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Diophantus of Alexandria; a study in the history of Greek algebra

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DIOPHANTUS OF ALEXANDRIA

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DIOPHANTUS OF ALEXANDRIA

A STUDY IN THE HISTORY

OF

GREEK ALGEBRA

BY

SIR THOMAS L. HEATH, K.C.B.,

SC.D., SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE

SECONT> EDITION

WITH A SUPPLEMENT CONTAINING AN ACCOUNT OF FERMAT'S

THEOREMS AND PROBLEMS CONNECTED WITH DIOPHANTINE

ANALYSIS AND SOME SOLUTIONS OF DIOPHANTINE

PROBLEMS BY EULER

Cambridge :

at the University Press

1910

(ffamtm&ge:

PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS

Library

PREFACE

r I "'HE first edition of this book, which was the first English •*• Diophantus, appeared in 1885, and has long been out of print. Inquiries made for it at different times suggested to me that it was a pity that a treatise so unique and in many respects so attractive as the Arithmetica should once more have become practically inaccessible to the English reader. At the same time I could not but recognise that, after twenty-five years in which so much has been done for the history of mathematics, the book needed to be brought up to date, Some matters which in 1885 were still subject of controversy, such as the date of Diophantus, may be regarded as settled, and some points which then had to be laboured can now be dismissed more briefly. Practically the whole of the Introduction, except the chapters on the editions of Diophantus, his methods of solution, and the porisms and other assumptions found in his work, has been entirely rewritten and much shortened, while the chapters on the methods and on the porisms etc., have been made fuller than before. The new text of Tannery (Teubner 1893, 1895) has enabled a number of obscure passages, particularly in Books V and VI, to be cleared up and, as a basis for a reproduction of the whole work, is much superior to the text of Bachet. I have taken the opportunity to make my version of the actual treatise somewhat fuller and somewhat closer to the language of the original. In other respects also I thought I could improve upon a youthful work which was my first essay in the history of Greek mathematics. When writing it I was solely concerned to make Diophantus himself known to mathematicians,

833088

vi PREFACE

and I did not pay sufficient attention to Fermat's notes on the various problems. It is well known that it is in these notes that many of the great propositions discovered by Fermat in the theory of numbers are enshrined ; but, although the notes are literally translated in Wertheim's edition, they do not seem to have appeared in English ; moreover they need to be supple- mented by passages from the correspondence of Fermat and from the Doctrinae analyticae Inventum Novum of Jacques de Billy. The histories of mathematics furnish only a very inadequate description of Fermat's work, and it seemed desirable to attempt to give as full an account of his theorems and problems in or connected with Diophantine analysis as it is possible to compile from the scattered material available in Tannery and Henry's edition of the Oeuvres de Fermat (1891 — 1896). So much of this material as could not be conveniently given in the notes to particular problems of Diophantus I have put together in the Supplement, which is thus intended to supply a missing chapter in the history of mathematics. Lastly, in order to make the book more complete, I thought it right to add some of the more remarkable solutions of difficult Diophantine problems given by Euler, for whom such problems had a great fascination ; the last section of the Supplement is therefore devoted to these solutions.

T. L. H. October, 1910.

CONTENTS

INTRODUCTION

PAGES

CHAP. I. Diophantus and his Works !_i3

„ II. The MSS. of and writers on Diophantus . . 14 31

„ III. Notation and definitions of Diophantus . . . 32—53

„ IV. Diophantus' methods of solution . . - . . 54 98

„ V. The Porisms and other assumptions in Diopbantus 99—110

„ VI. The place of Diophantus m 127

THE ARITHMETICA

BOOK L 129-143

» IL 143-156

» IIL • • 156—168

» IV- 168—199

\7

V* 200 — 225

VI.

• . - . 226—246

On Polygonal numbers 247 2^9

Conspectus of the Arithmetica 260—266

SUPPLEMENT

SECTIONS I— V. NOTES, THEOREMS AND PROBLEMS BY FERMAT.

I. On numbers separable into integral squares . . 267—277

II- Equation 3*-Ayi=\ ....... 277—292

III. Theorems and Problems on rational right-angled

triangles 293_3,g

IV. Other problems by Fermat 318—320

V. Fermat's Triple-equations 321—328

SECTION VI. SOME SOLUTIONS BY EULER 329—380

INDEX: I. GREEK 381—382

„ II. ENGLISH 382 — 387

INTRODUCTION CHAPTER I

DIOPHANTUS AND HIS WORKS

THE divergences between writers on Diophantus used to begin, as Cossali said1, with the last syllable of his name. There is now, however, no longer any doubt that the name was Diophantar, not Diophanter2.

The question of his date is more difficult Abu'lfaraj, the Arabian historian, in his History of tlte Dynasties, places Diophantus under the Emperor Julian (A.D. 361-3), but without giving any authority; and it may be that the statement is due simply to a confusion of our Diophantus with a rhetorician of that name, mentioned in another article of Suidas, who lived in the time of Julian*. On the other hand, Rafael Bombelli in his Algebra,

1 Cossali, Origine, trasporto in Italia, primi progressi in essa ddr Algebra (Parma, 1797-9), I. p. 61 : "Su la desinenza del nome comincia la diversita tra gli scrittori."

1 Greek authority is overwhelmingly in favour of Diophant<w. The following is the evidence, which is collected in the second volume of Tannery's edition of Diophantus (henceforward to be quoted as "Dioph.," "Dioph. n. p. 36" indicating page 36 of Vol. ii., while "Dioph. II. 20" will mean proposition 20 of Book II.): Suidas s.v. 'TraTia (Dioph. n. p. 36), Theon of Alexandria, on Ptolemy's Syntaxis Book I. c. 9 (Dioph. II. p. 35), Anthology, Epigram on Diophantus (Ep. xiv. 126; Dioph. II. p. 60), Anonymi prolegomena in Introductionem arithmeticam Nicomachi (Dioph. II. p. 73), Georgii Pachymerae paraphrasis (Dioph. n. p. 122), Scholia of Maximus Planudes (Dioph. n. pp. 148, 177, 178 etc.), Scholium on larublichus /« Nicomachi arithm. inirod., ed. PisteUi, p. 127 (Dioph. n. p. 72), a Scholium on Dioph. n. 8 from the MS. "A" (Dioph. n. p. 260), which is otherwise amusing (H ifwxn aw> Ai6#arre, eft; perk TOV Sarai-a frexa Trft Siv/coXi'as TUV re iXXwr ffov tfewpij/uiTwr KCU Si) ecu TOV rttporros ffcwpj- ftATm, " Your soul to perdition, Diophantus, for the difficulty of your problems in general and of this one in particular ") ; John of Jerusalem ( loth c.) alone ( Vita loannis Damas- ceni xi. : Dioph. n. p. 36), if the reading of the MS. Parisinus 1559 is right, wrote, in the plural, wj UvOaydpcu. 1} AIO^OJTOI, where however Aio^oirai is dearly a mistake for

3 \i3dvios, ffoQurriis 'Arrtox«/s, rwr tfl 'louXtoyoO TOV /ScurtAlcds Qeodoffiov TOV -rpfff^vripov • Qajryariov xarp6j, /ua^TTjj AIOC><UTOI/.

H. D.

2 INTRODUCTION

published in 1572, says dogmatically that Diophantus lived under Antoninus Pius (138-161 A.D.), but there is no confirmation of this date either.

The positive evidence on the subject can be given very shortly. An upper limit is indicated by the fact that Diophantus, in his book on Polygonal Numbers, quotes from Hypsicles a definition of such a number1. Hypsicles was also the writer of the sup- plement to Euclid's Book XIII. on the Regular Solids known as Book XIV. of the Elements ; hence Diophantus must have written later than, say, 150 B.C. A lower limit is furnished by the fact that Diophantus is quoted by Theon of Alexandria2; hence Diophantus wrote before, say, 350 A.D. There is a wide interval between 150 B.C. and 350 A.D., but fortunately the limits can be brought closer. We have a letter of Psellus (nth c.) in which Diophantus and Anatolius are mentioned as writers on the Egyptian method of reckoning. " Diophantus," says Psellus3, " dealt with it more accurately, but the very learned Anatolius collected the most essential parts of the doctrine as stated by Diophantus in a different way (reading erepwf) and in the most succinct form, dedicating (irpoae^xav^a-e) his work to Diophantus." It would appear, therefore, that Diophantus and Anatolius were contem- poraries, and it is most likely that the former would be to the latter in the relation of master to pupil. Now Anatolius wrote about 278-9 A.D., and was Bishop of Laodicea about 280 A.D. We may therefore safely say that Diophantus flourished about 250 A.D. or not much later. This agrees well with the fact that he is not quoted by Nicomachus (about 100 A.D.), Theon of Smyrna (about 130 A.D.) or lamblichus (end of 3rd c.).

1 Dioph. I. p. 470-2.

2 Theo Alexandrinus in primum librum Ptolemaei Mathematicae Compositionis (on c. IX.) : see Dioph. II. p. 35, Ka.0' a KOI &i6<pavr6s 0ij<rr TTJS yap parados d/uera^Tou o&o-ijs /cat fffTWffrjs irai/Tore, TO iro\\aTT\a(na£'6/j.ei>ov elSos CTT' avrrjv avrb TO eZSoj &TTCU K.r.e.

3 Dioph. II. p. 38-9 : irepl de rrjs aiyvimaKrjs fj,e668ov Tatirrjs Ai6<f>ai>Tos (j.ev di^\a^€i> aKpi^ffTfpov, 6 54 \oyiibraTos 'AvctToXios TCI ffweKTiKurara /J.^pij rrjs KOT' iKfivov (iriffTrnj.?)* airo\el;a/J.evos ertpw (PfWpws or eraifnf) Aio^di'Ty ffwoTTTiKurara Trpoffe<p&vr]ffe. The MSS. read rr^pw, which is apparently a mistake for erfyws or possibly for eralpip. Tannery con- jectures rif fralpif, but this is very doubtful ; if the article had been there, Aio^avTy T£ fralpq would have been better. On the basis of eratpy Tannery builds the further hypothesis that the Dionysius to whom the Arithmetica is dedicated is none other than Dionysius who was at the head of the Catechist school at Alexandria 232-247 and was Bishop there 248-265 A.D. Tannery conjectures then that Diophantus was a Christian and a pupil of Dionysius (Tannery, "Sur la religion des derniers mathematicians de 1'antiquite," Extrait des Annales de Philosophic Chretienne, 1896, p. 13 sqq.). It is however difficult to establish this (Hultsch, art. "Diophantos aus Alexandreia" in Pauly- Wissowa's Real-Encyclopddie der classischen Altertumswissenchaften}.

DIOPHANTUS AND HIS WORKS 3

The only personal particulars about Diophantus which are known are those contained in the epigram-problem relating to him in the Anthology x. The solution gives 84 as the age at which he died. His boyhood lasted 14 years, his beard grew at 21, he married at 33; a son was born to him five years later and died, at the age of 42, when his father was 80 years old. Diophantus' own death followed four years later2. It is clear that the epigram was written, not long after his death, by an intimate personal friend with knowledge of and taste for the science which Diophantus made his life-work3.

The works on which the fame of Diophantus rests are :

(1) The Arithmetica (originally in thirteen Books).

(2) A tract On Polygonal Numbers.

Six Books of the former and part of the latter survive. Allusions in the Aritkmetica imply the existence of

(3) A collection of propositions under the title of Porisms; in three propositions (3, 5 and 16) of Book V. Diophantus quotes as known certain propositions in the Theory of Numbers, prefixing to the statement of them the words " We have it in the Porisms that " (e^o/iey eV rot? TLopt,afj,acriv ort K.r.e.}.

A scholium on a passage of lamblichus where he quotes a dictum of certain Pythagoreans about the unit being the dividing line (fj,e06piov) between number and aliquot parts, says "thus

Diophantus in the Moriastica* for he describes as 'parts' the

progression without limit in the direction of less than the unit." Tannery thinks the Moptacrrucd may be ancient scholia (now lost) on Diophantus I. Def. 3 sqq.5; but in that case why should Diophantus be supposed to be speaking ? And, as Hultsch

1 Anthology, Ep. xiv. 126; Dioph. n. pp. 60-1.

2 The epigram actually says that his boyhood lasted | of his life; his beard grew after T\ more ; after f more he married, and his son was born five years later ; the son lived to half his father's age, and the father died four years after his son. Cantor ( Gesch. d. Math. I3, p. 465) quotes a suggestion of Heinrich Weber that a better solution is obtained if we assume that the son died at the time when his father's age was double his, not at an age equal to half the age at which his father died. In that case

is would substitute lof for 14, i6£ for 21, 25$ for 33, 30! for 42, 6i£ for 80, and 65^ for 84 above. I do not see any advantage in this solution. On the contrary, 4 think the fractional results are an objection to it, and it is to be observed that the ^nholiast has the solution 84, derived from the equation \x + T^X + f x + 5 + \x + 4 = x.

3 Hultsch, art. Diophantos in Pauly-Wissowa's Real-Encyclopadie.

4 lamblichus In Nicomachi arithm. introd. p. 127 (ed. Pistelli) ; Dioph. II. p. 72. jjs 5 Dioph. n. p. 72 note.

I — 2

4 INTRODUCTION

remarks, such scholia would more naturally have been quoted as a-yoXia and not by the separate title Mopiavriicd1. It may have been a separate work by Diophantus giving rules for reckon- ing with fractions ; but I do not feel clear that the reference may not simply be to the definitions at the beginning of the Arithmetica.

With reference to the title of the Arithmetica, we may observe that the meaning of the word dpid^TiKd here is slightly different from that assigned to it by more ancient writers. The ancients drew a marked distinction between dpidpijTiKij and \OJIO-TIKIJ, though both were concerned with numbers. Thus Plato states that dpid/j,rjTiKrj is concerned with the abstract properties of numbers (as odd and even, etc.), whereas \oyia-TiKtj deals with the same odd and even, but in relation to one another2. Geminus also distinguishes the two terms3. According to him dpiO^riK^ deals with numbers in themselves, distinguishing linear, plane and solid numbers, in fact all the forms of number, starting from the unit, and dealing with the generation of plane numbers, similar and dissimilar, and then with numbers of three dimensions, etc. \oyiariKij on the other hand deals, not with the abstract properties of numbers in themselves, but with numbers of concrete things (ala-drjTwv, sensible objects), whence it calls them by the names of the things measured, e.g. it calls some by the names /^XtTi?? and <£iaXtT?794. But in Diophantus the calculations take an abstract form (except in V. 30, where the question is to find the number of measures of wine at two given prices respectively), so that the distinction between XO^IO-TIKIJ and dpiQ/jLijTucr/ is lost.

We findjthe Arithmetica quoted under slightly different titles. Thus the anonymous author of prolegomena to Nicomachus" Introductio Arithmetica speaks of Diophantus' " thirteen Books of Arithmetic5." A scholium on lamblichus refers to " the last theorem of the first Book of Diophantus' Elements of Arithmetic

1 Hultsch, loc. cit.

2 Gorgias, 451 B, C : rd fjxv &\\a Kadairep 17 dpiB '^T/TIKT; i] hoyicrTiKri Hxfi' iff pi TO avrb ydp dffTi, rb re apriov Kal TO irepiTTbv • dia<p4pei 5£ roffovrov, OTL Kal irpos avra Kal irpds d\\ri\a TTWJ ?xfi TXTjtfous tiriffKoirel TO irepiTrov Kal TO apTiov i) \oyiaTiK7].

3 Proclus, Comment, on Euclid I., p. 39, 14-40, 7.

4 Cf. Plato, Laws 819 B, c, on the advantage of combining amusement with instruction in arithmetical calculation, e.g. by distributing apples or garlands (^Xow T£ TLVUV diavofMl Kal ffT€<pdvuv) and the use of different bowls of silver, gold, or brass etc. (tfudXas a/Mi xpvffov Kal X&XKOU *cai apytipov Kal rotoi/raw TIV&V aXXwc KepawuvTes, ol 5£ 6'Xas TTWS 5ia5t56»rcj, oirep elirov, eij iraidiav tvapubrrovTes ras TUV dva

5 Dioph. II. p. 73, 26.

DIOPHANTUS AND HIS WORKS 5

)1." A scholium on one of the epigrams in Metrodorus' collection similarly speaks of the " Elements of Diophantus2."

None of the MSS. which we possess contain more than the first six Books of the Arithmetic^ the only variation being that some few divide the six Books into seven3, while one or two give the fragment on Polygonal Numbers with the number vm. The idea that Regiomontanus saw, or said he saw, a MS. containing the thirteen Books complete is due to a misapprehension. There is no doubt that the missing Books were 'lost at a very early date. Tannery4 suggests that Hypatia's commentary extended only to the first six Books, and that she left untouched the remaining seven, which accordingly were first forgotten and then lost ; he compares the case of Apollonius' Conies, the first four Books of which were preserved by Eutocius, who wrote a commentary on them, while the rest, which he did not include in his commentary, were lost so far as the Greek text is concerned. While, however, three of the last four Books of the Conies have fortunately reached us through the Arabic, there is no sign that even the Arabians ever possessed the missing Books of Diophantus. Thus the second part of an algebraic treatise called the Fakhrl by Abu Bekr Muh. b. al-Hasan al-Karkhl (d. about 1029) is a collection of problems in determinate and indeterminate analysis which not only show that their author had deeply studied Diophantus, but in many cases are taken direct frcjmjihe Arithmetica, with the change, occasionally, of some of the constants. In the fourth section of this work, which begins and ends with problems corresponding to problems in Diophantus Books II. and in. respectively, are 25 problems not found in Diophantus ; but the differences from Diophantus in essential features (e.g. several of the problems lead to equations giving irrational results, which are always avoided by Diophantus), as well as other internal evidence, exclude the hypothesis that we have here a lost Book of Diophantus5. Nor is there any sign that more of the work than we possess was known

1 Dioph. II. p. 72, 17 ; lamblichus (ed. Pistelli), p. 132, 12.

2 Dioph. II. p. 62, 25.

3 e.g. Vaticanus gr. 200, Scorialensis 0-1-15, ar»d the Broscius MS. in the University Library of Cracow ; the two last divide the first Book into two, the second beginning immediately after the explanation of the sign for minus (Dioph. I. p. 14, i).

4 Dioph. II. p. xvii, xviii.

5 See F. Woepcke, Extrait du Fakhrl, traiti cTAlgtbre par Abou Bekr Mohammed ben Alhafan AlkarkhT (tnanuscrit 952, supplement arabe de la bibliotheque fnipdriale), Paris, 1853.

6 INTRODUCTION

to Abu'l Wafa al-Buzjam (940-998 A.D.), who wrote a "commentary (tafslr) on the algebra of Diophantus " as well as a " Book of proofs of the propositions used by Diophantus in his work..." These facts again point to the conclusion that the lost Books were lost before the loth c.

Tannery's suggestion that Hypatia's commentary was limited to the six Books, and the parallel of Eutocius' commentary on Apollonius' Conies, imply that it is the last seven Books, and the most difficult, which 'are lost. This view is in strong contrast to that which Jiad previously found most acceptance among com- petent authorities. The latter view was most clearly put, and most ably supported, by Nesselmann1, though Colebrooke2 had already put forward a conjecture to the same effect ; and historians of mathematics such as Hankel, Moritz Cantor, and Giinther have accepted Nesselmann's conclusions, which, stated in his own words, are as follows: (i) that much less of Diophantus is wanting than would naturally be supposed on the basis of the numerical proportion of 6 to 13; (2) that the missing portion is not to be looked for at the end but in the middle of the work, and indeed mostly between the first and second Books. Nesselmann's general argument is that, if we carefully read the last four Books, from the third to the sixth, we find that Diophantus moves in a rigidly defined and limited circle of methods and artifices, and that any attempts which he makes to free himself are futile ; " as often as he gives the impression that he wishes to spring over the magic circle drawn round him, he is invariably thrown back by an invisible hand on the old domain already known ; we see, similarly, in half-darkness, behind the clever artifices which he seeks to use in order to free himself, the chains which fetter his genius, we hear their rattling, whenever, in dealing with difficulties only too freely imposed upon himself, he knows of no other means of extricating himself except to cut through the knot instead of untying it." Moreover, the sixth Book forms a natural conclusion to the whole, in that it consists of exemplifications of methods explained and used in the preceding Books. The subject is the finding of right- angled triangles in rational numbers such that the sides and area satisfy given conditions, the geometrical property of the right-angled triangle being introduced as a fresh condition additional to the purely arithmetical conditions which have to be satisfied in the

1 Algebra der Griechen, pp. 264-273.

2 Algebra of the Hindus, Note M, p. Ixi.

DIOPHANTUS AND HIS WORKS 7

problems of the earlier Books. But, assuming that Diophantus' resources are at an end in the sixth Book, Nesselmann has to suggest possible topics which would have formed approximately adequate material for the equivalent of seven Books of the Arithmetical. The first step is to consider what is actually wanting which we should expect to find, either as foreshadowed by the author himself or as necessary for the elucidation or completion of the whole subject. Now the first Book contains problems leading to determinate equations of the first degree ; the remainder of the work is a collection of problems which, with few exceptions, lead to indeterminate equations of the second degree, beginning with simpler cases and advancing step by step to more complicated questions. There would have been room therefore for problems involving (i) determinate equations of the second degree and (2) indeterminate equations of the first. There is indeed nothing to show that (2) formed part of the writer's plan ; but on the other hand the writer's own words in Def. 1 1 at the beginning of the work promise a discussion of the solution of the complete or adfected quadratic, and it is clear that he employed his method of solution in the later Books, where in some cases he simply states the solution without working it out, while in others, where the roots are " irrational," he gives approximations which indicate that he was in possession of a scientific method. Pure quadratics Diophantus regarded as simple equations, taking no account of the negative -root. Indeed it would seem that he adopted as his ground for the classification of quadratics, not the index of the highest power of the unknown quantity contained in it, but the number of terms left in it when reduced to its simplest form. His words are1: " If the same powers of the unknown occur on both sides, but with different coefficients (prj 6fjW7r\r)0i) Se), we must take like from like until we have one single expression equal to another. If there are on both sides, or on either side, any terms with negative coefficients (ev eXXefy-eo-t riva eiS?)), the defects must be added on both sides until the terms on both sides have none but positive coefficients (evvTrap^ovra), when we must again take like from like until there remains one term on each side. This should be the object aimed at in framing the hypotheses of propositions, that is to say, to reduce the equations, if possible, until one term is left equated to one term. But afterwards I will

1 Dioph. I. Def. u, p. 14.

8 INTRODUCTION

show you also how, when two terms are left equal to one term, such an equation is solved." That is to say, reduce the quadratic, if possible, to one of the forms ax*= bx, axz=c, or bx = c\ I will show later how to solve the equation when three terms are left of which any two are equal to the third, z.^.'the complete quadratic ax*± bx ± c — o, excluding the case ax"- + bx + c = o. The exclusion of the latter case is natural, since it is of the essence of the work to find rational and positive solutions. Nesselmann might have added that Diophantus' requirement that the equation, as finally stated, shall contain only positive terms, of which two are equated to the third, suggests that his solution would deal separately with the three possible cases (just as Euclid makes separate cases of the equations in his propositions VI. 28, 29), so that the exposition might occupy some little space. The suitable place for it would be between the first and second Books. There is no evidence tending to confirm Nesselmann's further argument that the six Books may originally have been divided into even more than seven Books. He argues from the fact that there are often better natural divisions in the middle of the Books (e.g. at II. 19) than between them as they now stand ; thus there is no sign of a marked division between Books I. and II. and between Books II. and in., the first five problems of Book II. and the first four of Book in. recalling similar problems in the preceding Books respectively. But the latter circumstances are better explained, as Tannery explains them, , by the supposition that the first problems of Books II. and III. are interpolated from some ancient commentary. Next Nesselmann points out that there are a number of imperfections in the text, Book V. especially having been " treated by Mother Time in a very stepmotherly fashion " ; thus it seems probable that at V. 19 three problems have dropped out altogether. Still he is far from accounting for seven whole Books; he has therefore to press into the service the lost "Porisms" and the tract on Polygonal Numbers.

If the phrase which, as we have said, occurs three times in Book V., "We have it in the Porisms that...," indicates that the "Porisms" were a definite collection of propositions concerning the properties of certain numbers, their divisibility into a certain number of squares, and so on, it is possible that it was from the same collection that Diophantus took the numerous other pro- positions which he assumes, either explicitly enunciating them, or implicitly taking them for granted. May we not then, says

DIOPHANTUS AND HIS WORKS 9

Nesselmann, reasonably suppose the " Porisms " to have formed an introduction to the indeterminate and semi-determinate analy- sis of the second degree which forms the main subject of the Arithmetica, and to have been an integral part of the thirteen Books, intervening, probably, between Books I. and II. ? Schulz* on the other hand, considered this improbable, and in recent years Hultsch1 has definitely rejected the theory that Diophantus filled one or more Books of his Arithmetica exclusively with Porisms. Schulz's argument is, indeed, not conclusive. It is based on the consideration that " Diophantus expressly says that his work deals with arithmetical problems*" \ but what Diophantus actually says is " Knowing you, O Dionysius, to be anxious to learn the solution (or, nerhaps, ' discovery,' evpeviv) of problems in numbers, I have endeavoured, beginning from the foundations on which the study is bufit up, to expound (v-rrofTrfja-ai = to lay down) the nature and force subsisting in numbers," the last of which words would easily c°ver 'propositions in the theory of numbers, while " propositions," °ot T: pr££Jems," is the word used at the end of the Preface, where

he says, "let us now proceed to the propositions (7rpoT«<ra<?)

which have been treated in thirteen Books."

On reconsideration of the whole matter, I now agree in the view of Hultsch that the Porisms were not a separate portion of the Arithmetica or included in the Arithmetica at all. If they had been, I think the expression " we have it in the Porisms " would have been inappropriate. In the first place, the Greek mathe- maticians do not usually give references in such a form as this to propositions which they cite when they come from the same work as that in which they are cited ; as a rule the propositions are quoted without any references at all. The references in this case would, on the assumption that the Porisms were a portion of the thirteen Books, more naturally have been to particular pro- positions of particular Books (cf. Eucl. XII. 2, " For it was proved

1 Hultsch, loc. cit,

2 The whole passage of Schulz is as follows (pref. xxi): " Es ist daher nicht unwahr- scheinlich, dass diese Porismen eine eigene Schrift unseres Diophantus waren, welche vorziiglich die Zusammensetzung der Zahlen aus gewissen Bestandtheilen zu ihrem Gegenstande hatten. Konnte man diese Schrift als einen Bestandtheil des grossen in dreizehn Biichern abgefassten arithmetischen Werkes ansehen, so ware es sehr erklarbar, dass gerade dieser Theil, der den blossen Liebhaber weniger anzog, verloren ging. Da indess Diophantus ausdriicklich sagt, sein Werk behandele arithmetisfhe Probleme, so hat wenigstens die letztere Annahme nur einen geringen Grad von \Vahrscheinlichkeit."

io INTRODUCTION

in the first theorem of the loth Book that..."). But a still vaguer reference would have been enough, even if Diophantus had chosen to give any at all ; if the propositions quoted had preceded those in which they are used, some expression like TOVTO yap irpo- yeypaTTTai, " for this has already been proved," or SeSeitcrai yap TOVTO, " for this has been shown," would have sufficed, or, if the propositions occurred later, some expression like J>? efr;? Sei^BijareTai or Set%#/7£reTat v$> fipwv va-repov, "as will be proved in due course" or "later." The expression "we have it in the Porisms" (in the plural) would have been still more inappropriate if the "Porisms" had been, as Tannery supposes1, not collected together as one or more Books of the Arithmetica, but scattered about in the work as corollaries to particular propositions2. And, as Hultsch says, it is hard, on Tannery's supposition, to explain why the three partv.iiar theorems quoted from " the Porisms " were lost, while a ','air number of other additions survived, partly under the title iropia-^. (cf. I. 34, I. 38), partly as "lemmas to what follows," Xf///,/*r e^9 (cf. lemmas before IV. 34, 35, 36, V. 7, 8, VI. 12, 15). O> other hand, there is nothing improbable in the supposition that Diophantus was induced by the difficulty of his problems to give place in a separate work to the " porisms " necessary to their solution.

The hypothesis that the Porisms formed part of the Arithmet- ica being thus given up, we can hardly hold any longer to Nesselmann's view of the contents of the lost Books and their place in the treatise; and I am now much more inclined to the opinion of Tannery that it is the last and the most difficult Books which are lost. Tannery's argument seems to me to be very attractive and to deserve quotation in full, as finally put in the preface to Vol. II. of his Diophantus3. He replies first to the assumption that Diophantus could not have proceeded to problems more difficult than those of Book V. " But if the fifth or the sixth Book of the Arithmetica had been lost, who, pray, among us would have believed that such problems had ever been attempted by the Greeks? It would be the greatest error, in any case in which a

1 Dioph. II. p. xix.

2 Thus Tannery holds (loc. cit.) that the solution of the complete quadratic was given in the form of corollaries to I. 27, 30; and he refers the three "porisms" quoted in v. 3, 5, 16 respectively to a second (lost) solution of III. io, to III. 15, and to iv. i, 2.

3 Dioph. II. p. xx.

DIOPHANTUS AND HIS WORKS n

thing cannot clearly be proved to have been unknown to all the ancients, to maintain that it could not have been known to some Greek mathematician. If we do not know to what lengths Archimedes brought the theory of numbers (to say nothing of other things), let us admit our ignorance. But, between the famous problem of the cattle and the most difficult of Diophantus' problems, is there not a sufficient gap to require seven Books to fill it? And, without attributing to the ancients what modern mathematicians have discovered, may not a number of the things attributed to the Indians and Arabs have been drawn from Greek sources ? May not the same be said of a problem solved by Leonardo of Pisa, which is very similar to those of Diophantus but is nc now to be found in the Arithmetical In fact, it may fairly be said that, when Chasles made his reasonably probable restitution of the Porisms of Euclid, he, notwithstanding the fact that he had Pappus' lemmas to help him, undertook a more difficult task than he would have undertaken if he had attempted to fill up seven Diophantine Books with numerical problems which the Greeks may reasonably be supposed to have solved."

On the assumption that the lost portion came at the end of the existing six Books, Schulz supposed that it contained new methods of solution in addition to those used in Books I. to VI., and in particular extended the method of solution by means of the double equation (Bi7r\rj tcrori;? or SiTrXoia-oTT)?). By means of the double equation Diophantus shows how to find a value of the unknown which will make two expressions (linear or quadratic) containing it simultaneously squares. Schulz then thinks that he went on, in the lost Books, to make three such expressions simultaneously squares, i.e. advanced to a triple equation. But this explanation does not in any case take us very far.

Bombelli thought that Diophantus went on to solve deter- minate equations of the third and fourth degree1; this view, however, though natural at that date, when the solution of cubic and biquadratic equations filled so large a space in contemporary investigations and in Bombelli's own studies, has nothing to support it.

Hultsch2 seems to find the key to the question in the fragment of the treatise on Polygonal Numbers and the developments to

1 Cossali, I. pp. 75, 76. 2 Hultsch, loc. fit.

12 INTRODUCTION

which it might have been expected to lead. In this he differs from Tannery, who says that, as Serenus' treatise on the sections of cones and cylinders was added to the mutilated Conies of Apollonius consisting of four Books only, in order to make up a convenient volume, so the tract on Polygonal Numbers was added to the remains of the Arithmetical, though forming no part of the larger work1. Thus Tannery would seem to deny the genuineness of the whole tract on Polygonal Numbers, though in his text he only signalises the portion beginning with the enunciation of the problem " Given a number, to find in how many ways it can be a polygonal number " as a " vain attempt by a commentator " to solve this problem. Hultsch, on the other hand, thinks we may conclude that Diophantus really solved the problem. He points out moreover that the beginning of the tract is like the beginning of Book I. of the Arithmetica in containing definitions and pre- liminary propositions. Then came the difficult problem quoted, the discussion of which breaks off in our text after a few pages ; and to this it would be easy to tack on a great variety of other problems. Again, says Hultsch, the supplementary propositions added by Bachet may serve to give an approximate idea of the difficulty of the problems which were probably treated in Books VII. and the following. And between these and the bold combination of a triangular and a square number in the Cattle-Problem stretches, as Tannery says, a wide domain which was certainly not unknown to Diophantus, but was his hunting-ground for the most various problems. Whether Diophantus dealt with plane numbers, and with other figured numbers, such as prisms and tetrahedra, is uncertain.

The name of Diophantus was used, as were the names of Euclid, Archimedes and Heron in their turn, for the purpose of palming off the compilations of much later authors. Tannery prints in his edition three fragments under the head of "Diophantus Pseudepigraphus." The first2, which is not " from the Arithmetic of Diophantus " as its heading states, is worth notice as containing some particulars of one of " two methods of finding the square root of any square number"; we are told to begin by writing the number " according to the arrangement of the Indian method," t.e. according to the Indian numerical notation which reached us through the Arabs. The fragment is taken from a Paris MS.

1 Dioph. ii. p. xviii. 2 Dioph. n. p. 3, 3-14.

DIOPHANTUS AND HIS WORKS 13

(Supplem. gr. 387), where it follows a work with the title 'Ap^») 7779 fi€ryd\r)<; KOI 'Iv8itcfj<; "^r}(pi(f)opia<; (i.e. ifnj<f>o<j>opia<;), written in 1252 and raided about half a century later by Maximus Planudes. The second fragment1 is the work edited by C. Henry in 1879 as Opusculum de multiplicatione et divisione sexagesimalibus Diophanto vel Pappo attribuendum. The third2, beginning with Aio(j>dvTov eTri7re8ofj,6TpiKd, is a compilation made in the Byzantine period out of late reproductions of the yecof^erpovfieva and <nepeop,€rpovfi€va of Heron. The second and third fragments, like the first, have nothing to do with Diophantus.

1 Dioph. II. p. 3, 15-15, 17. 2 Dioph. n. p. 15, 18-31, 22.

CHAPTER II

THE MSS. OF AND WRITERS ON DIOPHANTUS

FOR full details of the various MSS. and of their mutual relations, reference should be made to the prefaces to the first and second volumes of Tannery's edition1. Tannery's account needs only to be supplemented by a description given by Gollob2 of another MS. supposed by Tannery to be non-existent, but actually rediscovered in the Library of the University of Cracow (Nr 544). Only the shortest possible summary of the essential facts will be given here.

After the loss of Egypt the work of Diophantus long remained almost unknown among the Byzantines ; perhaps one copy only survived (of the Hypatian recension), which was seen by Michael Psellus and possibly by the scholiast to lamblichus, but of which no trace can be found after the capture of Constantinople in 1 204. From this one copy (denoted by the letter a in Tannery's table of the MSS.) another MS. (a) was copied in the 8th or 9th century ; this again is lost, but is the true archetype of our MSS. The copyist apparently intended to omit all scholia, but, the distinction between text and scholia being sometimes difficult to draw, he included a good deal which should have been left out. For example, Hypatia, and perhaps scholiasts after her, seem to have added some alternative solutions and a number of new problems ; some of these latter, such as II. 1-7, 17, 18, were admitted into the text as genuine.

The MSS. fall into two main classes, the ante-Planudes class, as we may call it, and the Planudean. The most ancient and the best of all is Matritensis 48 (Tannery's A\ which was written in the 1 3th century and belongs to the first class; it is evidently a most faithful copy of the lost archetype (a). Maximus Planudes wrote a systematic commentary on Books I. and II., and his scholia,

1 Dioph. I. pp. iii-v, II. pp. xxii-xxxiv.

2 Eduard Gollob, "Ein wiedergefundener Diophantuscodex " in Zeitschrift filr Math. u. Physik, XLIV. (1899), hist.-litt. Abtheilung, pp. 137-140.

THE MSS. OF AND WRITERS ON DIOPHANTUS 15

which are edited by Tannery for the first time, are preserved in the oldest representative which we possess of the Planudean class, namely, Marcianus 308 (Tannery's B^, itself apparently copied from an archetype of the I4th century now lost, with the exception of ten leaves which survive in Ambrosianus Et 157 sup.

Tannery shows the relation of the MSS. in the following diagram :

(a) Lost copy of the Hypatian recension, (a) Lost copy, of eighth or ninth c.

(FIRST CLASS)

1

(PLANUDEAN CLASS)

i. Matritensis 48 = A, 1 3th c.

2. Vaticanusgr. 191= V, second half of i5th c.

3. Vaticanus gr. 304, beginning of i6th c.

9. Lost MS. of the i4th c. of which ten leaves are extant in Ambrosianus Et 157 sup.

i o. Marcianus 308 = B^ , beginning of i5th c.

1 1 . Guelferbytanus Gudianus i, i5th c.

14. Ambrosianus A 91 sup. (1545)

15. Vaticanus gr. 200 (1545)

4. Parisinus 2379 =C (after first two Books), middle of i6th c.

5. Parisinus 2 3 78 = />, middle of i6th c.

6. Neapolitanus IIIC 17, middle of i6th c.

7. Urbinas gr. 74, end of 1 6th c.

8. Oxon. Baroccianus 1 66 (part of Book I. only)

12. Palatinus gr. 391, end of 1 6th c.

13. Reginensis 128, end of 1 6th c.

1 6. Scorialensis T— I— 1 1 (1545)

17. Parisinus 2485 = A", middle of i6th c.

1 8. Scorialensis R-III-iS, middle of i6th c.

19. Ambrosianus Q 121 sup. (part of Book I.), middle of i6th c.

4. Parisinus 2379 = C (first two Books)

20. Taurinensis C III 16

21. Parisinus Ars. 8406 = X

22. Scorialensis fl-I-i 5, middle of 6th c.

23. ScorialensisR-II-3, end of 1 6th c.

? 24. Oxon. Savilianus, end of 1 6th c.

-^

Auria's recension made up out of MSS. 2, 3, 15 above and Xylander's translation: 25. Parisinus 2380 = Z>.

26. Ambrosianus E 5 sup.

27. MS. (Patavinus) of Broscius (Brozek) now at

28. Lost MS. of Cardinal du Perron.

Cracow.

1 6 INTRODUCTION

The addition of a few notes as regards the most important and interesting of the MSS., in the order of their numbers in Tannery's arrangement, will now sufficiently complete the story.

