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DIOPHANTOS OF ALEXANDRIA.


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Honlion : C. J. CLAY and SOX,CAMBEIDGE UNIVERSITY TRESS WAREHOUSE,

AVE :^rARIA LANE.

CAMBRIDGE: I>i:i(;ilToN. lil'.l.l.. .\M> co.LEIPZIG: r. A. liKocKllArs.


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CTambritigr

:

rniNTF.ri dy c. j. clay, m.a. and sox,AT THK rXIVERSITY PRKSS.


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PREFACE.The scope of the prosont book is sufficiently indicatod Itythe title and the Table of Contents. In the chapter on" Dioijhantos' notation and definitions" several suggestionsare made, which I believe to be new, with regard to theorigin and significance of the symbols employed by Diophantos.A few words may be necessary to explain the purp(.se of theAppendix. This is the result of the compression of a largebook into a very small space, and claims to have no inde-pendent value apart from the rest of my work. It is in-tended, first, as a convenient place of reference for mathe-maticians who may, after reading the account of Diophantos'methods, feel a desire to see them in actual operation, and,secondly, to exhibit the several instances of that variety ofpeculiar devices which is one of the most prominent of thecharacteristics of the Greek algebraist, but which cannot l)obrought under general rules and tabulated in the same wayas the processes described in Chapter V. The Appendix, then,is a necessary part of the whole, in that there is much inDiophantos which could not be introduced elsewhere ; it mustnot, however, be considered as in any sense an alternative tothe rest of the book: indeed, owing to its extremely con-densed form, I could hardly hope that, by itself, it wouldeven be comprehensible to the mathematician. I will merelyadd that I have twice carefully worked out the .;


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VI PREFACE.every problem from tlic proof-sheets, so that I hope and be-lieve that no mistakes will be found to have escaped me.

It would be mere tautology to enter into further detailshere. One remark, however, as to what the work does not,and does not profess to, include may not be out of place.No treatment of Diophantos could be complete without athorough revision of the text. I have, however, only cursorilyinspected one MS. of my author, that in the Bodleian Library,which unfortunately contains no more than a small part ofthe first of the six Books. The best Mss, are in Paris andRome, and I regret that I have had as yet no opportunity ofconsulting them. Though this would be a serious drawbackwere I editing the text, no collation of MSS. could afifect myexposition of Diophantos' methods, or the solutions of hisproblems, to any appreciable extent; and, further, it is morethan doubtful, in view of the unsatisfactory results of thecollation of three of the MSS, by three different scholars inthe case of one, and that the most important, of the few ob-scure passages which need to be cleared up, whether the textin these places could ever be certainly settled.

I should be ungrateful indeed if I did not gladly embracethis opportunity of acknowledging the encouragement whichI have received from Mr J. W, L. Glaisher, Fellow and Tutorof Trinity College, to whose prospective interest in the workbefore it was begun, and unvarying kindness while it wasproceeding, I can now thankfully look back as having beenin a great degree the " moving cause " of the whole. And,finally, I wish to thank the Syndics of the University Pressfor their liberality in undertaking to publish the volume.

T. L. HEATH.


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LIST OF BOOKS OH PAl'KKS KKAD ()I{ KKKKlMtKI) 'K >.SO FAR AS THEY CON'CERN OK AUK ISKFIL

TO THE SUBJECT.

1. Bookg directlif upon Dinphautois.Xylander, Diopliaiiti Alexambini Reruni Arithmetit-arum Libri sex

Item Liber de Numcri.s Polygonis. Opus incoiupiirabile Latinoredditum et Commeutariis explanatum Biusileae, 1575.

Bachet, Diophanti Alexandrini Arithmeticoioim Libri sex, et de niuueri.smultaugulis liber uiiu.s. Lutetiae Parisiorimi, 1G21.

Diophanti AJexandi-ini Ai-ithmeticorum libri sex, et de uumeris multaugu-lis liber unus. Cum commeutariis C. G. Bacheti V.C. et oWrua-tionibus D. P. de Fermat Senatoris Tolcsani. Tolosae, 1G70.

ScHULZ, Diophantus von Alexandria arithmetische Aufgaben nebst desseuSchrift liber die Polygon-zahlen. Aus dem Griecbi-scheu iibersetztund mit Anmerkungeu begleitet. Berlin, 18-22.

