplolAlIl) Lho illvil.uLion of Prof''IIIIor Ph. 1If. VOEIIIiliotJ. who prNlCnLe,cJ it. lit. 'l m'lCt.illg or tI,,· On"'I, AcocJrmy of ScitlncC'8 ill At.hllna on IG Moy II1G8. It. i8 rol'rillt.cci 1"'1"\' IH""UlIIO it. cOIlt.inue8 t.ho work hesun ill !.1m pro/lont. hook IlncJ poinLB out. II .lin·llI,illn fill' fut.uro rrsl!nl'd,. • H. I.. vnn ,1"1' \VII" ... I..... ";r"~"'//f'lIIlf! Jl'j~.'CII.~,./Ui/'. p. Hili. t Ihi.I., p. I!JR.'·hiH IIxplllnnLill1l gill'" hlll:k LeI 7.ullt.hull 111111 "vcm to 1'UJllII'I'Y (""1' II. Ii IM·III\\'). III.

in nat.mo3 Imel Lo hlLvo ol'iginlLLod ILlll0ngsL I.he B;LllylonilLIIR whu ILlso IlcwiKml pruccIlul'CS fur Imlvillg Lh11l1l.4 It, iH daillll!ll LluLL LIm .PyLhll.goI'CILIIH KllhRI!llIlontly gll.VO " W·"l11ol.l'il:al fUI'IIlIII:Lt.ioll of LhiR IllLrL of BlLhyluJliam lIuLLholl11LLic8 :LIlli ILI80 8ul'plil'cl t.l1O nl'oc8sI\ry l'roOfR.& I tlislLgrco wiLh Lho "hove inLcrl'rt!L'Lt.ion un t,ho following gl'UlIlIlls: (I) JI~V('II if we Mil I!onvillceti lIy Neugehalll'l"/l lLl'gUIIHlIILR 1I.l1d :1.c(·(·pL Llmt. Lhol'o WILfl Rllch IL Lhing 11.'1 'BILllylolliall algehm', LhiH tltlCS noL IIWILII LlmL Lho Gn!eli8 in pl'e-J~u,:lidClLn LilliCH act.ually Iwow ILllouL iL.lot ILIOIIIl LlmL they lool( iL UVCl'lLlll1 puL it illto getlllld.l'it· I'llI'm; in rad, IIU ono IUL'I yllL 8111:0Co(h:ll ill producing lLllY CllnCl'Cte (willonco to 8ul'pOI'L Lho view LlmL Lh"y uid. (Arwl' ILII LIm Ol'oul(s nov(!1' 1L(1opLed Lho 1'1ILCO-VILluo HYHUmt of noLILLion fur nUmht'I'R frolll Lho B:Lhyloni;LIIR, oven Lhough iL \~lIultl Imvo beon II1l1c:h ml.'1ier 1'111' Lhem lu havc! .Jllnn lhiK LImn lo h:wIl l'(lwol'lwd Lhe whole of lllLbylonilLn algclIl':L.) (2) Evor sinco tho Limo ofTILnllcryG iL ha..'1 bocn ctlsloll1ll.ry 1.0 rcgaml ctlrLILin of Euclid's pmpoRiLiollS II.'i 'nlgdll':~ic LlW()I'OIllS in geomcLric: fOI'III'. Howovcr, Lh('.8o proposiLiolls 1'1'0 ILlguhmie ollly in Lho sense UIILL we (:ILIl V(lI'Y oa.'Iily filldlLlgnllmil: I''-'Hulls whieh ILI'C C!qllivlLlellL UJ 1.11('111. It. t:1'l'l::.illly c:mnnt. ho IIl1Linl;Lillcll Llmt. 1.lwy HLarl.c·c\ 0111. n.'! rllljdmLil: Llworollls til' lIS Lho soluLiolls to IZlyebmic 1'I'Oblnlll!!, fUl' all or thOIl1 Imvc purdy geometric origins. Ono such prol'lIHil.ion is [1. 5 which. whcn illlnrprotetJ fwm IL lUodllrn IKlinL of view. c'all he l'ollll'ILrcd to lhe solll-

+ +

= ((I - b) . (a b) b2).1 Sinco Lhis clt.n bo slLid of ll. 6 us wl·lI. iL wlluld 1~J1l'lml' LluLL huth pl'Ol'OSiliolls ILtnnunt to Lho HlLIllO thing, '1'111' l'xl'lallation orr(~I'Cd fur I.his cUl'iuus flLCL2 is Lhl\(, II. 5 ILnu 11. 6 nrc 11111 ("('all), 1'l'IIl'llsiLitlllH, huL mUter the so11tLioll.3 to Cf'r(aiJ£ problc1ll3. 11. r, (III' II~

nr

lITrnlw:/ILI'" given, whm'(lIUi ill II. II iL iR Llw dif/r.I"l'.Iu:r. (fllrl/mltlltet tim Lwo lines which is givon. Thl~81! l'rnhl('IlIH al'e KIIJll'flRCc.1 Lo UO ILlgchmic

J 11",,1.1, (1~IIr1id'., J.:(c/IIcIlls. VIII. I, 1'1'. :IH:!IT) 1I1!!11 I'"inl", 1I11t. lImt. lUi "'111 IMI inl.o'rpr.,Le,cl lUI n "gnnrnnLricnl Rulut.illn of" fJl" .. lmLiu "que,Lion". Till! 'lilT."r. "11"11 IH't.",,,,," Villi ,11l1' \Vuunlo'lI IIIUI nu"t.h iH th,," Lilli I"LLe,r 1II111"'R 110 IIII'nt"1II of 'nllhylunilln Rcil'lIcn'j lIullth'" t.nuUllnt.ioll of t.ho Klr.IJICIII.9 npprnr£'fl h.·f..... • NnUH"III11I"r'R n'llllRrd, wun pllhliKllC!cJ. • 'I'h" Hl.4u"lnrcl rofert:llcu fCl .. I.hill iR n. N"ug.. I"''''· ... 'hilI' gnlllllllt. .. iR.:l.nll AIW· lorn . SLII.Ii,," 7.lIr G''fIchid,Le, clllr ulIl.il(l'l\ Alg£'hrn III'. Q"dlf:ll "'''' Slwlip.1l ... B :l (1!I:lIl) 241i-6!1. I B. L. VUII clur \VIl"r,lc'n. Krwnrllelule lI'i.911"".9,./',,/I. 1'. 20:\. , 1'. ·I·nllllnry. 'J)" I" RIII"I,itlu g.i,,,"ol,l'ifl"" .1,," 1,,·ohl,;IIII''' (III R"I:IUI.I ,I"~n; "VIlIII. I,:"tllid,,', 18R2 (1'I,l'd ..... ·.1 ill M,IlIIuirc., .9";,·/l/;{i./,,,.8 J, 21i.fHII). N"UU'·'IUII:·'· n'J:nnl"cI l.uIIU"m rlII tho ,IiR,~ov"I'I'r nf 'g"on,,·I.';,·,,1 nIH,·I,m·. wla"ro'''R il. W/IM III fllct 'I'unllnry Wlall "'·iHinR....,1 l.hiH i"on.

lioJl of ILII ILIgebmic problom. Nonethele88, it is a proposition which IIri~iJll\lIy hlullL geometric moaning, and we should not allow oursclve8 10 forget Lhis fn.ct.