1. The best and most ancient MS., that of Madrid (Tannery's A), was unfortunately spoiled at a late date by corrections made, especially in the first two Books, from some MS. of the Planudean class, in such a way that the original reading is sometimes entirely erased or made quite illegible. In these cases recourse must be had to the Vatican MS. 191.

2. The MS. Vaticanus graecus 191 was copied from A before it had suffered the general alteration by means of a MS. of the other class, though not before various other corrections had been made in different hands not easily distinguished ; thus V some- times has readings which Tannery found to have arisen from some correction in A. A appears to have been at Rome for a con- siderable period at the time when V was copied ; for the librarian who wrote the old table of contents1 at the beginning of V inserted in the margin in one place2 the word a/ofa/^evo?, which had been omitted, direct from the original (A).

3. Vat. gr. 304 was copied from V, not from A ; Tannery inferred this mainly from a collation of the scholia, and he notes that the word ap£a/iei/o<? above mentioned is here brought into the text by the erasure of some letters. This MS. 304, being very clearly written, was used thenceforward to make copies from. The next five MSS. do not appear to have had any older source.

4. The MS. Parisinus 2379 (Tannery's C) was that used by Bachet for his edition. It was written by one loannes Hydruntinus after 1545, and has the peculiarity that the first two Books were copied from the MS. Vat. gr. 200 (a MS. of the Planudean class), evidently in order to include the commentary of Planudes, while the MS. Vat. gr. 304 belonging to the pre-Planudes class was followed in the remaining Books, no doubt because it was con- sidered superior. Thus the class of which C is the chief repre- sentative is a sort of mixed class.

5. 6. Parisinus 2378 = P, and Neapolitans III C 17, were copied by Angelus Vergetius. In the latter Vergetius puts the

1 The MS. V was made up of various MSS. before separated. The old table of contents has Aio^dWou d/ji^ijTcm}- apfioviicii. didQopa. The appoviKa include the Intro- duction to Harmony by Cleonides, but without any author's name. This fact sufficiently explains the error of Ramus in saying, Schola mathematics Bk I. p. 35, "Scripserat et Diophantus harmonica. "

2 Dioph. i. p. 2, 5-6.

THE MSS. OF AND WRITERS ON DIOPHANTUS 17

numbers A, B, F, A, E, Z, H at the top of the pages (as we put headlines) corresponding to the different Books, implying that he regarded the tract on Polygonal Numbers as Book VII.

The other MSS. of the first class call for no notice, and we pass to the Planudean class.

9. Tannery, as he tells us, congratulated himself upon finding in Ambrosianus Et 157 sup. ten pages of the archetype of the class, and eagerly sought for new readings. So far, however, as he was able to carry his collation, he found no difference from the principal representative of the class (B^) next to be mentioned.

10. The MS. Marcianus 308 (= B^) of the 1 5th century formerly belonged to Cardinal Bessarion, and was seen by Regiomontanus at Venice in 1464. It contains the recension by Planudes with his commentary.

11. It seems certain that the Wolfenbuttel MS. Guelferbytanus Gudianus I (i5th c.) was that which Xylander used for his translation ; Tannery shows that, if this was not the MS. lent to Xylander by Andreas Dudicius Sbardellatus, that MS. must have been lost, and there is no evidence in support of the latter hypothesis. It is not possible to say whether the Wolfenbuttel MS. was copied from Marcianus 308 (B^) or from the com- plete MS. of which Ambrosianus Et 157 sup. preserves the ten leaves.

12. Palatinus gr. 391 (end of i6th c.) has notes in German in the margin which show that it was intended to print from it ; it was written either by Xylander himself or for him. It is this MS. of which Claudius Salmasius (Claude de Saumaise, 1588-1653) told Bachet that it contained nothing more than the six Books, with the tract on Polygonal Numbers.

13. Reginensis 128 was copied at the end of the i6th century from the Wolfenbuttel MS.

14. 15. Ambrosianus A 91 sup. and Vaticanus gr. 200 both come from B^ ; as they agree in omitting V. 28 of Diophantus, one was copied from the other, probably the latter from the former. They were both copied by the same copyist for Mendoza in 1545. Vat. gr. 200 has headings which make eight Books ; according to Tannery the first Book is numbered a', the fourth 8OV ; before V. 20 (in Bachet's numbering) — should this be IV. 20 ? — is the heading Ato<£ai/TOL> 6°", before the fifth Book Aio<j>dvTov r°", before the sixth At,o(j)dvTov f0", and before the tract on Polygonal Numbers &.io<f>dvTov 77°" ; this wrong division occurs in the next three MSS.

H. D. 2

i8 INTRODUCTION

(16, 17, 1 8 in the diagram), all of which seem to be copied from Vat. 200.

The MSS. numbered 20, 21, 22, 23 in the diagram are of the hybrid class derived from Parisinus 2379 (C). Scorialensis ft-I-15 and Scorialensis R-II-3, the latter copied from the former, have the first Book divided into two (cf. p. 5 above), and so make seven Books of the Arithmetica and an eighth Book of the Polygonal Numbers.

27. The Cracow MS. has the same division into Books as the MSS. last mentioned. According to Gollob, the collation of this MS., so far as it was carried in 1899, showed that it agrees in the main with A (the best MS.), B^ (Marcianus 308) and C (Parisinus 2379) ; but, as it contains passages not found in the two latter, it cannot have been copied from either of them.

25. Parisinus 2380 appears to be the copy of Auria's Diophantus mentioned by Schulz as having been in the library of Carl von Montchall and bearing the title " Diophanti libri sex, cum scholiis graecis Maximi Planudae, atque liber de numeris poly- gonis, collati cum Vaticanis codicibus, et latine versi a Josepho Auria1."

The first commentator on Diophantus of whom we hear is Hypatia, the daughter of Theon of Alexandria ; she was murdered by Christian fanatics in 415 A.D. According to Suidas she wrote commentaries on Diophantus, on the Astronomical Canon (sc. of Ptolemy) and on the Conies of Apollonius2. Tannery suggests that the remarks of Michael Psellus (nth c.) at the beginning of his letter about Diophantus, Anatolius, and the Egyptian method of arithmetical reckoning were taken bodily from some MS. of Diophantus containing an ancient and systematic commentary ; and he believes this commentary to have been that of Hypatia. I have already mentioned the attractive hypothesis of Tannery that Hypatia's commentary extended only to our six Books, and that this accounts for the loss of the rest.

Georgius Pachymeres (1240 to about 1310) wrote in Greek a paraphrase of at least a portion of Diophantus. Sections 25-44 of

1 Schulz, Diophantus, pref. xliii.

2 Suidas s.v. 'firaria: typa\f/ev virbnvwa els Aufyavrov, <ets> rbv dffrpovofUKbv Kavdva, efc TO. KuviKa 'AiroXXowfou vir&fu>7ifM. So Tannery reads, following the best MSS. ; he gives ample reasons for rejecting Kuster's conjecture ets bioQdvTov rbv dffrpovofj.tKi>v Kavova, viz. (i) that the order of words would have been TOV Aiotpavrov derpovofUKov Kavova, (i) that there is nothing connecting Diophantus with astronomy, while Suidas mentions, s.v. Qiuv, a commentary et's rbv UroXe^aiov -irpoxeipov Kavova.

THE MSS. OF AND WRITERS ON DIOPHANTUS 19

this survive and are published by Tannery in his edition of Diophantus1. The chapters lost at the beginning may have con- tained general observations and introductions to the first two paragraphs of Book I. ; section 25 begins with the third paragraph (Def. i), and the rest of the fragment takes us up to the problem in I. ii.

Soon afterwards Maximus Planudes (about 1260-1310) wrote a systematic commentary on Books I., II. This is also included by Tannery in his edition2.

There are a number of other ancient scholia, very few of which seemed to Tannery to be worth publication3.

But in the meantime, and long before the date of Georgius Pachymeres, the work of Diophantus had become known in Arabia, where it was evidently the subject of careful study. We are told in the Fihrist, the main part of which was written in the year 987 A.D., (i) that Diophantus was a Greek of Alexandria who wrote a book "On the art of algebra4/' (2) that Abu'l Wafa al-Buzjanl (940-998) wrote (a) a commentary (tafsir) on the algebra of Diophantus and (b} a book of " proofs to the pro- positions used by Diophantus in his book and to that which he himself (Abu'l Wafa) stated in his commentary5," (3) that Qusta. b. Luqa al-Ba'labakkl (died about 912) wrote a "com- mentary on three and a half Books of Diophantus' work on arithmetical problems6." Qusta b. Luqa, physician, philosopher, astronomer, mathematician and translator, was the author of works on Euclid and of an " introduction to geometry " in the form of question and answer, and translator of the so-called Books XIV., XV. of Euclid ; other Arabian authorities credit him with an actual " translation of the book of Diophantus on Algebra7." Lastly, we are told by Ibn abi Usaibi'a of " marginal glosses which Ishaq b. Yunis (died about 1077), the physician of Cairo, after Ibn al- ii aitham, added to the book of Diophantus on algebraic problems." The title is somewhat obscure ; probably Ibn al-Haitham (about 965-1039), who wrote several works on Euclid, wrote a commentary on the Arithmetic* and Ishaq b. Yunis added glosses to this commentary8.

Dioph. ii. pp. 78-122. 2 Dioph. II. pp. 125-255.

The few that he gives are in Vol. II. pp. 256-260; as regards the collection in general cf. Hultsch in Berliner philologisctie Wochenschrift, 1896, p. 615.

Fihrist, ed. Suter, p. 22. 5 ibid, p. 39. 6 ibid, p. 43.

Suter, Die Mathematiker und Astroiiomen der Araber, 1900, p. 41.

Suter, op. cit, pp. 107-8. Cf. Bibliotheca Malhematica iv3, 1903-4, p. 296.

2 2

20 INTRODUCTION

To Regiomontanus belongs the credit of being the first to call attention to the work of Diophantus as being extant in Greek. We find two notices by him during his sojourn in Italy, whither he journeyed after the death of his teacher Georg von Peurbach, which took place on the 8th April, 1461. In connexion with lectures on the astronomy of Alfraganus which he gave at Padua he delivered an Oratio introductoria in omnes scientias mathe- maticas1. In this he observed: "No one has yet translated from the Greek into Latin the fine thirteen Books of Diophantus, in which the very flower of the whole of Arithmetic lies hid, the ars rei et census which to-day they call by the Arabic name of Algebra2." Secondly, he writes to Bianchini, in answer to a letter, dated 5th February, 1464, that he has found at Venice "Diofantus," a Greek arithmetician, who has not yet been translated into Latin ; that in his preface Diophantus defines the various powers up to the sixth ; but whether he followed out all the combinations of these Regiomontanus does not know: "for not more than six Books are found, though in the preface he promises thirteen. If this book, which is really most wonderful and most difficult, could be found entire, I should like to translate it into Latin, for the knowledge of Greek which I have acquired while staying with my most reverend master [Bessarion] would suffice for this...." He goes on to ask Bianchini to try to discover a complete copy and, in the meantime, to advise him whether he should begin to translate the six Books3. The exact date of the Oratio is not certain. Regiomontanus made some astronomical observations at Viterbo in the summer and autumn of 1462. He is said to have spent a year at Ferrara, and he seems to have gone thence to Venice. Extant letters of his written at Venice bear dates from 27th July, 1463, to 6th July, 1464, and it may have been from Venice that he made his visit to Padua. At all events the Oratio at Padua must have been near in time to the discovery of the MS. at Venice.

Notwithstanding that attention was thus called to the work, it

1 Printed in the work Rudimenta astronomica Alfragani, Niirnberg, 1537.

2 As the ars rei et census, the solution of determinate quadratic equations, is not found in our Diophantus, it would seem that at the time of the Oratio Regiomontanus had only looked at the MS. cursorily, if at all.

3 The letter to Bianchini is given on p. 135 of Ch. Th. v. Murr's Memorabilia, Norimbergae, 1786, and partly in Doppelmayer's Historische Nachricht von den Niirn- bergischen Mathematicis und Kiinstlern (Niirnberg, 1730), p. 5, note 7.

THE MSS. OF AND WRITERS ON DIOPHANTUS 21

seems to have remained practically a closed book from the date of Maximus Planudes to about 1570. Luca Paciuolo, towards the end of the isth c., Cardano and Tartaglia about the middle of the 1 6th, make no mention of it. Only Joachim Camerarius, in a letter published in I5561, mentions that there is a MS. of Diophantus in the Vatican which he is anxious to see. Rafael Bombelli was the first to find a MS. in the Vatican and to conceive the idea of publishing the work. This was towards 1570, for in his Algebra2 published in 1572 Bombelli tells us that he had in the years last past discovered a Greek book on Algebra written by " a certain Diofantes, an Alexandrine Greek author, who lived in the time of Antoninus Pius " ; that, thinking highly of the contents of the work, he and Antonio Maria Pazzi determined to translate it ; that they actually translated five books out of the seven into which the MS. was divided ; but that, before the rest was finished, they were called away from it by other labours. Bombelli did not carry out his plan of publishing Diophantus in a translation, but he took all the problems of the first four Books and some of those of the fifth, and embodied them in his Algebra, interspersing them with his own problems. He took no pains to distinguish Diophantus' problems from his own ; but in the case of the former he adhered pretty closely to the original, so that Bachet admits his obligations to him, remarking that in many cases he found

1 De Graecis Latinisque numerorum ttotis et praeterea Saracenis seu Indicts, etc. etc., studio Joachimi Camerarii, Papeberg, 1556.

3 Nesselmann tells us that he has not seen this work but takes his information about it from Cossali. I was fortunate enough to find in the British Museum one of the copies dated 1579 (really the same as the original edition of 1572 except that the title-page and date are new, and a dedicatory letter on pp. 3-8 is reprinted ; there were not two separate editions). The title is L" Algebra, opera di Rafael Bombelli da Bologna diuisa in

tre Libri In Bologna, Per Giovanni Rossi, MDLXXIX. The original of the passage

from the preface is :

"Questi anni passati, essendosi ritrouato una opera greca di questa disciplina nella libraria di Nostro Signore in Vaticano, composta da un certo Diofante Alessandrino Autor Greco, il quale fu a tempo di Antonin Pio, e havendomela fatta vedere Messer Antonio Maria Pazzi Reggiano, publico lettore delle Matematiche in Roma, e giudicatolo con lui Autore assai intelligente de' numeri (ancorche non tratti de' numeri irrational!, ma solo in lui si vede vn perfetto ordine di operare) egli, ed io, per arrichire il mondo di cosl fatta opera, ci dessimo a tradurlo, e cinque libri (delli sette che sono) tradutti ne habbiamo ; lo restante non hauendo potuto finire per gli trauagli auenuti all' uno, e all' altro; e in delta opera habbiamo ritrouato, ch' egli assai volte cita gli Autori Indiani, col che mi ha fatto conoscere, che questa disciplina appo -gl' indiani prima fu, che a gli Arabi." The last words stating that Diophantus often quotes from Indian authors are no doubt due to Bombelli's taking for part of Diophantus the tract of Maximus Planudes about the Indian method of reckoning.

22 INTRODUCTION

Bombelli's translation better than Xylander's and consequently very useful for the purpose of amending the latter1.

It may be interesting to mention a few points of notation in this work of Bombelli. At the beginning of Book II. he explains that he uses the word "tanto" to denote the unknown quantity, not "cosa" like his predecessors ; and his symbol for it is -i, the square of the unknown (x^) is ,£., the cube d; and so on. ¥ or plus and -minus (piu and mend) he uses the initial letters / and m. Thus corresponding to x+6 we should find in Bombelli ii/>. 6, and for x* + $x — 4, \^ p. 5-1 m. 4. This notation shows, as will be seen later, some advance upon that of Diophantus in one important respect.

The next writer upon Diophantus was Wilhelm Holzmann who published, under the Graecised form of his name, Xylander, by which he is generally known, a work bearing the title : Diophanti Alexandrini Rerum Arithmeticarum Libri sex, quorum primi duo adiecta habent Scholia Maximi (tit coniectura esf) Planudis. Item Liber de Numeris Polygonis sen Multangulis, Opus incomparabile, uerae Arithmeticae Logisticae perfectionem continens, paucis adhuc uisum. A Gut/. Xylandro Augustano incredibili labore Latinc redditum, et Commentariis explanatum, inque lucem editum, ad Illustriss. Principem Ludovicum Vuirtembergensem. Basileae per Eusebium Episcopium, et Nicolai Fr. haeredes. MDLXX V. Xylander was according to his own statement a " public teacher of Aristotelian philosophy in the school at Heidelberg2." He was a man of almost universal culture3, and was so thoroughly imbued with the classical literature, that the extraordinary aptness of his quotations and his wealth of expression give exceptional charm to his writing whenever he is free from the shackles of mathematical formulae and techni- calities. The Epistola Nuncupatoria is addressed to the Prince Ludwig, and Xylander neatly introduces it by the line " Offerimus numeros, numeri sunt principe digni." This preface is very quaint and interesting. He tells us how he first saw the name of Diophantus mentioned in Suidas, and then found that mention

1 "Sed suas Diophanteis quaestionibus ita immiscuit, ut has ab illis distinguere non sit in promptu, neque vero se fidum satis interpretem praebuit, cum passim verba Diophanti immutet, hisque pleraque addat, pleraque pro arbitrio detrahat. In multis nihilominus interpretationem Bombellii, Xilandriana praestare, et ad hanc emendandam me adjuvisse ingenue fateor." Ad lectorem.

3 "Publicus philosophiae Aristoteleae in schola Heidelbergensi doctor." 3 Even Bachet, who, as we shall see, was no favourable critic, calls him " Vir omnibus disciplinis excultus."

THE MSS. OF AND WRITERS ON DIOPHANTUS 23

had been made of his work by Regiomontanus as being extant in an Italian library and having been seen by him. But, as the book had not been edited, he tried to think no more of it but, instead, to absorb himself in the study of such arithmetical books as he could obtain, and in investigations of his own1. Self-taught except in so far as he could learn from published works such as those of Christoff Rudolff (of the "Coss"), Michael Stifel, Cardano, Nunez, he yet progressed so far as to be able to add to, modify and improve what he found in those works. As a result he fell into what Heraclitus called oiija-iv, lepav VOGOV, that is, into the conceit of " being somebody " in the field of Arithmetic and "Logistic"; others too, themselves learned men, thought him an arithmetician of exceptional ability. But when he first became acquainted with the problems of Diophantus (he continues) right reason brought such a reaction that he might well doubt whether he ought previously to have regarded himself as an object of pity or of derision. He considers it therefore worth while to confess publicly his own ignorance at the same time that he tries to interest others in the work of Diophantus, which had so opened his eyes. Before this critical time he was so familiar with methods of dealing with surds that he had actually ventured to add something to the discoveries of others relating to them ; the subject of surds was considered to be of great importance in arithmetical questions, and its difficulty

1 I cannot refrain from quoting the whole of this passage : " Sed cum ederet nemo : cepi desiderium hoc paulatim in animo consopire, et eonim quos consequi poteram Arithmeticorum librorum cognitione, et meditationibus nostris sepelire. Veritatis porro apud me est autoritas, ut ei coniunctum etiam cum dedecore meo testimonium lubentissime perhibeam. Quod Cossica seu Algebrica (cum his enim reliqua comparata, id sunt quod umbrae Homerice in Necya ad animam Tiresiae) ea ergo quod non assequebar modo, quanquam mutis duntaxat usus preceptoribus caetera at/roSidaKTos, sed et augere, uariare, adeoque corrigere in loco didicissem, quae summi et fidelissimi in docendo uiri Christifer Rodolphus Silesius, Micaelus Stifelius, Cardanus, Nonius, aliique litteris mandauerant : incidi in otyaiv, Upav vbaov, ut scite appellauit Heraclitus sapientior multis aliis philoso- phis, hoc est, in Arithmetica, et uera Logistica, putaui me esse aliquid : itaque de me passim etiam a multis, iisque doctis uiris iudicatum fuit, me non de grege Arithmeticum esse. Verum ubi primum in Diophantea incidi : ita me recta ratio circumegit, ut flendusne mihi ipsi antea, an uero ridendus fuissem, haud iniuria dubitauerim. Operae precium est hoc loco et meam inscitiam inuulgare, et Diophantei operis, quod mihi nebulosam istam caliginem ab oculis detersit, immo eos in coenum barbaricum defossos eleuauit et repur- gauit, gustum aliquem exhibere. Surdorum ego numerorum tractationem ita tenebatn, ut etiam addere aliorum inuentis aliquid non poenitendum auderem, atque id quidem in rebus arithmeticis magnum habetur, et difficultas istarum rerum multos a rnathematibus deterret. Quanto autem hoc est praeclarius, in iis problematis, quae surdis etiam numeris uix posse uidentur explicari, rem eo deducere, ut quasi solum arithmeticum uertere iussi obsurdescant illi plane, et ne mentio quidem eorum in tractatione ingenio- sissimarum quaestionum admittatur. "

24 INTRODUCTION

was even such as to deter many from the study of mathematics. "But how much more splendid," says Xylander, "in the case of problems which seem to be hardly capable of solution even with the help of surds, to bring the matter to the point that, while the surds, when bidden (so to speak) to plough the arithmetic soil, become true to their name and deaf to entreaty, they are not so much as mentioned in these most ingenious solutions ! " He then describes the enormous difficulties which beset his work owing to the corruptions in his text. In dealing, however, with the mistakes and carelessness of copyists he was, as he says, no novice; for proof of which he appeals to his editions of Plutarch, Stephanus and Strabo. This passage, which is good reading, but too long to reproduce here, I give in full in the note1. Next Xylander tells us how he came to get possession of a manuscript of Dio- phantus. In October of the year 1571 he made a journey to Wittenberg ; while there he had conversations on mathematical subjects with two professors, Sebastian Theodoric and Wolfgang Schuler by name, who showed him a few pages of a Greek

1 " Id uero mihi accidit durum et uix superabile incommodum, quod mirifice deprauata omnia inueni, cum neque problematum expositio interdum integra esset, ac passim numeri (in quibus sita omnia esse in hoc argumento, quis ignorat?) tarn problematum quam solutionum siue explicationum corruptissimi. Non pudebit me ingenue fateri, qualem me heic gesserim. Audacter, et summo cum feruore potius quam alacritate animi opus ipsum initio sum aggressus, laborque mihi omnis uoluptati fuit, tantus est meus rerum arithmeti- carum amor, quin et gratiam magnam me apud omnes liberalium scientiarum amatores ac patronos initurum, et praeclare de rep. litteraria meriturum intelligebam, eamque rem mihi laudi (quam a bonis profectam nemo prudens aspernatur) gloriaeque fortasse etiam emolumento fore sperabam. Progressus aliquantulum, in salebras incidi : quae tantum abest ut alacritatem meam retuderint, ut etiam animos mihi addiderint, neque enim mihi novum aut insolens est aduersus librariorum incuriam certamen, et hac in re militaui, (ut Horatii nostri uerbis utar) non sine gloria, quod me non arroganter dicere, Dio, Plutarchus, Strabo, Stephanusque nostri testantur. Sed cum mox in ipsum pelagus monstris scatens me cursus abripuit : non despondi equidem animum, neque manus dedi, sed tamen saepius ad oram unde soluissem respexi, quam portum in quern esset euadendum cogitando prospicerem, depraehendique non minus uere quam eleganter ea cecinisse Alcaeum, quae (si possum) Latine in hac quasi uotiua mea tabula scribam.

Qui uela uentis uult dare, dum licet,

Cautus futuri praeuideat modum

Cursus. mare ingressus, marine

Nauiget arbitrio necesse est.

Sane quod de Echeneide pisce fertur, eum nauim cui se adplicet remorari, poene credibile fecit mihi mea cymba tot mendorum remoris retardata. Expediui tamen me ita, ut facile omnes mediocri de his rebus iudicio praediti, intellecturi sint incredibilem me laborem et aerumnas difficilimas superasse : pudore etiam stimulatum oneris quod ultro mihi impos- uissem, non perferendi. Paucula quaedam non plane explicata, studio et certis de causis in alium locum reiecimus. Opus quidem ipsum ita absoluimus ut neque eius nos pudere debeat, et Arithmeticae Logisticesque studiosi nobis se plurimum debere sint haud dubie professuri."

THE MSS. OF AND WRITERS ON DIOPHANTUS 25

manuscript of Diophantus and informed him that it belonged to Andreas Dudicius whom Xylander describes as "Andreas Dudicius Sbardellatus, hoc tempore Imperatoris Romanorum apud Polonos orator." On his departure from Wittenberg Xylander wrote out and took with him the solution of a single problem of Diophantus, to amuse himself with on his journey. This he showed at Leipzig to Simon Simonius Lucensis, a professor at that place, who wrote to Dudicius on his behalf. A few months afterwards Dudicius sent the MS. to Xylander and encouraged him to persevere in his undertaking to translate the Arithmetica into Latin. Accordingly Xylander insists that the glory of the whole achievement belongs in no less but rather in a greater degree to Dudicius than to himself. Finally he commends the work to the favour of Prince Ludwig, extolling the pursuit of arithmetical and algebraical science and dwelling in enthusiastic anticipation on the influence which the Prince's patronage would have in helping and advancing the study of Arithmetic1. This Epistola Nuncupatoria bears the date I4th August, I5743. Xylander died on the loth of February in the year following that of the publication, 1 576.

Tannery has shown that the MS. used by Xylander was Guelferbytanus Gudianus I. Bachet observes that he has not been able to find out whether Xylander ever published the Greek text, though parts of his commentary seem to imply that he had, or at least intended to do so. It is now clear that he intended to bring out the text, but did not carry out his intention. Tannery observes that the MS. Palatinus gr. 391 seems to have been written either by Xylander himself or for him, and there are German notes in the margin showing that it was intended to print from it.

Xylander's achievement has been, as a rule, quite inadequately appreciated. Very few writers on Diophantus seem to have studied the book itself: a fact which may be partly accounted for by its rarity. Even Nesselmann, whose book appeared in 1842, says that he has never been able to find a copy. Nesselmann however seems to have come nearest to a proper appreciation of the value of the work : he says " Xylander's work remains, in spite of the various

1 " Hoc non modo tibi, Princeps Illustrissime, honorificum erit, atque gloriosum ; sed te labores nostros approbante, arithmeticae studium cum alibi, turn in tua Academia et Gymnasiis, excitabitur, confirmabitur, prouehetur, et ad perfectam eiusscientiam multi tuis auspiciis, nostro labore perducti, magnam hac re tuis in remp. beneficiis accessionem factam esse gratissima commemoratione praedicabunt."

2 " Heidelberga. postrid. Eidus Sextiles CID ID LXXIV."

26 INTRODUCTION

defects which are unavoidable in a first edition of so difficult an author, especially when based on only one MS. and that full of errors, a highly meritorious achievement, and does not deserve the severe strictures which it has sometimes had passed upon it. It is true that Xylander has in many places not understood his author, and has misrepresented him in others ; his translation is often rough and un-Latin, this being due to a too conscientious adherence to the actual wording of the original ; but the result was none the less brilliant on that account. The mathematical public was put in possession of Diophantus' work, and the appearance of the translation had an immediate and enormous influence on the development and shaping of Algebra1." As a rule, the accounts of Xylander's work seem to have been based on what Bachet says about it and about his obligations to it. When I came to read Bachet myself and saw how disparaging, as a rule, his remarks upon Xylander were, I could not but suspect that they were unfair. His repeated and almost violent repudiation of obligation to Xylander suggested to me the very thing which he disclaimed, that he was under too great obligation to his predecessor to acknowledge it duly. I was therefore delighted at my good fortune in finding in the Library of Trinity College, Cambridge, a copy of Xylander, and so being able to judge for myself of the relation of the later to the earlier work. The result was to confirm entirely what I had suspected as to the unfair attitude taken up by Bachet towards his predecessor. I found it every- where ; even where it is obvious that Xylander's mistakes or difficulties are due only to the hopeless state of his solitary MS. Bachet seems to make no allowance for the fact. The truth is that Bachet's work could not have been as good as it was but for the pioneer work of Xylander; and it is the great blot in Bachet's otherwise excellent edition that he did not see fit to acknowledge the fact.

I must now pass to Bachet's work itself. It was the first edition published which contained the Greek text, and appeared in 1621 bearing the title: Diophanti Alexandrini Arithmeticorum libri sex, et de numeris multangulis liber unus. Nunc primiim Graece et Latine editi, atque absolutissimis Commentariis illustrati. Auctore Claudia Gaspare Baclieto Meziriaco Sebusiano, V.C. Lutetiae Parisiorum, Sumptibus Hieronymi Drovart"1, via Jacobaea, sub Scuto

1 Nesselmann, p. 279-80.

2 For " sumptibus Hieronymi Drovart etc. " some copies have " sumptibus Sebastiani

THE MSS. OF AND WRITERS ON DIOPHANTUS 27

Solari. MDCXXI. Bachet's Greek text is based, as he tells us, upon a MS. which he calls "codex Regius," now in the Bibliotheque Nationale at Paris (Parisinus 2379) ; this MS. is his sole authority, except that Jacobus Sirmondus had part of a Vatican MS. (Vat. gr. 304) .transcribed for him. He professes to have produced a good Greek text, having spent incalculable labour upon its emenda- tion, to have inserted in brackets all additions which he made to it, and to have given notice of all corrections, except those of an obvious or trifling nature ; a few passages he has left asterisked, in cases where correction could not be safely ventured upon. He is careful to tell us what previous works relating to the subject he had been able to consult. First he mentions Xylander (he spells the name as X/lander throughout), who had translated the whole of Diophantus, and commented upon him throughout, "except that he scarcely touched a considerable part of the fifth book, the whole of the sixth and the treatise on multangular numbers, and even the rest of his work was not very successful, as he himself admits that he did not thoroughly understand a number of points." Then he speaks of Bombelli (as already mentioned) and of the Zetetica of Vieta (in which the author treats in his own way a large number of Diophantus' problems : Bachet thinks that he so treated them because he despaired of restoring the book completely). Neither Bombelli nor Vieta (says Bachet) made any attempt to demonstrate the difficult porisms and abstruse theorems in numbers which Diophantus assumes as known in many places, or sufficiently explained the causes of his operations and artifices. All these omissions on the part of his predecessors he thinks he has supplied in his notes to the various problems and in the three books of "Porisms" which he prefixed to the work1. As regards his Latin translation, he says that he gives us Diophantus in Latin from the version of Xylander most carefully corrected, in which he would have us know that he has done two things in particular, first,

Cramoisy, via Jacobaea, sub Ciconiis." The copy (from the Library of Trinity College, Cambridge) which I used in preparing my first edition has the former words ; a copy in the Library of the Athenaeum Club has the latter.

1 On the nature of some of Bachet's proofs Nicholas Saunderson (formerly Lucasian Professor) remarks in Elements of Algebra, 1740, apropos of Dioph. ill. 15 : " M. Bachet indeed in the i6th and i7th props, of his second book of Porisms has given us demonstra- tions, such as they are, of the theorems in the problem : but in the first place he demonstrates but one single case of those theorems, and in the next place the demonstra- tions he gives are only synthetical, and so abominably perplexed withal, that in each demonstration he makes use of all the letters in the alphabet except I and O, singly to represent the quantities he has there occasion for. *

28 INTRODUCTION

corrected what was wrong and filled the numerous lacunae, secondly, explained more clearly what Xylander had given in obscure or ambiguous language ; " I confess however," he says, " that this made so much change necessary, that it is almost fairer to attribute the translation to me than to Xilander. But if anyone prefers to consider it as his, because I have held fast, tooth and nail, to his words when they do not misrepresent Diophantus, I have no objection1." Such sentences as these, which are no rarity in Bachet's book, are certainly not calculated to increase our respect for the author. According to Montucla2, "the historian of the French Academy tells us " that Bachet worked at this edition during the course of a quartan fever, and that he himself said that, disheartened as he was by the difficulty of the work, he would never have completed it, had it not been for the stubbornness which his malady generated in him.

As the first edition of the Greek text of Diophantus, this work, in spite of any imperfections we may find in it, does its author all honour.

The same edition was reprinted and published with the addition of Fermat's notes in 1670: Diophanti Alexandrini Arithmeticortim libri sex, et de numeris multangiilis liber unus. Cum commentariis C. G. Bacheti V.C. et obseruationibus D. P. de Fermat Senator is Tolosani. Accessit Doctrinae Analyticae inuentum nouum, collectum ex variis eiusdem D. de Fermat Epistolis. Tolosae, Excudebat Bemardus Bosc, e Regione Collegii Societatis Jesu. MDCLXX. This edition was not published by Fermat himself, but by his son after his death. S. Fermat tells us in the preface that this publication of Fermat's notes to Diophantus3 was part of an attempt to collect together from his letters and elsewhere his contributions to mathematics. The "Doctrinae Analyticae In- uentum nouum" is a collection made by Jacobus de Billy4

1 Deinde Latinum damus tibi Diophantum ex Xilandri versione accuratissime castigata, in qua duo potissimum nos praestitisse scias velim, nam et deprauata correximus, hiantesque passim lacunas repleuimus : et quae subobscure, vel ambigue fuerat interpretatus Xilander, dilucidius exposuimus; fateor tamen, inde tantam inductam esse mutationem, vt prope- modum aequius sit versionem istam nobis quam Xilandro tribuere. Si quis autem potius ad eum pertinere contendat, quod eius verba, quatenus Diophanto fraudi non erant, mordicus retinuimus, per me licet.'' - i. 323.

3 Now published in CEuvres de Fermat by P. Tannery and C. Henry, Vol. I. (1891), pp. 289-342 (the Latin original), and Vol. in. (1896), pp. 241-274 (French translation).

4 Now published in CEuvres de Fermat, in. 323-398 (French translation). De Billy had already published in 1660 a book under the title Diophantus geometra sive opus contextum ex arithmetica et geometria.

THE MSS. OF AND WRITERS ON DIOPHANTUS 29

from various letters which Fermat sent to him at different times. The notes upon Diophantus' problems, which his son hopes will prove of value very much more than commensurate with their bulk, were (he says) collected from the margin of his copy of Diophantus. From their brevity they were obviously intended for the benefit of experts1, or even perhaps solely for Fermat's own, he being a man who preferred the pleasure which he had in the work itself to any reputation which it might bring him. Fermat never cared to publish his investigations, but was always perfectly ready, as we see from his letters, to acquaint his friends and contemporaries with his results. Of the notes themselves this is not the place to speak in detail. This edition of Diophantus is rendered valuable only by the additions in it due to Fermat; for the rest it is a mere reprint of that of 1621. So far as the Greek text is concerned, it is very much inferior to the first edition. There is a far greater number of misprints, omissions of words, confusions of numerals ; and, most serious of all, the brackets which Bachet inserted in the edition of 1621 to mark the insertion of words in the text are in this later edition altogether omitted. These imperfections have been already noticed by Nesselmann2. Thus the reprinted edition of 1670 is untrust- worthy as regards the text.

In 1585 Simon Stevin published a French version of the first four books of Diophantus3. It was based on Xylander and was a free reproduction, not a translation, Stevin himself observing that the MS. used by Xylander was so full of mistakes that the text of

1 Lectori Seneuolo, p. iii : " Doctis tantum quibus pauca sufficiunt, harum obserua- tionum auctor scribebat, vel potius ipse sibi scribens, his studiis exerceri malebat quam gloriari; adeo autem ille ab omni ostentatione alienus erat, vt nee lucubrationes suas typis mandari curauerit, et suorum quandoque responsorum autographa nullo seruato exemplari petentibus vitro miserit ; norunt scilicet plerique celeberrimorum huius saeculi Geometrarum, quam libenter ille et quanta humanitate, sua iis inuenta patefecerit."

2 "Was dieser Abdruck an ausserer Eleganz gewonnen hat (denn die Bachet'sche Ausgabe ist mit ausserst unangenehmen, namentlich Griechischen Lettern gedruckt), das hat sie an innerm Werthe in Bezug auf den Text verloren. Sie ist nicht bloss voller Druckfehler in einzelnen Worten und Zeichen (z. B. durchgehends JT statt ~^, 900) sondern auch ganze Zeilen sind ausgelassen oder doppelt gedruckt (z. B. ill. 12 eine Zeile doppelt, iv. 25 eine doppelt und gleich hinterher eine ausgelassen, IV. 52 eine doppelt, v. ii eine ausgelassen, desgleichen v. 14, 15, 33, VI. 8, 13 und so weiter), die Zahlen verstiimmelt, was aber das Aergste ist, die Bachet'schen kritischen Zeichen sind fast uberall, die Klammer durchgangig weggefallen, so dass diese Ausgabe als Text des Diophant vollig unbrauchbar geworden ist," p. 283.

3 Included in L' Arithmelique de Simon Stevin de Bruges. ..A Leyde, De I'lmprimerie de Christophle Plantin, cio . ID . LXXXV.

3o INTRODUCTION

Diophantus could not be given word for word1. Albert Girard added the fifth and sixth books to the four, and this complete version appeared in i6252.

In 1810 was published an excellent translation (with additions) of the fragment upon Polygonal Numbers by Poselger : Diophantus von Alexandrien iiber die Polygonal-Zahlen. Uebersetzt mit Zusdtzen von F. Th. Poselger. Leipzig, 1810.

In 1822 Otto Schulz, professor in Berlin, published a very meritorious German translation with notes : Diophantus von Alexandria arithmetische Aufgaben nebst dessen Schrift iiber die Polygon-Zahlen. Aus dent Griechischen ilbersetzt und mit An- merkungen begleitet von Otto Schulz, Professor am Berlinisch- Colnischen Gymnasium zum grauen Kloster. Berlin, 1822. In der Schlesingerschen Buck- tind Mtisikhandlung. The work of Poselger just mentioned was with the consent of its author incorporated in Schulz's edition along with his own translation and notes upon the larger treatise, the Arithmetica. According to Nesselmann Schulz was not a mathematician by profession ; he produced, however, a thoroughly useful edition, with notes chiefly upon the matter of Diophantus and not on the text (with the exception of a very few emendations) : notes which, almost invariably correct, help much to understand the author. Schulz's translation is based upon the edition of Bachet's text published in 1670.