PoSELGER, Diophantus von Alexandrien iiber die Polygon-Zahlen.Uebersetzt, mit Zusiitzen. Leipzig, 1810.

Crivelli, Elementi di Fisica ed i Problemi aritlmietici di DiofantoAlessandrino analiticamente dimostrati. In Venczia, 1744.

P. Glimstedt, Forsta Boken af Diophanti Arithmetica algebraisk Ocfvcr-sattning. Lund, 1855.

Stevin and Girard, " Translation " in Les Oeuvres mathematiques deSimon Stevin. Leyde, 1684.

2. M'orha indirectly fluridati)i.Brassinne, Precis des Oeuvres mathematicpies de P. Fcrnuit et de I'Aritlj-

metique de Diophante. P'""is l''*-'>3-CossALi, Origine, traspoi-to in Italia, prinii progre.s.si in e-ssa dell' Algebni

Storia critica Parnm, 17U7.Nesselmanx, Die Algebra der Griechcn. Berlin, IM2.John Kersey, Elements of Algebra. London, 1674.Walms, Algebra (in Opera Mathematica. Ox.iiittC, 161)5 9 .Saundek.son, N., Elements of Algebra. >"''


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Vlll LIST OF AUTIlulUTIKS.

3. Buuks ic/tich iiifiitivii or (/ice infurmation about Dio^laiiUof,including historiiis of mathematics.

CuLEUHOOKE, AlgeVira with Arithmetic and ^Mensuration from the Sanscritof Brahmaguptii and Bhiiscara. London, 1817.

SriDAs, Lexicon (ed. G. Bernhardy). Ilalis et Brunsvigae, 1853.Fabricii.s, Bibliotheca Graeca (ed. Harless).AuCLEARAJ, History of the Dynasties (tr. Pococke). Oxon. 16C3.Ch. Th. v. Murr, Memorabilia Bibliothecarum publicarum Norimbergen-

sium et Universitatis Altdorfinae. Norimbergae, 1786.DoPPELMAYR, Historische Nachricht von den Xiirnbergischen Mathema-

ticis und Kiinstlern. (Nliruberg, 1730.)Vos.siis, De universae mathesius natiira et coustitutione

Amstelaedami, 16G0.Hkilbronneh, Historia matheseos universae. Lipsiae, 1742.MuNTLCLA, Histoire des Math(5matiques. Paris, An 7.IviAEUEL, Matheniatisches "\Vorterl)uch. Leipzig, 1830.Kaestner, Geschiclite der Matheniatik. Giittingen, 1796.BussuT, Histoire G(5uerale des Mathematiques. Paris, 1810.Hankel, Zur Geschichte der Mathematik in Altertlium und Mittelalter.

Leipzig, 1874.Cantor, Vorlesungen Uber Geschichte der Mathematik, Band L

Leipzig, 1880.Dr Heinrich Slter, Gesch. d. :^Lathematischen Wisseuschaften,

Zurich, 1873.Jame.s Gow, a short History of Greek Mathematics.

Camb. Univ. Press, 1884.

4. Papers or Pamphlets read in connection with Diophantos.Poselger, Beitriige zur Unbestimmten Analysis.

(Berlin xihhandhmgen, 1832.'iI.. RoDET, L'Algebre d'Al-Kharizmi et les methodes indienne et grecque.

{Journal AHiatitjite, Janvier, 1878.)WoEPCKE, Extrait du Faklni, traitc^ d'Algebrc par Abou Bekr ^[ohammed

ben Alhayan Alkarkhi, precede d'un memoiresurralgebre indeterminet;chez los Arabes. Paris, 1853

.

WoEi'CKE, Mathematiques chez les Orientaux.1. Journal Asiatique, Fdvrier-Mars, 1855.

2. Journal Asiatique, Avril, 1855.I'. Tanxehv, "A


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CONTENTS.CHAPTER I.