2 "IY OWN VIRW

III Illy opinion, II. 6 is I~ purely geomeLrielL1 lemma needed for the Hullltioll of I~ purely gcomctricn.1 problem (or to prove the corrcctncR." of thi!l solution). The l)roblem is sLn.ted lUI Proposition II. 14; it is "10 OllUJ/rllCI a .'iqllare equal 10 a given reclilineal figure". It iH obvillus thlLt 11. 14 depends upon 11.6, both from modern comIIwlIl.ILrieH ILIIlI Ll'lLIIRhLLiolls which almost a)wlLYs refer back to Lhe ItLttcr propusition in their discUSRiolls of tho fonner, ILnd even more so from t.Jw origilllLl Grccl( text. 'fo IL IrLrge extent II. 6 is ropeated word for word in the proof of II. 14, I~ suro sign that whoover composcd Lho text illiended to refcr lmcl, to this proposition. I Jldced, t.ho complicated ILlld awltward languago of II. 5 suggests thu.t ii, WII.." t.ailorcd cspccially U) fit tho needs of II. 14. The author appOILrR III h:we fOl"lnullLted it in such a compliCiLted way because ho Imow from Ihe 1I1art L1ml, t.hill seemingly I~wl{\vurd form of tho proposition WM t.ho IIIIC hest Imi(.ed for usc in tho proof of ll. ] 4. I shall show that this was i II f:~cL Lhe CII.80. 1"irNt, however, let 1110 t.:~1,0 this opportunity to point out that tho I,illll of eXl'hLJULtion offered ILbove is eqlllLlly convincing for other l~u­ "'idl'lul )II'OI'\II~itillns which up to now III\'yo hcon regr~rdod ,La I'I~rt uf IL ·~I~olllcLJ'icILIlLlgehrll.'. 11. 6, for oXlLlnplo, i8 IL purely geometrielLllomnm whil:h bl!aJ'H t!Xl\ct.Jy tim III~mo rellLLionship to Proposit.ion II. 11 ILR II. r. 1IIII'Ii In II. 14. Tho ron."4on why it I\,pl'~ars to bo I~ special CI\8O of II, fI, iii L1l1Lt J I. 11 is Im8iCl~lIy noLhing moro tlmn IL special eMC of II. 14. Y.,I. ltllll(.JUlr purely gnullloLrit:nl 10111 mIL which 111\.8 Imon v iowecl ILK pari. of a 'gculllotricILII~lgchm', ill l'w)losition 11. lO.7 'l'ho writingll of J'rudlls 1~t.tcSt to the fact tlmL it is in reality n. lemma8 needed for the I'rltllr of ILII ILnt:imll. Pyt.\lI\g(lrt"~n 1.\10(11'0111. }i'tuthorlllom, I.\my lu\.Yo II. 1-" VIlli .Inr WlUortl,,", J;:rl/l(l(:/"",,itl IVi,'IIIlIlIl"/ta/t, p. 202. • I" /'/c,wnilJ Rem 1'lIblir.mll CO''''"tJrJ/arii (ud. \V. I{roll), 190 I Chapter II, 1'1'. 2:1 ,,1111 21. 1 :-;""

335

AI'I'KNUIX

S7.AlIO: TJlR lIRIIINNINCIS 01' mmRK MATIIKlUTIC!!

onabled modern resen,rchern to reconlltruet 1,lIill theorem oven though it Wl\8 novor incorporated into thc 1~lcmeJlls.g I now want to give a moro deL."\.iled n.ceount of the genesis of Proposition II. 5. This willl'C'I"iro RODle tiiscuHSioJl of t.ho inu!I'Cstillg wl~y in which tho I 'ytllu,gol'oILllS dOl~lt with tllo ILI'l!ILK of l'a.mllulogrmlls. Tho next clllLl'uu' is (Ievoted to an outline of their theory. 3 ELEMENTS OF A PYTJIAOORRAN 'l'JlEORY AnOU'l' THE AREAS OF PARAI,[,JU.OORAMS

Our stlLrting point is the scene in Plrl.Lu's .AlcIlO (82L-850) whero Socmtcs !LSI(s ILn uneducl~ted ShLVO how to fintllL sqtJlLro Illwing t.wice tho Il.l'OIL of 0110 wiLh sides Lwo feot long. In oLhel' words, ho WILllt.." I.ho sln.vo tu find (t number n slIch t.hn.t Lho sides of I,ho rccluil'od sqllll.l·O will

r----'-----, I

I

I

I

I

I

~---tBl

I I

I

L____ J~ig.

I

-+I

12

IlILvo IL length of n feot. Tho sllwo's fil"Ht UIIIlIght ill L1mt tho way to dnuLlo tht! Inon. (If I~ Hlluaro ill t.tl d(luhlu till! length of its sidcR; so lit! IUlHWCrtl L1mt tho ro()lIirmllengUI is Inrtr l('.r1. HIII!mt.cH, hllwlwllr, .. /"Ocm~clH to show him Lhu.L this would yield I~ KC]UlLro h:wing fnur t.imes the ol'igirml ILrolL. (Tho ILrelL of SUt:h u. KljtllLm wuuld l!unHisL "f sixt.ccn unit S(IUILroR, wherclLa the I~rel~ of tho originlLl OJl(' 1!(llIsisteci of only four; sco Fig. 12.) Tho slll.vo's next suggestion iH LhlLt IL tUIUILI"O with .'1icir.'1lllrrp.lert lOllY might.lmvo twico t.ho urigiIllLlnml~. AW!mt.I'H 1.111'11 drn.ws IL c1ilLgmlll (Heo lrig. 1:1) showing thlLI. Lhis H111I1LJ'0 will nut tin hCCILURO its nrelL COJlsisLs "f nino unit SqUArc.". 111 rCtloulIl.ing UIO ';)IWO'S ILI.Lmnl'l." t.o ILIIHWOI' /:)t)(II'lI.WH' IJuoHtilln, 1'11~tu 14"011111 til 1111 hinl.ing L1I1LI., IIIICO tho sideH of IL givon sqUlLro hlLvo Looll a,'1siuneu. ,'illllte nllmlJCr (Ull/,ei,. lcny,h, tho Rll.tnO r.muwt 110 clUIIU for t.JlIl HiclnH IIf II. Hqll:LI"Il IlILvillg l,wit:1! Uw I~rm"

• n.

I,. vall c10r \VllOrclnll, ErllJ"e/'cmie lI'ill,"!Il"r/ur/', 1', 20(1,

SZAIIO: TIIK IIKOINNINOS Oil nnRI~K MATUKAlATICS

AI'I·I'.NUIX

3:l7

l!XI~CJ.ly

Lwi(:c Lim ILI'OI\ of Lho RCJUMO Oil Lilli side). HIlI1C(l LlUl cliRcovcry of lil/c",. illcommelUmrabilily weilL Imml ill halld wiLh n. Imowlcllge of

Jj·ig. 1:1

Ill' t.11Cl firsL ono. In flloct. Lho two slJulLl'es will hll,ve sidos which o.ro incomStmngcly onough. however. the prohlom of i ncomIItc!IIllumbilily is nevor monLioned explicitly in this scono. InsLctl.cl Hocm~ dmws ono lu.st. ditLgrn.m (~'ig. 14) which shows t.hn.L a squn.ro is clivillt'll inlo Lwo cCJulL1 illoscolcs LrilLngles by its diag01UZl. and honco UmL Lho squlLro construcLcd upon the dilLgonal of the original ono will )lILvn OXILCtly t.wico its area,

lIIr.II.m",blc illlc7I.gll,.

Fig. 14

Imliove thaL this Rcono from tho Jl CIlO gives us o.n idelL of huw tho Urc:c1ts may havo arrived aL the discovory of linear ineommensurlLhility hy way of a.n arithmetical problom which appoa.red in geomelry. Given a IIfIUILI'O whose sicloslULd beon /LSsignod a ntt7nber. they wanted to find t.ho Itltll/I'Cr corresponding to Lho sides of II. square wiLh t.wico its arello. 1~lIdicl'H fundamontn.1 dofinition of 'numbor' (VII. 2) procludod Lho posHihilil.y of Lhis }ll'ohloll1 h/Ioving nil arill,,"clical solutioll. nonotholeH8 it. J'(!nHLined of interest. from a ocolnetric Iloint of viow. The roo.liz/I.Lion (.h:~L ILlly square could ho doublod by constructing 1\llot.her one Oil its Ji"yulud. Logother wit.h thoir inlLbility to uscert.n.in Lho numeriall rlLLio IlUtwcmllhe side of ILSqUILro Imd its dia.gonal. must soon IULvo led Greek IIIILt.htlIllILLicil!.lUf to concludo LlIILL LhORt! Lwo sogmonLli WClrc, i".r.ommcu.~ltraMe in length. Of COUrsO, thoy must o.lso ho.vo been o.wo.re tlmli Lho lwo were mC48urable in aqu(lI'C (since tho square on tho dil'gonal hnd