Another German translation was published by G. Wertheim in 1 890 : Die A rithmetik und die Schrift fiber Polygonalzahlen des Diophantus von Alexandria. Ubersetzt tmd mit Anmerkungen begleitet von G. Wertheim (Teubner). Though it appeared before the issue of Tannery's definitive text, it is an excellent translation, the translator being thoroughly equipped for his task ; it is valuable also as containing Fermat's notes, also translated into German, with a large number of other notes by the translator elucidating both Diophantus and Fermat, and generalising a number of the problems which, with very few exceptions, receive only particular solutions from Diophantus himself. Wertheim has also included 46 epigram- problems from the Greek anthology and the enunciation of the famous Cattle-Problem attributed to Archimedes.

1 See Bibliotheca Mathematica Vii3, 1906-7, p. 59.

2 U Arithmetiqiu de Simon Stevin de Brvges, Reiteue, corrigee & augmenlee de phisieurs traictez et annotation par Albert Girard Samielois Mathematicien. A Leide, de I'lmprimerie des Elzeviers cio . 10 . cxxv. Reproduced in the edition of Les (Euvres Mathematiques de Simon Stevin de Bruges. Par Albert Girard. Leyde, cio . 10 . cxxxiv.

THE MSS. OF AND WRITERS ON DIOPHANTUS 31

No description is necessary of the latest edition, by Tannery, in which we at last have a definitive Greek text of Diophantus with the ancient commentaries, etc., Diophanti Alexandrini opera omnia cum Graecis commentaries. Edidit et Latine interpretatus est Paulus Tannery (Teubner). The first volume (1893) contains the text of Diophantus, the second (1895) the Pseudepigrapha, Testimonia veterum, Pachymeres' paraphrase, Planudes' com- mentary, various ancient scholia, etc., and 38 arithmetical epigrams in the original Greek with scholia. Any further edition will neces- sarily be based on Tannery, who has added all that is required in the shape of introductions, etc.

Lastly we hear of other works on Diophantus which, if they were ever written, are lost or remain unpublished. First, we find it asserted by Vossius (as some have understood him) that the Englishman John Pell wrote an unpublished Commentary upon Diophantus. John Pell (1611-1685) was at one time professor of mathematics at Amsterdam and gave lectures there on Dio- phantus, but what Vossius says about his commentary may well be only a recommendation to undertake a commentary, rather than a historical assertion of its completion. Secondly, Schulz states in his preface that he had lately found a note in Schmeisser's Orthodidaktik der Mathematik that Hofrath Kausler by command of the Russian Academy prepared an edition of Diophantus1. This seems however to be a misapprehension on the part of Schulz. Kausler is probably referring, not to a translation of Diophantus, but to his memoir of 1798 published in Nova Acta Acad. PetropoL XI. p. 125, which might easily be described as an Ausarbeitung of Diophantus' work.

I find a statement in the New American Cyclopaedia (New York, D. Appleton and Company), Vol. vi., that " a complete translation of his (Diophantus') works into English was made by the late Miss Abigail Lousada, but has not been published."

1 The whole passage of Schmeisser is : "Die mechanische, geistlose Behandlung der Algebra ist ins besondere von Herrn Hofrath Kausler stark geriigt worden. In der Vorrede zu seiner Ausgabe des Uflakerschen Exempelbuths beginnt er so : ' Seit mehreren Jahren arbeitete ich fur die Russisch-Kaiserliche Akademie der Wissenschaften Diophants unsterbliches Werk iiber die Arithmetik ans, und fand darin einen solchen Schatz von den feinsten, scharfsinnigsten algebraischen Auflosungen, dass mir die mechanische, geistlose Methode der neuen Algebra rait jedem Tage mehr ekelte u.s.w.' " (p. 33).

CHAPTER III

NOTATION AND DEFINITIONS OF DIOPHANTUS

As it is my intention, for the sake of brevity and per- spicuity, to make use of the modern algebraical notation in giving my account of Diophantus' problems and general methods, it is necessary to describe once for all the machinery which our author uses for working out the solutions of his problems, or the notation by which he expresses such relations as would be represented in our time by algebraical equations, and, in particular, to illustrate the extent to which he is able to manipulate unknown quantities. Apart, however, from the necessity of such a description for the proper and adequate comprehension of Diophantus, the general question of the historical development of algebraical notation possesses great intrinsic interest. Into the general history of this subject I cannot enter in this essay, my object being the elucidation of Diophantus ; I shall accordingly in general confine myself to an account of his notation solely, except in so far as it is interesting to compare it with the corresponding notation of his editors and (in certain cases) that of other writers, as, for example, certain of the early Arabian algebraists. -

First, as to the representation of an unknown quantity. The unknown quantity, which Diophantus defines as containing TrXrjQos /jiovdBcov dopio-rov, i.e. an undefined number of units (def. 2), is denoted throughout by what was printed in the editions before Tannery's as the Greek letter 9 with an accent, thus <?', or in the form 9°. This symbol in verbal description he calls o dpi&fj,6<>, " the number," i.e.t by implication, the number par excellence of the problem in question. In the cases where the symbol is used to denote in- flected forms, e.g., the accusative singular or the dative plural, the terminations which would have been added to the stem of the full word dpiOpos were printed above the symbol 9 in the manner of an

NOTATION AND DEFINITIONS OF DlOPHANTUS 33

exponent, thus ?xv (for apidpov, as TVV for TOJ/), s°5, the symbol being in addition doubled in the plural cases, thus 55°*," w0*** ??""> 5?°**, f°r dpiO/jLOi K.r.e. When the symbol is used in practice, the coefficient is expressed by putting the required Greek numeral immediately after it, thus s<?°' ta corresponds to I ix, 9' a to x and so on.

Tannery discusses the question whether in the archetype (a) of the MSS. this duplication of the sign for the plural and this addition of the terminations of the various cases really occurred1. He observes that any one accustomed to reading Greek MSS. will admit that the marks of cases are common in the later MSS. but are very frequently omitted in the more ancient. Further, the practice of duplicating a sign to express the plural is more ancient than that of adding the case-terminations. Tannery concludes that the case-terminations (like the final syllables of abbreviations used for other words) were very generally, if not always, wanting in the archetype (a). If this seems inconsistent with the regularity with which they appear in our MSS., it has to be remembered that A and Bl do not represent the archetype (a) but the readings of a, the copyist of which probably took it upon himself to substitute the full word for the sign or to add the case-terminations. Tannery's main argument is the frequent occurrence of instances where the wrong case-ending has been added, e.g., the nominative for the genitive ; the conclusion is also confirmed by instances in which different cases of the word dpi0/j,6<;, e.g. dpiffpov, dptffpov, and even aptBpAv written in full are put by mistake for xai owing to the resemblance between the common abbreviation for icai and the sign for dpiOpo*;, and of course in such cases the abbreviation would not have had the endings. As regards the duplication of the sign for the plural, Tannery admits that this was the practice of the Byzantines ; but he considers that the evidence is against sup- posing that Diophantus duplicated the sign ; he does not do so with any other of his technical abbreviations, those for p.ovds, Svvapis, etc. Accordingly in his text of Diophantus Tannery has omitted the case-endings and written the single sign for dpi0/j,6<; whether in the singular or in the plural ; in his second volume, however, containing the scholia, etc., he has retained the duplicated sign.

On the assumption that the sign was the Greek final sigma, it was natural that Nesselmann should explain it by the supposition

1 Dioph. II. pp. xxxiv-xxxix. H. D. 3

34 INTRODUCTION

that Diophantus, in search of a convenient symbol for his unknown quantity, would select the only letter of the Greek alphabet which was not already appropriated as a numeral1. But he made the acute observation2 that, as the symbol occurred in many places (of course in Bachet's text) for dpi0/j,6<t used in the ordinary un- technical sense, and was therefore, as it appeared, not exclusively used to designate the unknown quantity, the technical apropos, it must after all be more of the nature of an abbreviation than an algebraical symbol like our x. It is true that this uncertainty in the use of the sign in the MSS. is put an end to by Tannery, who uses it for the technical dpiO/j>6f alone and writes the untechnical dpiQjjLos in full ; but, even if Diophantus' practice was as strict as this, I do not think this argues any difference in the nature of the abbreviation. There is also a doubt whether the final sigma, ?, was developed as distinct from the form a- so early as the date of the MSS. of Diophantus, or rather so early as the first copy of his work, if the author himself really gave the explanation of the sign as found in our text of his second definition. These considerations suggested to me that the sign was not the final sigma at all, but must be explained in some other way. I had to look for con- firmation of this to the precise shape of the sign as found in extant MSS. The only MS. which I had the opportunity of inspecting personally was the MS. of the first ten problems of Diophantus in the Bodleian ; but here I found strong confirmation of my view in the fact that the sign appeared as '^, quite different in shape from, and much larger than, the final sigma at the end of words in the same MS. (There is in the Oxford MS. the same irregularity as was pointed out by Nesselmann in the use of the sign sometimes for the technical, and sometimes for the untechnical, aptfyio?3.) But I found evidence that the sign appeared elsewhere in some- what different forms. Thus Rodet in the Journal Asiatique of January, 1878, quoted certain passages from Diophantus for the purpose of comparison with the algebra of Muhammad b. Musa al-Khuwarazml. Rodet says he copied these passages exactly from Bachet's MS. ; but, while he generally gives the sign as the final sigma, he has in one case iji]01 for dpiffftoi. In this last case

1 Nesselmann, pp. 290-1. 2 ibid. pp. 300-1.

3 An extreme case is lra£a TO rov devrtpov'^o api8/j,ovtt>6$, where the sign (contrary to what would be expected) means the untechnical <i/H0/u6s, and the technical is written in full. Also in the definition 6 5e nydef rotiruv T&V Idiw/A&TUv KTriffd/j.evos...apit)[j.bs KaXerrai the word dpify^s is itself denoted by the symbol, showing that the word and the symbol are absolutely convertible.

NOTATION AND DEFINITIONS OF DIOPHANTUS 35

Bachet himself reads 9?ot. But the same form ijip which Rodet gives is actually found in three places in Bachet's own edition, (i) In his note to IV. 3 he gives a reading from his own MS. which he has corrected in his own text and in which the signs ija and LJLjr) occur, evidently meaning dpi0/j,o<; a and dpiO^ol rj, though the sign should have been that for dpid^oarov (= \\x). (2) In the text of IV. 1 3 there is a sentence (marked by Bachet as interpolated) which contains the expression LjLjr, where the context again shows that i]q is for dpiOfjuoL (3) At the beginning of V. 9 there is a difficulty in the text, and Bachet notes that his MS. has prjre 6 8nr\a<ria)v avrov LJ where a Vatican MS. reads dpid^ov (Xylander notes that his MS. had in this place fjurjre 6 BnrXaalcov avrov dp po a ...). It is thus clear that the MS. (Paris. 2379) which Bachet used sometimes has the sign for dpiQ/jios in a form which is at least sufficiently like i| to be taken for it. Tannery states that the form of the sign found in the Madrid MS. (A) is tj, while B1 has it in a form ($) nearly approaching Bachet's reproduction of it.

It appeared also that the use of the sign, or something like it, was not confined to MSS. of Diophantus ; on reference to Gardthausen, Griechische Palaeographie, I found under the head " hieroglyphisch-conventionell " an abbreviation 9, 99 for aptfytd?, -oL, which is given as occurring in the Bodleian MS. of Euclid (D'Orville 301) of the 9th century. Similarly Lehmann1 notes as a sign for dpi0/j,6<; found in that MS. a curved line similar to that which was used as an abbreviation for icai. He adds that the ending is placed above it and the sign is doubled for the plural. Lehmann's facsimile is like the form given by Gardthausen, but has the angle a little more rounded. The form iji|ot above mentioned is also given by Lehmann, with the remark that it seems to be only a modification of the other. Again, from the critical notes to Heiberg's texts of the Arenarius of Archimedes it is clear that the sign for dpidpos occurred several times in the MSS. in a form approximating to that of the final sigma, and that there was the usual confusion caused by the similarity of the signs for dpidpbs and teal2. In Hultsch's edition of Heron, similarly, the critical notes to the Geodaesia show that one MS. had an abbreviation for

1 Lehmann, Die tachygraphischen Abkiirzungen der griechischen Handschriften, 1880, p. 107 : " Von Sigeln, welchen ich auch anderwarts begegnet bin, sincl zu nennen dpi6/j.6s, das in der Oxforder Euclidhandschrift mit einer der Note xal ahnlichen Schlangenlinie bezeichnet wird."

2 Cf. Heiberg, Quaestiones Archimedean, pp. 172, 174, 187, 188, 191, 192; Archimedis opera omnia, II., pp. 268 sqq.

3—2

36 INTRODUCTION

oi in various forms with the case-endings superposed ; some- times they resembled the letter £, sometimes p, sometimes O and once £J. Lastly, the sign for dpiBfjios resembling the final sigma evidently appeared in a MS. of Theon of Smyrna2.

All these facts strongly support the assumption that the sign was a mere tachygraphic abbreviation and not an algebraical symbol like our x, though discharging much the same function. The next question is, what is its origin ? The facts (i) that the sign has the breathing prefixed in the Bodleian MS., which writes '^-7 for dptOfjios, and (2) that in one place Xylander's MS. read dp tor the full word, suggested to me the question whether it could be a contraction of the first two letters of dpidpos ; and, on con- sideration, this seemed to me quite possible when I found a contraction for ap given by Gardthausen, namely cf. It is easy to see that a simplification of this in different ways would readily produce signs like the different forms shown above. This then was the hypothesis which I put forward twenty-five years ago, and which I still hold to be the easiest and best explanation. Two alternatives are possible, (i) Diophantus may not have made the contraction himself. In that case I suppose the sign to be a cur- sive contraction made by scribes ; and I conceive it to have come about through the intermediate form <p. The loss of the downward stroke, or of the loop, would produce a close approximation to the forms which we know. (2) Diophantus may have used a sign approximately, if not exactly, like that which we find in the MSS. For it is from a papyrus of 154 A.D., in writing of the class which Gardthausen calls the "Majuskelcursive," that the contraction c?f> for the two letters is taken. The great advantage of my hypothesis is that it makes the sign for aptfytov exactly parallel to those for the powers of the unknown, e.g., Jr for Svvafjus and KY for /cu/8o?, and

o

to that for the unit fjwvds which is denoted by M, with the sole difference that the letters coalesce into one instead of being written separately.

Tannery's views on the subject are, I think, not very con- sistent, and certainly they do not commend themselves to me. He seems to suggest that the sign is the ancient letter Koppa, perhaps slightly modified ; he first says that the sign in Diophantus is peculiar to him and that, although the word dpi6p,b<i is very often

1 Heron, ed. Hultsch, pp. 146, 148, 149, 150.

2 Theon of Smyrna, ed. Hiller, p. 56, critical notes.

NOTATION AND DEFINITIONS OF DIOPHANTUS 37

represented in mathematical MSS. by an abbreviation, it has much oftener the form $ or something similar, closely resembling the ancient Koppa. In the next sentence he seems to say that " on the contrary the Diophantine abbreviation is an inverted di- gamma " ; yet lower down he says that the copyist of a (copied from the archetype a) got the form i] by simplifying the more complicated Koppa. And, just before the last remark, he has stated that in the archetype a the form must have been 5 or very like it, as is shown by the confusion with the sign for ical. (If this is so, it can hardly have been peculiar to Diophantus, seeing that the same confusion occurs fairly often in the MSS. of other authors, as above shown.) I think the last consideration (the con- fusion with ical} is very much against the Koppa-hypothesis ; and, in any case, it seems to me very unlikely that a sign would be used by Diophantus for the unknown which was already appro- priated to the number 90. And I confess I am unable to see in the sign any resemblance to an inverted digamma.

Hultsch1 regards it as not impossible that Diophantus may have adopted one of the signs used by the Egyptians for their unknown quantity hau, which, if turned round from left to right, would give V; but here again I see no particular resemblance. Prof. D'Arcy Thompson2 has a suggestion that the sign might be the first letter of o-«p6<?, a heap. But, apart from the fact that the final sigma (<?) is not that first letter, there is no trace whatever in Diophantus of such a use of the word o-twpo? ; and, when Pachymeres3 speaks of a number being crwpeia /j,ovdSa>v, he means no more than the 7r\fj0os fAovdSwv which he is explaining : his words have no connexion with the Egyptian hau.

Notwithstanding that the sign is not the final sigma, I shall not hesitate to use 9 for it in the sequel, for convenience of printing. Tannery prints it rather differently as =>.

We pass to the notation which Diophantus used to express the different powers of the unknown quantity, corresponding to x*, x3, and so on. He calls the square of the unknown quantity 8vz>a/u<?, and denotes it by the abbreviation Jr. The word Svvapts, literally " power," is constantly used in Greek mathematics for

1 Art. Diophantus in Pauly-Wissowa's Keal-Encyclopadie der classischen Altertums- wis sense h aften .

2 Transactions of the Royal Society of Edinburgh, Vol. xxxvm. (1896), pp. 607-9.

3 Dioph. n. p. 78, 4. Cf. lamblichus, ed. Pistelli, p. 7, 7 ; 34, 3 ; 81, 14, where <rw/>efa is similarly used to elucidate TrXTjflos.

38 INTRODUCTION

square*. With Diophantus, however, it is not any square, but only the square of the unknown ; where he speaks of any particular square number, it is Terpdytovos dpid/Mos. The higher powers of the unknown quantity which Diophantus makes use of he calls Kvftos, &vva,f*,o8vva/j,t<;, Swa/jLotcvfios, KvftoKvftos, corresponding respectively to Xs, x*, Xs, x6. Beyond the sixth power he does not go, having no occasion for higher powers in the solutions of his problems. For these powers he uses the abbreviations KY, AYA, AKY, KYK re- spectively. There is a difference between Diophantus' use of the word * 8vva/j,i<; and of the complete words for the third and higher powers, namely that the latter are not always restricted like 8iW/u<? to powers of the unknown, but may denote powers of ordinary known num- bers as well. This is no doubt owing to the fact that, while there are two words Bvva/j,i<; and rerpdyaivo^ which both signify " square," there is only one word for a third power, namely /cu/3o?. It is important, however, to observe that the abbreviations KY, AYA, AKY, KYK, are, like Svvafw and AY, only used to denote powers of the unknown. The coefficients of the different powers of the unknown, like that of the unknown itself, are expressed by the addition of the Greek letters denoting numerals, e.g., AKV *r cor- responds to 26x*. Thus in Diophantus' system of notation the signs AY and the rest represent not merely the exponent of a power like the 2 in xz, but the whole expression x*. There is no obvious connexion between the symbol AY and the symbol 9 of which it is the square, as there is between x* and x, and in this lies the great inconvenience of the notation. But upon this notation no advance was made by Xylander, or even by Bachet and Fermat. They wrote A7 (which was short for Numerus) for the 9 of Diophantus, Q (Quad- ratus) for Jr, C (Cubus) for KY, so that we find, for example, I Q + 5^= 24, corresponding to x"- + $x = 24. Other symbols were however used even before the publication of Xylander's Diophantus, e.g. in Bombelli's Algebra. Bombelli denotes the unknown and its powers by the symbols -i 1, £, and so on. But it is certain that up to this time (1572) the common symbols had been R (Radix or Res), Z (Zensus, i.e. square), C (Cubus). Apparently the first important step towards x*, x*, etc., was taken by Vieta (1540 —

1 In Plato we have SiW/uj used for a square number (Titnaeus, 31) and also (TTieaetetus , 147 D) for a square root of a number which is not a complete square, i.e. for a surd ; but the commonest use is in geometry, in the form 8vvA.fj.tL, " in square," e.g. "AB is dw6/j.ei double of £C" means " AB2= 2.5C2."

NOTATION AND DEFINITIONS OF DIOPHANTUS 39

1603), who wrote Aq, Ac, Aqq, etc. (abbreviated for A quadratus and so on) for the powers of A. This system, besides showing the con .exion between the different powers, has the infinite advantage that by means of it we can use in one and the same solution any number of unknown quantities. This is absolutely impossible with the notation used by Diophantus and the earlier algebraists. Diophantus in fact never uses more than one unknown quantity in the solution of a problem, namely the dpi6/j.6<; or <?.

Diophantus has no symbol for the operation of multiplication ; it is rendered unnecessary by the fact that his coefficients are all definite numbers or fractions, and the results are simply put down without any preliminary step which would call for the use of a symbol. On the ground that Diophantus uses only numerical expressions for coefficients instead of general symbols, it might occur to a superficial observer that there must be a great want of generality in his methods, and that his problems, being solved with reference to particular numbers only, would possess the attraction of a clever puzzle rather than any more general interest. The answer to this is that, in the first place, it was absolutely impossible that Diophantus should have used any other than numerical coefficients, for the reason that the available symbols of notation were already employed, the letters of the Greek alphabet always doing duty as numerals, with the exception of the final 9. In the second place, it is not the case that the use of none but numerical coefficients makes his solutions any the less general. This will be clearly seen when I come to give an account of his problems and methods.

Next as to Diophantus' expressions for the operations of addition and subtraction. For the former no symbol at all is used: it is expressed by mere juxtaposition, thus KYa^Y^y^€ corresponds to x* + \yc- + $x. In this expression, however, there is no absolute term, and the addition of a simple numeral, as for instance /8, directly after e, the coefficient of ?, would cause confusion. This fact makes it necessary to have some expression to distinguish the absolute term from the variable terms. For this purpose Diophantus uses the word /u,ovaSe<?, or units, and denotes

o

them after his usual manner by the abbreviation M. The number of units is expressed as a coefficient. Thus corresponding to the expression Xs + \y? + $x+2 we should find in Diophantus

KYa Jrt79eJ/£. As Bachet uses the sign + for addition, he

40 INTRODUCTION

has no occasion for a distinct symbol to mark an absolute term. He accordingly writes iC+ i^Q+^N+2. It is worth observing, however, that the Italians do use a symbol in this case, namely N (Numero), the first power of the unknown being with them R (Radice). Cossali1 makes an interesting comparison between the terms used by Diophantus for the successive powers of the unknown and those employed by the Italians after their instructors, the Arabians. He observes that Fra Luca (Paciuolo), Tartaglia, and Cardano begin their scale of powers from the power o, not from the power i, as does Diophantus, and he compares the scales thus :

Scala Diofantea. Scala Araba.

i. Numero.. .il Noto.

x \. Numero... 1' Ignoto. i. Cosa, Radice, Lato.

X* i. Podesta. 3. Censo.

x3 3. Cubo. 4. Cubo.

x* 4. Podesta-Podesta. 5. Censo di Censo.

xs 5. Podesta-Cubo. 6. Relate i°.

x6 6. Cubo-Cubo. 7. Censo di Cubo, o Cubo di Censo.

xi 7 8. Relate 2°.

Xs 8 9. Censo di Censo di Censo.

x9 9 10. Cubo di Cubo.

and so on.* So far, however, as this is meant to be a comparison between Diophantus and the early Arabian algebraists themselves (as the title " Scala Araba" would seem to imply), there appears to be no reason why Cossali should not have placed some term to express Diophantus' /ioi>«Se? in the same line with Numero in the Arabian scale, and moved the numbers i, 2, 3, etc. one place upward^ in the first scale, or downwards in the second. As Diophantus does not go beyond the sixth power, the last three places in the first scale are left blank. An examination of these two scales will show also that the evolution of the successive powers differs in the two systems. The Diophantine terms for them are based on the addition of exponents, the Arabic on

1 Upon Wallis' comparison of the Diophantine with the Arabian scale Cossali remarks: "ma egli non ha riflettuto a due altre differenze tra le scale medesime. La prima si e, che laddove Diofanto denomina con singolarita Numero il nuinero ignoto, denominando Monade il numero dato di comparazione : gli antichi italiani degli arabi seguaci denominano questo il Numero ; e Radice, o Lato, o Cosa il numero sconosciuto. La seconda e, che Diofanto comincia la scala dal numero ignoto ; e Fra Luca, Tartaglia, Cardano la incominciano dal numero noto. Ecco le due scale di rincontro, onde meglio risaltino all' occhio le differenze loro ", I. p. 195.

NOTATION AND DEFINITIONS OF DIOPHANTUS 41

their multiplication^. Thus the "cube-cube" means in Diophantus x?, while the Italian and Arabian system uses the expression " cube of cube " and applies it to x9. The first system may (says Cossali) be described as the method of representing each power by the product of the two lesser powers which are the nearest to it, the method of multiplication; the second the metliod of elevation, i.e. the method which forms by the process of squaring and cubing all powers which can be so formed, as the 4th, 6th, 8th, 9th, etc. The intermediate powers which cannot be so formed are called in Italian Relati. Thus the fifth power is Relate i°, x1 is Relato 2°, x10 is Censo di Relato i°, x" is Relato 3°, and so on. Another name for the Relati in use among European algebraists in the 1 6th and I7th centuries was sursolida, with the variants super- solida and surdesolida.

It is interesting to compare with these systems the Egyptian method described by Psellus2. The next power after the fourth (8vi>a/ioSiW/u9), i.e. x6, the Egyptians called " the first undescribed " ((1X0709 here apparently meaning that of which no account can be given), because it is neither a square nor a cube ; alternatively they called it " the fifth number," corresponding to the fifth power of x. The sixth power they apparently called " cube-cube " ; but the seventh was " the second undescribed " (0X0709 Sevrepos), as being the product of the square and the " first undescribed," or, alternatively, the "seventh number." The eighth power was the "quadruple-square" (rerpaTrXfj 8ui>a/u9), the ninth the "extended cube " (/eu/8o9 e£eXt*T09). Thus the " first undescribed " and the "second undescribed" correspond to "Relato i°" and "Relato 2°" respectively, but the "quadruple-square" exhibits the additive principle.

For subtraction Diophantus uses a symbol. His full term for negation or wanting is Xenjrt9, corresponding to inrap%i<; which denotes the opposite. The symbol used to denote it in the MSS., and corresponding to our — for minus, is (Def. 9 KOI 7-^9 Xetye<»9 eXXi7T€9 Kara) vevov, A) " an inverted ¥ with the top

1 This statement of Cossali's needs qualification however. There is at least one Arabian algebraist, al-Karkhi (died probably about 1029), the author of the Fakkri, who uses the Diophantine system of powers of the unknown depending on the addition of exponents. Al-Karkhl, namely, expresses all powers of the unknown above the third by means of mat, his term for the square, and kab, his term for the cube of the unknown, as follows. The fourth power is with him mdl mal, the fifth mal kab, the sixth kab ka'b, the seventh mdl mal kab, the eighth mdl kab kab, the ninth kab kab kab, and so on. Among the Italians too there was an exception, Leonardo of Pisa, who proceeded on the additive principle (Bibliotheea Mathematica, vis, 1905-6, p. 310). 2 Dioph. H. p. 37-38.

42 INTRODUCTION

shortened, /ft." As Diophantus uses no distinct sign for +, it is clearly necessary, in order to avoid confusion, that all the negative terms in an expression should be placed together after all the positive terms. And so in fact he does place them. Thus corresponding to x3 — $*? + &tr — i, Diophantus would write

KY a 9 T; A^l1 e Ma. With respect to this curious sign, given in the MSS. as T and described as an inverted truncated M*", I believe that I was the first to suggest that it could not be what it is represented as being. Even when, as in Bachet's edition, the sign was printed as ^ I could not believe that Diophantus used so fantastic a sign for minus as an inverted truncated ty. In the first place, an inverted ^ seems too far-fetched ; to one who was looking for a symbol to express minus many others more natural and less fantastic than jp> must have suggested themselves. Secondly, given that Diophantus used an inverted M/", why should he truncate it ? Surely that must have been unnecessary ; we could hardly have expected it unless, without it, confusion was likely to arise; but ./p. could not well have been confused with anything. This very truncation itself appears to throw doubt on the description of the symbol as we find it in the MS. I concluded that the con- ception of this symbol as an inverted truncated M* was a mistake, and that the description of it as such is not Diophantus' description, but an explanation by a scribe of a symbol which he did not understand1. I believe that the true explanation is the following. Diophantus here took the same course as in the case of the other symbols which we have discussed (those for apiOftos, Svvafjus, etc.). As in those cases he took for his abbreviation the first letter of the word with such an addition as would make confusion with numbers impossible (namely the second letter of the word, which in each of the cases happens to come later in the alphabet than the corre- sponding first letter), so, in seeking an abbreviation for A,et-\Ja<? and cognate inflected forms developed from \ITT, he began by taking the initial letter of the word. The uncial2 form is A. Clearly A by itself would not serve his purpose, since it denotes a number. Therefore an addition is necessary. The second letter is E, but AE is equally a number. The second letter of the stem

1 I am not even sure that the description can be made to mean all that it is intended to mean. AXiTr^s scarcely seems to be sufficiently precise. Might it not be applied to /]\ with any part cut off, and not only the top ?

2 I adhere to the uncial form above for clearness' sake. If Diophantus used the " Majuskelcursive " form, the explanation will equally apply, the difference of form being for our purpose negligible.

NOTATION AND DEFINITIONS OF DIOPHANTUS 43

\ITT is I, but Al is open to objection when so written. Hence Diophantus placed the I inside the A, thus, A. Of the possibility of this I entertain no doubt, because there are undoubted cases of combination, even in uncial writing, of two letters into one sign. I would refer in particular to X, which is an uncial abbreviation for TAAANTON. Now this sign, A, is an inverted and truncated W (written in the uncial form, y) ; and we can, on this assumption, easily account for the explanation of the sign for minus which is given in the text.

The above suggestion, made by me twenty-five years ago, seems to be distinctly supported by what Tannery says of the form in which the sign appears in the MSS.1 Thus he remarks (i) that the sign in the MSS. is often made to lean to the right so that it resembles the letter Lambda, (2) that Planudes certainly wrote fc as if he meant to write the first letter of Xen/ret, and (3) that the letter A appears twice in A where it seems to mean XotTro?. Yet in his edition of Diophantus Tannery did not adopt my explanation or even mention it, but explained the sign as being in reality adapted from the old letter Sampi (~>>), the objection to which suggestion is the same as that to which the identification of <? with Koppa is open, namely that ~^ represented the number 900, as ? represented 90. Tannery however afterwards2 saw reason to abandon his suggestion that the symbol was originally an archaic form of the Greek Sampi rather than "un monogramme se rattachant a la racine de Xen/rt?." The occasion for this change of view was furnished by the appearance of the same sign in the critical notes to Schone's edition of the Metrica of Heron3, which led Tannery to re-examine the evidence of the MSS. of Diophantus as to the sign and as to the exact word or words which it re- presented in different places, as well as to search for any similar expressions denoting subtraction which might occur in the works of other Greek mathematicians. In the MSS. of Diophantus, when the sign is resolved by writing a full word instead of it, it is generally resolved into Xen/ret, the dative of Xen/rt<? ; in such cases the only grammatical possibility is to construct it with the genitive case of the quantity subtracted, the meaning then being "with the wanting, or deduction, of ...". But the best MS. (A)

1 Dioph. n. p. xli.

- Bibliotheca Mathematica v3, 1904-5, pp. 5-8.

3 Heronis Alexandrini opera, Vol. in., 1903, pp, 156, 8, 10. The MS. reading is t'd', the meaning of which is 74 -TV

44 INTRODUCTION

has in some places the nominative Xetyv 9, while in others it has the symbol instead of parts of the verb \ei-7reiv, namely \nr(i)v or XeiS/ra? and once even XITTOJO-I ; hence we may conclude that in the cases where A and B^ have \etyei followed by the accusative (which is impossible grammatically) the sign was wrongly resolved, and the full word should have been a participle or other part of the verb \eliretv governing the accusative. The question therefore arises whether Diophantus himself used the dative \efyet at all or whether it was introduced into the MSS. later. Certain it is that the use is foreign to Classical Greek ; but, even if it began with Diophantus, it did not finally hold the field before the time of Planudes. No evidence for it can be found in Greek mathe- maticians before Diophantus. Ptolemy has in two places Xenjrai/ and \ei7rovaav respectively, followed by the accusative, and in one case TO airo rrjs FA \ei<l>9ev VTTO TOV aVo -n)9 Zf (where the meaning is ZP- FA2). Consequently we cannot suppose that the sign where it occurs in the Metrica of Heron represents the dative Xenjr«; it must rather stand for a participle, active or passive. Tannery suggests that the full expression in that passage was fjiovdSwv oB \ei<f)0evTo$ re(r<TapaKaiS€Kdrov, the participle being passive and the construction being the genitive absolute ; but I think a perhaps better alternative would be povaSmv 08 \en/rao-<wz/ reo-a-apaKaiSeicaTov, where the active participle would govern the accusative case of the term subtracted. From all this we may infer that the sign had no exclusive reference to the sub- stantive \enjrt9, still less to the dative case of that substantive, but was a conventional abbreviation associated with the root of the verb \ei7reiv. In these circumstances I think I may now fairly claim Tannery as, substantially, a convert to my view of the nature of the sfgn.

For division it often happens that no symbol is necessary, i.e. in the cases where the divisor divides the dividend without a remainder. In other cases the quotient has to be expressed as a fraction, whether the divisor is a specific number or contains the variable. The case of division comes then under that of fractions.

Fractions are represented in different ways according as they are submultiples (fractions with unity as numerator) or not. In the case of submultiples the Greeks did not write the numerator, but only the denominator, distinguishing the submultiple from the cardinal number itself by affixing a certain sign. In more recent

NOTATION AND DEFINITIONS OF DIOPHANTUS 45

MSS. a double accent was used for this purpose: thus 7" = ^. Diophantus follows this plan in the hypothesis and analysis of his problems, though in the solutions he seems to have written the numerator a and assimilated the notation to that used for other fractions. The sign, however, added to the cardinal number to express the submultiple takes somewhat different forms in A : sometimes it is a simple accent, sometimes more elaborate, as /" above the letter and to the right, or actually forming a continuation of the numeral sign, e.g. fr' = ^. Tannery adopts as the genuine mark in Diophantus the affix x in place of the accent : thus 7X = 3. For £ he writes L ' as being most suitable for the time of Diophantus, though A has <~*/, sometimes without the dot.

Of the other class of fractions (numerator not unity) f stands by itself, having a peculiar sign of its own ; curiously enough it occurs only four times in Diophantus. A has a sign for it which was confused with that for dpidpos in one place ; Tannery judges from the Greek mathematical papyrus of Achmlm1 that its original form was <y ; he himself writes in his text the common form iff'. In the rare cases where the first hand in the oldest MS. (A) has fractions as such with numerator and denominator written in full, the denominator is written above the numerator. Tannery therefore adopts, in his text, this way of writing fractions, separating the

numerator and denominator by a horizontal line: thus pica = —^.

PACT?

It is however better to omit the horizontal line (cf. p in Kenyon Papyri II. No. cclxv. 40; also the fractions in Schone's edition of Heron's Metrica). Once we find in the same MS. (A) in the first hand the form ies = 1f-. In this latter method of writing fractions the denominator is written as we write exponents ; and this is the method adopted by Planudes and by Bachet in his edition. Another alternative is to write the numerator first, and then the denominator after it in the same line, marking the denominator with the submultiple sign in some form ; thus jB' would mean £ ; this is the most convenient method for purposes of printing. Or the de- nominator may be written as an abbreviation for the ordinal number, and the case-termination may be added higher up ; e.g. v K>fv = 50 twenty-thirds. But the denominators are nearly always omitted

1 Published by Baillet in Memoire s publih par Its Membres de la Mission archeologique franfaise au Caire, T. Ix, Fascicule i, pp. 1-88. Paris, 1892.

46 INTRODUCTION

altogether in the first hand of A ; in the first two Books B^ and the second hand of A give the denominator in the place in which we write an exponent, following the method of Planudes ; in the last four Books both MSS. almost invariably omit the denominator. In some cases the omission is not unnatural, i.e. where the denominator has once been given, and it is almost superfluous to repeat it in other fractions immediately following which have the same denominator ; in other cases it was probably omitted because the superposed denominator was taken by the copyist to be an inter- linear scholium. A few examples of fractions from Diophantus may be added :

(v. 9) ;

A ff>

'

= (IV. 16) ; pKa-XS = (IV. 39) ;

(V.2).

152 V

Diophantus however often expresses fractions by putting eV or popiov between the numerator and denominator, i.e. he

r

says one number divided by another. Cf. Mpiv . g^irb poplov _ _ r

_

*r . ,/8/3/iS = 1507984/262144 (IV. 28), where of course M =

(tens of thousands); (3 . ,e% ev ^opiw picfS . /aice= 25600/1221025 (v. 22). As we said, the most orthodox way of writing a sub- multiple was to omit the numerator (unity) and use the denominator with a distinguishing sign attached, e.g. rx or r' = £. But in his solutions Diophantus often uses the form applicable to fractions

W i

other than submultiples ; e.g. he writes a for - (IV. 28).

Numbers partly integral and partly fractional, where the fraction is a submultiple or the sum of submultiples, are written much as we write them, the fraction simply following the integer ; e.g. a 7X = i^ ; in the Lemma to V. 8 we have ft L ' r' = 2 \ % or 2§, where f is decomposed into submultiples as in Heron. Cf. also (m. ii)roZ.Vx=37oiTV.

Before leaving the subject of numerical notation, it may be convenient to refer to the method of writing large numbers.

r Myriads (tens of thousands) are expressed by M, myriads to the

NOTATION AND DEFINITIONS OF DIOPHANTUS 47

second power by MM or, in words, Bevrepa fj.vpia<f. The de- nominator 1 87474560 in V. 8 would thus be written /j,opiov

a teal [ivpi(i8(i)v TrpcaTfav fij^n^ ical M ,£<£>£, and the fraction 131299224/1629586560 would be written Bevrepa pvpias a

M 0(TK?) fioptov Bevrepfav pup id Sow if M~r&£\

But there is another kind of fraction, besides the purely numerical one, which is continually occurring in the Arithmetica, such fractions namely as involve the unknown quantity in some form or other in their denominators. The simplest case is that in which the denominator is merely a power of the unknown, 9. Concerning fractions of this kind Diophantus says (Def. 3) : " As fractions named after numbers have similar names to those of the numbers themselves (thus a third is named from three, a fourth from four), so the fractions homonymous with the numbers just defined are called after them ; thus from a/>ifyto<? we name the fraction dpidfioarov \t£. ijx from x\, TO Swapoarov from is, TO KvftoffTov from >ti>/8o?, TO BwafioBwapoa-rov from , TO Bwa/AOKV/Soa'Tov from SvvafjLoicvftos, and TO from KV&OHV&OS. And every such fraction shall have, above the sign for the homonymous number, a line to indicate the species." Thus we find, for example, IV. 3, ?x 17 cor- responding to 8/x and, IV. 15, ?x Xe for 35/4:. Cf. Jrxor for 250^.