HISTORICAL INTRODUCTION'.PAGES

1. Diophantos' name and particulars of his life .... i 2. His date. Different views 3

() Internal evidence considered 4_S{b) External evidence 8 IG

3. Results of the preceding investigation 1617

CHAPTER II.THE WORKS OF DIOPHANTOS ; THEIR TITLES AND GENERAL

CONTENTS; THE PORTIONS WHICH SURVIVE. 1. Titles : no real evidence that 13 books of Aritliiiietics ever existed

corresponding to the title IS23No trace of lost books to be restored from Arabia. Corruptionmust have taken place before 11th cent, and probably before950 A.D 23 "iC,

Poiisms lost before 10th cent. a.d. 2 3.">

Other views of the contents of thf lost Books .... 3J37Conclusion 37CHAPTER III.

THE WlllTEKS UPON JlolMIA.\ T ),s. 1. (heck 38-39 2. Arabian 3912 3. European gencially 42 5('>


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CONTENTS.

CHAPTER IV.\OT.\TI(N AND DEFINITIONS (F DlOPH.\NTOS.

VAC.KS 1. Introduction ,57 2. Sign for the unknown quantity discubsed 5767 3. Notation for powers of the unknown G709 i. Objection that Diophantos loses generality by the want of

more algebraic symbols answered 69Other questions of notation : operations, fractions, dc. . . 6976 5. General remarks on the historical development of algebraic

notation : three stages exhibited 7680 6. Ou the influence of Diophantos' notation on his work . . 8082

CHAPTER V.

1.

3.

diophantos' METHODS OF SOLUTION.General remarks. Criticism of the positions of Hankcl andNcsselmann

Diophantos' treatment of equations .....(A) Determinate equations of different degrees.

(1) Pure equations of different degrees, i.e. equations containing only one power of tlie unknown

(2) Mixed quadratics(3) Cubic equation .......Indeterminate equations.'.. Indeterminate equations of the first and second degrees.

(li)

(1)

(2)

Single equation (second degree) 1. Those which can always be rationally solved2. Those which can be rationally solved onlyunder certain conditions

II,

Double equations.1. First general method (first degree) .

Second method (first degree) .2. Double equation of the second degrei

Indeterminate equations of liigher degrees.(1) Single ecjuations (first class)

,, (second class)(2) Double equations .

Summary of the prerediiiji incestiijntioii

88114

88-


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CONTKNTS.

CHAPTER VI.PAOEH

1. The PonsHis of Diophantos 121 I2.'i2. Other theorems assumed or implied 12.>132

('/) Numbers as the sum of two squares 127 1:


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EREATUM.On p. 78, last line but one of note, for " Targalia" read "Tartaglia"


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DIOPHANTOS OF ALEXANDRIA.CHAPTER I.

Historical Introduction. 1. The doubts about lîophantos begin, as has been

remarked by Cossali^ with his very name. It cannot be posi-tively decided whether his name was Diophanfos or Diophanês.The preponderance, however, of authority is in favour of theview that he was called Diophantos.

(1) The title of the work which has come down to us underhis name gives us no clue. It is Aiocpdvrov 'A\âvBpQ}


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2 DIOPIIAXTOS OF ALEXANDRIA.(3) In the only quotation from Diophantos which we

know Tlioon of Alexandria (fl. 3G5390 A.D.) speaks of himas At6


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HISTORICAL IXTUODUCTIOX. 3The solution of this epigram-problem gives 84 as the age

at Avhich Diophautos died. His boyhood lasted 14 years, hisbeard grew at 21, he married at 33/; a son was born to him5 years later and died at the age of 42, when his father was80 years old. Diophanto.s' own death followed 4 years laterat the age of 84. Diophantos having lived to so great an age,an approximate date is all that we can expect to find forthe production of his works, as we have no means of judg-ing at what time of life he would be likely to write hisAiithmetics.

2. The most important statements upon the date ofDiophantos which we possess are the following

:

(1) Abu'lfaraj, whom Cossali calls "the courageous compilerof a universal history from Adam to the 13th century," in hisHistory of the Dynasties before mentioned, places Diophantos,without giving any reason, under the Emperor Julian (3G1368 A.D.). This is the view which has been ordinarily held.It is that of Montucla.

(2) We find in the preface to Rafael Bombclli's Algebra,published 1572, a dogmatic statement that Diophantos livedunder Antoninus Pius (138161 A.D.). This view too hasmet with considerable favour, being adopted by Jacobus deBilly, Blancanus, Vossius, Heilbronner, and others.