quadraLic commonRurabiliLy Ilond Lho clifliculLics prescnted by the fnr1111'1' ('ould be cil'eulllvonLc(L by mCILIIS of Lim i;I.UCI', 'fhe hiHlul'icl\1 intm'CRt of S()C!rl\t.e~R ' conversaLiulI wiLh tho HIILVl! iM IIf~ighlcllecl Ily tho flLct. LlIt!.L Lho 'lrillwwlical prob/cm 0/ dOltbUII(I fi,e ,wIIUII'C iR equivlLlonL to Lhe 1'1'CI1JloIII of filltlillg (t mClm lJro]Jorliollll/ beIwren tll:O 7utmbers. onc 0/ which i., twice (t., la.rge fl..'J lite ollte,..lo '.I.'ho PyLlmglll'n/\IIR lIIuHI.lmvo Imown fmlll I.heir Ht.ully of l\I'iI.hmeLie LJmL Lhe I,LUCI' I'I'IIIJlmn Imc(llo llulllcl'ic:\1 RollltiollR. Thill fact WIL'J cerl;\inly f;LlIlili,\r lo t.hCIIl, Sillt'tl iL follows from 0. theorcm (ProposiLion 3) ill the Serlio r:",wllis whi(~h 8t.:\tc~: "Tlml't! clll(m IIut ('xilll,:\ IIlmLn proport.illll:~IIlUIII1,,:1' "eLwecn Lwn nUlllhefH in 1\ mliu RUl'l!l'l':LrtieuI:Lris (i.c. in II. ral iu of t.he furm (n I) : n)." BelLring ill milld thiH pl'uposiLion nnel Lltn 8c:ono fwm Lim AI ClIO cliR('1II480cl ILitOVO, wo co.n recolIRLnu:L LIIJ'(~fl inlflreHLin~ R(.:LgCH ill Lhe clevol"I"III,'IIL of I'm-l~tldicll!ILII nml.llIlIlmt.il!H, (u.) 'rho finil, atlLgo SIloW Lho omol'gollC(l orl'crl(Lillnril,hmetioal Pl'OltlcIIl8 whoRIl HollilillllH c()ulclnut he fOtllll1. On dURcr illSPI'I:tioll, it. l,ul'IWII OUL t.IIILL 1>01110 of lh(:80 were cOIllIJ1"iely ill.~oltlble. ",llUrOn!l oLh('I'II Imel .~ollt­ I;"" ... onl!l tHuler r.c,.'llin romlili(JlI,~. IndUflc'cI :LIllOlIg lilt! flll'llll:r \\'1I.'t Lim I'roblmn of finding IL mea.1I pl'oporLiUlIILllIlIlIIhl'I', x. Itelw(!el1 Rome 1111111lIlli', (I. ILlld 2n. The discovery Llmt. there W:L'J 110 stwh x fot, ILlly fl, repreRent'('d (If COUl'RO n. cOllsidomLlo IwhievClIllcnl, (") SolltewlllLL l:Lt.cr it. WIL'l 1'('alizc~11 I.hat. Lhese Sl\lIIe prohiems could c:I.'!ily Le sulved if they wel'O given gellmct.rical int.crprctuliuflS, For eXlLlltple, the monl1 prolJortiOlULI L.IOLweol1 a lLml 2a. was found by taltillg II III Ito I\n n,l'hitrllol'y stmight.lino il18lCl~1 of IL lIt1mLer, ILIIII cOllstructing II. HqlllLI'O 011 iL, The tliugOlIlLI, cl, of LhiR HCIIIlLI'll WIL'J till) I'nlJllired mOllon 1"'III" I1 ' I,iolllLl Hilll:e n : d = d : 211.• This oCI'IILt.ion C:LIl ho 1"'uvc'c1 wiLh I,Ito

+

11i1'1'",wuIA'" .. I' ChicOlt \\,1 ... ""!;Il'·""',1 I,hnl. U... , ..'.... 1.·111 .. r rI"""';III' '"~' 1111'1111 1"'''INH'LiuJln'8 h"",W""" " 11111111,,'1' fII,,1 il.H .I .. (cr, 0, B""lcl'''. /),,.. /I",t/,r/l",';,,,'''r IJrl/h:u ,I"r A IIlil.',-. lliiLtiIlW'" '!Jli7. p, 7r.), II. ill ill"' .... 'fItiJlI~ ..... ·.. IIHi.'.·', ,.,,\\, ,,,. Jlli~hL IOIl v" C'''lI\n "1'0" (.hiN i""II, lin kllnw. IIr C!OIlI1lf', tlml. LI,,, "lrlllimr'ri" ,,,'ol"mll or c1f1I1 .. lillg LlII' "(jIll'"'' .·",,1,1 Ion H"lvl'.1 hy fitulillJ{ lIIall 111""" ...... ,,, .. I.i,,"nl .... 1.\\'... ·11 " IIlIml ..... nllrl il.A .'""101,,, III Illy .. "illi .. ". 1,'lill " .... ""'.'.\' '1'" 'Iilll I,,, ""111,1",1 .. !.I III I. 1,1". 1,,·1111,,'" .",!I'coII/ctrir. 1lmllle". IIr c1""hlill~ Lim ""'In w.. lIl .. I." Rlllv"ll if 0110 cOllI. I find two Hilda 1111'1111 propurl.iollulH, I"

II,

Wlm

' '1' 1'It1,e cOIII.1 h" 8ulv...1 lIy fillclillg "h'."

22 1".1.\

:J3R

91.AII6: TnK IIItIJlNNIN(lS OJ! ORRltK ~IATJlE"IATIC8

help of simillLr right-angled tl'i!Lllgles, l)rovided t!tat tlte notion o/'being in the ,~(tme ratio' is defined lor incommenaurable magnitudes, i.e. provided Ihat somclTtin,g like tlte so-called Eudoxian definition 0/ proportionality (V. 5 in t.he Elements) is available. So WIl Ell~O that at this slin.go of devolol'monli, problems which hn.d HeclIlcU tu 1..10 insoluble from lihe arithmetical viewpoint ofst.ngc (a) woro given gl~()motrielLl solutions which depended bOUl on IL lcnowledge of i Ilcommellsurability and on Eu(loxll~' defil~il.ion of proport.ionalit.y. (c) JL WII.8 found at. an 'intermeditLLo st.age'll (,lULl, lihe origiJml prohl~llls could be transformed intu essontilLlly oquivn.lcllt ones which were cal'lLlJle of being solved geolllct.rica.lly withouli t.ILldng probloms of cOlllmellsurabilit.y (or incommensurabilit.y) into account. and wit.hout Lhe U80 of proport.ions. For example, t.he problem t.reo.ted ILbove (to find t.he mClLIl }ll'OportionlLl between 80me number or magnitude, a, ILnd 2a) was lmllsformed into one of finding tile straiuld line required to double a square lViiI, sides o/lel/gt" a. The significanco of this intermediate st.age liml in the fILet t.hat questions of proportiollality and incommensurahility wore, in 0. manner of spelLking, eliminalied. It wlLSl'robahly tho discovcry tlmt this could be accomplished lJy a transformation of tho problems concerned, which led tho Pythagoroo.lls t.o develop an inter('sting 'geometry 0/ arrfts'. 'J'his t.heory, which earlier 8(lhollLl'8 interpreted ILS 'geomotrical u.lgehm', will be diseusscd below.

• First of all it should be noted that there are 0. grent many examplp.8 to be found in the Elements, which iIIustmto the t.hree stages of development ouLlined above ill (a), (b) ILIlU (c). I wiII quoLo three of those

hero.

Slaye (a) Consider the following ILrit.hmoticlLl ProllOsitions from tho Elemellls: 1\ 1 do 1l0L wallt. to claim t.hut. 8Lugo (c) must II/wo procodod 8LI~go (b) in timo. QIIl'8Liollfl of chronology Bro not my primo concnrn horo. \VIIBL does Boom t.o III" to "" illlJlorl.unt is "hut. sLago (,:) "IltJwB how Lho problOlIl8 rni80d ut. Btagn (,,) could ho given solutiona which mudo no mnnt.ion of proport.ionB or the problom tlf incollllllllnaurabilit.y.