Where the denominator is a compound expression involving the unknown and its powers, Diophantus uses the expedient which he often adopts with numerical fractions when the numerators and denominators are large numbers, namely the insertion of ev popiy or /jiopiov between the expressions for the numerator and de- nominator. Thus in VI. 12 we have

= (6ar2 + and in VI. 14

For to-o?, equal, connecting the two sides of an equation, the sign in the archetype seems to have been iff • but copyists intro-

1 Hultsch, he. tit.

48 INTRODUCTION

duced a sign which was sometimes confused with the sign i| for dpidpos ; this was no doubt the same abbreviation Lj as that shown (with terminations of cases added above) in the list given at the end of Codex Parisinus 2360 (Archimedes) of contractions found in the " very ancient " MS. from which it was copied and which was at one time the property of Georgius Valla1.

Diophantus evidently put down his equations in the ordinary course of writing, i.e. they were written straight on, as are the steps in the propositions of Euclid, and not put in separate lines for each step in the process of simplification. In the scholia of Maximus Planudes however we find conspectuses of the problems with steps in separate lines which, except for the slightly more cumbrous notation, make the work scarcely more difficult to follow than it is in our notation2. Though in the MSS. we have the abbreviation t0" to denote equality, Bachet makes no use of any symbol for the purpose in his Latin translation. He uses throughout the full Latin word. It is interesting however to observe that in the notes to his earlier translation (1575) Xylander had already used a symbol to denote equality, namely ||, two short vertical parallel lines. Thus we find, for example (p. 76),

which we should express by x*1 + 12 =x*+ 6x+ 9.

Now that we have described in detail Diophantus' method of expressing algebraical quantities and relations, it is clear that it is essentially different in its character from the modern notation. While in modern times signs and symbols have been developed

1 Heiberg, Quaestiones Archimedeae, p. 115.

2 One instance will suffice. On the left Planudes has abbreviations for the words showing the nature of the steps or the operations they involve, e.g. &r0. = ftcflecns (setting - out), rerp. = rer/scry wpwy^s (squaring), wuvO. —ff^vOecrts (adding), a<f>. = d.<j>alpfffis (subtrac- tion), pep. = fj.epiffiJ.6s (division), OTT. = virap% is (resulting fact).

Dioph. I. 28. Planudes. " Modern equivalent.

(K&. rerp.

atvd. AYp(J.°ff I" M0^?

AYa

[Given numbers] 20, 208 Put for the numbers .*+ 10, 10 - * Squaring, we have *2+2a*+ 100, #2+ 100-20*. Adding, 2jr2+2OO = 2o8. Subtracting, 2.*2=8. Dividing, Jf2 = 4.

Result: [the numbers are] 12, 8.

NOTATION AND DEFINITIONS OF DIOPHANTUS 49

which have no intrinsic relationship to the things which they represent, but depend for their use upon convention, the case is quite different in Diophantus, where algebraic notation takes the form of mere abbreviation of words which are considered as pronounced or implied.

In order to show in what place, in respect of systems of algebraic notation, Diophantus stands, Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development, well marked and easily discernible, (i) The first stage Nessel- mann represents by the name Rhetorical Algebra or "reckoning by complete words." The characteristic of this stage is the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming in fact continuous prose. As representatives of this first stage Nesselmann mentions lambli- chus (of whose algebraical work he quotes a specimen in his fifth chapter) "and all Arabian and Persian algebraists who are at present known." In their works we find no vestige of algebraic symbols; the same may be said of the oldest Italian algebraists and their followers, and among them Regiomontanus. (2) The second stage Nesselmann proposes to call the Syncopated A Igebra. This stage is essentially rJielorical, and therein like the first in its treatment of questions ; but we now find for often-recurring operations and quantities certain abbreviational symbols. To this stage belong Diophantus and, after him, all the later Europeans until about the middle of the seventeenth century (with the exception of Vieta, who was the first to establish, under the name of Logistica speciosa, as distinct from Logistica numerosa, a regular system of reckoning with letters denoting magnitudes and not numbers only). (3) To the third stage Nesselmann gives the name Symbolic Algebra, which uses a com- plete system of notation by signs having no visible connexion with the words or things which they represent, a complete language of symbols, which supplants entirely the r/ietorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a connecting word or two here and there, and so on1. Neither

1 It may be convenient to note here the beginnings of some of our ordinary algebraical symbols. The signs + and - first appeared in print in Johann Widman's arithmetic (1489), where however they are scarcely used as regular symbols of operation ; next they are found in the Rechenbuch of Henricus Grammateus (Schreiber), written in 1518 but perhaps not published till 15.21, and then regularly in Stifel's Arithmetica Integra (1544)

5°

INTRODUCTION

is it the Europeans from the middle of the seventeenth century onwards who were the first to use symbolic forms of Algebra. In this they were anticipated by the Indians.

Nesselmann illustrates these three stages by three examples, quoting word for word the solution of a quadratic equation by Muhammad b. Musa as an example of the first stage, and the solution of a problem from Diophantus as representing the second.

First Stage, Example from Muhammad b. Musa (ed. Rosen, p. 5). "A square and ten of its roots are equal to nine and thirty dirhems, that is, if you add ten roots to one square, the sum is equal to nine and thirty. The solution is as follows. Take half the number of roots, that is in this case five; then multiply this by itself, and the result is five and twenty. Add this to the nine and thirty, which gives sixty-four; take the square root, or eight, and subtract from it half the number of roots, namely five, and there remain three: this is the root of the square which was required, and the square itself is nine1."

Here we observe that not even are symbols used for numbers, so that this example is even more "rhetorical" than the work of lamblichus who does use the Greek symbols for his numbers.

as well as in his edition of RudolfFs Coss (1553). Vieta (1540-1603) has, in addition, = for ~. Robert Recorde (1510-1558) had already in his Algebra (The Whetstone of Witte, 1557) used =(but with much longer lines) to denote equality (" bicause noe.2. thynges, can be moare equalle"). Harriot (1560-1621) denoted multiplication by a dot, and also by mere juxtaposition of letters; Stifel (1487-1567) had however already expressed the product of two magnitudes by the juxtaposition of the two letters represent- ing them. Oughtred (1574-1660) used the sign x for multiplication. Harriot also introduced the signs > and < for greater and less respectively, -f- for division is found in Rahn's Algebra (1659). Descartes introduced in his Geometry (1637) our method of writing powers, as a3, a* etc. (except a2, for which he wrote aa) ; but this notation was practically anticipated by Pierre Herigone (Cours matktmatique, 1634), who wrote ai, a$, 04, etc., and the idea is even to be found in the Rechenbuch of Grammateus above mentioned, where the successive powers of the unknown are denoted by pri, se, ter, etc. The use of x for the unknown quantity began with Descartes, who first used 2, then y, and then x for this purpose, showing that he intentionally chose his unknowns from the last letters of the alphabet. ^/ for the square root is traceable to Rudolff, with whom it had only two strokes, the first (down) stroke being short, and the other relatively long. 1 Thus Muhammad b. Musa states in words the following solution.

^+10^ = 39, xz+ 10^+25=64; therefore x + 5 = 8,

NOTATION AND DEFINITIONS OF DIOPHANTUS 51

Second Stage. As an example of Diophantus I give a trans- lation word for word of II. 8. So as to make the symbols correspond exactly I use 5 (Square) for Jr (Svvafus), N (Number) for 9, U (Units) for M (^o^aSe?).

"To divide the proposed square into two squares. Let it be proposed then to divide 16 into two squares. And let the first be supposed to be \S\ therefore the second will be 16 U— 1 5. Thus 16 U — i 5 must be equal to a square. I form the square from any number of N's minus as many (7's as there are in the side of 1 6 £/'s. Suppose this to be 2/V — ^U. Thus the square itself will be 4S i6U- \6N. These are equal to i6U- I S. Add to each the negative term (77 Xen/rt?, the deficiency) and take likes from likes. Thus 56" are equal to i6N, and the N is 16 fifths. One [square] will be ^, and the other l-gg, and the sum of the two makes up 4^°, or \6U, and each of the two is a square."

Of the third stage any exemplification is unnecessary.

To the form of Diophantus' notation is due the fact that he is unable to introduce into his solutions more than one unknown quantity. This limitation has made his procedure often very dif- ferent from our modern work. In the first place we can begin with any number of unknown quantities denoted by different symbols, and eliminate all of them but one by gradual steps in the course of the work ; Diophantus on the other hand has to perform all his eliminations beforehand, as a preliminary to the actual work, by expressing every quantity which occurs in the problem in terms of only one unknown. This is the case in the great majority of questions of the first Book, which involve the solu- tion of determinate simultaneous equations of the first degree with two, three, or four variables; all these Diophantus expresses in terms of one unknown, and then proceeds to find it from a simple equation. Secondly, however, this limitation affects much of Diophantus' work injuriously; for, when he handles problems which are by nature indeterminate and would lead with our notation to an indeterminate equation containing two or three unknowns, he is compelled by limitation of notation to assume for one or other of these some particular number arbitrarily chosen, the effect of the assumption being to make the problem a determinate one. How- ever, it is but fair to say that Diophantus, in assigning an arbitrary value to a quantity, is careful to tell us so, saying, "for such and such a quantity we put any number whatever, say such and such a

4—2

52 INTRODUCTION

number." Thus it can hardly be said that there is (as a rule) any loss of generality. We may say, then, that in general Diophantus is obliged to express all his unknowns in terms, or as functions, of one variable. He compels our admiration by the clever devices by which he contrives so to express them in terms of his single unknown, 9, as to satisfy by that very expression of them all conditions of the problem except one, which then enables us to complete the solution by determining the value of 9. Another consequence of Diophantus' want of other symbols besides 9 to express more variables than one is that, when (as often happens) it is necessary in the course of a problem to work out a subsidiary problem in order to obtain the coefficients etc. in the functions of 9 which express the numbers to be found, the unknown quantity which it is the object of the new subsidiary problem to find is also in its turn denoted by the same symbol 9 ; hence we often have in the same problem the same variable 9 used with two different meanings. This is an obvious inconvenience and might lead to confusion in the mind of a careless reader. Again we find two cases, II. 28 and 29, where for the proper working-out of the problem two unknowns are imperatively necessary. We should of course use x andjp; but Diophantus calls the first 9 as usual; the second, for want of a term, he agrees to call "one unit" i.e. I. Then, later, having completed the part of the solution necessary to find 9, he substitutes its value, and uses 9 over again to denote what he had originally called " I " — the second variable — and so finds it. This is the most curious case of all, and the way in which Diophantus, after having worked with this " I " along with other numerals, is yet able to put his finger upon the particular place where it has passed to, so as to substitute 9 for it, is very remark- able. This could only be possible in particular cases such as those which I have mentioned; but, even here; it seems scarcely possible now to work out the problem by using x and I for the variables as originally taken by Diophantus without falling into confusion. Perhaps, however, in working out the problems before writing them down as we have them Diophantus may have given the " I " which stood for a variable some mark by which he could recognise it and distinguish it from other numbers.

Diophantus will have in his solutions no numbers whatever except "rational" numbers; and in pursuance of this restriction he excludes not only surds and imaginary quantities, but also negative quantities. Of a negative quantity per se, i.e. without some positive

NOTATION AND DEFINITIONS OF DIOPHANTUS 53

quantity to subtract it from, Diophantus had apparently no con- ception. Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases aSui/aro?, impossible. So we find him (v. 2) describing the equation 4 = 4^+20 as aroTros, absurd, because it would give x = — ^. Diophantus makes it his object throughout to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, the conditions which must be satisfied in order to secure a result rational in his sense of the word. In the great majority of cases, when Diophantus arrives in the course of a solution at an equation which would give an irrational result, he retraces his steps and finds out how his equation has arisen, and how he may, by altering the previous work, substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Dio- phantus has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in V. 30 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.

CHAPTER IV

DIOPHANTUS' METHODS OF SOLUTION

BEFORE I give an account in detail of the different methods which Diophantus employs for the solution of his problems, so far as they can be classified, it is worth while to quote some remarks which Hankel has made in his account of Diophantus1. Hankel, writing with his usual brilliancy, says in the place referred to, "The reader will now be desirous to become acquainted with the classes of indeterminate problems which Diophantus treats of, and with his methods of solution. As regards the first point, we must observe that included in the 130 (or so) indeterminate problems, of which Diophantus treats in his great work, there are over 50 different classes of problems, strung together on no recognisable principle of grouping, except that the solution of the earlier problems facili- tates that of the later. The first Book is confined to determinate algebraic equations; Books II. to v. contain for the most part indeterminate problems, in which expressions involving in the first or second degree two or more variables are to be made squares or cubes. Lastly, Book VI. is concerned with right-angled triangles regarded purely arithmetically, in which some linear or quadratic function of the sides is to be made a square or a cube. That is all that we can pronounce about this varied series of problems without exhibiting singly each of the fifty classes. Almost more different in kind than the problems are their solutions, and we are completely unable to give an even tolerably exhaustive review of the different turns which his procedure takes. Of more general comprehensive methods there is in our author no trace discoverable : every question requires a quite special method, which often will not serve even for the most closely allied problems. It is on that

1 Zur Geschichte der Mathematik in Alterthum und Mittelaller, Leipzig, 1874, pp. 164-5.

DIOPHANTUS' METHODS OF SOLUTION 55

account difficult for a modern mathematician even after studying 100 Diophantine solutions to solve the icist problem; and if we have made the attempt, and after some vain endeavours read Diophantus' own solution, we shall be astonished to see how suddenly he leaves the broad high-road, dashes into a side-path and with a quick turn reaches the goal, often enough a goal with reaching which we should not be content; we expected to have to climb a toilsome path, but to be rewarded at the end by an extensive view; instead of which our guide leads by narrow, strange, but smooth ways to a small eminence; he has finished! He lacks the calm and concentrated energy for a deep plunge into a single important problem ; and in this way the reader also hurries with inward unrest from problem to problem, as in a game of riddles, without being able to enjoy the individual one. Diophantus dazzles more than he delights. He is in a wonderful measure shrewd, clever, quick-sighted, indefatigable, but does not penetrate thoroughly or deeply into the root of the matter. As his problems seem framed in obedience to no obvious scientific necessity, but often only for the sake of the solution, the solution itself also lacks completeness and deeper signification. He is a brilliant performer in the art of indeterminate analysis invented by him, but the science has nevertheless been indebted, at least directly, to this brilliant genius for few methods, because he was deficient in the speculative thought which sees in the True more than the Correct. That is the general impression which I have derived from a thorough and repeated study of Diophantus' arithmetic."

It might be inferred from these remarks of Hankel that Diophantus' object was less to teach methods than to obtain a multitude of mere results. On the other hand Nesselmann observes1 that Diophantus, while using (as he must) specific numbers for numbers which are " given " or have to be arbitrarily assumed, always makes it clear how by varying our initial as- sumptions we can obtain any number of particular solutions of the problem, showing "that his whole attention is directed to the explanation of the method, to which end numerical examples only serve as means"; this is proved by his frequently stopping short, when the method has been made sufficiently clear, and the remainder of the work is mere straightforward calculation. The truth seems to be that there is as much in the shape of general

1 Algebra der Griechen, pp. 308-9.

56 INTRODUCTION

methods to be found in Diophantus as his notation and the nature of the subject admitted of. On this point I can quote no better authority than Euler, who says1 : " Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats, and he is generally content with indicating numbers which furnish one single solution. But it must not be supposed that his method was restricted to these very special solutions. In his time the use of letters to denote undetermined numbers was not yet established, and consequently the more general solutions which we are now enabled to give by means of such notation could not be expected from him. Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use today; nay, we are obliged to admit that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Dio- phantus."

In his 8th chapter, entitled "Diophantus' treatment of equations2," Nesselmann gives an account of Diophantus' solutions of (i) Deter- minate, (2) Indeterminate equations, classified according to their kind. In chapter 9, entitled "Diophantus' methods of solution3," he classifies these " methods " as follows4: (i) " The adroit assump- tion of unknowns," (2) "Method of reckoning backwards and auxiliary questions," (3) "Use of the symbol for the unknown in different significations," (4) "Method of Limits," (5) "Solution by mere reflection," (6) "Solution in general expressions," (7) "Arbi- trary determinations and assumptions," (8) "Use of the right- angled triangle."

At the end of chapter 8 Nesselmann observes that it is not his solutions of equations that we have to wonder at, but the art, amounting to virtuosity, which enabled Diophantus to avoid such equations as he could not technically solve. We look (says Nessel- mann) with astonishment at his operations, when he reduces the most difficult problems by some surprising turn to a quite simple

1 Nffvi Commentarii Academiae Petropolitanae, 1756-7, Vol. VI. (1761), p. 1 55 = Com- mentationes arithmtticae collectae (ed. Fuss), 1849, I. p. 193.

2 " Diophant's Behandlung der Gleichungen."

3 " Diophant's Auflbsungsmethoden."

4 (r) "Die geschickte Annahme der Unbekannten," (i) " Methode der Zuriick- rechnung und Nebenaufgabe," (3) " Gebrauch des Symbols fur die Unbekannte in verschiedenen Bedeutungen," (4) "Methode der Grenzen," (5) " Auflosung durch blosse Reflexion," (6) "Auflosung in allgemeinen Ausdriicken," (7) " Willkiihrliche Bestim- mungen und Annahmen," (8) "Gebrauch des rechtwinkligen Dreiecks."

DIOPHANTUS' METHODS OF SOLUTION 57

equation. Then, when in the pth chapter Nesselmann passes to the "methods," he prefaces it by saying: "To give a complete picture of Diophantus' methods in all their variety would mean nothing else than copying his book outright. The individual characteristics of almost every problem give him occasion to try upon it a peculiar procedure or found upon it an artifice which cannot be applied to any other problem.... Mean while, though it may be impossible to exhibit all his methods in any short space, yet I will try to describe some operations which occur more often or are particularly re- markable for their elegance, and (where possible) to bring out the underlying scientific principle by a general exposition and by a suitable grouping of similar cases under common aspects or characters." Now the possibility of giving a satisfactory account of the methods of Diophantus must depend largely upon the meaning we attach to the word "method." Nesselmann's arrangement seems to me to be faulty inasmuch as (i) he has treated Diophantus' solutions of equations, which certainly proceeded on fixed rules, and therefore by "method" separately from what he calls "methods of solution," thereby making it appear as though he did not look upon the "treatment of equations" as "methods"; (2) the classification of the "Methods of solution" seems unsatisfactory. Some of the latter can hardly be said to be methods of solution at all; thus the third, " Use of the symbol for the unknown in different significations," might be more justly described as a "hindrance to the solution"; it is an inconvenience to which Diophantus was subjected owing to the want of notation. -Indeed, on the as- sumption of the eight "methods," as Nesselmann describes them, it is really not surprising that no complete account of them could be given without copying the whole book. To take the first, "the adroit assumption of unknowns." Supposing that a number of essentially different problems are proposed, the differences make a different choice of an unknown in each case absolutely necessary. That being so, how could a rule be given for all cases? The best that can be done is to give a number of typical instances. Precisely the same remark applies to "methods" (2), (5), (6), (7). The case of (4), " Method of Limits," is different ; here we have a " method " in the true sense of the term, Le. in the sense of an instrument for solution. And accordingly in this case the method can be exhibited, as I hope to show later on; (8) also deserves to some extent the name of a " method."

58 INTRODUCTION

In one particular case, Diophantus formally states a method or rule ; this is his rule for solving what he calls a " double-equation," and will be found in II. 1 1, where such an equation appears for the first time. Apart from this, we do not find in Diophantus' work statements of method put generally as book-work to be applied to examples. Thus we do not find the separate rules and limitations for the solution of different kinds of equations systematically arranged, but we have to seek them out laboriously from the whole of his work, gathering scattered indications here and there, and to formulate them in the best way that we can.

I shall now attempt to give a short account of those methods running through Diophantus which admit of general statement. For the reasons which I have stated, my arrangement will be different from that of Nesselmann ; I shall omit some of the heads in his classification of "methods of solution"; and, in accordance with his remark that these "methods" can only be adequately described by a transcription of the entire work, I shall leave them to be gathered from a perusal of my reproduction of Diophantus' book.

I shall begin my account with

I. DIOPHANTUS' TREATMENT OF EQUATIONS.

This subject falls naturally into two divisions: (A) Determinate equations of different degrees, (B) Indeterminate equations.

(A) Determinate equations.

Diophantus was able without difficulty to solve determinate equations of the first and second degrees; of a cubic equation we find in his Arithmetica only one example, and that is a very special case. The solution of simple equations we may pass over; we have then to consider Diophantus' methods of solution in the case of (i) Pure equations, (2) Adfected, or mixed, quadratics.

(i) Pure determinate equations.

By pure equations I mean those equations which contain only one power of the unknown, whatever the degree. The solution is effected in the same way whatever the exponent of the term in the

DIOPHANTUS' METHODS OF SOLUTION 59

unknown; and Diophantus treats pure equations of any degree as if they were simple equations of the first degree.

He gives a general rule for this case without regard to the degree1: "If a problem leads to an equation in which any terms are equal to the same terms but have different coefficients, we must take like from like on both sides, until we get one term equal to one term. But, if there are on one side or on both sides any negative terms, the deficient terms must be added on both sides until all the terms on both sides are positive. Then we must take like from like until one term is left on each side." After these operations have been performed, the equation is reduced to the form Axm = B and is considered solved. The cases which occur in Diophantus are cases in which the value of x is found to be a rational number, integral or fractional. Diophantus only recognises one value of x which satisfies this equation; thus, if m is even, he gives only the positive value, excluding a negative value as "impossible." In the same way, when an equation can be reduced in degree by dividing throughout by any power of x, the possible values, x=o, thus arising are not taken into account. Thus an equation of the form x* = ax, which is of common occurrence in the earlier part of the book, is taken to be merely equivalent to the simple equation x=a.

It may be observed that the greater proportion of the problems in Book I. are such that more than one unknown quantity is sought. Now, when there are two unknowns and two conditions, both unknowns can easily be expressed in terms of one symbol. But, when there are three or four quantities to be found, this reduction is much more difficult, and Diophantus shows great adroitness in effecting it: the ultimate result being that it is only necessary to solve a simple equation with one unknown quantity.

(2) Mixed quadratic equations.

After the remarks in Def. 1 1 upon the reduction of equations until we have one term equal to another term, Diophantus adds2: "But we will show you afterwards how, in the case also when two terms are left equal to a single term, such an equation can be solved." That is to say, he promises to explain the solution of a mixed quadratic equation. In the Arithmetica, as we possess the book, this promise is not fulfilled. The first

1 Def. ii.

* fcrrepo*' & eroi Sfl&nev /tot TWJ Suo eiSwv tffuv evl KaraXfi^vruv TO TOIOVTOV Xi/erot.

60 INTRODUCTION

indications we have on the subject are a number of cases in which the equation is given, and the solution written down, or stated to be rational, without any work being shown. Thus

(IV. 22) x* = 4JF — 4, therefore x = 2 ;

(IV. 31) 325^ = 3^+18, therefore x = $fc or &;

(vi. 6) 84*2 + 7.^ = 7, whence x=\;

(vi. 7) 84*2 - 7-*-= 7, hence .*• = £ ;

(VI. 9) 630.^2 - fix = 6, therefore .ar = -^ ;

and (vi. 8) 630.^2 + 73^ = 6, and x is rational.

These examples, though proving that Diophantus had somehow arrived at the result, are not in themselves sufficient to show that he was necessarily acquainted with a regular method for the solution of quadratics ; these solutions might (though their variety makes it somewhat unlikely) have been obtained by mere trial. That, however, Diophantus' solutions of mixed quadratics were not merely empirical is shown by instances in V. 30. In this problem he shows that he could approximate to the root in cases where it is not " rational." As this is an important point, I give the substance of the passage in question: "This is not generally possible unless we contrive to make x. > | (x°- — 60) and < \ (x^ — 60). Let then x* — 60 be > 5*, but x* — 60 < $>x. Since then xz-6o> $x, let 60 be added to both sides, so that & > $x + 60, or x*=$x + some number > 60; therefore x must not be less than n." In like manner Diophantus concludes that "x* = Sx+ some number less than 60 ; therefore x must be found to be not greater than 1 2."

Now, solving for the positive roots of these two equations, we have

x > 2 (5 + V265) and x < 4 + ^76, or x> 10*6394. .. and x< 127177....

It is clear that x may be < 1 1 or > 1 2, and therefore Dio- phantus' limits are not strictly accurate. As however it was doubtless his object to find integral limits, the limits u and 12 are those which are obviously adapted for his purpose, and are a fortiori safe.

In the above equations the other roots obtained by prefixing the negative sign to the radical are negative and therefore would be of no use to Diophantus. In other cases of the kind occurring

DIOPHANTUS' METHODS OF SOLUTION 61

in Book V. the equations have both roots positive, and we have to consider why Diophantus took no account of the smaller roots in those cases.

We will take first the equations in V. 10 where the inequalities to be satisfied are

17 ........................ (i).

19 ........................ (2).

Now, if a, /8 be the roots of the equation

x- —px + q = o (p, q both positive), and if a > fi, then

(a) in order that x"2- — px + q may be > o

we must have x> a or < /3, and (b) in order that x* — px + q may be < o

we must have x < a and > /9. (i) The roots of the equation

— J2X+ 17 =o

36 + V 1007 , 6773... ,4-26...

are — - -- - ; that is, ' ' J — and - — ;

17 17 17

and, in order that ijx* — 72^+ 17 may be.< o, we must have

(2) The roots of the equation

19=0

that . 66-577... and

19 19 19

and, in order that ic^tr2 — J2X + 19 may be > o, we must have

x> 66^77^ Qr< 5-422^ 19 19

Diophantus says that x must not be greater than f| or less than ff . These are again doubtless intended to be a fortiori limits ; but ff should have been f£, and the more correct way of stating the case would be to say that, if x is not greater than ff and not less than \\, the given conditions are a fortiori satisfied.

Now consider what alternative (if any) could be obtained, on Diophantus' principles, if we used the lesser positive roots of the

62 INTRODUCTION

equations. If, like Diophantus, we were to take a fortiori limits, we should have to say

^<T5gbut>^,

which is of course an impossibility. Therefore the smaller roots are here useless from his point of view.

This is, however, not so in the case of another pair of in- equalities, used later in V. 30 for finding an auxiliary x, namely

X* + 60 > 22X,

The roots of the equation

X* — 22X + 6O = O

are ii + V6i ; that is, i8'8i... and 3'i8...; and the roots of the equation x* - 2<\x + 60 = o

are 12 + \/84; that is, 2ri6... and 2-83.... In order therefore to satisfy the above inequalities we must have

x> i8'8i ... or < 3'i8..., and x< 2i'i6 ... but > 2-83.

Diophantus, taking a fortiori integral limits furnished by the greater roots, says that x must not be less than 19 but must be less than 21. But he could also have obtained from the smaller roots an integral value of x satisfying the necessary conditions, namely the value x = 3 ; and this would have had the advantage of giving a smaller value for the auxiliary x than that actually taken, namely 2O1. Accordingly the question has been raised2 whether we have not here, perhaps, a valid reason for believing that Diophantus only knew of the existence of roots obtained by using the positive sign with the radical, and was unaware of the solution obtained by using the negative sign. But in truth we can derive no certain knowledge on this point from Diophantus' treatment of the particular equations in question. Thus, e.g., if he chose to use the first of the two equations

17, 19, for the purpose of obtaining an upper limit only, and the second

1 This is remarked by Loria (Le scienze esatte delP antica Grecia, V. p. 128). But in fact, whether we take 20 or 3 as the value of the auxiliary unknown, we get the same value for the original .r of the problem. For the original x has to be found from x*-6o=(x-m)* where m is the auxiliary.*; and we obtain x= n£ whether we put .r2 - 60 = (.*•- 2 o)2 or x'2-6o = (x-3)2.

2 Loria, op. cit. p. 129.

DIOPHANTUS' METHODS OF SOLUTION 63

for the purpose of obtaining a lower limit only, he could only use the values obtained by using the positive sign. Similarly, if, with

the equations

x2 + 6o> 22.x,

x* + 60 < 24*,

he chose to use the first in order to find a lower limit only, and the second in order to find an upper limit only, it was not open to him to use the values corresponding to the negative sign1.

For my part, I find it difficult or impossible to believe that Diophantus was unaware of the existence of two real roots in such cases. The numerical solution of quadratic equations by the Greeks immediately followed, if it did not precede, their geometrical solution. We find the geometrical equivalent of the solution of a quadratic assumed as early as the fifth century B.C., namely by Hippocrates of Chios in his Quadrature of limes*, the algebraic form of the particular equation being ^2 + Vf -ax = a'i. The complete geometrical solution was given by Euclid in VI. 27-29: and the construction of VI. 28 corresponds in fact to the negative sign before the radical in the case of the particular equation there solved, while a quite obvious and slight variation of the con- struction would give the solution corresponding to the positive sign. In VI. 29 the solution corresponds to the positive sign before the radical; in the case of the equation there dealt with the other sign would not give a "real" solution3. It is true that we do not find the negative sign taken in Heron any more than in Diophantus, though we find Heron4 stating an approximate solution of the equation

x ( 1 4 - x) = 6/2O/ 144,

without showing how he arrived at it; x is, he says, approximately equal to 8£. It is clear however that Heron already possessed a scientific method of solution. Again, the author of the so-called Geometry of Heron5 practically states the solution of the equation

212

r A/ ( 154 X 212 + 841) — 20

in the form x= v - — — - -,

ii

1 Enestrom in Bibliotheca Mathematica IX3, 1908-9, pp. 71-2.

2 Simplicius, Comment, in Aristot. Phys., ed. Diels, p. 64, t8; Rudio, Der Bericht des Simplicius iiber die Quadraluren des Antiphon und des Hippokrates, 1907, p. 58,8-11.

3 The Thirteen Books of Euclid's Elements, Cambridge, 1908, Vol. II. pp. 257-267.

4 Heron, Metrica, ed. Schdne, pp. 148-151. The text has 8 as the approximate solu- tion, but the correction is easy, as the inference immediately drawn is that \\-x=§\.

6 Heron, ed. Hultsch, p. 133, 10-23. See M. Cantor, Geschichte der Math. I3, p. 405.

64 INTRODUCTION

showing pretty clearly the rule followed after the equation had been written in the form

121^ + 638^ = 212 x 154.

We cannot credit Diophantus with less than a similar uniform method ; and, if he did not trouble to give two roots where both were real, this seems quite explicable when it is remembered that he did not write a text-book of algebra, and that his object through- out is to obtain a single solution of his problems, not to multiply solutions or to show how many can be found in each case.

In solving such an equation as

ax1 — bx + c = o,

it is our modern practice to divide out by a in order to make the first term a square. It does not appear that Diophantus divided out by a\ rather he multiplied by a so as to bring the equation into the form

cPx? — abx + ac = o ; then, solving, he found

and wrote the solution in the form

a

wherein his method corresponds to that of the Pseudo-Heron above referred to.

From the rule given in Def. 1 1 for removing by means of addition any negative terms on either side of an equation and taking equals from equals (operations called by the Arabians aljabr and almukd- bala) it is clear that, as a preliminary to solution, Diophantus so arranged his equation that all the terms were positive. Thus, from his point of view, there are three cases of mixed quadratic equations.

Case I. Form mx?+px=q; the root is

m

according to Diophantus. An instance is afforded by vi. 6. Dio- phantus namely arrives at the equation 6x* + yc = 7, which, if it is to be of any service to his solution, should give a rational value of x ; whereupon he says " the square of half the coefficient1 of x

1 For " coefficient " Diophantus simply uses ir\rj6os, multitude or number ; thus "number of dpiO/ioi " = coeff. of x. The absolute term is described as the "units."

DIOPHANTUS' METHODS OF SOLUTION 65

together with the product of the absolute term and the coefficient of x? must be a square number ; but it is not," i.e. \jp + mq, or in this case (|)2 + 42, must be a square in order that the root may be rational, which in this case it is not.

Case 2. Form mx* =px + q. Diophantus takes

An example is IV. 39, where 2^2>6>+i8. Diophantus says: "To solve this take the square of half the coefficient of x, i.e. 9, and the product of the absolute term and the coefficient of x*, i.e. 36. Adding, we have 45, the square root1 of which is not2 < 7. Add half the coefficient of x, [and divide by the coefficient of x*\ ; whence x is not < 5." Here the form of the root is given completely ; and the whole operation of finding it is revealed. Cf. IV. 31, where Diophantus remarks that the equation 5^r2=3^r+ 18 "is not rational. But 5, the coefficient of x*, is a square plus I, and it is necessary that this coefficient multiplied by the 18 units and then added to the square of half the coefficient of x, namely 3, that is to say 2$, shall make a square."

Case 3. Form mx* + q =px. Diophantus' root is

Cf. in V. 10 the equation already mentioned, 1 7^r2 + 1 7 Diophantus says: " Multiply half the coefficient of x into itself and we have 1296; subtract the product of the coefficient of x* and the absolute term, or 289. The remainder is 1007, the square root of which is not3 > 31. Add half the coefficient of x, and the result is not > 67. Divide by the coefficient of ;r2, and x is not > 67/17." Here again we have the complete solution given. Cf. VI. 22, where, having arrived at the equation 172^=336^+24, Diophantus remarks that "this is not always possible, unless half the coefficient of x multiplied into itself, minus the product of the coefficient of x* and the units, makes a square."

For the purpose of comparison with Diophantus' solutions of quadratic equations we may refer to a few of his solutions of

1 The " square root" is with Diophantus TrXeupd, or "side."

2 7, though not accurate, is clearly the nearest integral limit which will serve the purpose.

3 As before, the nearest integral limit.

H. D. 5

66 INTRODUCTION

(3) Simultaneous equations involving quadratics. Under this heading come the pairs of equations

I use Greek letters to distinguish the numbers which the problem requires us to find from the one unknown which Dio- phantus uses and which I shall call x.

In the first two of the above problems, he chooses his x thus. Let, he says,

Then it follows, by addition and subtraction, that

£ = a + x, r) = a — x. Consequently, in I. 27,

£7 = (a + x) (a -x) = a>-x* = B, and x is found from this "pure" quadratic equation.

If we eliminate £ from the original equations, we have

if - 2ai) + B = o, which we should solve by completing the square (a — T/)2, whence

(a-^^tf-B, which is Diophantus' ultimate equation with a — tj for x.

Thus Diophantus' method corresponds here again to the ordi- nary method of solving a mixed quadratic, by which we make it into a pure quadratic with a different x.

In I. 30 Diophantus puts £ + i\ = 2;r, and the solution proceeds in the same way as in I. 27.

In I. 28 the resulting equation in x is

x?= 2 (a* +*2) =B.

(4) Cubic equation.

There is no ground for supposing that Diophantus was acquainted

with the algebraical solution of a cubic equation. It is true that there

is one cubic equation to be found in the Arithmetica, but it is only

a very particular case. In VI. 17 the problem leads to the equation

x* + 2x + 3 = xs + ix - 3** - i,

DIOPHANTUS' METHODS OF SOLUTION 67

and Diophantus says simply "whence x is found to be 4." All that can be said of this is that, if we write the equation in true Diophantine fashion, so that all the terms are positive,

x3 + x = 4*2 + 4. This equation being clearly equivalent to

Diophantus no doubt detected the presence of the common factor on both sides of the equation. The result of dividing by it is x= 4, which is Diophantus' solution. Of the other two roots x= ± v'(— i) no account is taken, for the reason stated above.

It is not possible to judge from this example how far Dio- phantus was acquainted with the solution of equations of a degree higher than the second.

I pass now to the second general division of equations.

(B) Indeterminate equations.

As I have already stated, Diophantus does not, in his Arithmetica as we have it, treat of indeterminate equations of the first degree. Those examples in Book I. which would lead to such equations are, by the arbitrary assumption of a specific value for one of the required numbers, converted into determinate equations. Nor is it likely that indeterminate equations of the first degree were treated of in the lost Books. For, as Nesselmann observes, while with indeterminate quadratic equations the object is to obtain a rational result, the whole point in solving indeterminate simple equations is to obtain a result in integral numbers. But Diophantus does not exclude fractional solutions, and he has therefore only to see that his results are positive, which is of course easy. Inde- terminate equations of the first degree would therefore, from Diophantus' point of view, have no particular significance. We take therefore, as our first division, indeterminate equations of the second degree.

(a) Indeterminate equations of tJte second degree. The form in which these equations occur in Diophantus is invariably this: one or two (but never more) functions of the unknown quantity of the form Ax* + Bx + C or simpler forms are to be made rational square numbers by finding a suitable value for x. Thus the most general case is that of solving one or two equations of the form Ax* + Bx + C=y*.

5—2

68 INTRODUCTION

(i) Single equation.

The single equation takes special forms when one or more of the coefficients vanish or satisfy certain conditions. It will be well to give in order the different forms as they can be identified in Diophantus, premising that for " = j2" Diophantus simply uses the formula ia~ov rerpayoovo), " is equal to a square," or iroiel rerpdytavov, " makes a square."

I. Equations which can always be solved rationally. This is the case when A or C or both vanish.

Form Bx=y*. Diophantus puts for j2 any arbitrary square number, say m*. Then x = m*\B.

Ex. III. 5: 2x=yz, y is assumed to be 16, and ;r= 8.

Form Bx+ C=y*. Diophantus puts for y* any square nP, and x=(m*—C)IB. He admits fractional values of x, only taking care that they are "rational," i.e. rational and positive.

Ex. III. 6: 6x+ i =y* = 121, say, and x= 20.

Form Axz + Bx=y*. Diophantus substitutes for y any multiple of x, as — x; whence Ax + B = —^ x, the factor x disappearing and

the root x = o being neglected as usual. Thus x — — - — -^.

Exx. II. 21: 43?+ yc =jj/2 = ($xf, say, and x = f . II. 33 : i6x* + *]x =y* = ($x)*, say, and x = $.

2. Equations which can only be rationally solved if certain conditions are fulfilled.

The cases occurring in Diophantus are the following.