Besides these views we may mention Bachet's conjecture,which identifies the Diophantos of the Arithmetics with anastrologer of the same name, who is ridiculed in an epigramattributed to Lucilius ; whence Bachet concludes that helived about the time of Nero (5468) (not under Tiberius,as Nesselmann supposes Bachet to say). The three viewshere mentioned will be discussed later in detail, as they areall worthy of consideration. The same cannot be said of anumber of other theories on the subject, of which I will quoteonly one as an example. Simon Stevin* places Diophantoslater than the Arabian algebraist Mohammed ibn Miisa

1 Les Oeuvrcs Mathcin. de Sim. Stevin, augm. par Alh. Girard, Loyden, 1634,"Quant h, Diophant, il semblc iiu'cn son temps los inventions de Mahometayent seulement tsto cognues, commc bo poult colligcr de sea six premierslivres." 12


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DIOPHANTOS OF ALEXANDRIA.Al-Kliarizmi who lived in the first half of the 9th century, theabsurdity of which view will appear.We must now consider in detail the

(a) Internal evidence of the date of Diophantos.

(1) It would be natural to hope to find, under this head,references to the works of earlier or contemporary mathema-ticians. Unfortunately there is only one such reference trace-able in Diophantos' extant writings. It occurs in the fragmentupon Polygonal Numbers, and is a reference to a definitiongiven by a certain Hypsikies\ Thus, if we knew the date ofHypsikles, it would enable us to fix with certainty an upperlimit, before which Diophantos could not have lived. It isparticularly unfortunate that we cannot determine accuratelyat what time Hypsikles himself lived. Now to Hypsikles isattributed the work on Regular Solids which forms BooksXIV. and xv, of the Greek text of Euclid's Elements. In theintroduction to this work the author relates'^ that his fatherknew a treatise of Apollonius only in an incorrect form, whereashe himself afterwards found it correctly worked out in anotherbook of ApoUonios, which was easily accessible anywhere inhis time. From this we may with justice conclude that Hypsikles'father was an elder contemporary of ApoUonios, and must havedied before the corrected version of ApoUonios' treatise wasgiven to the world. Hypsikles' work itself is dedicated to afriend of his father's, Protarchos by name. Now ApoUoniosdied about 200 B.C.; hence it follows that Hypsikles' treatise

' Polyg. Numbers, prop. 8."Kal iirtdelxOri t6 waph. 'typiKkeX iv 8p(p Xeyd/J-evov.^''^


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HISTORICAL INTRODUCTION. 5on Regular Solids was probably written about 180 B.C. Itwas clearly a youthful productiou. Besides this we have anotherwork of Hypsikles, of astronomical content, entitled in Greekdva^opLK6


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6 DIOPHANTOS OF ALEXANDRIA.form in Nikomachos and Theon of Smyrna ; from this heargues that Hypsikles must have been later than both thesemathematicians, adducing as further evidence that Theon (whois much given to quoting) does not quote him. Doubtless, asTheon lived under Hadrian, about 130 A.D., this would give adate for Hypsikles which would agree with that drawn fromFabricius' conjecture ; but it is not possible to regard eitherpiece of evidence as in any way trustworthy, even if it w^erenot contradicted by the evidence before adduced on the otherside.We may say then with certainty that Hypsikles, and there-fore afortiori Diophantos, cannot have written before 180 B.C!.

(2) The only other name mentioned in Diophantos' writingsis that of a contemporary to whom they are dedicated. Thisname, however, is Dionysios, which is of so common occurrencethat we cannot derive any help from it whatever.

(3) Diophantos' work is so UTiique among the Greek trea-tises which we possess, tliat he cannot be said to recal the styleor subject-matter of any other author, except, indeed, in thefragment on Polygonal Numbers ; and even there the referenceto Hypsikles is the only indication we can lay hold of.