AI'I'I~N()JX

330

IX. lG: "l/two numher.<1 be prime to one (£Ito/her,/he .'lecollfl will not be to any otherltl£mber as ate firllt is to tlte secOIul." (l.o. if a and b nro rol'Ltivdy prime, then thero i8 no x such L1ULt a : II = b : x.) IX. 18: "Given two 7U£llIbers, to investigate wltrlT,er it i81)0.'1$ible 10 filld 1£ If,irrll)ml)orlimllll10 litem." (l.e. 1.11 find (~lIll1lil.ionH on It n.lld b whieh ILI'C IUlCCaSILl'Y ILnd sulli(!ient to emiliI'll the llxiaLlmco of '\11 x sl1ch tlmt a: b = b : x.) IX. H}: "Given tlu'ee numbers, to i lIVellI igale ",Twi/writ i.'i lJO.'l.'1ible 10 fir&d rl, /mutlt ]lmportiol/allo t"('/II.." (1.11. til fillll (~lIl1diLi()nH on II, () lLIul c whit!h IU'C II(weHHI~ry 1t.1lI1Hullieient. to l'IIf.C\lI'I~ t.lw llxistelwo of ILlI X sunh tlmt. (l : b = C : :1:.) These I'rul'"Hitions were ubviolJldy wl'iLtnn dllwlI n.t. IL timc when it WILS lwowli that t.he prohlem of finding IL t.hird proport.iOlHLI t.o t.wo givcn IHllIlbcra (or a. fmlrt.h proportionl,,1 t.o t.I'I'CCl givcn oncs) could olily he sui vcd wul('r ccrtrtin r.omlitiolls. Hlaytl (Il) Now ifct,bmlll c:t.rc tlLlwlI to be ILriJitmry sl.might lines inlltelLd of numbers, ueometrical flolutiuns t.o the abovo problell1s CI\Il always ho found. 'I'his is proved by Euclid in Propositiolls VI. 11 ILnd VI. 12 where ho showH how to conlltruct the l"(l(luir('d third (01' four!.h pl'Oport.ionnl). II iH cunHl.ruct.ionH, howeVl~r, fre11llent.ly yil'lcl illcolltmr!nsumflie magnituti('s. III'IICO it CILnllot be chLinwtl (.hlLt t.Iwy ILlwlLyH I'l'Ilvidc COl"l'c('t HolutionH, unlell8 tho Eudo:t:ian (lefinilion o/l'rtJ]lorliollalily ill ILSSUIllCII. Siage (e) It. is p088iblc til solve t.hcso 1111.lIIe pmblems wit.hout considering proport.iOllR or i".commen,'lltl'Clbility, if t.hey ILre firHt tmnsformed i II the following wa.y. 'I'hn problem of finding Lht! third prol'"rl.iIllULI (i.c. 1.110 x HIII~1l llllLt a : b -_ - b : x) hccomcs thll.t uf finding tho rm:Llt.ngle (ax) which hfLS 'one sido lIf length (J, 1\1ll1 tho SILIlIll 11.rClL n..'i II. givell H'l"Me, b2 • Similnrly, illHtelLd of t.ry iug to find tho fourth )ll'IIl'urt.iOluLI (i.e. the :1: such thlLt a : b = c : x), ono loohs for tho l"t'ctlLllgle which hILS one side of IilOgLh a allli t.ho samo rt.rCIt.Il08 IL given fl~I:!.ILIIgle, be. Euclid's Proposition I. 44 providcs the meatlH for solving IlOl.h problems. n l'uns 11.8 follows: "'l'o a give" slrair!ltt line 10 apply, in a yiven recti/ineal angle, rt parallelogram, equal to a uiven triangle." (J~uclitl sl'olLks of 'IL givllfl trilLnglo' ILIlt! II. 'I'ILmlldogmlU in a givlm rm!tilineal ILngle'. instead of dealing with rectangles, for the saItO of grelLtR.r gonomlil.y.) The )ll'Oof of t.hiH prol'llHil.ion URCH '''Pll/imlio'M 0/ arcrul', 0. met.hod developeu by tho PythagorclLnH which hILS nothing to do with incommonElur;i.lJility or proportiuns.

A I'I'I~N

87.AII6: TIII~ ImOINNINU~ OV OItKKK "IA1'1JI(&lATII~

:I·IIJ 1~;mmlI1e

III X

:' .. 1

I'ruhll~1ll

2

Slage (et) The ncxt two of Euclid's proposit.ions which wo WILIlt to I' x1\111 ilie ILJ'O: 12 V III. 18: "Belwcen two similar plane num.bers tl,ere is olle m,ean proJlor/io/l(lt number." V Ill. 20: "1/ one mcanpropm'tioll.al J<11lber /allsbelween two nttmber.'1, 'he ,,,mwers will be similar IJlmu: numbers." (BoLh 1'l'OpOsit.ion8 ".ssumo Definitions VII. 16 ILnd VII. 21.) Ho, 'Lcc()l'cling Lo theso two pl'Opositions, 1I.llIonn }lropoI'LionILlnumbm' l'xiHls het.wcen t.wo numbers (a o.nd b) if, ILnd only if, a = cd ILnd b ful' somc c, cI, e and / such that. c: e = d : /. In other words, this "lIl1lliLon i8 ncccssary 'Lnd sufficient for the pl'Oulem of finding II. melLn I'I'IIPIIJ't.iOlULI to hn.vo n.n arithmetical solulion. Slaue (b) Now Proposit.ion V 1. 13 show8 how to const.ruct. I~ mOIl.n Jll'IIl'ol'LiolllLI to any two stro.ight. lines, a ILnd b. As in t.he provioull example, however, tho construct.ion i8 ono whichdo(JendsuJlon Eudoxu_<;' til'/; /tilion o/lJrOportimwJily bCCILUSO of t.he f'Lct. \.Imt. it. yields lion inc-omnU'l/sumble mu!/nihuie in mn.ny CII..qC8. S'tI!/e (e) Thc ILbovo problcm (\.0 find lion x Ruch \.Imt. (' : x = x : b) is equivalcnt. to UULt. of finding II. SCJUILl'C (:r) which hILS tho SILme I\J'CI\ a.'l IL givcn J'ecLlmgle, abo Onco it. hiLS beon rcformulaLt.cd in this w,t.y, it CILn he Rol\'cd without using llroporlions and witholtt botltering elbout inr,omnll'"surability. This i8 cXlLctly what I~uclid dol'S in Pl'Ol'osit.ion II. 14.

= ('/

which hll8 11.11 arithml'tir.at ,90/ltliolt only untie,' "I'r!l ,·c.~//'i"'l'd emulition." Slll!/e (II) The UIIIIILI (lroofll of t.lw t.heorelll IIl1Ll<e IIM'~ of l,r()fl ol'limM, fur !.Imy rl'ly UII the flt(·t. t.lmt t.he per(l{!lIllicIIIsLr frulll til" I·i~ht. all~I(! to t.he hYl'ot,mllJRc llivilh'R n. righl. t.rhLllglc illto t.WO RIlI:dlm' IIIlI'R whil:h an! Millliiar both to eIL(:h lIt.hel' 'LillI to tho origilllLl t.rilLlIglll. ThiM Il1CILIIM I h:,(' Lim l:Ol'rcRl'ClIuling HidcR of t.hese trilLllglcM Me proportiollal, fl'OJIl whil:h ii, ill e:I.'1)' 10 (lerivI'lho fllnnu!:L (t'! b'l. = (;'1.. It. JIlUSt. he :ulllliU'(~ll th:LI. 1101'1'11"1' of t.hillldl,,1 iH to he f01l1l(1 ill 1~lIdill. NUl1cthd('HH, it ill 1'1'11.'1011:",,11\ to l'Olljl'Ct.III'C Llmt. OliO WILlI Imowll ill 1'1'Il-I~lIC\i"CILII thlWS, RilH~(1 IIlI hilllfmlf IIl1l'S llro]lortimu; to provc 1\ muJ'(~ gml(,1'IL1 furm of Py\.lmgol'U..<;' theon'm (V l. 31). Of COIll'SO IIlIch ILl'l"Ouf ill lIoL con\'iawillg 1I11IcII!~ pro)IC11'\'iOlmliLy hn," SOIJlchow been defhwti fol' ineollllllCllslImhlc IImglli(,tHlcR. (Thill is ILc(~omplished hy Defillition V. !) 118 f".,· 'L!~ J~lIdid ill (:m wu'ncd. ) Rla!!1' (('.) A Idnd of 'gcometry of ,\rcas' ill used to pJ'Ove PytlmgoJ'a.q' theorcm (I. 47) in Lhe J!lIClllcnt.'1. 'I'ho 1'1'00f, liIw LhoRc ill our pl'cviuu!I exnmpills of KLILgO (e), is disLinguishcd hy Mill fILCt. tlmt it (woids I'nlirely bot" 1I1Oll0l'tio"s amilhe problem 0/ incomllll'll.mrability. Figures I!) mlil IG ILI'C inlcndcd to iIIust.rat.c Euclid's ILrgllllnml., whieh rUlls roughly 1\8 fnlluWR:

+

,....