Form Ax* + C=y*. This can be rationally solved according to Diophantus

(a) When A is positive and a square, say «2. Thus cPx* + C=yi. In this case f is put = (ax ± mf ; therefore a^x- + C = (ax ± mf,

^- and x= + — — ,

" 2ma

(m and the doubtful sign being always assumed so as to give x a positive value).

(/?) When C is positive and a square number, say c*. Thus Axz + c* =_/. Here Diophantus puts y = (mx ± c) ;

DIOPHANTUS' METHODS OF SOLUTION 69

therefore Ax* + £* = (mx ± cj,

2mc

and x = + -; -- , .

~ A — m*

(7) When one solution, is known, any number of other solutions can be found. This is enunciated in the Lemma to vi. 1 5 thus, though only for the case in which C is negative: "Given two numbers, if, when one is multiplied by some square and the other is subtracted from the product, the result is a square, then another square also can be found, greater than the aforesaid square which has the same property." It is curious that Diophantus does not give a general enunciation of this proposition, inasmuch as not only is it applicable to the cases + Ax* ± C=y3, but also to the general form Ax- + Bx + C=y\

Diophantus' method of finding other greater values of x satisfy- ing the equation Ax*- — C=y* when one such value is known is as follows.

Suppose that x0 is the value already known and that q is the corresponding value of y.

Put ;r = jF0 + £ in the original expression, and equate it to (q — k%)-, where k is some integer.

Since A(x. + &-C=(q-k&t

it follows (because by hypothesis Ax£— C = q*) that

2 (Ax0 + kq) whence g= #_

2(A and *«.+

In the second Lemma to VI. 12 Diophantus does prove that the equation Ax* + C=y* has an infinite number of solutions when A +C is a square, i>. in the particular case where the value x= I satisfies the equation. But he does not always bear this in mind; for in III. 10 the equation S2x* + I2=y2 is regarded as impossible of solution although 52+12 = 64, a square, and a rational solution is therefore possible. Again in III. 12 the equation 266r2— 10 =y* is regarded as impossible though x = i satisfies it.

The method used by Diophantus in the second Lemma to VI. 12 is like that of the Lemma to VI. 15.

Suppose that A + C = q11.

Put i + f for x in the original expression Ax* + C, and equate it to (q — kgf, where k is some integer.

70 INTRODUCTION

Thus A(i + £?+C=(q-k%}\

and it follows that 2£ (A + kq} = ? (k- -A),

so that £

It is of course necessary to choose k? such that & > A.

It is clear that, if X — G satisfies the equation, C is a square, and therefore this case (7) includes the previous case (yS).

It is to be observed that in VI. 14 Diophantus says that a rational solution of the equation

Ax* - & = j/8

is impossible unless A is the sum of two squares.

[In fact, if x-=p\q satisfies the equation, and Ax*- c2 = &, we have Ap* = c*q* + k-q-,

Lastly, we have to consider

Form Ax* + Bx+C=y*.

This equation can be reduced by means of a change of variable to the preceding form wanting the second term. For, if we put

D

x = z -- -j , the transformation gives

Diophantus, however, treats ' this form of the equation quite separately from the other and less fully. According to him the rational solution is only possible in the following cases.

(a) When A is positive and a square, or the equation is

tfxn- + Bx + C=y\ Diophantus then puts y* = (ax — mf, whence

= ™?-c ^Exx 22 ^

2am + B (#) When C is positive and a square, or the equation is

Ax* + Bx + c* = y\ Diophantus puts_y2 = (c — mxf, whence

(Exx.IV.8,9etc.)

DIOPHANTUS' METHODS OF SOLUTION 71

(7) When ^B* - AC is positive and a square number. Dio- phantus never expressly enunciates the possibility of this case; but it occurs, as it were unawares, in IV. 31. In that problem

is to be made a square. To solve this Diophantus assumes

which leads to the quadratic $x+ 18 — $xa = o; but "the equation is not rational." Accordingly the assumption 4** will not do; "and we must find a square [to replace 4] such that 18 times (this square + i) +(f)2 may be a square." Diophantus then solves the auxiliary equation i8(?«2 + 1) + £ =fy finding m=i8. He then assumes 3*+ 18 -^a = (i8)2^2, which gives $2$x* — 3*- 18 = 0, "and x=-^, that is ^.'Jl

1 With this solution should be compared the much simpler solution of this case given by Euler (Algebra, tr. Hewlett, 1840, Part n. Arts. 50-53), depending on the separation of the quadratic expression into factors. (Curiously enough Diophantus does not separate quadratic expressions into their factors except in one case, vi. 19, where however his purpose is quite different : he has made the sum of three sides of a right-angled triangle 4^ + 6^ + 2, which has to be a cube, and, in order to simplify, he divides throughout by x+ i, which leaves 4^ + 2 to be made a cube.)

Since \EP-AC is a square, the roots of the quadratic Ax* + Bx + C=o are real, and the expression has two real linear factors. Take the particular case now in question, where Diophantus actually arrives at 3*+ 18- x* as the result of multiplying 6-x and 3+jr, but makes no use of the factors.

We have 3*+ i8-xn- = (6-x) (3+*).

Assume then (6 - x) (3 + x) =^ (6 - *)2,

and we have /J(6-*

where/, q may be any numbers subject to the condition that ip*>q*. If ^ = 9, q*= 16, we have Diophantus' solution x = — .

In general, if Ax? + £x+C= (f+gx) (A+Jtx),

we can put (f+gx) (h + kx) = £ (f+gx)*,

whence ?2 (A + >br) =/(/+£*),

»££

This case, says Euler, leads to a fourth case in which Ax3 + Bx + C=y* can be solved, though neither A nor C is a square, and though ^-^Cis not a square either. The fourth case is that in which Ax* + Bx + C is the sum of two parts, one of which is a square and the other is the product of two factors linear in x. For suppose

Ax* + Bx +C=Z* + XY, where Z=dx + e, X=/x+g, Y=hx + k.

72

INTRODUCTION

It is worth observing that from this example of Diophantus we can deduce the reduction of this general case to the form

Ax* + C=y* wanting the middle term.

Assume, with Diophantus, that Ax* + Bx+ C=m2x2: therefore by solution we have

A-m2

and x is rational provided that IB* — AC+ Cm* is a square. This condition can be fulfilled if \B* - AC is a square, by the pre-

We can then put Z2 + X Y= ( Z + ^ JfY,

whence F=2^Z + 4/A',

q q>

,

that is, x (p2f+ ipqd -fA) = kq* - ipqe -p2g.

Ex. i. Equation ix2- i=y2. Put •2X*-i

* A£

Therefore x- i = it-x+^(x + i),

and x (f + ipq -g*)=-(p* + q\

As x2 is alone found in our equation, we can take either the positive or negative sign and we may put

Ex. 2. Equation ix2 + 'i=y2. Here we put

Equating this to -J2 + - (x+ i)| ,

P P* , we have i(x- i) = 4 <- + -„(*+ i),

1 ?2V or x (/•> - iq2) =-(iqt+4j>q +/),

It is to be observed that this method enables us to solve the equation

Ax'1-c2=y2

whenever it can be solved rationally, i.e. whenever A is the sum of two squares (d2 + e2, say). For then

Ax2 -£2 = d^x2 + (ex - c) (ex + c).

In cases not covered by any of the above rules our only plan is to try to discover one solution empirically. If one solution is thus found, we can find any number of others; if we cannot discover such a solution by trial (even after reducing the equation to the simplest form A'x'2+ C=ya), recourse must be had to the method of continued fractions elaborated by Lagrange (cf. Oeuvres, II. pp. 377-535 and pp. 655 — 726 ; additions to Euler's Algebra).

DIOPHANTUS' METHODS OF SOLUTION 73

ceding case. If \B'i — AC is not a square, we have to solve (putting, for brevity, D for \B* — AC) the equation

D + Cm2 =y\ and the reduction is effected.

(2) Double-equation.

By the name "double-equation" Diophantus denotes the pro- blem of finding one value of the unknown quantity which will make two different functions of it simultaneously rational square numbers. The Greek term for the "double-equation" occurs variously as SwrXoi'- croT?;?, St7rX?7 tVor??? or 8frr\rj to-axrt?. We have then to solve the equations

mx2 + a.x + a = u2}

nx2 + fix + & = w2}

in rational numbers. The necessary preliminary condition is that each of the two expressions can severally be made squares. This is always possible when the first term (in x2) is wanting. This is the simplest case, and we shall accordingly take it first.

I. Double-equation of the first degree.

Diophantus has one general method of solving the equations

ax + a = u2'

taking slightly different forms according to the nature of the coefficients.

(a) First method of solution of

ax + a = u2

This method depends upon the identity

If the difference between the two expressions in x can be separated into two factors p, qt the expressions themselves are equated to {^(p+q)}2 respectively. Diophantus himself (II. u) states his rule thus.

• "Observing the difference [between the two expressions], seek two numbers such that their product is equal to this difference; then equate either the square of half the difference of the two factors to the lesser of the expressions or the square of half the sum to the greater."

74 INTRODUCTION

We will take the general case and investigate to what particular classes of cases the method is applicable, from Diophantus' point of view, remembering that his cases are such that the final quadratic equation in x always reduces to a simple equation.

Take the equations

Q.X + a =

Subtracting, we have

(a - £) x + (a - b) = if - w\

We have then to separate (a — $)x + (a-b} into two factors; let these be/, {(a - /3)* + (a - b]}lp. We write accordingly

.

u ± w = -

P

u + w Thus u2

4

therefore {(a-@)x + a-6 +/2}2 = 4/2 (ax + a),

or (a - /3)2;tr2 + 2x {(a - ff) (a - b +/2) - 2p" a}+(a-b +/2)2 - 4«/2=o, that is, (a - /S)2*2 + 2x {(a -ft) (a- b} -/2 (a + £)}

+ (a- by -2^(a + b} +^ = o.

Now, in order that this equation may reduce to a simple equation, either

(i) The coefficient of x" must vanish, so that

«-A

or (2) The absolute term must vanish, that is,

or /-

so that ab must be a square number.

Therefore either a and b are both squares, in which case we may substitute & and d* for them respectively, / being then equal to c ± d, or the ratio of a to b is the ratio of a square to a square.

With respect to (i) we observe that on one condition it is not necessary that a — /8 should vanish, z>. provided we can, before solving the equations, make the coefficients of x the same in both expressions by multiplying either equation or both equations by some square number, an operation which does not affect the problem, since a square multiplied by a square is still a square.

DIOPHANTUS' METHODS OF SOLUTION 75

In other words, it is only necessary that the ratio of a to @ should be the ratio of a square to a square1.

Thus, if a/^ = m2/n2 or an?=@m*, the equations can be solved by multiplying them respectively by nz and m*; we can in fact solve the equations

like the equations

tax + a = u' a.x + b = w

in an infinite number of ways. Again, the equations under (2)

ax + c2 = u2

can be solved in two different ways according as we write them in this form or in the form

obtained by multiplying them respectively by d*, c1*, in order that the absolute terms may be equal.

I shall now give those of the possible cases which we find solved in Diophantus' own work. These are equations (i) of the form

b =

1 Diophantus actually states this condition in the solution of iv. 32 where, on arriving at the equations

he says : " And this is not rational because the coefficients of x have not to one another the ratio which a square number has to a square number."

Similarly in the second solution of III. 15 he states the same condition along with an alternative condition, namely that a has to b the ratio of a square to a square, which is the second condition arrived at under (2) above. On obtaining the equations

Diophantus observes "But, since the coefficients in one expression are respectively greater than those in the other, neither have they (in either case) the ratio which a square number has to a square number, the hypothesis which we took is useless." Cf. also iv. 39 where he says that the equations

are possible of solution because there is a square number of units in each expression.

76 INTRODUCTION

a case which includes the more common one where the coefficients of x in both are equal \

(2) of the form

ax + c* = u2 }

solved in two different ways according as they are written in this form or in the alternative form

General solution of Form ( i ) or

am2x + a = u2 a.n2x + b = w2\'

Multiply by n2, m2 respectively, and we have to solve the equations

am2n'2x+an2 = tt'2\

The difference is an2 — dm2; suppose this separated into two factors p, q.

Let u' ± w' =/,

u' 4- w = q ;

therefore u'2 = ^(p + qj, w'* — i (P — #)2>

and a m2n2x + an2 = i (/ + q)2,

or a.m2n2x + bm2 — \(p — q)*.

Either equation gives the same value of x, and

a.m2n2 since pq = an2 — bm2.

Any factors /, q may be chosen provided that the resulting value of x is positive.

Ex. from Diophantus :

65 -

65-24*

therefore , 260 - 24^ = u'2 }

65 - 24^ = w2 j '

The difference = 195 = 15 . 13, say; therefore £(15 — I3)2 = 65 — 24^; that is, 24^=64, and

=*2l. =^}'

DIOPHANTUS' METHODS OF SOLUTION 77

•

General solution (first method) of Form (2), or ax + & = u'z \ /3x + d* = w'2)'

In order to solve by this method, we multiply by d2, c2 respectively and write

u being supposed to be the greater.

The difference = (ad* - fic^x. Let the factors of this be/*, q. Therefore u2 = £ ( px + qf,

w> = \(px-qf. Thus x is found from the equation

This equation gives

fx* + 2x (pq - 2ad2) + q^-^d- = O, or, since pq = (ad2 — fie2),

p*x* - 2x (ad2 + &<*) + f-4c2d2^ o.

In order that this may reduce to a simple equation, as Dio- phantus requires, the absolute term must vanish,

and q — 2cd.

Thus our method in this case furnishes us with only one solution of the double-equation, q being restricted to the value 2cd, and the solution is

_ 2 (ad2 + &")

Ex. from Diophantus. This method is only used in one par- ticular case (IV. 39), where c* = d* as the equations originally stand, the equations being

6x + 4 =

The difference is 2x, and q is necessarily taken to be 2\/4, or 4; the factors are therefore ^x, 4.

Therefore %x + 4 = \ (%x + 4)2,

and x= 112.

General solution (second method) of Form (2) or

ax + c2 =

78 INTRODUCTION

The difference = (a - /8) x + (<? - d*).

Let the factors of this be p, {(a - /3) ^ + cz -

Then, as before proved (p. 74), / must be equal to (c ± d).

Therefore the factors are

and we have finally

which equation gives two possible values for x. Thus in this case we can find by our method two values of x, since one of the factors p may be either (c + d) or (c - d).

Ex. from Diophantus. To solve the equations

(ill. 15.)

The difference is here 5^+ 5, and Diophantus chooses as the factors 5, x+ i. This case therefore corresponds to the value c + d of/. The solution is given by

(\x + 3)2 = \QX + 9, whence x = 28.

The other value, c — d, of / is in this case excluded, because it would lead to a negative value of x.

The possibility of deriving any number of solutions of a double- equation when one solution is known does not seem to have been noticed by Diophantus, though he uses the principle in certain cases of the single equation (see above, pp. 69, 70). Fermat was the first, apparently, to discover that this might always be done, if one value a of x were known, by substituting x + a for x in the equa- tions. By this means it is possible to find a positive solution, even if a is negative, by successive applications of the principle.

But, nevertheless, Diophantus had certain peculiar artifices by which he could arrive at a second value. One of these artifices (which is made necessary in one case by the unsuitableness of the value of x found by the ordinary method) gives a different way of solving a double-equation from that which has been explained, and is used only in one special case (IV. 39).

DIOPHANTUS' METHODS OF SOLUTION 79

(£) Second method of solving a double-equation of the first degree.

Consider only the special case

Take these expressions, and »2, and write them in order of magnitude, denoting them for convenience by A, B, C.

A -B f , A-B Therefore ~ — -^ =-j , and _ _

Suppose now that Jix + «2 = (y + »)* ; therefore hx = y* + 2ny,

and

or

thus it is only necessary to make this expression a square. Assume therefore that

and any number of values for y, and therefore for x, can be found, by varying/.

Ex. from Diophantus (the only one), IV. 39.

In this case there is the additional condition of a limit to the value of x. The double-equation

6x + 4 = has to be solved in such a manner that x < 2.

Here „ -~= £, and B is taken1 to be (y + 2f.

Therefore A - B = $ (f + 47) ;

therefore A

which must be made a square.

1 Of course Diophantus uses the same variable JT where I have for clearness used y. Then, to express what I have called m later, he says: "I form a square from 3 minus some number of JT'S, and x becomes some number multiplied by 6 and then added to 12, divided by the difference by which the square of the number exceeds 3."

8o INTRODUCTION

If we multiply by f , we must make

3j2 + 1 2y + 9 = a square, where y must be < 2. Diophantus assumes

6m + 12

whence y = — — ,

*»'- 3

and the value of m is then taken such as to make j> < 2.

It is in a note on this problem that Bachet shows that the double-equation

ax + a =

can be rationally solved by a similar method provided that the coefficients satisfy either of two conditions, although none of the coefficients are squares and neither of the ratios a : /3 and a : b is equal to the ratio of a square to a square. Bachet's conditions are :

(1) That, when the difference between the two expressions is multiplied or divided by a suitably-chosen number, and the expression thus obtained is subtracted from the smaller of the original expressions, the result is a square number, or

(2) That, when the difference between the two expressions is multiplied or divided by a suitably-chosen number, and the smaller of the two original expressions is subtracted from the expression obtained by the said multiplication or division, the result is a number bearing to the multiplier or divisor the ratio of a square to a square1.

1 Bachet of course does not solve equations in general expressions (his notation does not admit of this), but illustrates his conditions by equations in which the coefficients are specific numbers. I will give one of his illustrations of each condition, and then set the conditions out more generally.

Case (i). Equations

difference 2.r + 6

The suitably-chosen number (to divide by in this case) is 2 ;

\ (difference) =x + 3,

and (lesser expression) - | (difference) = x + 7 - (x + 3) = 4, that is, a square.

We have then to find two squares such that

their difference— 2 (difference between lesser and 4). Assume that the lesser = (y + 2)2, 2 being the square root of 4. Therefore (greater square) = 3 (lesser) - 8

DIOPHANTUS' METHODS OF SOLUTION 81

2. Double-equation of the second degree, or the general form

These equations are much less thoroughly treated in Diophan- tus than those of the first degree. Only such special instances

To make 3;c2 + 127 + 4 a square we put

3J2+I2J|/ + 4 = (2-/^)2,

where p must lie between certain limits which have next to be found. The equation gives

In order that y may be positive, fp must be > 3 ; and in order that the second of the original expressions, assumed equal to (jy+2)2, may be greater than 7 (it is in fact x+j), we must have 0'+2)>2f (an a fortiori limit, since 2§ >«/7), or_y>|.

Therefore 4/ + 12 > f (/2 - 3) ,

Suppose that 3/2=i6/ + 53|, which gives/ = 7f.

Therefore / < 7$, but /2 > 3.

Put / = 3 in the above equation ; therefore

3/+t2;/ + 4 = (2-37)2,

and j/ = 4.

Therefore .r=0/ + 2)2- 7 = 29.

Case (2). Equations 6#+25 = «2|

wH-j^f ;

difference 4^+22

The suitable-number (again to divide by) in this case is 2 ;

\ (difference) = ix + 1 1,

and £ (difference) - (lesser expression) = 8=2.4,

where 2 is the divisor used, and 4 is the ratio of a square to a square. Hence two squares have to be found such that

(their difference) = 2 (sum of lesser and 8). If the lesser is j2, the greater is 3jj/2+ i6 = (4 -pyf1, say. Bachet gives, as limits for /,

/Mil /*>3» and puts p = 3. This gives 7 = 4, so that jr= 6 \.

Let us now state Bachet's conditions generally. Suppose the equations to be

The difference is (a - /3) * + (a - b). This has to be multiplied by — ^— which is the "suitable" factor in this case, and, if we subtract the product from fix + b, we obtain

ab-afi

'-^<*-*>. - T^T-

82 INTRODUCTION

occur as can be easily solved by the methods which we have described for equations of the first degree.

The following types are found.

(i) p

The difference is (a — /3) x + (a — b\ and, following the usual course, we may, e.g., resolve this into the factors

(1) The first of Bachet's conditions is that

ab-aS a_B = a square =/2/?2, say.

(2) The second condition is that

aB-ab _p ft

; a ratio of a square to a square.

It is to be observed that the first of these conditions can be obtained by considering the equation

obtained on page 74 above.

Diophantus only considers the cases in which this equation reduces to a simple equation ; but the solution of it as a mixed quadratic gives a rational value of x provided that

{ (a - j8) (a - b) -f (a + B) } 2 - (a - 0)2{ (a - £)2 - 2/>2 (a + b) +/4 } is a square, that is, if

^{(a + ^-(a-^}+^{(a + l>)(a-^-(a2-^)(a-/>)} is a square, which reduces to a/3/2 + (a-/3) (ab-aB)~ a square ........................ (A).

This can be solved (cf. p. 68 above), if ab-aB

is a square. (Bachet's first condition.)

Again take Bachet's second condition

aB - ab r*

p = a square = -z say,

and substitute fir2/*2 for a/3 - a/> in the equation (A) above. Therefore a/3/2 - (a - ft) /3 -^ = a square,

or aj3/'2 -- (a - 18) /3 = a square.

This is satisfied by p'=i; therefore (p. 69) any number of other solutions can be found .

The second condition can also be obtained directly by eliminating x from the equations ax + a = «2 1

for the result is ^ w2 + -*-y- = «2>

which can be rationally solved if

aB - a/'

DIOPHANTUS' METHODS OF SOLUTION 83

<"-

as usual, we put

or

In order that x may be rational a condition is necessary; thus x is rational if

= 1 *~8

This is the case in the only instance of the type where a is not equal to b, namely (III. 13)

the difference is i6x+4, and the resolution of this into the factors 4, ^x + i solves the problem.

In the other cases of the type a = b\ the difference is then (a — ft}x, which is resolved into the factors

'

ifa — B \2

and we put p2x2 + ax + a = - — 4 2px ,

4\ 2p r j

4\ 2p

whence — —x =

Exx. from Diophantus :

xz- and ^ +

(2) The second type found in Diophantus1 is x2 + ax + a = u- )

0x + a = zv*)' where one equation has no term in x*, and p = i, a = b.

1 It is perhaps worth noting that the method of the "double-equation " has a distinct advantage in this type of cose. The alternative is to solve by the method of Euler (who does not use the " double-equation "), i.e. to put the linear expression equal to/* and then, substituting the value of JT (in terms of/) in the quadratic expression, to solve the

6—2

84 INTRODUCTION

The difference x* + (*—$)x is resolved into the factors

and we put &x + a = | (a -

which gives x.

resulting equation in /. But the difficulties would generally be great. Take the case of vi. 6 where

> have to be made squares.

If

(•fp — |)2

therefore x'2+ i = — — - -- h i has to be made a square,

or /4 - 2/2 + 1 97 = a square.

This does not admit of solution unless we could somehow discover empirically one value of / which would satisfy the requirement, and this would be very difficult.

Let us take an easier case for solution by this method,

which is solved by Euler (Algebra, Part II. Art. 222), and let us compare the working of the two methods in this case.

I. Enter's method. Assuming x+ i ==/>2 and substituting^2— i for x in the quadratic expression, we have

/4 - 2/J2 + 2 = a square.

This can only be solved generally if we can discover one possible value of / by trial ; this however is not difficult in the particular case, for/= i is an obvious solution.

To find others we put i + q instead of p in the expression to be made a square ; this gives

i + 4^2 + 4^3 + ij4 = a square.

This can be solved in several ways.

1. Suppose i + 4^2 + 4^8-i-^4=(i + ?*)2;

thus 4$r" + 4^=2^2, whence q— --, p = - and x= ---.

2. Suppose i+4?2 + 4?3 + ^=(I-?2)2;

thus 4^2 + 4^= -2?2, and q= -|, /=-- and *=--.

3. Suppose i+4^2 + 4^3 + ^4 = (i±2^±^2)a; and we find, in either case, that q= - i, so that/= -1,^ = 0.

4. Suppose i + 4^2 + 4^3 + y4 = ( i + 2^2)2 ;

and we have 4^ + ^4=4^4, whence <? = -, p = ~ and x=(-\ -i=^.

3 3 \3/ 9

This value of x satisfies the conditions, for

The above five suppositions therefore give only two serviceable solutions

x=-*, ,= *?. 4 9

To find another solution we take one of the values of q already found, say y= — , and

DIOPHANTUS' METHODS OF SOLUTION 85

Exx. from Diophantus :

3*- 12 = u*) *i ah (v- *•)

O*;r —12 = T& j

(vi. 6.)

-6144*+ 1048576 = «M ,yl 22>

jr + 64 = wir

substitute r - - for q. This gives / = i + ? = r + - , and we substitute this value for / in the expression /* - 2/a + 2.

We have then ^-^r-- i* + 'it3+t* to be made a square, or 16 2 2

25 - 24r- 8^+32^+ i6^=a square.

i. We take S+A*^ for the root, so that the absolute term and the term in r4 may disappear. We can make the term in r disappear also by putting iof= - 24 or

/= - — . We then have

(a) The upper sign gives

- 8 + 32r = 40 +/*+ 8A.

<* r

and r=(

thus P=— , and jr=/2_I=?r±.

20 400

(£) The lower sign gives

-8 + 32r= -40+/2-8A,

and r=(f

thus p= - — , and or=^i as before.

' 20 400

We have therefore .r+i = (^ , and

) . /

2. Another solution is found by assuming the root to be 5 +/r+g^ and determining /and g so that the absolute term and the terms in r, r3 may vanish ; the result is ,_ £2 £7» 2/g-3?= 1550

/- 5 ' ^^ 125' ' ~ i6-^r2 861 '

/-

1 1 . Method of " doubU-tqitation.

The difference =jca-jr.

(i) If we take as factors JT, x- i and, as usual, equate the square of half their difference, or - , to x + i, we have

"•=;•

86 INTRODUCTION

The absolute terms in the last case are made equal by multiply- ing the second equation by (i28)2 or 16384.

(3) One separate case must be mentioned which cannot be solved, from Diophantus' standpoint, by the foregoing method, but which sometimes occurs and is solved by a special artifice.

The form of double-equation is

our2 4- ax = uz \ (i),

/3;r2 + bx = w* \ ( 2 ) .

Diophantus assumes ?/2 = m^x*, whence, by ( I ), x = a/(mz — a),

(2) If we take -x, tx-t, as factors, half the sum of which is -*- i, so that the

4 absolute terms may disappear in the resulting equation, we have

U^-libA

16 2

and *-.

9

(3) To find another value by means of the first, namely x= - - , we substitute y - - for x in the original expressions. We then have to solve ,•-3 2| = 1<t

Multiply the latter by — so as to make the absolute terms the same, and we must have

a,+s.**

4' 16

Subtract from the first expression, and the difference is yt-^—y=y (y- — 1 5 then,

equating the square of half the difference of the factors to the smaller expression, we have

so that 961=400^+100.

Therefore

(4) If we start from the known value — and put j+— for j; in the equations, we

obtain Euler's fourth value of *, namely .

7 2965284

Thus all the four values obtained by Euler are more easily obtained by the method of the "double-equation."

DIOPHANTUS' METHODS OF SOLUTION 87

and, by substitution in (2), we derive that

ba

must be a square,

m2 — a.

a2 ft + ba (m2 - a) or

We have therefore to solve the equation

abmz + a (aft — a.b) =y2,

and this form can or cannot be solved by the methods already given according to the nature of the coefficients1. Thus it can be solved if (a^ — a.b}ja is a square or if a\b is a square.

Exx. from Diophantus :

*••••• 4*--, (VL I2.)

(£) Indeterminate equations of a degree higher than the second.

(i) Single equations,

These are properly divided by Nesselmann into two classes ; the first comprises those problems in which it is required to make a function of x, of a degree higher than the second, a square ; the second comprises those in which a rational value of x has to be found which will make any function of x, not a square, but a higher power of some number. The first class of problems requires the solution in rational numbers of

Axn + Bxn~l + . . . + Kx + L =/>, the second the solution of

Axn + Bxn~l + . . . + Kx -f L = y3,

for Diophantus does not go beyond making a certain function of x a cube. In no instance, however, of the first class does the index n exceed 6, while in the second class n does not (except in a special case or two) exceed 3.

1 Diophantus apparently did not observe that the above form of double-equation can be reduced to one of the first degree by dividing by x2 and substituting^ for i/jr, when it becomes

Adapting Sachet's second condition, we see that the equations can be rationally solved if (/3a - ab)la is a square, which is of course the same as one of the conditions under which the above equation abttfi + a (aft — a&) can be solved.

88 INTRODUCTION

First Class. Equation

Axn + Bxn~l + . . . + Kx + L = y". The forms found in Diophantus are as follows : i . Equation Ax* + Bx* + Cx + d2 = y\

Here, as the absolute term is a square, we might put for y the expression mx + d, and determine m so that the coefficient of x in the resulting equation vanishes. In that case

2md = C, and m — C\2d\

and we obtain, in Diophantus' manner, a simple equation for x, giving

C* -^B

4tl* A

Or we might put for y the expression nPx* + nx + d, and deter- mine m, n so that the coefficients of x, x* in the resulting equation both vanish, in which case we should again have a simple equa- tion for x. Diophantus, in the only example of this form of equation which occurs (VI. 18), makes the first supposition. The equation is

and Diophantus assumes y = \x + i, whence x = *£.

2. Equation Ax* + Bx* + Cx* + Dx + E=y\ In order that this equation may be solved by Diophantus' method, either A or E must be a square. If A is a square and

D

equal to a2, we may assume y = ax* H -- x + n, determining n so that the term in x* vanishes. If E is a square (= ez), we may write y = mx* H — x + e, determining m so that the term in x* in the

resulting equation may vanish. We shall then, in either case, obtain a simple equation in x.

The examples of this form in Diophantus are of the kind

ayx4 + Bx3 + Cx* + Dx + e* =f,

where we can assume y=±ax* + kx±e, determining k so that in the resulting equation the coefficient of x3 or of x may vanish ; when we again have a simple equation.

Ex. from Diophantus (iv. 28) :

Diophantus assumes y= yc*— 6x+ i,and the equation reduces to 2jr3 — 6>2 = o, whence *=.

DIOPHANTUS' METHODS OF SOLUTION 89

Diophantus is guided in his choice of signs in the expression ± ax'1 + kx ± e by the necessity for obtaining a " rational " result.

Far more difficult to solve are those equations in which, the left-hand expression being bi-quadratic, the odd powers of x are wanting, i.e. the equations Ax* + Cx*1 + E =y* and Ax* + E=y*, even when A or E is a square, or both are so. These cases Diophantus treats more imperfectly.

3. Equation Ax* + Cx* + E =y*,

Only very special cases of this form occur. The type is

a*x* - c*x2 + e2 =y\ which is written

a*x*-(? + ?L=y2.

Here y is assumed to be ax or ejx, and in either case we have a rational value for x.

Exx. from Diophantus :

This is assumed to be equal to

where y*- is assumed to be equal to

4. Equation Ax* + E =y\

The case occurring in Diophantus is x* + 97 =yz (V. 29). Dio- phantus tries one assumption, y = xz — 10, and finds that this gives X* = -£Q, which leads to no rational result. Instead, however, of investigating in what cases this equation can be solved, he simply drops the equation x* + 97 =j2 and seeks, by altering his original assumptions, to obtain, in place of it, another equation of the same type which can be solved in rational numbers. In this case, by altering his assumptions, he is able to replace the refractory equa- tion by a new one, ^4 + 337=j2, and at the same time to find a suitable substitution for j, namely y = x*— 25, which brings out a rational result, namely x = ^-. This is a good example of his characteristic artifice of " Back-reckoning1/' as Nesselmann calls it.

5. Equation of sixth degree in the special form

x6 — Ax* + J3x + c2 =y-.

" Methode der Zuriickrechnung und Nebenaufgabe."

90 INTRODUCTION

It is only necessary to put j> = x3 + c, and we have -Ax* + B = 2cx\

-,—

A + 2C*

which gives a rational solution if B/(A + 2c) is a square.

6. If, however, this last condition does not hold, as in the case occurring iv. 18, x6 — i6xs + x + 64 =j/2, Diophantus employs his usual artifice of "back-reckoning," which enables him to replace the equation by another, Xs — 128^ + ^ + 4096 =/2, where the condition is satisfied, and, by assuming y = xs + 64, x is found to be^.

Second Class, Equation of the form

Axn + Bxn~l + . . . + Kx + L = j3.

Except for such simple cases as Ax2=y3, Ax* = y3t where it is only necessary to assume y = mx, the only cases occurring in Diophantus are of the forms

i. Equation Ax* + Bx + C=y*.

There are only two examples of this form. First, in VI. I the expression x* — ^x + 4 is to be made a cube, being already a square. Diophantus naturally assumes x — 2 = a cube number, say 8, and x= 10.

Secondly, in VI. 17 a peculiar case occurs. A cube is to be found which exceeds a square by 2. Diophantus assumes (x — i)3 for the cube and (x+ i)2 for the square, and thus obtains the equation

•** ~ 3^2 + Zx ~ i =*2 + 2* + 3) or x3 + x=4xz + 4,

previously mentioned (pp. 66-7), which is satisfied by ^ = 4. The question arises whether it was accidentally or not that this cubic took so simple a form. Were x-i, x+i assumed with knowledge and intention? Since 27 and 25 are, as Fermat observes1, the only integral numbers which satisfy the conditions, it would seem that Diophantus so chose his assumptions as to lead back to a known result, while apparently making them arbitrarily with no particular reference to the end desired. Had this not

1 Note on vi. 17, Oeuvi-es, I. pp. 333-4, II. p. 434. The fact was proved by Euler (Algebra, Part II. Arts. 188, 193). See note on vi. 17 infra for the proof.

DIOPHANTUS' METHODS OF SOLUTION 91

been so, we should probably have found him, here as elsewhere in the work, first leading us on a false tack and then showing us how we can correct our assumptions. The fact that he here makes the right assumptions to begin with makes us suspect that the solution is not based on a general principle but is empirical merely.

2. Equation Ax3 + Ex* + Cx+D =f.

If A or D is a cube number, this equation is easy of solution.

jy

For, first, if A = a?, we have only to write y = ax + — a , and we obtain a simple equation in x.

Secondly, if D = d3, we put y = — ^ x + d.

If the equation is a3xs + JBx2 + Cx+ ds=y3, we can use either assumption, or we may put y = ax + d, obtaining a simple equation as before.

Apparently Diophantus used the last assumption only ; for in IV. 27 he rejects as impossible the equation

because y = 2x — i gives a negative value x= — ^, whereas either of the other assumptions gives a rational value1.

( 2 ) Double-equations.

There are a few examples in which, of two functions of x, one is to be made a square, and the other a cube, by one and the same rational value of x. The cases are for the most part very simple ; e.g. in vi. 19 we have to solve

2X+ I =

thus j/3 = 2z*, and z = 2.

A rather more complicated case is VI. 21, where we have the double-equation

2X

Diophantus assumes y = mx, whence x = 2/(m2 — 2), and we have

2 V / 2 \2 2 2

2m*

(m* - 2)3

1 There is a special case in which C and D vanish, Ay?+ Bx~—y*. Here y is put equal to mx, and x=BI(mz - A). Cf. IV. 6, 28 (i).

92 INTRODUCTION

To make 2m* a cube, we need only make 2m a cube or put m = 4. This gives x=\. The general case

Ax9 + Bx" + Cx =

would, of course, be much more difficult; for, putting y = mx, we have

x=cl(m?-b),

and we have to solve

or Ccm* + c(Bc- 2bC) m? + be (bC -£c) + Ac3 = u3,

of which equation the above corresponding equation is a very particular case.

Summary of the preceding investigation,

1. Diophantus solves completely equations of the first degree, but takes pains to secure beforehand that the solution shall be positive. He shows remarkable address in reducing a number of simultaneous equations of the first degree to a single equation in one variable.

2. For determinate equations of the second degree he has a general method or rule of solution. He takes, however, in the Arithmetica^ no account of more than one root, even where both roots are positive rational numbers. But, his object being simply to obtain some solution in rational numbers, we need not be surprised at his ignoring one of two roots, even though he knew of its existence.

3. No equations of a degree higher than the second are solved in the book except a particular case of a cubic.

4. Indeterminate equations of the first degree are not treated of in the work. Of indeterminate equations of the second degree, as Ax* + Bx + C=y*> only those cases are fully dealt with in which A or C vanishes, while the methods employed only enable us to solve equations of the more general forms

Ax* + C=y> and Ax* + Bx+ C=f

when A, or C, or \B? -AC is positive and a square number, or (in the case of Ax* ± C=y2) when one solution is already known.

DIOPHANTUS' METHODS OF SOLUTION 93

5. For double-equations of the second degree Diophantus has a definite method when the coefficient of x* in both expressions vanishes ; the applicability of this method is, however, subject to conditions, and it has to be supplemented in one or two cases by another artifice. Of more complicated cases we find only a few examples under conditions favourable for solution by his method.

6. Diophantus' treatment of indeterminate equations of degrees higher than the second depends upon the particular conditions of the problems, and his methods lack generality.

7. More wonderful than his actual treatment 9f equations are the clever artifices by which he contrives to avoid such equations as he cannot theoretically solve, e.g. by his device of "back- reckoning," instances of which would have been out of place in this chapter and can only be studied in the problems themselves.

I shall not attempt to class as "methods" certain headings in Nesselmann's classification of the problems, such as (a) " Solution by mere reflection," (£) " Solution in general expressions," of which there are few instances definitely so described by Diophantus, or (c) "Arbitrary determinations and assumptions." The most that can be done by way of describing these " methods " is to quote a few characteristic instances. This is what Nesselmann has done, and he regrets at the end of his chapter on " Methods of Solution" that it must of necessity be so incomplete. To under- stand and appreciate the various artifices of Diophantus it is in fact necessary to read the problems themselves in their entirety.

With regard to the " Use of the right-angled triangle," all that can be said of a general character is that only " rational " right- angled triangles (those namely in which the three sides can all be represented by rational numbers) are used in Diophantus, and accordingly the introduction of the " right-angled triangle " is merely a convenient way of indicating the problem of finding two square numbers, the sum of which is also a square number. The general form used by Diophantus (except in one case, VI. 19, q.v.) for the sides of a right-angled triangle is (at + i?), (a'—P), 2ab, which expressions clearly satisfy the condition

The expression of the sides of a right-angled triangle in this form Diophantus calls "forming a right-angled triangle from the numbers a and b" His right-angled triangles are of course formed from particular numbers. " Forming a right-angled

94 INTRODUCTION

triangle from 7, 2 " means taking a right-angled triangle with sides (7s + 2"), (7s ~ 22), 2 . 7 . 2, or 53, 45, 28.