Tiie epigram-problem, which forms the last question of the5th book of Diophantos, has been used in a way which is rathercurious, as a means of determining the date of the Arithmetics,by M. Paul Tannery \ The enunciation of this problem, whichis different from all the rest in that (a) it is in the form of anepigram, (6) it introduces numbers in the concrete, as appliedto things, instead of abstract numbers (with which alone allthe other problems of Diophantos are concerned), is doubtlessborrowed by him from some other source. It is a questionabout wine of two different qualities at the price respectively of8 and 5 drachmae the %o{;9. It appears also that it was wine ofinferior quality as it was mixed by some one as drink for hisservants. Now M. Tannery argues (a) tliat the numbers 8 and5 were not hit upon to suit the metre, for, as these are the onlynumbers which occur in the epigram, and both are found in

* lUilh'tin (ten Sciences mathnnntiqiiis et astronomif/ucs, 1879, p. 201.


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HISTORICAL INTRODrCTION. 7the same line in the compounds 6KTa8pd^^f^ov


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8 DIOPHAXTOS OF ALEXANDRIA.On the vexed question as to how far Diophantos was original

we shall have to speak later.Wo pass now to a consideration of the(b) External evidence as to the date of Diophantos.

(1 ) We have first to consider the testimony of a passage ofSuidas, which has been made much of by writers on the ques-tion of Diophantos, to an extent entirely disproportionate to itsintrinsic importance. As however it does not bear solely uponthe question of date, but upon another question also, it cannotbe here passed over. The passage in question is Suidas' article'TTTartaV The words which concern us apparently stood inthe earliest texts thus, eypay^rev vTro/xvrj/jLa et? AiocfxivrrjvTov darpovofiiKov. Kavova et


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HISTORICAL INTRODUCTION. f)Diophantos, as representing that science. However, Baclicthas proposed to identify our Diophantos with an astrologer ofthe same name, who is ridiculed in an epigram' supposed to hewritten by Lucilius. Now the ridicule of the epigram wouldbe clearly out of place as applied to the subject of the epigrammentioned above, even supposing that Lucilius' ridiculoushero is not a fictitious personage, as it is not unreasonable tosuppose.

Bachet's reading of the passage is vTro/ivrjfia eh AioâfToz/,t6i> darpovofiLKOv Kavova, etf ra koovlku AttoWcoviov vTro^vr,-fia'^. He then proceeds to remark that it shows that Hypatiawrote a Canon Astronomicus, so that she evidently was versedin Astronomy as well as Geometry (as shown by the Commen-tary on Apollonios), two of the three important branches ofMathematics. It is likely then, argues Bachet, that she wasacquainted with the third. Arithmetic, and wrote a commentaryon the AritJtmetics of Diophantos. But in the first place weknow of no astronomical work after that of Claudius Ptolemy,and from the way in which 6 da-rpovoîLKO'; Kavwv is mentionedit would be necessary to suppose that it had been universallyknown, and was still in common use at the time of Suidas, andyet was never mentioned by any one else whom we knjULUUiinexplicable hypothesis.

' 'ISipixoyivt) Tov larpov 6 affrpoXoyoi Ai6ai'T(XEln-e /xovovi ^wfjs ivvia pLrjvas ^X^'"-

KcLKeivos ycXdaai, Ti /jl(v 6 KpSvos ivvia. /xrjvwy,^qal, \^yL, (TV voef Ta/xa 5i ci'inofxa. aoc

Elwe Kal ^KTslvas fwvov Tjxj/aro' Kal AiO(pain-os'AWov dve\iriû)v, avrbs awf(TKapKrev.

"Ludit non innenustus poeta turn in Diopbantum AstroloRum, turn in niccli-cum Hermogenera, quem et alibi saepe false admodum perstringit, qniVl soloattactu non aegros modo, sed et ben(^ valentes, velut pestifero sidere afflntoarepente necaret. Itaque nisi Diopbantum nostrum Astrologiae iieritum fuisscnegemus, nil prohibet, quo minus eum aetate Lucillij extitisse dicanius."

Bacbet, Ad Urtorrm.- From tbis reading it is clear that Bachet did not rest his view of the

identity of our Diophantos with the astrologer upon the i)as8age of Suidas.M. Tannery is therefore mistaken in supposing this to be the case, "Bachet,ayant lu dans Suidas qu'Hy^mtia avait commentu le Canon astronomique dnotre auteur..."; that is precisely what Bacbet did rmt read there.