"'....

3

'A

Singe (tt) As my Lilinl cXlLIlll'lc I would liko to tlLlm tho s()"cl~lIed "l'hc'ol'ulII of Pyl.IIILJ,tIll·IU4'. 11. ill wl'll 1(J\(lWIl Umt. t.hiH pl'Upllllit.inn ('!LII 110 vcwified uritlwU!lically nnly for right.-ILIlglod tl'iunglcs with mt.iUlULI Hid.'H. Tho pi()lIecl's of (h'oel, III1LLhonut.t.ics seolll to luwo hoon intOl'cRLcd ill Lhi.'l 1'1LJ't.icullLr CIL'iO. (They oven wOllt so fILl' u.s to doviso rules for ~'~1I1'11LLillg Ll'il'lcs (If IItllllOOI'H which could Lo tho sides tlf right Lril\n~I.'H.I:I) Su t.ho vll\·ifi'!ILt.iol1 (If Pyt.!IILJ,tIll·IL
Fill'

til" "1I1Ul

of Aillll,lidLy, I'l'ul'clllitioll V I (['1 R io 'I"O"':c1 ill

lUI

RbhI'IIvi ..... ·,1

li,nll.

u

ar. o.

1'1" fi211.

Be'elter, Daa lII(ltitclllldi.,cl,e IJ'lIIkell tlcr Antike, OiiLJ.ingulI

l!lli7,

'-

'-


l~x(£mll/e

. . . :/

/

/

/

10

1/)

0,

1 I

'-,-

/

18 I I I I

I I I I I ", I n, I I 1 I L_.L ___ .J F.

Iii". Hi

liiJ!. Ifl

Tim P('I'I,cmclicmIILr (U f)) rJ'llIll t.l1I~ righl. :LI1~I(! 1.11 !.hll h),l'uI4mlllu~ of ILl! n.rhi!.I·ILI·y right. Ll'iILIIglc (AUJI) elm Ill' I~xt.mlcll'd (h)' I)N) so 11.'1 III divido t.ho SfjlllLJ'O on thc hYl'"\.cntJHI', ililo two loedlLI1W'II, III :LIlli U 2 (HI'" Ii'ig. 15). NowQI = Ill' ILIlIHJz = il 2.'I'lu: HhuAII'cll,ri'LnglcMGAJ.'1L1II1 11 All (soo Fig. 16) ILJ'O ohviomdy c()lI~nwl1l.. li'tJl'thormoro, II A /l hlL'I

3012

87.AII0: TilE IIROINNINOS OF OIlR~K alATJI~MATIC9

IlILlf Lho I~roa of QI' ILnd GAll' ho.a half tho aroa of III (uy l)roposiLion I. 01 l). Hcnco Rl and Ql must havo tho samo aroa. In a similn.r manner ono can prove that R z = Q2' 'fho proof is completed uy combining t.hese two equations. Tho preceding examples together with my discussion of Socratos' mo.LhoIDatieal demonstration in the M eno Imvo, I hope, lent sorno pln.usi hility to the claim that the kind of 'geometry of arelLS' used at stage (c) was designed to eliminate the application of proporLions and the problem of incommensul'l1.uility. The most elementary porLions of this Pyt.llllgorClLn 'gcometry of !LI'C:L'I' (or 'geomeLriclLI algeum', ILa Tannery called it)" arc sUJ"Veyed in (I), (2), and (3) below. (I) Once again our stlLrting poinL is thll.t part of Lhe .Meno which tlll:Lls with squares and their InolLS. The pMSILgO in quesLion descrihes how Socrates, having drawn a square with sides two leet long, bi.,ects

Ji'ig. 17

Lhese sides Lo show that it is mado up of four unit squares, and how he gocR on to indicate that the squ"ro's area is Italved by its diagonal, (See Ji'ig. 17. I should mention hero the trivial fact that tho diagonal and the 1incs which hisect tho si(les intersoct at a point. This point ofintorsection will pllLY ILIl imporLant 1'010 ill the soquel.) III short, Socmt.cs dO\JlUIIsLru.tcs Lwo ways of splitting up the area of 0. square. Doth of these are t.nlltt.ml il~ (L more uclltr.rallorm hy Euclid. For his Proposition I. 34, "In 1l1lrallelo(lra1ll1ltic arCfUJ... lite diameter bisecl8 tlte areas", is just Sucrn.tes' remark about the diugollal of a Bquare generalized to the clLSe of l'"rallelograms: whorea.s Prol'osiLion II. 4 <UscU88es tho way in which . tho area of 0. square eo.n be split up when its aide, inBteo.d of being ltiHI'I:(.(I
n. 6 above.

AI'I'P'NJ)J X

3013

Fig. 18

tion, which is perhaps ILn evon moro illtel'eHLing gencmlization than Lhe fOl'mOl', l'uns ILS follows: "II (t stmigltt line be C1U at rmuiom, Ihe .
+

+

.

Fig. 10

•

:1·1·1

345

'\I'I'KHIII)(

R1.AIIO: TIIK IIKfIINHINOS 01' OItKKK )IATIIKalATU:9

to cach other and to tho origilu~l pamllclogl'll.m, whoren.s tho other two, t.IlC so-called na(!a.n).'Il!wf'aTa (shnded ill lrig, 19), will ho equal. I must ndmit that the above theorem is In,rgoly my own roconstruction. l~uclid docs not state it in this form, but tren.ts it instead a.q two R(~pamto propositions. One, which WSCUSRCS the similarity of tho 'p"",l. /rf0!lI"lIllM abortlilte diameler', docs not nppcar until Boult V J; the other deals wil.h the eqluzlily of their 'cOInplemenls' (pamplerOlnnla) lI,nd is indutll'11 in Book I. This nrml1gclllent wn.Ii forced UpOIl Euclid by tho raid. LlmL tho l,roof of tho fUrlnor (Propositioll V L 2·.) dopends in t~n l');.~(,IlLial wn.y on Eruloxrta' de/iniliolt (V. 5) 0/ proportiOlll,liIy. On the ollwr IU\Ild, tho latter (Proposition J. 43) is proved simply by sub· LmcLing l~ron8; its proof hna nothing to do with proportionality 01' in· (·ommclllluru.bility. I should cmphll8izo here tlmt the theorem which I hnvo roconstructed I'IIL}'{'d l\ \'ery important. l'olo in the Pythngorcall 'geometry of ILrel\S' or, t .. 1IH1.~ UI(! IIl1forturmto nmI millieading Imme which 'L';~llIlC1ry gawo this thellry, 'gcofllelricl~lalgebru.'. If, fOl' exnmple, the PytJmgoreuIls I5 had not Iwown that tho paraplel'Omata wore oqUlLI, thoy would not IULVe lmcll nhle to develop thoir method of (tl'1,licatiolt 0/ area8. 'fhis method Wa.ll mentioned in Example 1 above, whore it wns pointed out t1mt. fillilillg 1\ fourth proportional to three givon 7~umbcr8 or m"!lnitruics (i.e. I~II x Imch that (l. : b = c : x) could bo cOllstrued ns tho problem of finding IL rectlLngle with ~ givon side (a) which hns the SILInO o.rea. lUI IL givell rectangle (be). PI'IIJlOIlitioll I. 44 of tho ElcmcnLs provide-OJ the following solution lCl thiM 1'I'OilIell1 (Reo Figa 20 and 21). If we thinl, of tho given rectangle (be) ILOJ I~ 1,araplermna and the rect.u.nglo whieh the line a makes with IIlIn of its sides (c or b) ns 0. 'pn.rallelogmrn allout the din.gOIll~I', then tho HI'(~OIlll 'PILI'Il,llologram about tho diagonn,l' (bx or ex) olLn be conatructcd Ity extending tho side (c or b) of the origillal rect.u.nglo until it moots tho ('tllllilllmlion of the din.gonlLl of ar. (Fig. 20) or of ab (Fig. 21). Now these three l·c<~tILllglcs together determine uniquely the second paraplerorna ILlld, in IlILrt.ioull\r, its sido x which wo sct out to find. (This is sometimr,s callcd parabolic applicn.tioll of arOM, beclLuse tho givon arelL be is applied lo the lino a; cf. the word 1I0(!u{lrU1E&V.)