II. METHOD OF LIMITS.

As Diophantus often has to find a series of numbers in ascending or descending order of magnitude, and as he does not admit negative solutions, it is often necessary for him to reject a solution which he has found by a straightforward method because it does not satisfy the necessary condition ; he is then very frequently obliged to find solutions which lie within certain limits in place of those rejected.

1. A very simple case is the following : Required to find a value of x such that some power of it, xn, shall lie between two given numbers. Let the given numbers be a, b. Then Diophantus' method is to multiply both a and b by 2n, 3", and so on, successively, until some wth power is seen which lies between the two products. Thus suppose that cn lies between apn and bpn ; then we can put x = c\p, in which case the condition is satisfied, for (<://)n lies between a and b.

Exx. In IV. 31 (2) Diophantus has to find a square between i \ and 2. He multiplies both by a square, 64 ; this gives 80 and 128, and 100 is clearly a square which lies between them; there- fore (±gf or f| satisfies the prescribed condition.

Here, of course, Diophantus might have multiplied by any other square, as 16. In that case the limits would have become 20 and 32 , between these lies the square 25, which gives the same square ff as that before found.

In VI. 21 a sixth power ("cube-cube") is sought which shall lie between 8 and 16. The sixth powers of the first four natural numbers are I, 64, 729, 4096. Multiply 8 and 16 by 26 or 64, and we have as limits 512 and 1024, between which 729 lies. There- fore -7^j9- is a sixth power satisfying the given condition. To multiply by 729 in this case would not give us a solution.

2. Sometimes a value of x has to be found which will give some function of x a value intermediate between the values of two other functions of x.

Ex. i. In IV. 25 it is necessary to find a value of x such that 8/(;r2 +.*•) shall lie between x and x + i. The first condition gives 8 >x3 + x*.

DIOPHANTUS' METHODS OF SOLUTION 95

Diophantus accordingly assumes that

which is greater than xs+x*.

Thus # = f satisfies one condition. It is also seen to satisfy

o

the second condition, or — - <x+ I. Diophantus. however, says x* + x

nothing about the second condition being satisfied ; his method is, therefore, here imperfect.

Ex. 2. In V. 30 a value of x has to be found which shall make * > | (*2 - 60) but <l(;tr2-6o),

that is, x* — 60 >

x* - 60 < &r

Hence, says Diophantus, x is not less than 1 1 and not greater than 12. We have already spoken (p. 60 sqq.) of his treatment of such cases.

Next, the problem in question requires that x* — 6o shall be a square. Assume then that

x* — 60 = (x — mf, and we have x — (m* + 6o)/2m.

Since, says Diophantus,^ is greater than 11 and less than 12, it follows that

m2 + 60 > 22m but < 247/2 ;

and m must therefore lie between 19 and 21 (cf. p. 62 above). He puts m = 20, and so finds x=n^.

III. METHOD OF APPROXIMATION TO LIMITS.

We come, lastly, to a very distinctive method called by Diophantus TrapHrorrjs or Trapia-orijTos dytoyrj. The object of this is to solve such problems as that of finding two, or three, square numbers the sum of which is a given number, while each of them approximates as closely as possible to one and the same number.

This method can be best exhibited by giving Diophantus' two instances, in the first of which two such squares, and in the second three, are required. In cases like this the principles cannot be so well indicated with general symbols as with concrete numbers, which have the advantage that their properties are immediately

96 INTRODUCTION

obvious, and the separate expression of conditions is rendered unnecessary.

Ex. I. Divide 13 into two squares each of which >6 (v. 9). Take half of 13, or 6^, and find what small fraction ijx^ added to it will make it a square : thus

6^ + — , or 26 H — - , must be a square. Diophantus assumes

26+ -3 «Y$ + i) i or 26j2 + i = ($y -f i)a,

whence y= 10 and 1/7* = ^, or i/^ = ^; and

6^+^= a square, (W-

[The assumption of (i^+i)4 is not arbitrary, for assume 26?*+ i —(Py-\- i)s> and_y is then 2//(26— /"); since ijy should be a small proper fraction, $ is the most suitable and the smallest possible value for/, inasmuch as 26 -f < 2p or/8 + 2p + i > 27.]

It is now necessary, says Diophantus, to divide 13 into two squares the sides of which are both as near as possible to f^.

Now the sides of the two squares into which 13 is naturally decomposed are 3 and 2, and

3 is > ft by &, 2 is < ft by ^

But, if 3 — -fa , 2 + ^ were taken as the sides of two squares, the sum of the squares would be

which is > 13.

Accordingly Diophantus puts 3 — gx, 2 + i ix for the sides of the required squares, where therefore x is not exactly ^ but near it.

Thus (3 - 9*)' + (2 + i \xj = 1 3,

and Diophantus obtains ^r=_^T.

The sides of the required squares are f£f, f$f.

[It is of course a necessary condition that the original number, here 13, shall be a number capable of being expressed as the sum of two squares.]

DIOPHANTUS' METHODS OF SOLUTION 97

Ex. 2. Divide 10 into three squares such that each of them is > 3 (v. n).

[The original number, here 10, must of course be expressible as the sum of three squares.]

Take one-third of 10, or 3^, and find what small fraction of the form i/x* added to 3^ will make a square; i*. we have to make

30 + — , a square, or 30^+ i a square, where 3/r = i/y.

Diophantus assumes

307*+ i =(57+ i)*, whence .7 = 2 and therefore i/*» = & ; and 3^ + ^ff = W» a square.

[As before, if we assume y*)P = (py+ i)J,^' = 2//(3O— /*); and, since ijy must be a small proper fraction, 30— p* should be < 2p, or /* + 2/-f i >3i. Accordingly Diophantus chooses 5 for/ as being the smallest possible integral value.]

We have now, says Diophantus, to make each of the sides of our required squares as near as may be to ty.

Now 10

and 3, f, ± are the sides of three squares the sum of which is 10. Bringing (3, f, §) and -^ to a common denominator, we have

And

If now we took 3 — f$, f + §£, f + f£ for sides of squares, the sum of the squares would be 3 (*£f or ^^, which is > 10.

Accordingly Diophantus assumes as the sides of the three required squares

3-35*. f + 37*. 1 + 31*. where x must therefore be not exactly ^ but near it

Solving (3 - 3^r)« + (f + 37^ + (| + 31*)* = 10, or 10-116*+ 3555**= 10,

we find x = j^ ;

the required sides are therefore

and the required squares

1745041 lesigys 1658944 505681 » 505521 » 50B6X1 *

98 INTRODUCTION

Other instances of the application of the method will be found in V. 10, 12, 13, 14, where, however, the squares are not required to be nearly equal, but each of them is subject to limits which may be the same or different ; e.g. sometimes each square is merely required to be less than a given number (10, say), sometimes the squares lie respectively between different pairs of numbers, some- times they are respectively greater than different numbers, while they are always subject to the condition that their sum is a given number.

As it only lies within the scope of this Introduction to explain what we actually find in Diophantus, I cannot do more than give a reference to such investigations as those of Poselger in his " Beitrage zur unbestimmten Analysis" published in the Abhand- lungen der Koniglichen Akademie der Wissenscliaften zu Berlin aus dem Jahre 1832, Berlin, 1834. One section of this paper Poselger entitles " Annaherungs-methoden nach Diophantus," and obtains in it, on Diophantus' principles, a method of approximation to the value of a surd which will furnish the same results as the method of continued fractions, with the difference that the " Diophantine method " is actually quicker than the method of continued frac- tions, so that it may serve to expedite the latter.

CHAPTER V

THE PORISMS AND OTHER ASSUMPTIONS IN DIOPHANTUS

I HAVE already mentioned (in Chapter I.) the three explicit refer- ences made by Diophantus to " The Porisms " and the possibility that, if these formed a separate work, it may have been from that work that Diophantus took a number of other propositions relating to properties of numbers which he enunciates or tacitly takes for granted in the Arithmetica.

I begin with the three propositions for which he expressly refers to " The Porisms."

Porism i. In V. 3 he says, "We have it in the Porisms that, ' If each of two numbers and their product when severally added to the same given number produce squares, the squares with which they are so connected are squares of two consecutive numbers1.'"

That is to say, if x + a = mz, y + a = n*, and if xy + a is also a square, then m~n=i.

The theorem is not correctly enunciated, for it would appear that m ~ n = i is not the only condition under which the three expressions may be simultaneously squares.

For suppose

x + a = m*, y + a = n\ xy + a =/2.

By means of the first two equations we have xy + a = m*ri* — a (m2 + n2 - i) + a\

In order that

nfif- - a (m* + «2 — i ) + a*

may be a square certain conditions must be satisfied. One suffi- cient condition is

or m ~ n = ± i ,

which is Diophantus' condition.

1 Literally "(the numbers) arise from two consecutive squares" (yeybvaffiv avb 8i5o r(av Kara TO (%?}*)•

7—2

INTRODUCTION But we may also regard

as an indeterminate equation in m of which we know one solution, namely m = n± i .

Other solutions are then found by substituting z + (n ± i) for m, whence we obtain the equation

(nz-a)z*+2{n*(n± i)-a(n±i)}z

+ O2 - a)(n ± i )2 - a (nz - i ) + «2 =/2,

or (n- -a) z2 + 2 (n* -a)(n± i)z + {n(n ± i)-a}2=p\

which is easy to solve in Diophantus' manner, since the absolute term is a square.

But in the problem V. 3 three numbers are required, such that each of them, and the product of each pair, when severally added to a given number, produce squares. Thus if the third number be z, three additional conditions have to be satisfied, namely

z 4- a = it2, zx + a = v2, zy + a = w*. The two last conditions are satisfied, if m+i=n, by putting

z = 2 (x +y) — i = 4.m* -f 40? + I - 4a, when xz + a = {m(2m + i) - 2a}2

and

and perhaps this means of satisfying the conditions may have affected the formulation of the Porism1.

The problem V. 4 immediately following assumes the truth of the same Porism with — a substituted for -f a.

Porism 2. In V. 5 Diophantus says, " Again we have it in the Porisms that, ' Given any two consecutive squares, we can find in addition a third number, namely the number greater by 2 than the double of the sum of the two squares, which makes, the greatest of three numbers such that the product of any pair of them added to either the sum of that pair or the remaining number gives a square."'

That is, the three numbers

1 Euler has a paper describing and illustrating a general method of finding such "porisms" the effect of which is to secure that, when some conditions are satisfied, the rest are simultaneously satisfied ("De problematibus indeterminatis quae videntur plus quam determinata" in Novi Commentarii Acad. Petropol. 1756-57, Vol. vi. (1761), p. 85 sqq. = Commentationes arithmeticae collectae, I. pp. 245 — 259). This particular porism of Diophantus appears as a particular case in § 13 of the paper.

THE PORISMS AND OTHER ASSUMPTIONS 101

have the property that the product of any two plus either the sum of those two or the remaining number gives a square. In fact, if X, Y, Z denote the numbers respectively,

XY+X+ Y=( m* + m+i)*, XY+Z=(m* + m + 2)*,

\T 7 i y i 7 f ^ jj/- J- "2ff* -I- 2^* V '7 -I- y — /•?••/* _L 24** _l_ ^\*

ZJf + Z + ^ = (2/«s + /« + 2)2, Z^+ K=(2*»s + »+!)«.

Porism 3 occurs in v. 16. Unfortunately the text is defective and Tannery has had to supply three words1 ; but there can be no doubt that the correct statement of the Porism here in question is " The difference of any two cubes is also the sum of two cubes," i.e. can be transformed into the sum of two cubes, or two cubes can be found the sum of which is equal to the difference between any two given cubes. Diophantus contents himself with the enuncia- tion of the proposition and does not show how to prove it or how he effected the transformation in practice. The subject of the transformation of sums and differences of cubes was investigated by Vieta, Bachet and Fermat

Vieta (Zetetica, IV. 18-20) has three problems on the subject

(i) Given two cubes, to find in rational numbers two other cubes such that their sum is equal to the difference of the given cubes8.

As a solution of a* — fc=x*+y*, he finds a(a?-2P)

a* + P ' y~

rotj Uoplff fount on " Tarrwr Suo Kvfiur i) irrepaxh Kvfkai> <

3 The solution given by Vieta is obtainable thus. The given cubes being a3, A3, where a> b, we assume x - b, a - kx as the sides of the required cubes.

Thus whence

This reduces to a simple equation if we assume

• lP-a?Jk=o, or Jk = in which case

and the sides of the cubes are therefore

Vieta's second problem is similarly solved by taking a+x, kx-b as the sides of the required cubes, and the third problem by taking x - 6, kx - a as the sides of the required cubes respectively.

102 INTRODUCTION

(2) Given two cubes, to find in rational numbers two others such that their difference is equal to the sum of the given cubes.

Solving a3 + & = x3 —}>3, we find that

(3) Given two cubes, to find in rational numbers two cubes such that their difference is equal to the difference of the given cubes.

For the equation a3 - d3 = x3 — j3, Vieta finds

_ b(2a*-P) _ a (26s -a9)

; at + b3 ' y= tf + b3 as a solution1.

In the solution of (1} x is clearly negative if 2& >a3; therefore, in order that the result may be " rational," a3 must be > 2^. But for a " rational " result in (3) we must, on the contrary, have a3 < 2b3. Fermat was apparently the first to notice that, in consequence, the processes in (i) and (3) exactly supplement each other, so that by employing them successively we can effect the transformation required in (i) even when a3 is not > 2b3.

The process (2) is always possible ; therefore, by a suitable combination of the three processes, the transformation of a sum of two cubes into a difference of two cubes, or of a difference of two cubes into a sum or a difference of two other cubes is always

1 Vieta's formulae for these transformations give any number of very special solutions (in integers and fractions)of the indeterminate equation jcs+y3 + z? = v3, including solutions in which one of the first three cubes is negative. These special solutions are based on the assumption that the values of two of the unknowns are given to begin with. Euler observed, however, that the method does not give all the possible values of the other two even in that case. Given the cubes 33 and 43, the method furnishes the solution

33 + 43+(— Y=(— Y, but not the simpler solution 33 + 43+53 = 63. Euler ac-

cordingly attacked the problem of solving the equation x?+y3 + zs = v3 more generally. He began with assuming only one, instead of two, of the cubes to be given, and, on that assumption, found a solution much more general than that of Vieta. Next he gave a more general solution still, on the assumption that none of the cubes are given to begin with. Lastly he proceeded to the problem To find all the sets of three integral cubes the sum of which is a cube and showed how to obtain a very large number of such sets including sets in which one of the cubes is negative (Novi Commentarii Acad. Petropol- 1756—57, Vol. VI. (1761), p. 155 sq. = C omni enta done s arithmeticae, I. pp. 193 — 207). The problem of solving xs+y3=z3 + 7p in integers in any number of ways had occupied Frenicle, who gave a number of solutions (Oeuvres de Fermat, ill. pp. 420, 535) ; but the method by which he discovered them does not appear,

THE PORISMS AND OTHER ASSUMPTIONS 103

practicable1. Fermat showed also how, by a repeated use of the several processes as required, we can transform a sum of two cubes into a sum of two other cubes, the latter sum into the sum of two others and so on ad infinitum*.

Besides the " Porisms " there are many other propositions assumed or implied by Diophantus which are not definitely called

1 Fermat (note on IV. 2) illustrates by the following case :

Given two cubes 125 and 64, to transform their difference into the sum of two other cubes.

Here a=s, 6=4, and so 26s > a3', therefore we must first apply the third process by which we obtain

'-"(*)' -(*)'•

As f -7 J > 2 / -~- J , we can, by the first process, turn the difference between the

cubes ( -—- ) and ( ?- ) into the sum of two cubes. \<>3/ \63/

" In fact," says Fermat, "if the three processes are used in turn and continued ad infinitum, we shall get a succession ad infinitum of two cubes satisfying the same condition ; for from the two cubes last found, the sum of which is equal to the difference of the two given cubes, we can, by the second process, find two more cubes the difference of which is equal to the sum of the two cubes last found, that is, to the difference between the two original cubes; from the new difference between two cubes we can obtain a new sum of two cubes, and so on ad infinitum"

As a last illustration, to show how a difference between cubes can be transformed into the difference between two other cubes even where the condition for process (3) is not satisfied, Fermat takes the case of 8- i, i.e. the case where a = 2, d=i and o?>ilP.

First use process (i) and we have

--=®'+©

Then use process (2), and

2 Suppose it required to solve the fourth problem of transforming the sum of two cubes into the sum of two other cubes.

Let it be required so to transform 23+ i3 or 9.

First transform the sum into a difference of two cubes by process (2). This gives

The latter two cubes satisfy the condition for process (3) and, applying that process, we get

/2oV_ /i7\» /I88479V _ f 365*oV \ II \7 / \90391 / \9°391/

The cubes last found satisfy the condition for process ( i), and accordingly the difference between the said last cubes, and therefore the sum of the original cubes, is at last trans- formed into the sum of two other cubes.

io4 INTRODUCTION

porisms, though some of them are of the same character as the three porisms above described.

Of these we may distinguish two classes.

I. The first class of theorems or facts assumed without ex- planation by Diophantus are more or less of the nature of identical formulae. Some are quite simple, e.g. the facts that the expressions [%(a+b}}z — ab and a* (a + i)2 + a2 + (a + i)2 are respectively squares, that the expression a (a2 — a) + a + (a* - a) is always a cube, and the like.

Others are more difficult and betoken a certain facility in work- ing with quasi-algebraical expressions. Examples of this kind are the following :

(a) If X=a2x+2a, Y=(a + i)*x + 2(a + i ), or, in other words, if xX + i =(ax+ i)2, xY+ i=[(a+ i)x+ i}2, then XY + i is a square [IV. 20]. As a matter of fact,

(/3) 8 times a triangular number plus \ gives a square [IV. 38]. In fact, 8 . £(£±1) + j = (2x + i)2.

(7) If X±a = m\ Y±a = (m + i)2, and Z=2(X+Y)-i, then the expressions YZ ± a, ZX ± a, X Y ± a are all squares. (The upper signs refer to the assumption in V. 3, the lower to that in V. 4.)

In fact, YZ ± a = {(m + i)(2m + i) + 2a}\

ZX ± a= [m(2m + i)+ 2a}2, XY±a={m(m+ i) + a}\

then the six expressions

FZ-( F+Z), ZX-(Z+X), XY-(X+ F) YZ-X, ZX-Y, XY-Z

are all squares [v. 6]. In fact, YZ - ( F+ Z) = (2m* + $m + 3)2, FZ - X = (2m* + $m + 4)2, etc.

2. The second class is much more important, consisting of a number of propositions in the Theory of Numbers which we find

THE PORISMS AND OTHER ASSUMPTIONS 105

first stated or assumed in the Arithmetica. It was, in general, in explanation or extension of these that Fermat's most famous notes were written. How far Diophantus possessed scientific proofs of the propositions which he assumes, as distinct from a merely empirical knowledge of them, must remain to a great extent matter of speculation.

(a) Theorems in DiopJtantus respecting the composition of num- bers as the sum of two squares.

(1) Any square number can be resolved into two squares in any number of ways, II. 8.

(2) Any number which is the sum of two squares can be resolved into two other squares in any number of ways, II. 9,

N.B. It is implied throughout that the squares may be frac- tional as well as integral.

(3) If there are two whole numbers each of which is the sum of tivo squares, their product can be resolved into the sum of two squares in two ways, III. 19.

The object of III. 19 is to find four rational right-angled triangles having the same hypotenuse. The method is this. Form two right-angled triangles from (a, b) and (c, d) respectively, i.e. let the sides of the triangles be respectively

and c- + d\ ? - d*, 2cd.

Multiplying all the sides in each by the hypotenuse of the other, we have two triangles with the same hypotenuse, namely

(a* + &}((* + dz), (a2 - P)(<* + d*\ 2ab (t* + d*\ and (*» + P)(e? + d-\ (a2 + P)((* - d*), 2cd (a* + #).

Two other triangles having the same hypotenuse are obtained by using the theorem enunciated. In fact,

(a2 + b*)(c* + d*) = (ac ± bd^ + (ad + be?

and the triangles are formed from ac ± bd, ad + be, being the triangles

(a2 + P)(c- + d*\ \abcd + (a2 - b*)(c- - d*), 2(ac-\- bd)(ad - bc\ (a* + &)((? + d"-\ ^abcd - (a2 - &*)(<;* - d*\ 2 (ac - bd)(ad + be).

106 INTRODUCTION

In the case taken by Diophantus

and the four triangles are respectively

(65, 52, 39), (65, 60, 25), (65, 63, 16), (65, 56, 33).

(If certain relations1 hold between a, b, c, d, this method fails. Diophantus has provided against them by taking two triangles " in the smallest numbers" (VTTO eXa%to-T&>i> dpid/j,oov), namely 3,4, 5 and 5, 12, 13.)

Upon this problem III. 19 Fermat has a long and important note which begins as follows2 :

" [i] A prime number of the form 4^+1 is the hypotenuse of a right-angled triangle in one way only, its square is so in two ways, its cube in three, its biquadrate in four ways, and so on ad infinitum.

"[2] The same prime number 4« + I and its square are the sum of two squares in one way only, its cube and its biquadrate in two ways, its fifth and sixth powers in three ways, and so on ad infinitum.

"[3] If a prime number which is the sum of two squares be multiplied into another prime number which is also the sum of two squares, the product will be the sum of two squares in two ways ; if the first prime be multiplied into the square of the second

1 (i) We must not have a\b — c\d or a\b = d\c, for in either case one of the perpendiculars of one of the resulting triangles vanishes, making that triangle illusory. Nor (2) must c\d be equal to (a + b)l(a-b) or to (a-l>)j(a + l>), for in the first case ac-bd=ad+bc, and in the second case ac + bd=ad-bc, so that one of the sums of squares equal to (a? + l>2) (c2 + cf2) is the sum of two equal squares, and consequently the triangle formed from the sides of these equal squares is illusory, one of its perpendicular sides vanishing.

3 G. Vacca (in Bibliotheca Mat hematic a, H3. 1901, pp. 358-9) points out that Fermat seems to have been anticipated, in the matter of these theorems, by Albert Girard, who has the following note on Diophantus v. 9 (Oeuvres mathhnatiques de Simon Stevin par Albert Girard, 1634, p. 156, col. i):

" ALB. GlR. Determinaison d'un nombre qui se peut diviser en deux quarrez entiers.

I. Tout nombre quarre.

II. Tout nombre premier qui excede un nombre quaternaire de 1'unite.

III. Le produict de ceux qui sont tels.

IV. Et le double d'un chacun d'iceux.

Laquelle determinaison n'estant faicte n'y de PAutheur n'y des interpretes, servira tant en la presente et suivante comme en plusieurs autres. "

Now Girard died on 9 December, 1632 ; and the Theorems of Fermat above quoted are apparently mentioned by him for the first time in his letter to Mersenne of 25 December, 1640 (Oeuvres de Fermat^ II. p. 213). Was the passage of Girard known to Fermat ?

THE PORISMS AND OTHER ASSUMPTIONS 107

prime, the product will be the sum of two squares in three ways ; if the first prime be multiplied into the cube of the second, the product will be the sum of two squares in four ways, and so on ad infinitum*."

It is not probable that Diophantus was aware that prime num- bers of the form 4# + I and numbers arising from the multiplication of such numbers are the only classes of numbers which are always the sum of two squares ; this was first proved by Euler2. But it is remarkable that Diophantus should have selected the first two prime numbers of the form 4«+ I, namely 5 and 13, which are both sums of two squares, as the hypotenuses of his first two right- angled triangles and then made their product, 65, the hypotenuse of other right-angled triangles, that product having precisely the property of being, as in Fermat's [3], the sum of two squares in two ways. Diophantus may therefore have had an inkling, whether obtained empirically or otherwise, of some of the properties enunci- ated by Fermat.

(4) Still more remarkable is a condition of possibility of solution prefixed to the problem V. 9. The object of this problem is " to divide I into two parts such that, if a given number is added to either part, the result will be a square." Unfortunately, the text of the added condition is uncertain. There is no doubt about the first few words, " The given number must not be odd" />. No number of tlie form 4« + 3 [or 4*1 — i] can be tJte sum of two squares.

The text, however, of the latter half of the condition is corrupt. The true condition is given by Fermat thus : " The given number must not be odd, and the double of it increased by one, when divided by tJte greatest square which measures it, must not be divisible by a prime number of the form %n— I." (Note upon V. 9; also in a letter to Roberval3.) There is room for any number of conjectures as to what may have been Diophantus' words4.

1 For a fuller account of this note see the Supplement, section I.

8 Novi Commentarii Acad. Petrofol. 1751-3, Vol. IV. (1758), pp. 3-40, and 1754-5, Vol. v. (1760), pp. 3-58= Commentationes arithmcticae, I. pp. 155-173 and pp. 110—233 '•> cf. Legendre, Zahlentheorie, tr. Maser, I. p. 108; Weber and Wellstein's EncyclopdtSe der Ekmentar-Mathematik, I2. pp. 285 sqq.

3 Ofttvres de Fermat, II. pp. 203-4, See the Supplement, section I.

4 Bachet's text has 5e? 5^ rbv di86tJKvov /n^re re/xerffdr flrcu, fir/re 6 SwXcurfuH' oirrou q' n° a. neifova. txi P*(KK 5. 17 fifrpeiTai irrb rov a0". t°*.

He also says that a Vatican MS. reads nijre 6 5ir\affiw afoov aptOnov /xortWa a. fjLti^ova txV M^pos TfTaprov, rj fjLerpeirai intb rov ffxarov apiffpov.

Neither does Xylander help us much. He frankly tells us that he cannot understand

io8 INTRODUCTION

There would seem to be no doubt that in Diophantus' condition there was something about " double the number " (i.e. a number of the form \n\ also about " greater by unity " and " a prime number." It seems, then, not unlikely that, if Diophantus did not succeed in giving the complete sufficient and necessary condition stated by Fermat, he made an approximation to it ; and he certainly knew that no number of the form 4# + 3 could be the sum of two squares l.

(b) On members which are the sum of three squares.

In the problem v. 1 1 a condition is stated by Diophantus re- specting the form of a number which, added to three parts of unity, makes each of them square. If a be this number, clearly 30+1 must be divisible into three squares.

Respecting the number a Diophantus says, " It must not be 2 or any multiple of 8 increased by 2."

That is, a number of the form 2^n + 7 cannot be the sum of three squares. Now the factor 3 of 24 is irrelevant here, for with respect to three this number is of the form ^m + I, and this, so far as 3 is concerned, might be a square or the sum of two or three squares. Hence we may neglect the factor 3 in 24^.

We must therefore credit Diophantus with the knowledge of

the passage. " Imitari statueram bonos grammaticos hoc loco, quorum (ut aiunt) est multa nescire. Ego vero nescio heic non multa, sed paene omnia. Quid enim (ut reliqua taceam) est ju^re 6 diirXafflwv O.VTOV op fju> a etc., quae causae huius irpoffSiopifffj,ov, quae processus? immo qui processus, quae operatic, quae solutio?"

Nesselmann discusses an attempt made by Schulz to correct the text, and himself suggests Atiyre TOV SurXaaiova avTov &pi9/j.bv fj.ovdSi /j.ei£ova ^xet") & perpetrat vvo TIVOS irp&Tov dpi6fwv. But this ignores frfpos rtraprov and is not satisfactory.

Hankel, however (Gesch. d. Math. p. 169), says: " Ich zweifele nicht, dass die von den Msscr. arg entstellte Determination so zu lesen ist : Aei 5^ rbv diS6/j.evov n^re wepiffffbv elva.i, /tijre ^bv oiirXafftova. O.VTOV api8fj.bv fiovASi d (Jifl^ova /j.fTptiff6ai vir6 TOV irp&Tov dpiOfiov, 6s &v fi.ovA.dt. d ndfav fyv V-tpos rtrapTov." This correction seems a decidedly probable one. Here the words ptpos T^raprov find a place ; and, secondly, the repetition of fj.ov&di d fj.d$wv might well confuse a copyist. TOV for TOV is of course natural enough ; Nesselmann reads rtvos for TOV.

Tannery, improving on Hankel, reads A« 5r; TOV $i56u.fvov yu^re wepurffov elvai, //.ijre trip SurXturioj' avrov Kal fj.ov6.8i M'? /xeiforo fjierpeiffOai viro TOV irp&TOv dpiB/J.ov <ov 6 fiovddt, fdq. fj.eifav> ?x?7 M^pos T^Taprov t, " the given number must not be odd, and twice it plus i must not be measured by any prime number which, when increased by i, is divisible by 4."

1 A discussion of the text and a suggestion as to the considerations which may have led to the formulation of the condition will be found in Jacobi, " Ueber die Kenntnisse des Diophantus von der Zusammensetzung der Zahlen" (Berliner Monatsberichte, 1847; Gesammelte Werke, vil., 1891, pp. 332-344).

THE PORISMS AND OTHER ASSUMPTIONS 109

the fact that no nttmber of the form 8« + 7 can be the sum of three squares*.

This condition is true, but does not include all the numbers which cannot be the sum of three squares, for it is not true that all numbers which are not of the form 8# + 7 are made up of three squares. Even Bachet remarked that the number a might not be of the form 32^ + 9, or a number of the form 96^ + 28 cannot be the sum of three squares.

Fermat gives the conditions to which a must be subject thus2.

Write down two geometrical series (common ratio of each 4), the first and second series beginning respectively with i, 8,

i 4 16 64 256 1024 4096 8 32 128 512 2048 8192 32768;

then a must not be (i) any number obtained by taking twice any term of the upper series and adding all the preceding terms, or (2) the number found by adding to the numbers so obtained any multiple of the corresponding term of the second series. Thus a must not be

128/^+2.16 + 4+1 =128/^ + 37, 512^+ 2.64+ 16 + 4+ i = 512^+ 149, and so on, where k = o or any integer.

That is, since i + 4 + 42 + . . . + 4n-1 = £(4" — i ), a cannot be either

therefore 30 + I cannot be of the form 4" (24^ + 7) or 4n (8£ + 7).

Again, there are other problems, e.g. v. 10 and V. 20, in which, though conditions are necessary for the possibility of solution, none are mentioned ; but suitable assumptions are tacitly made, without explanation. It does not follow, from the omission to state the conditions, that Diophantus was ignorant of even the minutest points connected with them ; as, however, we have no definite statements, we must be content to remain in doubt.

1 Legendre proved (Zahlentheorie, tr. Maser, I. p. 386), that numbers of this form are the only odd numbers which are not divisible into three squares. 3 Note on Diophantus v. 1 1 .

no INTRODUCTION

(c) Composition of numbers as the sum of four squares.

Every number is either a square or the sum of two, three or four squares. This well-known theorem, enunciated by Fermat1, and proved by Lagrange2 (who followed up results obtained by Euler) shows at once that any number can be divided into four squares either integral or fractional, since any square number can be divided into two other squares, integral or fractional. We have now to look for indications in the Arithmetica as to how far Diophantus was acquainted with the properties of numbers as the sum of four squares. Unfortunately, it is impossible to decide this question with anything like certainty. There are three problems, iv. 29, 30 and V. 14, in which it is required to divide a number into four squares, and from the absence of mention of any condition to which the number must conform, considering that in both cases where a number is to be divided into three or two squares, v. 1 1 and V. 9, he does state a condition, we should probably be right in inferring that Diophantus was aware, at least empirically, that any number could be divided into four squares. That he was able to prove the theorem scientifically it would be rash to assert. But we may at least be certain that Diophantus came as near to the proof of it as did Bachet, who takes all the natural numbers up to 120 and finds by trial that all of them can actually be expressed as squares, or as the sum of two, three or four squares in whole numbers. So much we maybe sure that Diophantus could do, and hence he might have empirically satisfied himself that it is possible to divide any number into four squares, integral or fractional, even if he could not give a rigorous mathematical demonstration of the general theorem.

1 See note on Diophantus IV. 29 ; cf. also section I. of the Supplement.

z " Demonstration d'un theoreme d'arithmetique " in Nouveaux Memoires de PAcad. royale des sciences de Berlin, annee 1770, Berlin 1772, pp. 123-133= Oeuvres de Lagrange, in. pp. 187-201 ; cf. Wertheim's account of the proof in his Diophantus, pp. 324-330.

CHAPTER VI

THE PLACE OF DIOPHANTUS

IN algebra, as in geometry, the Greeks learnt the beginnings from the Egyptians. Familiarity on the part of the Greeks with Egyptian methods of calculation is well attested. Thus (i) Psellus in the letter published by Tannery1 speaks of "the method of arithmetical calculations used by the Egyptians, by which problems in analysis are handled " (77 /car' Alyvrrrtovf rtov aptdfjbwv /j,edoBo<i, 81 779 oiKovofj^eirat ra Kara rrjv dva\VTiicr)v 7rpo/3A,?7/4aTa) ; the details which he goes on to give respecting the technical terms for different kinds of numbers, including the powers of the unknown quantity, in use among the Egyptians are doubtless taken from Anatolius. (2) The scholiast to Plato's Charmides 165 E may be drawing on the same source when he says that " parts of \oyia-TtKij (the science of calculation) are the so-called Greek and Egyptian methods in multiplications and divisions, and the additions and subtractions of fractions.... The aim of it all is the service of common life and utility for contracts, though it seems to deal with things of sense as if they were perfect or abstract." (3) Plato himself, in the Laws2, says that free-born boys should, as is the practice in Egypt, learn, side by side with reading, simple mathematical calculations adapted to their age, which should be put into a form such as to give amusement and pleasure as well as instruction ; e.g. there should be different distributions of such things as apples, garlands, etc., different arrangements of numbers of boys in contests of boxing or wrestling, illustrations by bowls of different metals, gold, copper, silver, etc., and simple problems of calculation of mixtures ; all of which are useful in military and civil life and " in any case make men more useful to themselves and more wide-awake."

1 Dioph. II. pp. 37-42. 2 Laws, VII. 819 A-c.

ii2 INTRODUCTION

The Egyptian calculations here in point (apart from their method of writing and calculating in fractions, which differed from that of the Greeks in that the Greeks worked with ordinary fractions, whereas the Egyptians separated fractions into sums of submultiples, with the exception of | which was not decomposed) are the ^^-calculations. Hau, meaning a heap, is the term used to denote the unknown quantity, and the calculations in terms of it are equivalent to the solutions of simple equations with one unknown quantity1. Examples from the Papyrus Rhind2 corre- spond to the following equations:

= 19,

(jr+f*)-i (* + !*) =10-

Before leaving the Egyptians, it is right to mention their anticipation, though in an elementary form, of a favourite method of Diophantus, that of the " false supposition " or " regula falsi " as it is sometimes called. An arbitrary assumption is made as to the value of the unknown, and the value is afterwards corrected by a comparison of the result of substituting the wrong value in the original expression with the actual fact. Two instances mentioned by Cantor3 may be given. The first, taken from the Papyrus Rhind, is the problem of dividing 100 loaves among five persons in numbers forming an arithmetical progression and such that one- seventh of the sum of the first three shares is equal to the sum of the other two. If a + qd, a + ^d, a+2d, a + d, a are the shares, we have

or d = $^a.

Ahmes merely says, without explanation, " make the difference, as it is, 5^," and then, assuming a=i, writes the series 23, 17^, 12, 6^, i. The addition of these gives 60, and 100 is if times 60. Ahmes says simply "multiply if times" and thus gets the correct values 38^, 294, 20, iof £, if. The second instance (taken from the Berlin Papyrus 6619) is the solution of the equations

x*+y*=ioo, x \y = i : f , or y = £ x.

1 For a complete account of the subject the reader is referred to Moritz Cantor's Geschichte der Mathematik, I3. Chapter II., especially pp. 74-81.

2 Eisenlohr, Ein mathematisches Handbuch der alten Agypter {Papyrus Rhind des British Museum) , Leipzig, 1877.

3 Geschichte der Math. I3. pp. 78-9 and p. 95.

THE PLACE OF DIOPHANTUS 113

x is first assumed to be I, and x*+jP is then found to be 25/16. In order to make 100, 25/16 has to be multiplied by 64 or S2. The true value of x is therefore 8 times I, or 8.

The simple equations solved in the Papyrus Rhind are just the kind of equations of which we find numerous examples in the arithmetical epigrams included in the Greek Anthology. Most of these appear under the name of Metrodorus, a grammarian, who probably lived about the time of the Emperors Anastasius I. (491-518 A.D.) and Justin I. (518-527 A.D.). They were obviously only collected by Metrodorus, from ancient as well as more recent sources ; none of them can with certainty be attributed to Metro- dorus himself. Many of the epigrams (46 in number) lead to simple equations, with one unknown, of the type of the hau- equations, and several of them are problems of dividing a number of apples or nuts among a certain number of persons, that is to say, the very type of problem alluded to by Plato. For example, a number of apples has to be determined such that, if four persons out of six receive one-third, one-eighth, one-fourth and one-fifth respectively of the total number of apples, while the fifth person receives ten apples, there remains one apple as the share of the sixth person, i.e.

We are reminded of Plato's allusion to problems about bowls (<f>id\at) of different metals by two problems (Antliol. Palat. XIV. 12 and 50) in which the weights of bowls have to be found. We can now understand the allusions of Proclus1 and the scholiast on Charmides 165 E to p,rj\lTac and (f>ia\iTai dpt0(j,oi, the adjectives being respectively formed from p.fi\ovy an apple, and </>taXi;, a bowl. It is clear from Plato's allusions that the origin of such simple algebraical problems dates back, at least, to the fifth century B.C.

I have not thought it worth while to reproduce at length the problems contained in the Anthology2, but the following is a classification of them, (i) Twenty-three are simple equations containing one unknown and of the type shown above ; one of these is the epigram on the age of Diophantus and incidents in his life (XIV. 126). (2) Twelve more are easy simultaneous

1 Proclus, Comment, on Eucl. /., ed. Friedlein, p. 40, 5.

2 They are printed in Greek, with the scholia, in Tannery's edition of Diophantus (il. pp. 43-72 and x), and they are included in Wertheim's German translation of Diophantus, pp. 331-343.

