The Beginnings of Greek Mathematics (Synthese Historical Library)

SYNTHESE HISTORICAL LIBRARY

ARPAD SZABO Matl,ematicallnstilute, Hungari4R .Amdem!l

0/ Science4

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

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

THE BEGINNINGS OF

a1J

GREEK l\fATHEMATICS

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

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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-

INTllODuarloN

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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).

INTILOOUcrlON

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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

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

f

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

I

B.C.).

[

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

---------_

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

-......

51

llEl~ z ,

and

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

3

1:2.3:6

1:2 -2:4

Fig. 1

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

-1

-

52

8ZAUO: TUK DKOINNINOS Off UllmUt MATUEUATICB

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

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

53

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

54

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

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

(~8 oquivalent to the problem of finding

0.

1. 7

meo.n proportiono.1 betwcen

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

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

f

I'itn}'l

55

'!'HB MATlfBMATlCS LECTURB DBLlVBRBD BY THEODORUS

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

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

'0

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56

8~AUc):

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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

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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):

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

TUK DUU1NN1NUli 011' UltKKK AlATUJoaIAT1CII

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

i

1

I t

61

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

uo

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

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

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

l'Altl' 1.

'l'U~

I>AIU.\ 111:.1
114

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

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

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

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

115

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

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

8-

llti

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

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

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

(Goleborg 1932), p. 66, 2411.

DUring

117

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

...

118

•

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

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

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

, V..

-

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

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

VIU1

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

no

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

-~

2.5

A mORE88JON ON TIm 'rnRORY 011' MUSIC

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

120

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

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

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

121

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

1 ,),) ~~

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

'''I'here is 0. nice story found in Nichomachus (p. 10 Meibom), Gaudentills (p. 13 Meibom) and Doothius (pp. 10-1 Friedlein) whioh tells how the Pythagoreana came to represent intervals by means of numerical ratios. It cannot p088ibly be true !lowever. According to the story, Pyt~llgo~_~ns plL88ing by ~m?d heard the tones of the falling..!!amme~produoe various m rvals, in particular tho octave, fourth arul"fifth. Ho oarefully weighed. tho hammers ILIlyrho.gorean namo for this in1crvn.1 wu.s til'toltov 6tacn'1pa63 i.e. 'line 1~ (units) in length'. This dCBcription also refers to the whole monochord, for in this co.so IUf well, tho scction of string which produced the second sound (tho length '0-8' on the co.non) WIUf tho unit. This WIUf regarded as 0. 'wholo' (8.tov) and in rolation to it, the COII\1,lete monochord (i.e. tho length '0-12' on the canon) WIUf 'one whole together with half of tho wholo', namoly 0. 'lino 1~ (units) in length' (tip,o).tOV lJtaO'T1lpa). So if tho monochord is treated IUf a 'line 1~ (units) in longth', i.e. if the whole string is pluokod first, followed by tho unit (two. thirds of the string) relu.tive to which tho whole string is 0. 'line 1~ (unita) in length', then ono obtuinll 0. fifth. The Pythagorean namo for the numerical rutio of the fourth (4 : 3), lnlT(!&Tov 6taO'T'1l'a," co.n be explained in 0. similur way. On the monochord 0. fourth is obtaincd by kooping u quarter of the string from vibruting when it is plucked for the second time. ThUll tho second noto is produced by three quarters of the string (12 : 9). If this length (0-9) is reglLnled IUf tho unit, then tho whole monochord (0-12) becomes 0. 'line l~ (units) in length', brlT(!&TOV 6tat1nJpa. (The Grecle word briTetTOJI mco.ns 'a third in addition to it', i.o. 'in addition to tho unit'.) -----~-~- .. - - ~---~--~-~ .. -~ ,'., Sul'},riaing confirmation Cor the above eXl,lanation of tho terms til'tOAIOV o.nd briTetTOV 6taO'T1lpa is to be found in Theorem 8 of tho Seelio Canonis. To prove its correctness, however, it would bo ncces8IlJ'Y to fCI,rint here almost the entire Greek text of this thcorem together with it.s u(l('.()mI'ILIlying dingmms. (Tho only thing which should nut be Corgotlun when interpreting this ProllOsition [8eclio Canonis 8], is thlLt o.di(J8lema, the ratio hotwoon two numbcl'H, is always illustrated

Soo n. 24. "bovo.

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

umt, but t.his 1'0"

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

dldl11''Ipa.

BOOmB

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

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

132

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

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

hy two 8traigl.t linea in tho Bedia OanoniB. Hence tbe whole monochord ILnu the piece of IItring which produces the second noto, the unit, 11.1'0 ropresented by two distinct straight linu.} 'fhe etymological investigation conducted above hIlS led us to two concl usions: (1) The introduction of the terms dipla8ion, hemiolion and epitriloll I " diastema into the theory of music ajlt.edates lhe division 0/ the canoll f' --0n1o twelve parts, although tho pfuportionOl numbers of the three most important consonances woro known a.l!eady at tho oarlier time -, [:.? anu_.!~~__ g~ve~ as !_a..tios belween lenutll8. The-quest.ion 'of-whliUitir /WlClie ratios were' diScovere 4/3, 2/3 < 3/4.) Now this ClLl1 lio shown very easilY on a cl\non with twelve divisions. In thiH cu.