•

It scorns to me, however, Llmt thc iIllJltlr\'ILIwe of our theorem fm' Pytlmgul'C"" goomotry elLII IIIl illuRt_mLl~(1 (lvon more I!onvincingly, Let us loolt onco more I~t the theorom itself. J t. flta.tea t.Imt. two lillcs drawn Jlamllcl t.o tho Rides of a pn.mllclogmm through I~n ILl'hitml'Y point 011 ita din.gOlULI will spliLlhe area of tho plLnLlIPlogmm int.o rOil!' plLI'ls; twu of tlwse will he etjulLl, ILlld the other two will he simill~1' (huth to each

-;, . - - - - - - - - it

'I 'I

I I

b

II

I

b

/1 I I I

I

I

I

c

I

I

1 I Q

Q

1 I

I

I

/

1 I I I II_____ .JI __ ~

Fig. 20

1\ I I I I I I I I I I I I I,' I II 1 I

II I k.. __ L_ Fig. 21

uthor ;\lId lu tho origilULI 1':~J'IL\ldogmlll). My ('IILilll ill UIILL I hill lIillll'lu yet LelLutiful discovery led tho WILy to n famouB pmhlem of t.he en,rly Pythagoreans ahout whieh PlutlLrch wro(.(l the following (Symposium. VIII. 2.4): Amoll~ Lhe mORt gcometl'iclLl thUClI'CIII II , til' m.thm' pro IJI III 1111 , ill UII! following: given two figu rCK , t.o apply ;L Lhil'
(3) I CILllnot ccmt:\ulic this 1~(:l'(lUllt or tho dmnclIls or tim Pytlmglll'C1\1l 'gCCIlllotl'Y of arcn.,,' wil.llUllt. ment.inllillg l~llclicl'lI definiLioll flf Lhn ~c(lmoLricl~1 (~Clnccl't !I'WlIlotL:

'"I'rnclu8 (cd. Fricdlcin), 41D.16-420.23. Roo WHO T. [-:trlllen/,

Vol. I, 1'1" 34311.

T~.

IlORth, EfU'lill'tI II Qlllltmi IIY llunLh, Iflt!. cit.. ([':lIr/i," .• ftJI""II'"h, V,,1. I, 1'1'. :I·I:IIT).

346

SZABO: TilE DEOINNINOS OP OllltRK !IATllEBIATICS

Definilion II, 2: "In any paralleiogrammic area let anyone whatever 0/ lite parallelograms about its diagonal with tile two complements be called (t u,wmon,"

x

Fig. 22

In ono clI.8e (Fig. 23) t.he aron, consisting of 0110 of Lho ·pILrll.llclogrILlIls lI.bout Lho diagonal' (Lho smlLllest BCJIIII.m ill tho iIIustraLion, which for convenience wo will denote by 'c') and one of Lhe lmrttplero"uzt(L is, in a mn.l1ncr of spen.king, joincd onLo the lefL- hand side of the sccond l/arlL1,leroma,' whereas in the oLher CILSe (li'ig. 2·1) it is IL po.ml'leromn. which iR joined OlltO tho loft-llIl.nd side of LIm 11.1'l~IL consist.ing of the snULlIcsL sqUl1.re amI t.ho ot.her parapleroma. The I'Clmlt.ing recLangle (t.ho shaded pn.rt ill FigR 23 and 2·1) is Lho SI\llle in hoth CILSCS; it. ha.OJ sidc.OJ (x + y) n.nd (x - y) (theRo leLters refer to Llw RILme lengths n.s in Fig. 22). 1.'he only diffcrenee hot.wccn tho two cClns\.r'uctiolls has to do with flie square 0, a compollenl 01 tlte oriui7Ud u,wm01I. III Lhe fil"Ht case, 11. rccLlmglu .

As we can see, the definition of this concept presupposes that the ILrca of tho parallelogram has already boen split into lour parts in t.he nmnner described previously. A gnomon consists of tl&Tee of these po.rts, namely Lhe Lwo paraplermnata toget.her wit.h either ono of the 'pamlIclogm.ms about the diagonal'. The gnomon of 0. square is illustrated in Ii'ig. 22, where its three components are in()jcatod by different Idnda of Hhading. The figuro n.lso showfl LlULt. Lho ol'iginll.l SqUILro (x2 ) co.n bo obt.aincd 11.8 the sum of a smaller squn.ro (1/) and the gIWIIWIt. (In ot.her WOI'ds, subt.ro.cting tho smaller square from the greater leaves a gnmllo1~ 11..'1 t.ho remo.inder_) This fact is important; because the gnolllo" itself can bo very eMily t.ransformed into 0. rectangle. Thero are two ways of co.rrying out the tmllsfllrlllILLion; those are iIIustmt.ed in Figll. 23 and 24 respectively.

347

Al'l'ltNOIX

p

hyperballej

parapleroma

Fig.

2·'

hlLYing Lho SI1.me arelL IL.OJ Lho gnomon itt ILJlplic(llo Lim line 2x Imd ,fllI.'i sltort (llle{1letv) by 0 (of tho rect.o.ngle on t.his line); henee Lhis ill SOlI10times (lalled ellil)tic appli(,Afion 01 arOO3. Jn Lhe seconc) cn.c;c, t.he I'Ccto.nglo is n.pplied t.o 2y and exceetls (V1lE(1{loAJI) by 0 (Lhe I'Cl~LILngle Oil Lhislino); 80 thiR iR BomeLilllCR roferred to M hyperbolic. "Jlpli('.(l/iorl of arC/M. Now hi not. t.he Lime Lu go into ILlly lIlom uot,ILil'LIICIIIL p.lleip.'ii.'i ILlltlltyper/m/n,17 8uffico it to say tlmt. Euclid cmploys elliI,/ic ttl'lJlicalion in Lhe pl"Oof III' ProJlOl-liLioll 11. /) and ItY1Jerbolic "PI/lim/ilm in II. n.

4

HOW '1'0 lrlNII A SQUAItN WI'I'U 'I'IIN RAMI'] AIUCA AR A mVltN IUCC'rA Nor.le

l!·Y~O

• - Va b

Wo 11.11) now in 11. JlosiLioll 1.0 I'OLIlI'll to 11. 5, Lhe pl'oposition whoHU genesis we 8ct out. to oXl'llLin.

•• I!.!..k 2

Y o-b

=ZFig. 23

11

Son tho diRcU88ioll in H(lilth's edition !If

r.i1Cl

RlcIII611t., (Vol. I, I'Jl. :l.j:1-4)