H. D. 8

ii4 INTRODUCTION

equations containing two unknowns, and of the same sort as Diophantus I. 1-6 ; or, of course, they can be reduced to a simple equation with one unknown by means of an easy elimination. One other (XIV. 51) gives simultaneous equations in three un- knowns

and one (XIV. 49) gives four equations in four unknowns

With these may be compared Diophantus I. 16-21. (3) Six more are problems of the usual type about the filling of vessels by pipes : e.g. (XIV. 130) one pipe fills the vessel in one day, a second in two, and a third in three ; how long will all three running together take to fill it? Another about brickmakers (XIV. 136) is of the same sort

The Anthology contains (4) two indeterminate equations of the first degree which can be solved in positive integers in an infinite number of ways (xiv. 48 and 144); the first is a distribution of apples satisfying the equation x — ^y =y, where y is not less than 2, and the original number of apples is $x ; the second leads to the following three equations between four unknown quantities :

the general solution of which is x = 4^, y = £, xl = $k, y^ = 2k. These very equations, made however determinate by assuming that x+y = xl +j/i = 100, are solved in Diophantus I. 12.

We mentioned above the problem in the Anthology (XIV. 49) leading to the following four simultaneous linear equations with four unknown quantities,

x + y = a,

x + ti = c,

The general solution of any number of simultaneous linear equations of this type with the same number of unknown quantities was given by Thymaridas, apparently of Paros, and an early Pythagorean. He gave a rule, e</>oSo<?, or method of attack (as

THE PLACE OF DIOPHANTUS 115

lamblichus1, our informant, calls it) which must have been widely known, inasmuch as it was called by the name of the e-n-avdrffia, " flower" or "bloom," of Thymaridas. The rule is stated in general terms, but, though no symbols are used, the content is pure algebra. Thymaridas, too, distinguishes between what he calls dopia-Tov, the undefined or unknown quantity, and the atpio-pevov, the definite or known, therein anticipating the very phrase of Diophantus, TrXijOos povdSow dopta-rov, "an undefined number of units," by which he describes his dpidfjios or x. Thymaridas' rule, though obscurely expressed, states in effect that, if there are n equations between n unknown quantities ;r, xlt x^...xn_l of the following form,

x*+ ... + x then the solution is given by

« — 2

lamblichus goes on to show that other types of equations can be reduced to this, so that the rule does not leave us in the lurch (ov Tra/aeX/cet) in those cases either. Thus we can reduce to Thymaridas' form the indeterminate problem represented by the following three linear equations between four unknown quantities :

b(u + y\

From the first equation we obtain

x + y + z + u = (a + i) (z + u),

from which it follows that, if x, y, z, u are all to be integers, x+y + z + u must contain a+i as a factor. Similarly it must contain b + I and c + I as factors.

Suppose now that x+y+z+u = (a+ i)(£+ i)fc+ i). There- fore, by means of the first equation, we get

(x+y} i + =(«+ !)(*+!)(*+ I),

1 lamblichus, /« Nicomachi arithmeticam introductionem (ed. Pistelli), pp. 62, 18-68, 26.

8—2

n6 INTRODUCTION

or

Similarly x+ z = b (c + i)(a+ i),

x + u = c(a+ i)(b+ i),

and the equations are in the form to which Thymaridas' rule is applicable.

Hence, by that rule,

_ a(b+ i)(V+i) +...-(<?+ I) (b 4-

In order to ensure that x may always be integral, if is only necessary to assume

x+y+z + u = 2(a+ i)(b + i)(c+ i).

The factor 2 is of course determined by the number of un- knowns. If there are n unknowns, the factor to be put in place of 2 is n — 2.

lamblichus has the particular case corresponding to a = 2, b = 3, c = 4. He goes on to apply the method to the equations

^0"M),

for the case where £//= f , mjn = |, pjq = f .

Enough has been said to show that Diophantus was not the inventor of Algebra. Nor was he the first to solve indeterminate problems of the second degree.

Take, first, the problem of dividing a square number into two squares (Diophantus II. 8), or of finding a right-angled triangle with sides in rational numbers. This problem was, as we learn from Proclus1, attributed to Pythagoras, who was credited with the discovery of a general formula for finding such triangles which may be shown thus :

- i\2 _ /;?2H

\ 2

where n is an odd number. Plato again is credited, according to the same authority, with another formula of the same sort,

Comment, on Euclid, Book /. pp. 428, 7 sqq.

THE PLACE OF DIOPHANTUS 117

Both these formulae are readily connected with the geometrical proposition in Eucl. II. 5, the algebraical equivalent of which may be stated as

The content of Euclid Book II. is beyond doubt Pythagorean, and this way of stating the proposition quoted could not have escaped the Pythagoreans. If we put I for b and the square of any odd number for a, we have the Pythagorean formula ; and, if we put a = 211*, b = 2, we obtain Plato's formula. Euclid finds a more general formula in Book X. (Lemma following X. 28). Starting with numbers u = c + b and v = c — b, so that

uv = cz- P,

Euclid points out that, in order that uv may be a square, u and v must be " similar plane numbers " or numbers of the form mnfP, mngr*. Substituting we have

But the problem of finding right-angled triangles in rational numbers was not the only indeterminate problem of the second degree solved by the Pythagoreans. They solved the equation

2x*-y*=± i .

in such a way as to prove that there are an infinite number of solutions of that equation in integral numbers. The Pythagoreans used for this purpose the system of "side-" and "diagonal-" numbers1, afterwards fully described by Theon of Smyrna 2. We begin with unity as both the first "side" and the first " diagonal";

thus

#1= i, d^= i.

We then form (a2, d2), (a3, d3), etc., in accordance with the following law,

and so on. Theori states, with reference to these numbers, the general proposition that

dn2=2an*± i, and observes that (i) the signs alternate as successive d's and a's

1 See Proclus, In Platonis rempublicam commtntarii (Teubner, Leipzig), Vol. II. c. 27, p. 27, 11-18.

2 Theon of Smyrna, ed. Hiller, pp. 43, 44.

n8 INTRODUCTION

are taken, d? — 2a? being equal to — i, d?-2a? equal to+i, d? - 2#32 equal to - I and so on, (2) the sum of the squares of all the d's will be double of the sum of the squares of all the a's. For the purpose of (2) the number of successive terms in each series, if finite, must of course be even. The algebraical proof is easy.

and so on, while df — 2a? = — I. Proclus tells us that the property was proved by means of the theorems of Eucl. II. 9, 10, which are indeed equivalent to

(2.X +j)2 - 2 (X +_X)2 = 2*2 -f.

Diophantus does not particularly mention the indeterminate equation 2x*— i =j2, still less does he mention "side-" and "diagonal-" numbers. But from the Lemma to VI. 15 (quoted above, p. 69) it is clear that he knew how to find any number of solutions when one is known. Thus, seeing that x= i, /= i is one solution, he would put

2 (i + xf — i = a square

= (px- i)2 say, whence # = (4 + 2/)/(/2 - 2).

Take the value p=2, and we have ^ = 4, or ;tr+i=5; and 2 . 52 — i = 49 = 72. Putting x + 5 in place of x, we find a still higher value, and so on.

In a recent paper Heiberg has published and translated, and Zeuthen has commented on, still further Greek examples of in- determinate analysis1. They come from the Constantinople MS. (probably of I2th c.) from which Schone edited the Metrica of Heron. The first two of the thirteen problems had been published before (though in a less complete form) 2 ; the others are new.

The first bids us find two rectangles such that the perimeter of the second is three times that of the first, and the area of the first is three times that of the second (the first of the two con- ditions is, by some accident, omitted in the text). The number 3

1 Bibliotheca Mathematica, vm3, 1907-8, pp. 118-134.

2 Hultsch's Heron, Geeponicat 78, 79. The two problems are discussed by Cantor, Agrimensoren, p. 62, and Tannery, Mem. dc la soc. des sc. de Bordeaux, IV2, 1882.

THE PLACE OF UIOPHANTUS 119

is of course only an illustration, and the problem is equivalent to the solution of the equations

xy = n.uv }

the solution given in the text is equivalent to x = 2n*-i, y = 2n* ) u = n (4>/s — 2), v = n)

Zeuthen suggests that this solution may have been arrived at thus. As the problem is indeterminate, it would be natural to make trial of some hypothesis, e.g. to put v = n. It would follow from the first equation in (i) that u is a multiple of n, say nz. We have then

x +y =1+2,

xy = ns2,

whence xy = n3 (x +y) — n3,

or (x - n3) (y - «3) = n3 (n3 - i ).

An obvious solution of this is

x — n3 = n3 — i, y — 1? = n3.

The second problem is equivalent to the solution of the equations

I ........................ (I);

xy = n . uv)

and the solution given in the text is

i ..................... (2),

In this case trial may have been made of the assumption

v = nx, y = «2#, when the first equation in (i) would give

(«-i);r = («2-i)/,, a solution of which is x= w2 — i, u = n — i.

The fifth problem is of interest in one respect. We are asked to find a right-angled triangle (in rational numbers) with area of 5 feet. We are told to multiply 5 by some square containing 6 as a factor, e.g. 36. This makes 180, and this is the area of the triangle (9, ^b, til). Dividing each side by 6, we have the triangle required. The author, then, is aware of the fact that the area of a right-angled triangle with sides in whole numbers is divisible

120 INTRODUCTION

by 6. If we take the Euclidean formula for a right-angled triangle, thus making the sides

m2 — #2 m* 4- #2

a . mn. a . - — , a . — — ,

2 2

where a is any number, and m, n are numbers which are both odd or both even, the area is

2 mn (m — n} (m + n)

4

and, as a matter of fact, the numerator mn(m — n)(m + n) is divisible by 24, as was proved later (for another purpose) by Leonardo of Pisa1. ' There is no sign that Diophantus was aware of the proposition ; this however may be due to the fact that he does not trouble as to whether his solutions are integral, but is satisfied with rational results.

The last four problems (numbered 10 to 13) are of great interest. They are different particular cases of one problem, that of finding a rational right-angled triangle such that the numerical sum of its area and all its three sides is a given number. The author's solution depends on the following formulae, where a, b are the perpendiculars, and c the hypotenuse, of a right-angled triangle, 6" its area, r the radius of its inscribed circle, and

S = rs = \ab> r + s = a + b, c = s — r.

(The proof of these formulae by means of the usual figure, that used by Heron to prove his formula for the area of a triangle in terms of its sides, is easy.)

Solving the first two equations, in order to find a and b, we have

a] r + s + <J{(r + s)2 - 8rs]

which formula is actually used by the author for finding a and b. The method employed is to take the sum of the area and the three sides 5 + 2s, separated into its two obvious factors s (r + 2), to put s(r+2) = A (the given number), and then to separate A into suitable factors to which s and r+ 2 may be equated. They must obviously be such that sr, the area, is divisible by 6. To take the first example where A is equal to 280 : the possible factors are

1 Scritti, ed. B. Boncompagni, n. (1862), p. 264. Cf. Cantor, Gesch. d. Math. iilt p. 40.

THE PLACE OF DIOPHANTUS 121

2x140, 4x70, 5x56, 7x40, 8x35, 10x28, 14x20. The suitable factors in this case are r + 2 = 8, s = 35, because r is then equal to 6, and rs is a multiple of 6. The author then says that

a = 6 + 35- v1(6+35)2-8. 6.35)^41- i ^2Q>

2 2

and ^=35-6 = 29.

The triangle is therefore (20, 21, 29) in this case. The triangles found in the other cases, by the same method, are (9, 40, 41), (8, 15, 17) and (9, 12, 15).

Unfortunately there is no guide to the date of the problems just given. The form, however, cannot be that in which the discoverer or discoverers of the methods indicated originally explained those methods. The probability is that the original formulation of the most important of the problems belongs to the period between Euclid and Diophantus. This supposition best agrees with the fact that the problems include nothing taken from the great collection in the Arithmetica. On the other hand, it is strange that none of the seven problems above mentioned is found in Diophantus. The five of them which relate to rational right- angled triangles might well have been included by him ; thus he finds rational triangles such that the area plus or minus one of the perpendiculars is a given number, but not the rational triangle which has a given area ; and he finds rational triangles such that the area plus or minus the sum of two sides is a given number, but not the rational triangle such that the sum of the area and the three sides is a given number. The omitted problems might, it is true, have come in the lost Books ; but, on the other hand, Book VI. is the place where we should have expected to find them. Nor do we find in the above problems any trace of Diophantus' peculiar methods.

Lastly, the famous Cattle-Problem attributed to Archimedes1 has to be added to the indeterminate problems propounded before Diophantus' time. According to the heading prefixed to the epigram, it was communicated by Archimedes to the mathe- maticians at Alexandria in a letter to Eratosthenes. The scholiast

1 Archimedes, ed. Heiberg, Vol. II. p. 450 sqq.

122 INTRODUCTION

on Charmides 165 E also refers to the problem "called by Archi- medes the Cattle-Problem." Krumbiegel, who discussed the arguments for and against the attribution to Archimedes, con- cluded apparently that, while the epigram can hardly have been written by Archimedes in its present form, it is possible, nay probable, that the problem was in substance originated by Archimedes1. Hultsch2 has a most attractive suggestion as to the occasion of it. It is known that Apollonius in his WKVTOKIOV had calculated an approximation to the value of TT closer than that of Archimedes, and he must therefore, have worked out more difficult multiplications than those contained in the Measurement of a circle. • Also the other work of Apollonius on the multipli- cation of large numbers, which is partly preserved in Pappus, was inspired by the Sand-reckoner of Archimedes ; and, though we need not exactly regard the treatise of Apollonius as polemical, yet it did in fact constitute a criticism of the earlier book. That Archimedes should then reply with a problem involving such a manipulation of immense numbers as would be difficult even for Apollonius is not altogether outside the bounds of possibility. And there is an unmistakable vein of satire in the opening words of the epigram, " Compute the number of the oxen of the Sun, giving thy mind thereto, if thou hast a share of wisdom," in the tran- sition from the first part to the second, where it is said that ability to solve the first part would entitle one to be regarded as " not unknowing nor unskilled in numbers, but still not yet to be counted among the wise," and again in the. last lines. Hultsch concludes that in any case the problem is not much later than the time of Archimedes and dates from the beginning of the second century B.C. at the latest.

I have reproduced elsewhere3, from Amthor, details regarding the solution of the problem, and I need do little more than state here its algebraical equivalent. Eight unknown quantities have to be found, namely, the numbers of bulls and cows respectively of each of four colours (I use large letters for the bulls and small letters for the cows). The first part of the problem connects the eight unknowns by seven simple equations ; the second part adds two more conditions.

1 Zeitschrift fur Math. n. Physik (Hist. lilt. Abtheilung), xxv. (1880), p. 121 sq. Amthor added (p. 1 53 sq. ) a discussion of the problem itself.

2 Art. Archimedes in Pauly-Wissowa's Real-Encyclopcidie, n. i, pp. 534, 535.

3 The Works of Archimedes, pp. 319-326.

THE PLACE OF DIOPHANTUS 123

First part of Problem.

(I) W=($ + ^X+Y .................. (i),

.................. (2),

.................. (3).

(II) w = (i + i)(.T + *) ........ .......... (4),

) .................. (5),

+«0 .................. (7)-

Second part.

W+X = a. square ........................... (8),

F + /? = a triangular number ............ (9).

The solution of the first part gives

W = 10366482 n, w = 7206360 n, X = 74605 1 4 n, x = 4893246 n, Y= 4149387 w, y= 5439213 «, Z= 7358060 «, -s: = 35 1 5820 «,

where » is an integer. The solution given by the scholiast1 corre- sponds to n = 80.

The complete problem would not be unmanageable but for the condition (8). If for this were substituted the requirement that W + X shall be merely a product of two unequal factors (" Wurm's problem "), the solution in the least possible numbers is W= 1217263415886, w = 846 1 924 1 0280,

x= 876035935422, *= 574579625058,

F= 487233469701, 7 = 638688708099, Z= 864005479380, # = 412838131860.

But, if we include condition (8) and first of all find a solution satisfying the conditions (i) to (8), we have then, in order to satisfy condition (9), to solve the equation

q(q+\)J2 = 5 1285802909803 . f

If we multiply by 8, and put

2q + I = /, 2 . 4657 | = it, we have the equation

^-1 = 2.3.7. 1 1. 29. 353. «2, or f-- 4729494 «2= i.

1 Archimedes, ed. Heiberg, Vol. II. pp. 454, 455.

i24 INTRODUCTION

The value of W would be a number containing 206545 digits.

Such are the very few and scattered particulars which we possess of problems similar to those of Diophantus solved or propounded before his time. They show indeed that the kind of problem was not invented by him, but on the other hand they show little or no trace of anything like his characteristic alge- braical methods. In the circumstances, and in default of discovery of fresh documents, the question how much of his work represents original contributions of his own to the subject must remain a matter of pure speculation. It is pretty obvious that one man could not have been the author of all the problems contained in the six Books. There are also inequalities in the work ; some problems are very inferior in interest to the remainder, and some solutions may be assumed to be reproduced from other writers of less calibre, since they reveal none of the mastery of the subject which Diophantus possessed. Again, it seems probable that the problem V. 30, which is exceptionally in epigrammatic form, was taken from someone else. The Arithmetica was no doubt a collection, much in the same sense as Euclid's Elements were. And this may be one reason why so little trace remains of earlier labours in the same field. It is well known that Euclid's Elements so entirely superseded the works of the earlier writers of Elements (Hippocrates of Chios, Leon and Theudius) and of the great contributors to the body of the Elements, Theaetetus and Eudoxus, that those works have disappeared almost entirely. So no doubt would Diophantus' work supersede, and have the effect of con- signing to oblivion, any earlier collections of problems of the same kind. But, if it was a compilation, we cannot doubt that it was a compilation in the best sense, therein resembling Euclid's Elements; it was a compilation by one who was a master of the subject, who took account of and assimilated all the best that had been written upon it, arranged the whole of the available material in due and progressive order, but also added much of his own, not only in the form of new problems but also (and even more) in the mode of treatment, the development of more general methods, and so on.

It is perhaps desirable to add a few words on the previous history of the theory of polygonal numbers. The theory certainly goes back to Pythagoras and the earliest Pythagoreans. The triangle came first, being obtained by first taking I, then adding

THE PLACE OF DIOPHANTUS 125

2 to it, then 3 to the sum ; each successive number would be represented by the proper number of dots, and, when each number was represented by that number of dots arranged symmetrically under the row representing the preceding number, the triangular form would be apparent to the eye, thus :

etc.

Next came the Pythagorean discovery of the fact that a similar successive addition of odd numbers produced successive square numbers, the odd numbers being on that account called gnomons, and again the process was shown by dots arranged to re- present squares. The accompanying figure shows the successive squares and gnomons.

Following triangles and squares came the figured numbers in which the "gnomons," or the numbers added to make one number of a given form into the next larger of the same form, were numbers in arithmetical progression starting from I, but with common difference 3, 4, 5, etc., instead of I, 2. Thus, if the common difference is 3, so that the successive numbers added to i are 4, 7, 10, etc., the number is a pentagonal number, if the common difference is 4 and the gnomons 5, 9, 1 3, etc., the number is a hexagonal number, and so on. Hence the law that the common difference of the gnomons in the case of a «-gon is 11 — 2.

Perhaps these facts had already been arrived at by Philippus of Opus (4th c. B.C.), who is said to have written a work on polygonal numbers1. Next Speusippus, nephew and successor of Plato, wrote on Pythagorean Numbers, and a fragment of his book survives2, in which linear numbers, polygonal numbers, triangles and pyramids are spoken of: a fact which leaves no room for doubt as to the Pythagorean origin of all these con- ceptions3.

Hypsicles, who wrote about 170 B.C., is twice mentioned by Diophantus as the author of a " definition " of a polygonal number,

1 Bioypd(poi, Vitarum scriptores Graeci minores, ed. Westermann, 1845, p. 448.

2 Theologumena arithmcticae (ed. Ast), 1817, pp. 61, 62 ; the passage is translated with notes by Tannery, Pour Phistoire de la science hellene, pp. 386-390.

3 Cantor, Geschichte der Mathematik, I3 , p. 249.

i26 INTRODUCTION

which is even quoted verbatim1. The definition does not mention any polygonal number beyond the pentagonal ; but indeed this was unnecessary : the facts about the triangle, the square and the pentagon were sufficient to enable Hypsicles to pass to a general conclusion. The definition amounts to saying that the nth #-gon (i counting as the first) is

*» (2 +(»-!) (a -2)}.

Theon of Smyrna2, Nicomachus3 and lamblichus4 all devote some space to polygonal numbers. The first two, who flourished about 100 A.D., were earlier than Diophantus, and are accordingly of interest here. Besides a description of the successive polygonal numbers, Theon gives the theorem that two successive triangular numbers added together give a square. That is, (n-i)n n(n+ i)_

~ — 7£~.

2 2

The fact is of course clear if we divide a square into two triangles as in the figure.

Nicomachus gave various rules for transforming triangles into squares, squares into pentagons, etc.

1. If we put two consecutive triangles together we get a square (as in Theon's theorem).

2. A pentagon is obtained from a square by adding to it a triangle the side of which is i less than that of the square; similarly a hexagon from a pentagon by adding a triangle the side of which is i less than that of the pentagon ; and so on.

In fact,

\n {2 + (n-i)(a- 2)} +i (»-i)» = J« [2 + (n-i){(a + i) -2}]. Next Nicomachus sets out the first triangles, squares, pentagons, hexagons and heptagons in a diagram thus :

Triangles

... i

3

6

10

15

2 i

28

36

45

55

Squares

... i

4

9

16

25

36

49

64

81

100

Pentagons

i

5

12

22

35

51

70

92

117

H5

Hexagons

... i

6

15

28

45

66

91

120

153

190

Heptagons

... i

7

18

34

55

81

112

148

189

235

and observes

that

1 Dioph. i. pp. 470-472-

2 Expositio rerum mathematicanim ad legendum Platonem ut ilium, ed. Hiller, pp. 31-40.

8 Introductio arithmetica, ed. Hoche, II. 8-12, pp. 87-99.

4 In Nicomachi arithmeticam introd., ed. Pistelli, pp. 58-61, 68-72.

THE PLACE OF DIOPHANTUS 127

3. Each polygon is equal to the polygon immediately above it in the diagram plus the triangle with i less in its side, i.e. the triangle in the preceding column.

4. The vertical columns are arithmetical progressions, the common difference of which is the triangle in the preceding column.

But Plutarch, a contemporary of Nicomachus, mentioned another method of transforming triangles into squares: Every triangular number taken eight times and then increased by I gives a square*. That is,

Diophantus generalised this proposition into his theorem for transforming any polygonal number into a square. If P be a polygonal number, a the number of angles,

8P (a - 2) + (a - 4)2 = a square.

He deduces rules for finding a polygonal number when the side and the number of angles are given, and for finding the side when the number and the number of its angles are given. These fine results and the fragment of the difficult problem of finding the number of ways in which any given number can be a polygonal number no doubt represent part of the original contributions by Diophantus to the theory of that class of numbers.

1 Plat, quaest. V. 2, 4, 1003 F.

THE ARITHMETICA BOOK I

PRELIMINARY Dedication.

" Knowing, my most esteemed friend Dionysius, that you are anxious to learn how to investigate problems in numbers, I have tried, beginning from the foundations on which the science is built up, to set forth to you the nature and power subsisting in numbers.

" Perhaps the subject will appear rather difficult, inasmuch as it is not yet familiar (beginners are, as a rule, too ready to despair of success) ; but you, with the impulse of your enthusiasm and the benefit of my teaching, will find it easy to master ; for eagerness to learn, when seconded by instruction, ensures rapid progress."

After the remark that " all numbers are made up of some multitude of units, so that it is manifest that their formation is subject to no limit," Diophantus proceeds to define what he calls the different "species" of numbers, and to describe the abbreviative signs used to denote them. These " species " are, in the first place, the various powers of the unknown quantity from the second to the sixth inclusive, the unknown quantity itself, and units.

Definitions.

A square (=x-) is StW/u? (" power "), and its sign is a J with Y

superposed, thus Jr. A cube (=x3) is icvfios, and its sign KY.

A square-square (=#*) is 8vvafj,oBvva^i<;1, and its sign is JrJ. A square-cube (=x5) is 8vva/j,6Kvj3o<;, and its sign JATr. A cube-cube (= x6) is tcvftoicvfios, and its sign KYK.

1 The term Swa/uodiVa/us was already used by Heron (Metrica, ed. Schone, p. 48, n, 19) for the fourth power of a side of a triangle.

H. D. Q

1 30 THE ARITHMETICA

" It is," Diophantus observes, " from the addition, subtraction or multiplication of these numbers or from the ratios which they bear to one another or to their own sides respectively that most arithmetical problems are formed" ; and "each of these numbers... is recognised as an element in arithmetical inquiry."

" But the number which has none of these characteristics, but merely has in it an indeterminate multitude of units (7rX?}$o<? fi.ovd8ci)v dopia-Tov) is called dpiOpos, ' number I and its sign is 9 [=*]."

"And there is also another sign denoting that which is in- variable in determinate numbers, namely the unit, the sign being

M with o superposed, thus M."

Next follow the definitions of the reciprocals, the names of which are derived from the names of the corresponding species themselves.

Thus

from dpid/jLos [x~\ we derive the term dpidpoarov [= I/*]

[= i/^2] [= I/*3]

„ 8vvafj,o8vvafj,i<; \x*\ „ „ SwaftoSwa/jLoa-Tov [= I/*"4]

„ Suva/jioicvfios \X5~\ „ „ Suva/Motcvftoo-TOv [= I/^5]

„ /eu/3o/ei»/3o<? [x6] „ „ KvftoKvftoaTov [= I/*"6],

and each of these has the same sign as the corresponding original species, but with a distinguishing mark which Tannery writes in the form x above the line to the right. Thus JJ'X = ljx\ just as 7* = £.

Sign of Subtraction (minus}.

" A minus multiplied by a minus makes a plus^ ; a minus multiplied by a plus makes a minus ; and the sign of a minus is a truncated ^f turned upside down, thus fa."

Diophantus proceeds : " It is well that one who is beginning this study should have acquired practice in the addition, subtraction and multiplication of the various species. He should know how to add positive and negative terms with different coefficients to

1 The literal rendering would be "A wanting multiplied by a wanting makes a forthcoming." The word corresponding to minus is Xetfis ("wanting"): when it is used exactly as our minus is, it is in the dative Xetyei, but there is some doubt whether Diophantus himself used this form (cf. p. 44 above). For the probable explanation of the sign, see pp. 42-44. The word for "forthcoming" is Uirap!;is, from VTT&PXU, to exist. Negative terms are XeiTrovra eldrj, and positive virdpxoi>ra.

BOOK I ' 131

other terms \ themselves either positive or likewise partly positive and partly negative, and how to subtract from a combination of positive and negative terms other terms either positive or likewise partly positive and partly negative.

" Next, if a problem leads to an equation in which certain terms are equal to terms of the same species but with different coefficients, it will be necessary to subtract like from like on both sides, until one term is found equal to one term. If by chance there are on either side or on both sides any negative terms, it will be necessary to add the negative terms on both sides, until the terms on both sides are positive, and then again to subtract like from like until one term only is left on each side.

" This should be the object aimed at in framing the hypotheses of propositions, that is to say, to reduce the equations, if possible, until one term is left equal to one term ; but I will show you later how, in the case also where two terms are left equal to one term, such a problem is solved"

Diophantus concludes by explaining that, in arranging the mass of material at his disposal, he tried to distinguish, so far as possible, the different types of problems, and, especially in the elementary portion at the beginning, to make the more simple lead up to the more complex, in due order, such an arrangement being calculated to make the beginner's course easier and to fix what he learns in his memory. The treatise, he adds, has been divided into thirteen Books.

PROBLEMS

1. To divide a given number into two having a given difference.

Given number 100, given difference 40. Lesser number required x. Therefore 2^ + 4.0= loo, ^=30. The required numbers are 70, 30.

2. To divide a given number into two having a given ratio.

Given number 60, given ratio 3:1. Two numbers x, ^x. Therefore ;r= 15. The numbers are 45, 15.

1 eTSos, "species," is the word used by Diophantus throughout.

9—2

i32 THE ARITHMETTCA

3. To divide a given number into two numbers such that one is a given ratio of the other plus a given difference1.

Given number 80, ratio 3:1, difference 4. Lesser number x. Therefore the larger is "$x f 4, and 4* -f 4 = 80, so that x = 19. The numbers are 61, 19.

4. To find two numbers in a given ratio and such that their difference is also given.

Given ratio 5:1, given difference 20. Numbers $x, x. Therefore ^x= 20, x= 5, and the numbers are 25, 5.

5. To divide a given number into two numbers such that given fractions (not the same) of each number when added together produce a given number.

Necessary condition. The latter given number must be such that it lies between the numbers arising when the given fractions respectively are taken of the first given number.

First given number 100, given fractions \ and ^, given

sum of fractions 30.

Second part $x. Therefore first part = 3 (30 —x). Hence 90 + 2x — 100, and x = 5.

The required parts are 75, 25.

6. To divide a given number into two numbers such that a given fraction of the first exceeds a given fraction of the other by a given number.

Necessary condition. The latter number must be less than that which arises when that fraction of the first number is taken which exceeds the other fraction.

Given number 100, given fractions £ and £ respectively,

given excess 20.

Second part 6x. Therefore the first part is 4 (;r + 20). Hence lox + 80 = 100, x = 2, and the parts are 88, 12.

1 Literally "to divide an assigned number into two in a given ratio and difference (ev Xctycf) Kal inrepoxs r-ij 5o0ei<rij)." The phrase means the same, though it is not so clear, as Euclid's expression (Data, Def. 1 1 and passim) boOivrt pelfav 17 ev \kryif. According to Euclid's definition a magnitude is greater than a magnitude "by a given amount (more) than in a (certain) ratio" when the remainder of the first magnitude, after subtracting tlie given amount, has the said ratio to the second magnitude. This means that, if x, y are the magnitudes, d the given amount, and k the ratio, x-d=ky or

BOOK I 133

7. From the same (required) number to subtract two given numbers so as to make the remainders have to one another a given ratio.

Given numbers 100, 20, given ratio 3:1. Required number x. Therefore x — 20 = 3 (x — 100), and x = 140.

8. To two given numbers to add the same (required) number so as to make the resulting numbers have to one another a given ratio.

Necessary condition. The given ratio must be less than the ratio which the greater of the given numbers has to the lesser. Given numbers 100, 20, given ratio 3:1. Required number x. Therefore ~$x + 60 =x + 100, and

9. From two given numbers to subtract the same (required) number so as to make the remainders have to one another a given ratio.

Necessary condition. The given ratio must be greater than the ratio which the greater of the given numbers has to the lesser. Given numbers 20, 100, given ratio 6 : I. Required number x. Therefore 120 — 6x = 100 — x, and

10. Given two numbers, to add to the lesser and to subtract from the greater the same (required) number so as to make the sum in the first case have to the difference in the second case a given ratio.

Given numbers 20, 100, given ratio 4:1. Required numbers. Therefore (20 + x} = 4(100-.*'), and ;r=76.

11. Given two numbers, to add the first to, and subtract the second from, the same (required) number, so as to make the resulting numbers have to one another a given ratio.

Given numbers 20, 100, given ratio 3:1. Required number x. Therefore yc — 30x3 =x+ 20, and x = 160.

12. To divide a given number twice into two numbers such that the first of the first pair may have to the first of the second pair a given ratio, and also the second of the second pair to the second of the first pair another given ratio.

134 THE ARITHMETIC A

Given number 100, ratio of greater of first parts to lesser of second 2:1, and ratio of greater of second parts to lesser of first parts 3:1.

x lesser of second parts.

The parts then are

-<5r)

and

IOO—2X) X \

Therefore 300 - $x= 100, x= 40, and the parts are (80, 20), (60, 40).

1 3. To divide a given number thrice into two numbers such that one of the first pair has to one of the second pair a. given ratio, the second of the second pair to one of the third pair another given ratio, and the second of the third pair to the second of the first pair another given ratio.

Given number 100, ratio of greater of first parts to lesser of second 3:1, of greater of second to lesser of third 2:1, and of greater of third to lesser of first 4: i.

x lesser of third parts. Therefore greater of second parts = 2x, lesser of second

= loo- 2x, greater of first = 300 - 6>. Hence lesser of first = 6x— 200, so that greater of third

= 24*- - 800. Therefore 2$x — 800 = 100, x = 36, and

the respective divisions are (84, 16), (72, 28), (64, 36).

14. To find two numbers such that their product has to their sum a given ratio. [One is arbitrarily assumed.]

Necessary condition. The assumed value of one of the two must be greater than the number representing the ratio1.

Ratio $ : i, x one of the numbers, 12 the other (> 3). Therefore 1 2x — $x + 36, x = 4, and the numbers are 4, 12.

15. To find two numbers such that each after receiving from the other a given number may bear to the remainder a given ratio.

Let the first receive 30 from the second, the ratio being then 2:1, and the second 50 from the first, the ratio being then 3:1; take x + 30 for the second.

1 Literally "the number homonymous with the given ratio."

BOOK I 135

Therefore the first = 2x — 30, and

(x + 80) = 3 (2x - 80). Thus x = 64, and

the numbers are 98, 94.

16. To find three numbers such that the sums of pairs are given numbers.

Necessary condition. Half the sum of the three given numbers must be greater than any one of them singly.

Let (i) + (2) = 20, (2) + (3) = 30, (3) + (i) = 40.

x the sum of the three. Therefore the numbers are

x — 30, x — 40, x — 20. The sum x = *x — go, and x = 45. The numbers are 15, 5, 25.

17. To find four numbers such that the sums of all sets of three are given numbers.

Necessary condition. One-third of the sum of the four must be greater than any one singly.

Sums of threes 22, 24, 27, 20 respectively.

x the sum of all four. Therefore the numbers are

X—22, *— 24, X-27, X—2O.

Therefore ^x — 93 —x, x = 3 1, and the numbers are 9, 7, 4, n.

1 8. To find three numbers such that the sum of any pair exceeds the third by a given number.

Given excesses 20, 30, 40.

2.x the sum of all three.

We have (i) + (2) = (3) + 20.

Adding (3) to each side, we have : twice (3) + 20= 2x, and

(3) = *- 10. Similarly the numbers (i) and (2) are ^—15, x - 20

respectively.

Therefore yc — 45 = 2x, ^=45, and the numbers are 30, 25, 35. \OtJierwise thus1. As before, if the third number (3) is x,

(l) + (2)=;r+20.

Next, if we add the equations

(I) + (2) -(3) = 20) (2) + (3)-(l)=3OJ'

1 Tannery attributes the alternative solution of I. 18 (as of I. 19) to an old scholiast.

136 THE ARITHMETICS

we have (2) = | (20 + 30) = 25. Hence (i)=.r-5.

Lastly .u) + (i)-(2) = 40,

or 2r-5-25 = 4o.

Therefore x= 35.

The numbers are 30, 25, 35.]

19* To find four numbers such that the sum of any three exceeds the fourth by a given number.

Necessary condition. Half the sum of the four given differences must be greater than any one of them. Given differences 20, 30, 40, 50.

2r the sum of the required numbers. Therefore the numbers are

-r— 15, x — 20, x— 25, x— 10. Therefore 4-r — 70 = 2jr, and x — 35.

The numbers are 20, 15, 10, 25.

\Otkeronse tk*s\ If the fourth number (4) is x,

(I) + (2) + (3) = X + 20.

Put (2) + (3) equal to half the sum of the two excesses 20 and 30, i.e. 25 [this is equivalent to adding the two equations

It follows by subtraction that (I) = JT- 5. Next we add the equations beginning with (2) and (3) respectively, and we obtain

(3) + (4) = 1 (30 -I- 40) = 35, so that (3) = 35~^

It follows that (2) = jr- ia ' Lastly, since (4) + (i) + (2)-(3) = 50,

y- I5-(35--r>=50, and ^ = 25. The numbers are accordingly 20, 15, 10, 25.]

2Oi To divide a given number into three numbers such that the sum of each extreme and the mean has to the other extreme a given ratio.

Given number ioo; and let (i) + (2)= 3 .(3) and (2) + (3) = 4-(0-

19 (as of I. 1 8) to an old scholiast.

BOOK I 137

x the third number. Thus the sum of the first and second

= 3^r, and the sum of the three =^x= 100. Hence x = 25, and the sum of the first two = 75. Let y be the first1. Therefore sum of second and third

The required parts are 20, 55, 25.

21. To find three numbers such that the greatest exceeds the middle number by a given fraction of the least, the middle exceeds the least by a given fraction of the greatest, but the least exceeds a given fraction of the middle number by a given number.

Necessary condition. The middle number must exceed the least by such a fraction of the greatest that, if its denominator2 be multiplied into the excess of the middle number over the least, the coefficient of x in the product is greater than the coefficient of x in the expression for the middle number resulting from the assumptions made3.

Suppose greatest exceeds middle by \ of least, middle exceeds least by \ of greatest, and least exceeds \ of middle by 10. [Diophantus assumes the three given fractions or submultiples to be one and the same.] x + 10 the least. Therefore middle = 3-r, and greatest

= 6x - 30.

Hence, lastly, 6x - 30 — yc = £ (x + 10), or x+ 10 = 93- — 90, and x= 12^.

The numbers are 45, yj\, 22^.

1 As already remarked (p. 52), Diophantus does not use a second symbol for the second unknown, but makes d/M0/xos do duty for the second as well as for the first.

2 "Denominator," literally the "number homonymous with the fraction," i.e. the denominator on the assumption that the fraction is, or is expressed as, a submultiple.

3 Wertheim points out that this condition has reference, not to the general solution of the problem, but to the general applicability of the particular procedure which Diophantus adopts in his solution. Suppose X, Y, Z required such that X- Y=Z\m, Y-Z=X\n, Z-a=Y\p. Diophantus assumes Z=x + a, whence Y=px, X=n(j>x-x-a). The condition states that np — n >p. If we solve for x by substituting the values of X, Y, Z in the first equation, we in fact obtain

m {(np-n-#)x-na}=x + a, or x (mnp - mn - mp - i) = a (tnn+ i).

In order that the value of x may be positive, we must have mnp>mn + mp -f- 1, that is,

np>n+p + — or (if m, «, / are positive integers) np>n +/.

138 THE ARITHMETICA

[Another solution1.

