[Arpad Szabo] the Beginnings of Greek Mathematics (BookFi.org)

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

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D. REIDEL PUBLISHING .. COMPANY VOLUME 17

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DORDRECHT

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HOLLAND'/BOSTON

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U.S.A.

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CONTENTS

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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

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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

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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

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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)

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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

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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

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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

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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-

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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).

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"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.)

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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

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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

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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.

\

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32

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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'

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I

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,~':.

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

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,\. 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.

. 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"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

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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

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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):

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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.don 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.

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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.

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I>AIU.\ 111:.1KAN 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

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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. - ... --

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136

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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, 4/3, 2/3 < 3/4.) Now this ClLl1 lio shown very easilY on a cl\non with twelve divisions. In thiH cu.