3.fR

81.AII6: TIII~ IIIWINI'IINOS 01" OIlJ.:I'.K AIATIII~MAl'l~

1 lllLvo ILh'OILUY mentioned (in Clmptor 2 I~Love) th'Lt it ia n. 101111n1L which is needed to provo "nother importILnt proposition, namely II, 14, lU1I1 thn.t it is quoted word for word in the proof of the latter, l.et liS IlI'gill by taldng IL closor Innle ILt H, 14, whieh is com:orncd with the followillg problem: "'1'0 co1lsirur.l a square equullo a given rertilincal {igUl·e."1~he prupflsit iOIl ia really ahout tetragonis1Iles tho trnnsformfLtion of a rectangle into n. "qllltre of the SILl11e area. It is stll.ted for ILIl lI.rbitmry roctilineILr figul'O clIlly because Euclid ILlwlLyS sought aftor I. he grcl\tcst pOfl8ibIegenernlity. 'I'h(!rcfore his proof hegins with n. I'cfcrence to l)roposition I. 45, which IImltes it llossible for ILny auelh figure to be tmnsformed into a rectangle, In eXILml,le 2 (of ChapLel' 3) it WILS observed that the t.mnsfornmtion IIf IL rectangle into" squaro of Lhe same are II. is equi vILlent. to the prohlem of {illcliJl(J tlte mean prol,orlionul between fwo ntunber.'1 or mauniltu/es. I~lIclid disculiRcs the eOllst.rucl.ion of the mCILIl proportionILI to two slmight lines ill I>roposition VI. 13. Jt, turns out that n. 14 !Lilli VI. 1:1 Icad to the snme result. 11I Our interest, however, is in Lhe (lifferonccs be(.ween these two l)ropositions, Tho surprising thing ILbout the constructions in II. 14 and V1. 1:1 is IlmL aL first sight they appcar Lo Lo identiclLI in all illlportant l't'fll'CCts (Hec Fig, 25). In both CII.80S the siciCfl of the l"octlLnglo concr-med (01' tho two lines, a and b, to which Lhe mean proportional is being sought) "ro added together to form oile stmight line. 'l'his line is then bi.<Jecled, alld a , ' I 'tl _.1: a + b. ,~eIlUClrC e WI I l"iMllUS - 2 - 18 drawn around it, Finally a perpolldicullLr

is mised from tho 110il1t /l.t which a and b me:et to tho cireumfcrencc of lhe snmicirclo. Thisl)eTpCmlicular,d, is the side of a squ!Lre hlLving tho SILIIlO arelL as ab (or the required melLIl proportional botween a and b). Tho steps outlined ILLove mll.leo it seem ,1.8 though tho idont.iClLI conIMuction is used in both propositions. Y ct the eignificlLncc of thCElc RI'('PH in tho two cnses is lUI difforent ILS it could be. Completely clifforcnt hll!lLH lie behind ll. 14 "nd V1. ta. J~t us fil'st intel'prot. t.ho ILbovo fl'OJn t.he: pl/int of view of tho llLtlcr proposition. '"!:lUll A .. i8LoLlCl, De AnillUl 11.2,413ul:I-20 1\1It! 1I.1etal"ly8i~ H2. 01l1lb18-21. Th"RIl Lwo pllRRngr'8 cOllvincod both HoilJl'rg (,MIlLiulRlIltischoR hoi A"isoolAll('./I', .. II,I"/lIIIIIIII(len zIIr fl'!lIrMdltfl ,Ier Im"l"mllrtil!cl,f! lI'i.'IIf!".•rJlfI/tr.ra I R Hun., I.IIi l17.ig nlll.l) IIml JI""I.h (1Ifal/lfm,"li~ ill Ari"tolle, Ox (;'1'11 11)4 !I, )1)1, I!H-3) Llmt. VI.13 cnmll LI,ront 11.14.

349

A1'I'I'.NIIIX

A b

a

I·'ig. 25

Tho two sllgml'l1tK fl ILlld b

ILI'O

aclrlrn. together in ol'cll'r to ohtain the

"!lllo/emMe (AC ill Jt'ig, 2!l) of IL right·anglell t.rilmgle (with UIO though I. ILlrCll~(I'y ill milld Llmt a will evclltulLlly fOl'm t.ho IOllyer side n.llout I.he right. ILIlgio 01' ILlllltlllW t.I'iILnglo, Imel b LlIC .~/wrler siclel ILbollt the righl. nllgh: of IL third (1110). The segmcnL AG iH hisccteel a.lld a. semicircle iH (ImwlI ILrUlllul iL, Hincel thill givclK, hy TIIIL!c'/l' tlwClrmn, Lho lOCUli "r points whi(:h c:lLII he LllIl thinl vlwLex of lL right tl'ilLllglo with hYI'Ot.OIlIlKU AG, )~iI1lLlly, d ill the recl"il'ed lUoan proplll'timmi het.ween fl. ILnd fl, In'C'lLlIlIC thc Lri;LIIgllls II IlIJ lLJIII JIG/) ILI'Cl flilllill~r, Imel d ill Lhc. ,~"m'l(')' .'1idr ILhuut thn l'ight ILnglo in th(l lill'lIulI' Imcl Lhe lO)/(Jer ,'1itir ILhllll1 1.111' right. ILl1glo ill the IILLI.cr. Thill iIl1l1ulouht.I'r1ly Lim ILrglllllcllt IIpllll whi('h Lim construction in VI. I:' is hll.Reel. Whal appClLI'M to hc UIC HILll1e 1'011, Rtl'uction in 11. 14, howl'vel', J'('flt.S IIpOIl ILII ('lIlircly dilfercllt chaill (If rOll.'1olling, which wo Klmlt now ILttelllpl, 1.0 reproclw:e,11I ] II orclcr t.1I tJ'lLnHfurll1 IL givt'll l'('dl~llgh., "fl, into It "'IWLI'Cl of lltl) lIalllt' ILI'I"l;\, ILIl ILl'gulIlcnt whi(:h rtlllH ROlncthing lilw tile fullClwing iR re'luired. There iR, 811 IOllflClLlt, quiLl' IL di8t:LIH:(, to C'CI\'(W. OUI' RLnrting poillt, I hI' J'(','I.lmgle f,fl, iR givc'll, n.llcl 0111' lLilll iH I·" n"d It lIfllllLl'C', ,['1., whit·h hll.1I till' KlLllIn l~rC;L, Sill(ltl Ol1r gOILI iH so fiLl' removed frolll 0111' "tarl.iug poinl, ii, wonld hn clc~Himhln t" find HOlllll WlLy of hrillgillM t.h",n dOHIH·logc'lllI'l'. If, fut, eXILIllI'I(!, iL (!ellllcl 110 ,i!lIIwII \.IH~I. hot.h LIllI rc·C:I.:Lllglll 1~lId I.hll K'IU1LI'C' C:ILII II(! oht.lLilll!11 Ily 1',wful'millg f:rrl"in armllrlr;f'.Ill fll1C'ml;oll.~, IL lilli, would IIILV" hnc'lI C',d.'LI,liHI",c) 1..,I.wl"'" 1.111'111. 'VI' Hindi IIfIW !.ttl", "I' t.h(\ qtwst.ioll (If huw l.hiH iR to 1m ILI:l:UlIIl'lisltml ill t.ho llltHll of t.11C' rei!' I:L"gln,

,v (:r. Illy

n.

J;lIitln

!'uIYIl, IInrlt '" .~"/II{! it' 2nd ",I"., Nllw Y"rlt which li,lI1lwlt in 1.1," ""xl•.

ill 1.11 .. l·nt:UlIHI.I·Ut:LiIJlt

HI:'7.

ThiH lou"l< WitH

'tum

Uti

';amston.tOr • ·•• ••e.9

350

't' •

U

'S

•

S

rtd_,

,.,

12

)

+

fa ~ b)

Z

and 11 =

fa - b)

:&

(see Fig. 23). then the rocLurnglo

lie regarded ILB the di1Tore!ee of two 8(llHLrefl. since

P

er. sre

eo

me

r

351

API'KNDlX

I17.AIIO: TUR nE(1JNNIN08 OV ORltKK lIATURlIATIOS

Consider tho subtract.ion illustrated in l!'igs 22 and 23. A squlLre, yZ, is subtracted from JL larger one, :il, leaving 0. gnomon as the remo.indCl·; this '(Jlwmon' can easily be transformed into the rectn.ngle (x y) . . (x - y). Our present case ditTors from the preceding in thn.t the sides (It [Lnd b) of the rectn.nglo n.re given insteu.d of the two sqUlLf(!H. However, it is a simple matter to find xZ and yZ from a and b. Let :il = =

,'eit4r' _

(ll, Wlon

ab = fa + il)

2 _

2 a ._b)2 (- 2 - " hence it can be described ILB tho result of performing n.

~eometrico.l

operation. (l'his is actutLlly the transformn.tion which is Iwcomplished by the lommn. n. 5.) Similarly the square d'- co.n be viewed n.a the result of performing n.nother geometrical operation, the operation being 8ubtraction in this elise R.8 wcll. Given the right-angled trilLngle shown in Fig. 26, we know by Pythagoras' Theorem that :il = '!t d'-. So the squn.re on ono of tho sidcs (lobout the right-angle CUll he obtained by subtracting tho squn.re Oil the other from the square on the hypotenuse, n.nd in particulu,r cP :il - '!t. The solution to our problem is now almost at lumd.