Necessary condition. The given fraction of the greatest must be such that, when it is added to the least, the coefficient of x in the sum is less than the coefficient of x in the expression for the middle number resulting from the assumptions made2.

Let the least number be x + 10, as before, and the given

fraction \ ; the middle number is therefore $x. Next, greatest = middle + £ (least) = $%x + 3$. Lastly, 3* = x + i o + £ (fa + 3i)

Therefore x = \2\, and

the numbers are, as before, 45, 37^, 22^.]

22. To find three numbers such that, if each give to the next following a given fraction of itself, in order, the results after each has given and taken may be equal.

Let first give ^ of itself to second, second \ of itself to

third, third £ of itself to first. Assume first to be a number of x's divisible by 3, say

$x, and second to be a number of units divisible by

4, say 4.

Therefore second after giving and taking becomes -*"+ 3. Hence the first also after giving and taking must become

;tr+3; it must therefore have taken x + $—2x, or

3-;r; ^-x must therefore be £ of third, or third

= I$-5*

Lastly, 15 -5*- (3 -#)+!=*+ 3, or 13—4^ = ^ + 3, and x=2.

The numbers are 6, 4, 5.

23. To find four numbers such that, if each give to the next following a given fraction of itself, the results may all be equal.

Let first give \ of itself to second, second \ of itself

to third, third £ of itself to fourth, and fourth £ of

itself to first. Assume first to be a number of ;r's divisible by 3, say yc,

and second to be a number of units divisible by 4,

say 4.

1 Tannery attributes this alternative solution, like the others of the same kind, to an ancient scholiast.

2 Wertheim observes that the scholiast's necessary condition comes to the same thing as Diophantus' own.

BOOK I

139

The second after giving and taking becomes x + 3. Therefore first after giving x to second and receiving £ of fourth =x+ 3 ; therefore fourth

= 6(^+3 — 2x}= i8-6.tr. But fourth after giving $—x to first and receiving \ of

third =-*- + 3 ; therefore third =301: -60. Lastly, third after giving 6x— i? to fourth and receiving

I from second = .*•+ 3. That is, 24^— 47=;tr+3, and * = f$.

The numbers are therefore J^, 4, J^f, J^; or, after multiplying by the common de- nominator, 150, 92, 120, 114.

24. To find three numbers such that, if each receives a given fraction of the sum of the other two, the results are all equal.

Let first receive £ of (second + third), second \ of

(third + first), and third i of (first + second). Assume first =x, and for convenience' sake (rov irpo^Lpov eveicev) take for sum of second and third a number of units divisible by 3, say 3. Then sum of the three = x-\- 3,

and first -I- 1 (second + third) = x + I .

Therefore second + ^ (third + first) = x+ I ; hence 3 times second + sum of all = 4^ + 4, and therefore second = x + %.

Lastly, third + { (first + second) = x + I, or 4 times third + sum of all = 5^+5,

and third =x + ^.

Therefore x + (x + £) + (x + £) = x + 3, and ^ = T!-

The numbers, after multiplying by the common denominator, are 13, 17, 19.

25. To find four numbers such that, if each receives a given fraction of the sum of the remaining three, the four results are equal.

Let first receive £ of the rest, second 1 of the rest,

third i of rest, and fourth £ of rest. Assume first to be x and sum of rest a number of units

divisible by 3, say 3. Then sum of all =x+ 3.

Now first + £ (second -f third 4 fourth) = x + I .

i4o THE ARITHMETICA

Therefore second + | (third + fourth + first) =^+ I,

whence 3 times second + sum of all = 4.^ + 4,

and therefore second = x + \.

Similarly third =.*• + £,

and fourth = ;r+f.

Adding, we have 4^r + f§ = ^+3,

and x==$-

The numbers, after multiplying by a common denominator, are 47, 77, 92, 101.

26. Given two numbers, to find a third number which, when multiplied into the given numbers respectively, makes one product a square and the other the side of that square.

Given numbers 200, 5 ; required number x.

Therefore 2OOtr = ^r2, and

.27. To find two numbers such that their sum and product are given numbers.

Necessary condition. The square of half the sum must exceed the product by a square number, ecm Se roOro irXao-fiaTiicbv1. Given sum 20, given product 96. 2x the difference of the required numbers. Therefore the numbers are io+x, 10— x. Hence ioo-xz = g6. Therefore x=2, and

the required numbers are 12, 8.

28. To find two numbers such that their sum and the sum of their squares are given numbers.

Necessary condition. Double the sum of their squares must exceed the square of their sum by a square, ecrrt Se KOI TOVTO

1 There has been controversy as to the meaning of this difficult phrase. Xylander, Bachet, Cossali, Schulz, Nesselmann, all discuss it. Xylander translated it by "effictum aliunde." Bachet of course rejects this, and, while leaving the word untranslated, maintains that it has an active rather than a passive signification ; it is, he says, not something "made up" (effictum) but something "a quo aliud quippiam effingi et plasmari potest," " from which something else can be made up," and this he interprets as meaning that from the conditions to which the term is applied, combined with the solutions of the respective problems in which it occurs, the rules for solving mixed quadratics can be evolved. Of the two views I think Xylander's is nearer the mark. Tr\afffj.aTiK6v should apparently mean "of the nature of a ir\dfffj.a," just as 5pa/uariK<5j> means something connected with or suitable for a drama ; and ir\d<r/j.a means something

BOOK I 141

Given sum 20, given sum of squares 208. Difference 2x.

Therefore the numbers are lo+x, IQ-X. Thus 200 + 2x* = 208, and x = 2.

The required numbers are 12, 8.

29. To find two numbers such that their sum and the difference of their squares are given numbers.

Given sum 20, given difference of squares 80. Difference 2x.

The numbers are therefore lO + x, 10 — x. Hence (lo + xf- (io-xf= 80,

or 401: = 80, and x = 2. The required numbers are 12, 8.

30. To find two numbers such that their difference and product are given numbers.

Necessary condition. Four times the product together with the square of the difference must give a square, eo-ri Se real rovro

Given difference 4, given product 96. 2x the sum of the required numbers.

Therefore the numbers are x+2, x — 2\ accordingly x*— 4 = 96, and x= 10.

The required numbers are 12, 8.

31. To find two numbers in a given ratio and such that the sum of their squares also has to their sum a given ratio.

Given ratios 3 : I and 5 : I respectively. Lesser number x.

Therefore \ox- = 5 . 4-r, whence x = 2, and the numbers are 2, 6.

32. To find two numbers in a given ratio and such that the sum of their squares also has to their difference a given ratio.

Given ratios 3 : I and 10 : I.

Lesser number x, which is then found from the equation

ior2 = 10. ix. Hence* =2, and

the numbers are 2, 6.

"formed" or "moulded." Hence the expression would seem to mean "this is of the nature of a formula," with the implication that the formula is not difficult to make up or discover. Nesselmann, like Xylancler, gives it much this meaning, translating it "das lasst sich aber bewerkstelligen." Tannery translates irXaoytart/roi' by "formativum."

i42 THE ARITHMETICA

33. To find two numbers in a given ratio and such that the difference of their squares also has to their sum a given ratio.

Given ratios 3 : i and 6 : I. Lesser number x, which is found to be 3. The numbers are 3, 9.

34. To find two numbers in a given ratio and such that the difference of their squares also has to their difference a given ratio.

Given ratios 3 : I and 12 : I.

Lesser number x, which is found to be 3.

The numbers are 3, 9.

Similarly by the same method can be found two numbers in a given ratio and (i) such that their product is to their sum in a given ratio, or (2) such that their product is to their difference in a given ratio.

35. To find two numbers in a given ratio and such that the square of the lesser also has to the greater a given ratio.

Given ratios 3 : i and 6 : i respectively. Lesser numbers, which is found to be 18. The numbers are 18, 54.

36. To find two numbers in a given ratio and such that the square of the lesser also has to the lesser itself a given ratio.

Given ratios 3 : i and 6 : i . Lesser number x, which is found to be 6. The numbers are 6, 18.

37. To find two numbers in a given ratio and such that the square of the lesser also has to the sum of both a given ratio.

Given ratios 3 : i and 2:1. Lesser number x, which is found to be 8. The numbers are 8, 24.

38. To find two numbers in a given ratio and such that the square of the lesser also has to the difference between them a given ratio.

Given ratios 3 : i and 6 : i. Lesser number x, which is found to be 1 2. The numbers are 12, 36.

BOOK II 143

'Similarly can be found two numbers in a given ratio and

(1) such that the square of the greater also has to the

lesser a given ratio, or

(2) such that the square of the greater also has to the

greater itself a given ratio, or

(3) such that the square of the greater also has to the sum

or difference of the two a given ratio.

39. Given two numbers, to find a third such that the sums of the several pairs multiplied by the corresponding third number give three numbers in arithmetical progression. Given numbers 3, 5. Required number x.

The three products are therefore 3^ + 15, 5^+15, &r. Now 3-r + 1 5 must be either the middle or the least of the three, and $x+i$ either the greatest or the middle.

(i) 5^+15 greatest, 3^+15 least.

Therefore *>x + 1 5 + 3-*" + 1 5 = 2 . &tr, and

(2) 5^+15 greatest, 3.^+15 middle.

Therefore (5*+ 15) -(3^+ 15) = 3*+ IS -&r, and

*-?•

(3) %x greatest, yc + 1 5 least.

Therefore &r + 3* + 1 5 = 2 ($x + 1 5), and

BOOK II

[The first five problems of this Book are mere repetitions of problems in Book I. They probably found their way into the text from some ancient commentary. In each case the ratio of one required number to the other is assumed to be 2 : i. The enunciations only are here given.]

1. To find two numbers such that their sum is to the sum of their squares in a given ratio [cf. i. 31].

2. To find two numbers such that their difference is to the difference of their squares in a given ratio [cf. I. 34].

144 THE ARITHMETICA

3. To find two numbers such that their product is to their sum or their difference in a given ratio [cf. I. 34].

4. To find two numbers such that the sum of their squares is to their difference in a given ratio [cf. I. 32].

5. To find two numbers such that the difference of their squares is to their sum in a given ratio [cf. I. 33].

61. To find two numbers having a given difference and such that the difference of their squares exceeds their difference by a given number.

Necessary condition. The square of their difference must be less than the sum of the said difference and the given excess of the difference of the squares over the difference of the numbers.

Difference of numbers 2, the other given number 20. Lesser number x. Therefore x + 2 is the greater, and

4^+4 = 22. Therefore x = 4^, and

the numbers are 4^, 6£.

71. To find two numbers such that the difference of their squares is greater by a given number than a given ratio of their difference-. [Difference assumed.]

Necessary condition. The given ratio being 3:1, the square of the difference of the numbers must be less than the sum of three times that difference and the given number.

Given number 10, difference of required numbers 2. Lesser number x. Therefore the greater is x+ 2, and

4^ + 4 = 3.2+ 10. Therefore x = 3, and

the numbers are 3, 5.

8. To divide a given square number into two squares3.

1 The problems n. 6, 7 also are considered by Tannery to be interpolated from some ancient commentary.

2 Here we have the identical phrase used in Euclid's Data (cf. note on p. 132 above) : the difference of the squares is rfjs vTrepoxw avr&v doOtvTi. apid/nf /j.flfai> rj ev \6yif, literally "greater than their difference by a given number (more) than in a (given) ratio," by which is meant "greater by a given number than a given proportion or fraction of their difference."

3 It is to this proposition that Fermat appended his famous note in which he enunciates what is known as. the "great theorem" of Fermat. The text of the note is as follows :

"On the other hand it is impossible to separate a cube into two cubes, or a

BOOK II 145

Given square number 16.

x* one of the required squares. Therefore \6-x* must be equal to a square.

Take a square of the form1 (inx— 4)*, m being any integer and 4 the number which is the square root of 1 6, e.g. take (2^ — 4)*, and equate it to 16 — x*.

Therefore 4x*—\6x+i6=\(:>—x\

or 5** = \6x, and x = ±g-.

The required squares are therefore y-, ^.

9. To divide a given number which is the sum of two squares into two other squares2.

biquadrate into two biquadrates, or generally any power except a square into two pcnvers with the same exponent. I have discovered a truly marvellous proof of this, which however the margin is not large enough to contain."

Did Fermat really possess a proof of the general proposition that xm+ym = zl* cannot be solved in rational numbers where m is any number >2? As Wertheim says, one is tempted to doubt this, seeing that, in spite of the labours of Euler, Lejeune-Dirichlet, Kummer and others, a general proof has not even yet been discovered. Euler proved the theorem for m = $ and /» = 4, Dirichlet for *w = 5, and Kummer, by means of the higher theory of numbers, produced a proof which only excludes certain particular values of m, which values are rare, at all events among the smaller values of m ; thus there is no value of m below 100 for which Kummer's proof does not serve. (I take these facts from Weber and Wellstein's Encyclopddie der Elementar-Mathematik, I2, p. 284, where a proof of the formula for m = + is given.)

It appears that the Gottingen Academy of Sciences has recently awarded a prize to Dr A. Wieferich, of Miinster, for a proof that the equation xp+yp = gp cannot be solved in terms of positive integers not multiples of p, if 2P - 2 is not divisible by p*. " This surprisingly simple result represents the first advance, since the time of Kummer, in the proof of the last Fermat theorem " (Bulletin of the American Mathematical Society, February 1910).

Fermat says ("Relation des nouvelles decouvertes en la science des nombres," August 1659, Oeuvres, II. p. 433) that he proved that no cube is divisible into two cutesby a variety of his method of infinite diminution (descente infinie or indefinie) different from that which he employed for other negative or positive theorems ; as to the other cases, see Supplement, sections I., n.

1 Diophantus' words are: "I form the square from any number of dp<.0/j.oi minus as many units as there are in the side of 16." It is implied throughout that m must be so chosen that the result may be rational in Diophantus' sense, i.e. rational and positive.

2 Diophantus' solution is substantially the same as Euler's (Algebra, tr. Hewlett, Part n. Art. 219), though the latter is expressed more generally.

Required to find x, y such that

If x £ /, then y $ g.

Put therefore -r=/+/te, y=g-qz.

H. D.

146 THE ARITHMETICA

Given number 13 = 22 + 32.

As the roots of these squares are 2, 3, take (x + 2)* as the

first square and (mx— 3)2 as the second (where m is

an integer), say (2.x — 3)*.

Therefore (x* + 4-*" + 4) + (^x* + 9-1 2x} = 1 3, or 5-r2 + 13 -8x= 13.

Therefore x = f , and

the required squares are ~t £-.

10. To find two square numbers having a given difference.

Given difference 60.

Side of one number x, side of the other x plus any

number the square of which is not greater than 60,

say 3.

Therefore (* + 3)2 - ;r2 = 60 ;

x=%\, and

the required squares are 72^, 132^.

1 1. To add the same (required) number to two given numbers so as to make each of them a square.

(i) Given numbers 2, 3 ; required numbers.

X -4- 2 1

Therefore \ must both be squares.

This is called a double-equation (StTrXoiVo-n;?). To solve it, take the difference between the two expressions and resolve it into two factors1 ; in this case let us say

4, i- Then take either

(a) the square of half the difference between these factors

and equate it to the lesser expression, or (b) the square of half the sum and equate it to the greater.

hence iffz +/V - igqz + </2s2 = o,

and «=3az2#

so that *=^lj:\f', y=

in which we may substitute all possible numbers for/, q.

1 Here, as always, the factors chosen must be suitable factors, i.e. such as will lead to a "rational " result, in Diophantus' sense.

BOOK II t47

In this case (a) the square of half the difference is %££-. Therefore x + 2 = -2^, and x = g, the squares being ^5-, Jj/£.

Taking (b) the square of half the sum, we have x+ 3 = -^,

which gives the same result. (2) To avoid a double-equation1,

first find a number which when added to 2, or to 3,

gives a square. Take e.g. the number x~ — 2, which when added to 2 gives

a square.

Therefore, since this same number added to 3 gives a square,

x* + i = a square = (x — 4)", say, the number of units in the expression (in this case 4)

being so taken that the solution may give x- > 2. Therefore jr = -1g5-, and

the required number is ^, as before.

12. To subtract the same (required) number from two given numbers so as to make both remainders squares. Given numbers 9, 21.

Assuming 9 — x* as the required number, we satisfy one condition, and the other requires that 12 +x* shall be a square. Assume as the side of this square x minus some number

the square of which > 12, say 4. Therefore (x — 4)-=\2 + xz,

and x=\.

The required number is then 8f .

[Diophantus does not reduce to lowest terms, but says x = | and then subtracts |-|~from 9 or ^^.]

1 This is the same procedure as that of Euler, who does not use double-equations. Euler (Algebra, tr. Hewlett, Part II. Art. 214) solves the problem

Suppose x + 4 =

therefore jr=/2~4, and x + 7=

Suppose that /

therefore / = (3 -

Thus X— (Q-22

or, if we take a fraction r[s instead of q, x = (gs* -

148 THE ARITHMETICA

13. From the same (required) number to subtract two given numbers so as to make both remainders squares. Given numbers 6, 7.

(i) Let x be the required number. Therefore x ~ I are both squares.

The difference is I, which is the product of, say, 2 and and, by the rule for solving a double equation,

(2) To avoid a double-equation, seek a number which exceeds

a square by 6, say x* + 6.

Therefore x* — I must also be a square = (x — 2)2, say. Therefore x = f , and

the required number is ^.

14. To divide a given number into two parts and to find a square which when added to each of the two parts gives a square number.

Given number 20.

Take two numbers1 such that the sum of their squares < 20, say 2, 3.

1 Diophantus implies here that the two numbers chosen must be such that the sum of their squares <2O. Tannery pointed out (Bibliotheca Matkematua, 1887, p. 103) that this is not so and that the condition actually necessary to ensure a real solution in Diophantus' sense is something different. We have to solve the equations x+y=a, 22-t-jc = «2, 22+_y = z>2.

We assume u = z + m, z> = 2 + «, and, eliminating x, y, we obtain

i (m + n)

In order that z may be positive, we must have n& + »2 < a ; but z need not be positive in order to satisfy the above equations. What is really required is that x, y shall both be positive.

Now from the above we derive

Solving for x, y, we have

_ (m - n) (a + 2tnn)

m (a + mn - «2) _ n (a + mn - m2) m+n ' m+n

If, of the two assumed numbers, m>n, the condition necessary to secure that x, y shall both be positive is a + mn > m*.

BOOK II 149

Add x to each and square. We then have

and, if , [ are respectively subtracted, the remainders

are the same square. Let then x* be the required square, and we have only to

make £ . • I tne required parts of 20.

Thus 10* +13 = 20,

and X = IQ.

The required parts are then (£?, ^Y and the required square is — .

15. To divide a given number into two parts and to find a square which, when each part is respectively subtracted from it, gives a square.

Given number 20.

Take (x -I- m? for the required square \ where m* is not

greater than 20, e.g. take (x + 2)*.

This leaves a square if either AX + 4 } . ,

> is subtracted. or 2;r + 3J

Let these then be the parts of 20.

1 Here again the implied condition, namely that m- is not greater than 10, is not necessary ; the condition necessary for a real solution is something different.

The equations to be solved are x+y=a, z^ — x — u^, z*—y=iP.

Diophantus here puts (£ + m)* for zz, so that, if x=im$ + m?, the second equation is satisfied. Now (£ + w)2 -y must also be a square, and if this square is equal to (£ + m - «)*, say, we must have

Therefore, since x+y=a,

i (m + n) £ + mz + *mn -n*=a,

a — m? + w2 — imn

whence £= -- ; - r - ,

2 (/// + «)

and it follows that

_ m (a - mn + »8) _ » (a - mn + m2) m + tt ' y~ m + n '

If m>n, it is necessary, in order that x, y may both be positive, that a + «2 > mn, which is the true condition for a real solution.

150 THE ARITHMETICA

Therefore 6x + 7 = 20, and x = J^-.

The required parts are therefore f^-, ^ j , and

the required square is —~.

1 6. To find two numbers in a given ratio and such that each when added to an assigned square gives a square.

Given square 9, given ratio 3:1.

If we take a square of side (mx + 3) and subtract 9

from it, the remainder may be taken as one of the

numbers required.

Take, e.g., (x + 3)* — 9, or x* + 6x, for the lesser number. Therefore 3^2+ i8x is the greater number, and 3-r2+ 18^+9

must be made a square = (2.x — 3)2, say. Therefore x = 30, and

the required numbers are 1080, 3240.

17. To find three numbers such that, if each give to the next following a given fraction of itself and a given number besides, the results after each has given and taken may be equal1.

First gives to second \ of itself + 6, second to third £ of

itself + 7, third to first j of itself + 8. Let first and second be 5^ 6x respectively. When second has taken x + 6 from first it becomes 7^ + 6,

and when it has given x + J to third it becomes

6x-\. But first when it has given x + 6 to second becomes

^x — 6 ; and this too when it has taken \ of third

+ 8 must become 6x — i . Therefore f of third + 8 = 2x + 5, and

third = i^x — 21. Next, third after receiving £ of second + 7 and giving \ of

itself + 8 must become 6x— I. Therefore I ye — 19 = 6x — I, and x = ±£.

The required numbers are ^-, - — , — .

1 Tannery is of opinion that the problems II. 17 and 18 have crept into the text from an ancient commentary to Book I. to which they would more appropriately belong. Cf. I- 22, 23.

BOOK II 151

1 8. To divide a given number into three parts satisfying the conditions of the preceding problem1. Given number 80. Let first give to second £ of itself + 6, second to third

£ of itself + 7, and third to first f of itself + 8. [What follows in the text is not a solution of the problem but an alternative solution of the preceding. The first two numbers are assumed to be ^x and 12, and

the numbers found are ^^, - — -.1 19 19 19 J

19. To find three squares such that the difference between the greatest and the middle has to the difference between the middle and the least a given ratio.

Given ratio 3:1.

Assume the least square = x3, the middle =x* + 2x -\- I. Therefore the greatest = x> + 8x + 4 = square = (x + 3>2, say. Thus;tr=2£, and

the squares are 30^, 12^, 6£.

20. To find two numbers such that the square of either added to the other gives a square2.

1 Though the solution is not given in the text, it is easily obtained from the general solutjon of the preceding problem, which again, at least with our notation, is easy.

Let us assume, with Wertheim, that '-the numbers required in n. 17 are 5.*, 6y, "jz. Then by the conditions of the problem

4x-6 + z+8 = ;,y-7 + x + 6 = 6z-8+y + 7, from which two equations we can find x, z in terms of y.

In fact *=(20>-i8)/i9 and z=(\-jy- 3)/i9,

and the general solution is

In his solution Diophantus assumes x=y, whence y — — .

Now, to solve II. 18, we have only to equate the sum of the three expressions to 80, and so findy.

We have jgaj

y(i>. 26 + 6. 19 + 7. 17) -5. 18-7. 3 = 80. 19, ? = -^'>

and the required numbers are

944Q 9786 9814 363 ' 363 ' 363 '

2 Euler (Algebra, Part n. Art. 239) solves this problem more generally thus. Required to find x, y such that x3 +y and jp + x are squares.

If we begin by supposing x2+y=J?, so that y=f^-x^, and then substitute the value of y in terms of x in the second expression, we must have p\ _ 2/2jc2 + x* + x = square.

But, as this is difficult to solve, let us suppose instead that •*2 +y=(P~ *)2 =? ~ tfx -•- J:2,

152 THE ARITHMETICA

Assume for the numbers x, 2x + I, which by their form

satisfy one condition. The other condition gives

4^r2 + $x+ I = square = (2x — 2)2, say. Therefore x = -^¥, and

the numbers are — , — .

21. To find two numbers such that the square of either minus the other number gives a square.

x+i, 2x+i are assumed, satisfying one condition. The other condition gives

4^2 + yc = square = gx2, say. Therefore x = f , and

the numbers are -, — .

22. To find two numbers such that the square of either added to the sum of both gives a square.

Assume x, x + I for the numbers. Thus one condition is

satisfied. It remains that

xz + 4*+2 = square = (x — 2)2, say. Therefore x = ^, and

the numbers are -, -. [Diophantus has f, Jg°-.]

23. To find two numbers such that the square of either minus the sum of both gives a square.

Assume x, x + i for the numbers, thus satisfying one

condition.

Then x* — 2x — i = square = (x — 3)2, say. Therefore x = 2%, and

the numbers are 2^, 3^.

and that j/2 + x = (q -yf = q* - iqy +y2.

It follows that

whence

tfq - i 4/7 - i

Suppose, for example, /=2, ?=3, and we have * = — , y= — ; and so on. We

must of course choose /, q such that x, y are both positive. Diophantus' solution is obtained by putting p= - i, ^=3.

BOOK II 153

24. To find two numbers such that either added to the square of their sum gives a square.

Since x3 + $x*, x2 + Sx* are both squares, let the numbers

be $xz, 8x* and their sum x. Therefore I2ix* = x2, whence I ix* = x, and x = Jp

The numbers are therefore , .

25. To find two numbers such that the square of their sum minus either number gives a square.

If we subtract 7 or 12 from 16, we get a square.

Assume then \2x*t jx* for the numbers, and i6x- for the

square of their sum. Hence i$x* = 4*, and x = ^.

The numbers are ^2, **2.

26. To find two numbers such that their product added to either gives a square, and the sides of the two squares added together produce a given number.

Let the given number be 6.

Since x (4* — i ) + x is a square, let x, 4* - i be the numbers.

Therefore 4#2 + 3^—1 is a square, and the side of this square must be 6 — 2x [since 2x is the side of the first square and the sum of the sides of the square is 6].

Since 4^ + 3* — i = (6 — 2x)*,

we have x = fj, and

the numbers are §[. — . 27 27

27. To find two numbers such that their product minus either gives a square, and the sides of the two squares so arising when added together produce a given number.

Let the given number be 5.

Assume 4x+i,x for the numbers, so that one condition

is satisfied.

Also 4*2 - 3* — i = (5 — 2x)*.

Therefore * = ff , and

the numbers are *|, *^.

i54 THE ARITHMETICA

28. To find two square numbers such that their product added to either gives a square.

Let the numbers1 bex*,y2.

X& -1/2 i i/2)

Therefore * 2j- are both squares.

To make the first expression a square we make x* -f i a

square, putting

x* + i = (x - 2)2, say. Therefore x = f , and x* = ^. We have now to make ^(y* + i) a square [and y must be

different from x\

Put 9^ + 9 = (3y- 4)2> say,

and y = lt.

Therefore the numbers are -%. Q. io' 576

29. To find two square numbers such that their product minus either gives a square.

Let xz,y* be the numbers.

.£.2 ,,,2 _ »£\

Then * J \ are both squares. x y —XT)

A solution of x* — I = (a square) is x"1 = ff . We have now to solve

f f ^ - ft = a square. Put ^-i=(j-4)2,say.

Therefore y = *£, and

the numbers are $9 ??. 64 64

30. To find two numbers such that their product + their sum gives a square.

Now mz + n? ± 2mti is a square.

Put 2, 3, say, for m, n respectively, and of course

22 + 32 ± 2 . 2 . 3 is a square. Assume then product of numbers = (22 + 32)^2 or 1 3^, and

sum = 2 . 2 . 3^r2 or 1 2x*.

The product being 13^, let x, lye be the numbers. Therefore their sum 14^= i2x*, and ^ = ^. The numbers are therefore L .

1 Diophantus does not use two unknowns, but assumes the numbers to be xz and I until he has found x. Then he uses the same unknown (x) to find what he had first taken to be unity, as explained above, p. 52. The same remark applies to the next problem.

BOOK II 155

31. To find two numbers such that their sum is a square and their product ± their sum gives a square.

2 . 2m .m = a. square, and (2m)* + m* ± 2 . 2m .m = a. square.

If m = 2, 4* + 22 ± 2 . 4 . 2 = 36 or 4,

Let then the product of the numbers be (4* + 2f)x* or 2Ora,

and their sum 2.4.2JT2 or idr1, and take 2-r, lor for

the numbers.

Then \2x= idr2, and * = f.

The numbers are -, — .

32. To find three numbers such that the square of any one of them added to the next following gives a square.

Let the first be x, the second 2x -f i, and the third 2(2*+!)+ i or 4* +3, so that two conditions are satisfied.

The last condition gives (44: + 3)* + x = square = (44: — 4^, say.

Therefore x=^y and

the numbers are ^, g, &.

33. To find three numbers such that the square of any one of them minus the next following gives a square.

Assume x + i, 2x+ \, $x+ i for the numbers, so that two

conditions are satisfied. Lastly, 1 6r2 + "jx = square = 25*', say, and x = £.

The numbers are ^, ^, ^. 999

34. To find three numbers such that the square of any one added to the sum of all three gives a square.

\^(m — n^-\-mn is a square. Take a number separable into two factors (#/, «) in three ways, say 1 2, which is the product of (i, 12), (2,6) and (3, 4).

The values then of \ (m — n) are 5|, 2, £.

Take ^\x, 2xt ^x for the numbers, and for their sum \2x-.

Therefore &r = 1 2x*, and x = \. The numbers are — , *, -.

[Diophantus says |, and ^, f , |.]

156 THE ARITHMETICA

35. To find three numbers such that the square of any one minus the sum of all three gives a square.

{£ (m + «)}2 — mn is a square, Take, as before, a number

divisible into factors in three ways, as 12. Let then 6^, 4*, ^\x be the numbers, and their sum \2x*. Therefore \4x=i2x*, and;r = f.

The numbers are &, , .

BOOK III

1. To find three numbers such that, if the square of any one of them be subtracted from the sum of all three, the remainder is a square1.

Take two squares ;r2, ^x* ; the sum is 5;r2.

If then we take 5;r2 as the sum of the three numbers, and

x, 2x as two of them, we satisfy two conditions. Next divide 5, which is the sum of two squares, into two other squares ^, J^- [ll. 9], and assume \x for the third number. Therefore x + 2x -f \x = 5*2, and x = ^.

The numbers are ^, ^, ~^. [Diophantus writes -^ for x and T%\, ||g, -^ for the numbers.]

2. To find three numbers such that the square of the sum of all three added to any one of them gives a square.

Let the square of the sum of all three be x*, and the

numbers ^, 8x*, i$x\ Hence 26;r2 = x, x = -£%, and

the numbers are ~y. 7-i, — ,.

0/0 0/0 OyO

3. To find three numbers such that the square of the sum of all three minus any one of them gives a square.

Sum of all three 4*", its square i6.r2, the numbers "jx*,

I2XZ, l$X\

Then 34Jtr2 = 4*, x = &, and

the numbers are J?-, £, £.

1 The fact that the problems III. 1-4 are very like II. 34, 35 makes Tannery suspect that they have found their way into the text from some ancient commentary.

BOOK III 157

4. To find three numbers such that, if the square of their sum be subtracted from any one of them, the remainder is a square.

Sum x, numbers 2xzt $xz, lot:2. Then 17 x* = x, x = -fa, and

the numbers are ^, J-, Jj|.

5. To find three numbers such that their sum is a square and the sum of any pair exceeds the third by a square.

Let the sum of the three be (x+ i)2; let first + second = third + i, so that third = ^ + x ; let second •+ third = first + x*, so that first =x + $. Therefore second = ^x* + £.

It remains that first + third = second + a square. Therefore 2.x — square = 16, say, and x = 8.

The numbers are 8|, 32^, 40. Ot/ierwise thus\

First find three squares such that their sum is a square. Find e.g. what square number + 4 + 9 gives a square,

that is, 36 ;

4> 36, 9 are therefore squares with the required property. Next find three numbers such that the sum of each pair = the third + a given number ; in this case suppose first + second — third = 4, second + third — first = 9, third + first — second = 36. This problem has already been solved [I. 18].

The numbers are respectively 20, 6£, 22^.

1 We should naturally suppose that this alternative solution, like others, was inter- polated. But we are reluctant to think so because the solution is so elegant that it can hardly be attributed to a scholiast. If the solution is not genuine, we have here an illustration of the truth that, however ingenious they are, Diophantus' solutions are not always the best imaginable (Loria, Le scienze esatte nelf antica Greda, Libro v. pp. 138-9). In this case the more elegant solution is the alternative one. Generalised, it is as follows. We have to find JT, y, z so that

-x+y + z = a. square\ x —y + 2 = a square Y , x+y-z = a. square)

and also x +y + z = a square.

We have only to equate the first three expressions to squares a2, P, c* such that square, ^ say, since the sum of the first three expressions is itself

The solution is then

i58 THE ARITHMETICA

6. To find three numbers such that their sum is a square and the sum of any pair is a square.

Let the sum of all three be x* + 2x+ i, sum of first and second x2, and therefore the third 2.x + I ; let sum of second and third be (x - i)2.

Therefore the first = ^x, and the second =;r2 — <\x. But first + third = square, that is, 6x + i = square =121, say. Therefore x = 20, and

the numbers are 80, 320, 41.

[An alternative solution, obviously interpolated, is practically identical with the above except that it takes the square 36 as the value of 6x+i, so that x — ^-, and the numbers are ^Q- _?4o 385 456-, - 36 ' 36 ' 36 <J

7. To find three numbers in A.P. such that the sum of any pair gives a square.

First find three square numbers in A.P. and such that half their sum is greater than any one of them. Let ;r2, (x+ i)2 be the first and second of these ; therefore the third is x* + ^x + 2 = (x - 8)2, say.

Therefore x = f $ or f£ ;

and we may take as the numbers 961, 1681, 2401.

We have now to find three numbers such that the sums of pairs are the numbers just found.

The sum of the three = a^a = 25211, and

the three numbers are 120^, 840^, 1560^.

8. Given one number, to find three others such that the sum of any pair of them added to the given number gives a square, and also the sum of the three added to the given number gives a square.

Given number 3.

Suppose first required number + second =x"* + 4x+ i, second + third = x* + 6x + 6, sum of all three = x* + Sx + 1 3. Therefore third =^x + 12, second =x*-+2x-6, first = Also first + third + 3 = a square, that is, 6x + 22 = square = 100, suppose. Hence x= 13, and

the numbers are 33, 189, 64.

BOOK III 159

9. Given one number, to find three others such that the sum of any pair of them minus the given number gives a square, and also the sum of the three minus the given number gives a square.

Given number 3.

Suppose first of required numbers + second = x* + 3,

second + third =x*+2x + 4,

sum of the three = x* + ^x + 7.

Therefore third = 4^+4, second = x* — zx, first = 2x + 3. Lastly, first + third — 3 = 6;r + 4 = a square = 64, say. Therefore x = 10, and

(23, 80, 44) is a solution.

10. To find three numbers such that the product of any pair of them added to a given number gives a square.

Let the given number be 12. Take a square (say 25)

and subtract 12. Take the difference (13) for the

product of the first and second numbers, and let these

numbers be 13*, \\x respectively. Again. subtract 12 from another square, say 16, and let the

difference (4) be the product of the second and third

numbers.

Therefore the third number = 4*. The third condition gives 52^* 4- 12 = a square; now

52 = 4. 13, and 13 is not a square; but, if it were a

square, the equation could easily be solved1. Thus \ve must find two numbers to replace 13 and 4 such

that their product is a square, while either + 12 is

also a square. Now the product is a square if both are squares ; hence we

must find two squares such that either + 12 = a square. " This is easy2 and, as we said, it makes the equation easy

to solve." The squares 4, \ satisfy the condition.

1 The equation 52^+ n = «2 can in reality be solved as it stands, by virtue of the fact that it has one obvious solution, namely x = i . Another solution is found by substituting jr+i for jr, and so on. Cf. pp. 69, 70 above. The value JT= i itself gives (13, i, 4) as a solution of the problem.

2 The method is indicated in II. 34. We have to find two pairs of squares differing by 12. (a) If we put 12 = 6.1, we have

and 1 6, 4 are squares differing by n, or 4 is a square which when added to n gives a square. (£) If we put 12 = 4.3, we fi°d '-(4-3); or - to be a square which when added to 12 gives a square.

160 THE ARITHMETICA

Retracing our steps, we now put <\x, ijx and x\^ for the numbers, and we have to solve the equation

x*+i2 = square = (x + 3)", say. Therefore x = %, and

(2, 2, 1 j is a solution1.

ii. To find three numbers such that the product of any pair minus a given number gives a square. Given number 10. Put product of first and second = a square + 10 =4 -H 10,

say, and let first = 14*, second = \\x. Let product of second and third = a square + 10= 19, say ;

therefore third = 19*:. By the third condition, 266x'2 — 10 must be a square ; but

266 is not a square2. Therefore, as in the preceding problem, we must find two

squares each of which exceeds a square by 10. The squares 30^, 12\ satisfy these conditions3. Putting now 30^^, \\x, \2-\x for the numbers, we have,

by the third condition, 37OT95;r2 — 10 = square [for

370^ Diophantus writes 370 J^]; therefore 5929*2 - 160 = square = (77* - 2)2, say. Therefore x = } |, and

the numbers are ^, 22, ».

1 Euler (Algebra, Part n. Art. 232) has an elegant solution of this problem in whole numbers. Let it be required to find x, y, z such that xy + a, yz + a, zx + a are all squares. Suppose xy + a=jP, and make z = x+j> + $;

therefore xz + a = x* + xy + qy. + a = xz + qx + {P,

and yz + a=xy+y* + qy + a=jP + qy+lP',

and the right hand expressions are both squares if ^ = ±2/, so that z = x+y±ip.

We can therefore take any value for p such that /2>«, split p^-a into factors, take those factors respectively for the values of x and y, and so find 2.

E.g. suppose a— 11 and ^2 = 25, so that xy=\$\ let x=i,y=i^t and we have 2=14=^10=24 or 4, and (i, 13, 4), (i, 13, 24) are solutions.

2 As a matter of fact, the equation 266^- io = «2 can be solved as it stands, since it has one obvious solution, namely x=i. (Cf. pp. 69, 70 above and note on preceding problem, p. 159.) The value x=i gives (14, i, 19) as a solution of the problem.

3 Tannery brackets the passage in the text in which these squares are found, on the ground that, as the solution was not given in the corresponding place of in. 10, there was no necessity to give it here. 10 and i being factors of 10,

thus 3<>i is a square which exceeds a square by 10. Similarly {-(5 + 2) I or 12^ is such a square. The latter is found in the text by putting mz- 10 = square = (m -- 2)2, whence *» = 34i and a«2=n|.

BOOK III 161

12. To find three numbers