+

=

a (

~ ~) 2

will be the flqlllLro on ono of Lhe fiitlos about Lho righL ILnglc.

a+b

Sinco we know thn.t the "YIJotenU8l! C(llIlLls - - 0.1111 one of the other 2 8idc8 c(l'lILls (t - b , wo shoulcl be I~hlc t.n ohLILin Lhe third sid". rI. of I.h(\ 2

trilLugle (sco Fig. 27). The !I(IUIUC on d will Llll'lI havo the SILllle ILI'CIL u.."! tho giVllIl rctlLILnglo abo ThiR is the ILI'glllllclIL which undorlieR LIm COIlsLrucLioll in Pl'Ol'osiLioll J 1. 14. So wo 800 tho addition of Lllo two Ritlc.<; of (,Jill rocLILllglc pln,ya I\, dill'l'r· enL 1'010 ill I.llis cOIIRLructioll (sce Fig. 27) to 1.1100110 which it plll.YR in V J. 1:1 (for the hLLwr aee mg. 25). TIll' linl'R (£ and b ILI'O added LogoLhor in V1. 1:l beclLllse a b will form Lho IIYl'oi.flIlIIRe of II. righL triangle; Lhis hypowntlRe LImn IULB t.o he bi.w:lccl ill order to IL[lply Thn,ll's' theorem. 1n 11. 14. on Lho other }lILlld. a is cultled to b, and Llleil' Slllll

+

bi.9ccll'd heCILllRO a

~ I~

triILng"~

will iLself he Lho hYJlol.cnuso of II. I'ighL

(sec )i'ig. 27). 'fhm'e is OliO other interesting fILet whieh I should like hcre. 1 L collcerns the side of Lhe squlLre

b) -2-

It . (

Z

(,0

mentioll

. No sJleeilLI construe- .

Lion iR rcquil'cd t.o ohLILin thiK sido in the proof of II. l4. for lite lille

-----,

I I I

I I

y

(b7'77T.1'77'J'77T.1'77'J'7T.'Im,'77T."":ri---t-- _J

Fig. 26

b

o

Subtrn.cting olle square from another led us first to a rectangle and then to a 8quare,20 so wo should be able to transform the rectangle ab into a squa.re of the same area by combining these two results.

(a;- b)

2

Fig. 27

will be the square on tho hypotenuse of a right-angled triangle, wheren.a betw(!f!1t t/te midpoint 10

Tho lot.toring \lsod in Fig. 23 is int.onded to mo.lte t.ho diacu88inn easior t.o

u/IIlul"stund.

a- b exactly 2 = Y .

IIf

a -I- IJ (mel lIu? lJOi III at wltirll

II 1(lUi

II

11I1'(lt

i.'1

t

_.1

co h ¢

•

ad • • • ft ••

:Hi2

b

t 1*<

* rt •

•

·cw

I17.AIIO: l'lIJ( IIItUlNNINOS 01' Olll(J~K MATllltMATWI!

In conclusion, I would JiltO to Sll.y somothing moro I~Lt)lIt my cln.im tha.t I l. 5 is o.lommo. which wo.s especio.lly tailored to suit tho needs of 1'I'01'0sition II. 14. First of 0.11 I should note thn.t. Euclid, in his ox})ln.nn.tion and proof of Pl'tlposition n. 5, mo.ttcs uso of n. dingrn.1ll which corroRponds in all cR.'1cntiul respocts to my Fig. 23. This figure shows howthel>ytho.goreo.ns ohtn.incd n.n n.rbit.rary squaro o.s tho sum of 0. s11urller square o.nd 0. (1l1omon (or M tho sum of a square and 0. rcelangle). Ilnsicnlly this is t.ho Huhject. mattcr of II. 5 o.s well. Of courso Euclid docs not. start out. with IL square, but. with 0. straight liue whioh is cut into equal and u7&cqual ..~('gl/lents. Cutting the lino into unoquo.lsogmonts is tho revorso of adding the two sides of the recto.ngle in II. 14. Tho lo.ttcr operation converts 0. rectangle into 0. straight Iino, whoroo.s tho former turns a straight line into II. rectangle. Bisecting tho lino plo.ys the so.me rolo in hoth proposilions; it. furnishes tho sido of tho Iu.rgor sqtllLro (from whioh tho smo.llor IIlle is subtrnctcd). Tho following is oven moro intcrcsting. As I have already omphMized, ii, is J1(l('.cssn.ry to find !tal/tile diOerence between tIle two side.s 0/ tlte ,.eeIII/lgie heforo Pytlmgoras' Theorom cu.n bo n.ppliod in 11. 14. Howover, thcl'e is no necd to construct thislongth sim:o it is just Ille li.nc brJ.wcc7& lite mid1Joi1lt 0/ a band llle point at widell a and b meel. In 11. 5 this H;~lne Hegment is descril>ed as "lite atrai!!'" line belween tlte poims 0/ ,~('clion". Euclid's use of this clumsy mld awkwu.rd phrase, in my IIpinion, proves conclusively tho.t ho ho.d II. 14 in mind whon ho stated and proved Proposition II. 5.

.._--

.._

AI'I'ICNOIX

35:1

(2) As WM montioned in Chnpter II ;,hllve, the uther propoRitionH which lu'e usually rcgn.rdcd ILq 1'1LI't. of' PyUmgoreu.n gcollletricalo.lgebrll.' elm also bo given a purely geometrical expltLnation. On tho other hand. no tm(:ca of gClluine I\lgehmic ideM21 lmve yeL been discovered in t,he lnaUummtical LmdiLiun which clllmilmtetl in Euelid's Element.'i, (:1) There iR however one point. on whieh I agreo with Tu.nnery, Lhe sellOll\)' whose icien..'1 glt.VC rise to later sl'ccul:~tion n.\'out the supposed 'geollldl'i(:ILI algebm of UIC Gl'colul' ,%2 Ho WlLq of tho opinion Uml, this Idnd of gl'Olllctl'Y uWl!C1 its ex.illtcnce t(l the discovery IIf inoonllllOIl' sumLilil.y. In flLct, 11..'1 1 Imvo Lried to show. tho' Pythl\gol'onn gl'ollletl',r of lLl'tms' eliminated Llw l'whloJll (If inl'lIll1lnellRllmhility 1~lId consequenUy avoided the URO of propol'Lions, (4) Tho claim tho.t the 'geometl'iCl\1 ulge"rlL of the Pythngorenns' rcsulted from tho Oreol(s ciLher tl~kiJlg ovcr 0)' developing furLher an idca of tho nlLhylonilLna ho.s no basis in flu:L. No connootion III~ OVOI' Loon cHtahlisho(1 ho~woon this branch of ml~LhcIllI\LiC8 ILud 'Babylonian scioJlI:c', II. Heelllll muuh lIIoro likely tlll~t tim Urcckfl wcre solcly J"e~l'(JIl' aihlo fIll' tim cl"(ll~ti()n (If thiR 'gcolnutl'Y of MeM',

+

5 CONCLUSION

It scorns to me tlmt tho results of tho preceding invcsLigo.tion o.rc of interest from tho viowpoint of further resoaroh. ~rhey 1\1'0 8umIItlLrhmd below. (I) Proposit.ion II. 5 is I~ purely goomotricl\llemma needed for Lho )Il'Oof of 11. 14. Although it is e'luivl\lont to 'tho solution of ILIl algobraie NIIlIL!.ion', it should not. he interpreted in thia way. Such IUl intcrpretaI.ioll is misleading hocl\uSO it. ous(:uros the truo goometrie men,ning of tho prnpllsitinn "lid suggests tho fu.lHo hilltoriclLl ideo. thlLt tho Greeks I\ctllnl1y opcro.ted with o.lgebraio equations in pre-EuolidClt.n times.

1l1l1ll0

II VUII.lnr \VIII.r,11I1I (ill 8r.icllcc Awnkcniuf1. p. 118) givl'o ' l gUlllliullly ulgohrnie int.4!J'prot.nLioll of 1'I'0lK18iLion H.14, hut t.hin 1IIC11ll'rll iut.orpfcl.nl.ioll in foroign 1.0 1.110 npil'it of luwiullt IImLln'mllLica, II R,," fl, 6 nhovn.