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10 DIOPHANTOS OF ALEXANDRIA.Next, the expression ek Aio^avrov has been objected to by

Nesselmann as not being Greek. He maintains that the Greeksnever speak of a book by the name of its author, and thereforewe ought to have Atocfxivrov dpidfiijTiKa, if the reference wereto Diophantos of the Arithmetics. M. Tannery, however, de-fends the use of the expression, on the ground that similarones are common enough in Byzantine Greek. M. Tannery,accordingly, to avoid the difficulties which we have mentioned,supposes some words to have dropped out after ^lo^avrov, andthinks that we should read et? At6


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HISTORICAL INTRODUCTION. \\cujus sex libros, cum tamcn author ipso tredccim poUiceatur,graecos habemus de arithmcticis admirandac subtilitatis artcmconiplexis, quae vulgo Algebra arabico nomine appellatur : cumtamen ex authore hoc antique (citatur enim a Theone) anti-quitas artis appareat. Scripserat et Diophantus harmonica."This quotation was known to Montucla, who however draws anabsurd conchision from it* which is repeated by Klucrel in hisWorterbuchl The words of Theon which refer to Diopliantosare koI Ai6(})avT6


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12 DIOPHANTOS OF ALEXANDRIA.certainly incorrect .and due to a confusion on the part of Suidas,for Pappos probably flourished under Diocletian (a. D, 284305) ; but the date of a certain Commentary of Theon has beendefinitely determined' as the year 372 A. D. and he undoubtedlyflourished, as Suidas says, in the reign of Theodosius I. (379395 A. D.).

(4) The next authority who must be mentioned is theArabian historian Abu'lfaraj, who places Diophantos withoutremark under the emperor Julian. This statement is importantin that it gives the date which has been the most generally ac-cepted. The passage in Abu'lfaraj comes after an enumerationof distinguished men who lived in the reign of Julian, and isthus translated by Pococke : "Ex iis Diophantes, cuius liberA. B. quem Algebram vocat Celebris est."

It is a difiicult question to decide how much weight is to beallowed to Abu'lfaraj's dogmatic statement. Some great autho-rities have unequivocally pronounced it to be valueless. Cossaliattributes it to a confusion by Abu'lfaraj of our author withanother Diophantos, a rhetorician, who is mentioned in anotherarticle'^ of Suidas as having been contemporary with the em-peror Julian (361363); and assumes that Abu'lfaraj made thestatement solely on the authority of Suidas, and confused twopersons of the same name. Cossali remarks at the same timeupon a statement of Abu'lfaraj's translator, Pococke, to the effectthat the Arabian historian did not know Greek and Latin.Colebrooke too' {Algebra of the Hindus) takes the same view.Ncnv it certainly seems curious that Cossali should remark uponAbu'lfaraj's ignorance of Greek and yet suppose that he made astatement merely upon the authority of Suidas ; and the ques-tion suggests itself: had Abu'lfaraj no other authority? We

1 "On the date of Pappus," Ac., by Hermann Usener, Neues RheinischesMuseum, 1873, Bd. xxviii. 403.

* Ai^dvios, (TO(piaTr)i AvTioxf'y. twv ivl toO 'lovXiavov toO Uapa^aTov xp


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HISTORICAL INTR()incTR)X. l.Smust certainly, as was remarked by Schulz, admit that he musthave had ; for he gives yet another statement about Diophaiitos,which certainly comes from another source, that his work wastranslated into Arabic, or commented upon, by MohanuiicdAbu'1-Wafa. There would seem however to be but one possibilitywhich would make Abu'lfaraj's statement trustworthy. Is itpossible that the two persons, whom he is supposed to haveconfused, are identical ? Is it a sufficient objection that Liba-nius distinguished himself chiefly as a rhetor and not as amathematician ? In fact, in the absence of any evidence to thecontrary, why should the arithmetician Diophantos not havebeen a rhetorician also ? This question has given occasion tosome jests on the compatibility of the two accomplishments.M. Tannery, for example, quotes Fermat, who was " Conseillerde Toulouse " ; and Nesselmann mentions Aristotle, arrivingfinally at the conclusion that the two may be identical, and so,while Abu'lfaraj's statement has nothing against it, it has agreat deal in its favour. But M. Tannery thinks he has madethe identification impossible by finding Suidas' authority, namelyEunapios in the Lives of the Sophists, who mentions this otherDiophantos as an Arabian, not an Alexandrian, and professingat Athens \ Certainly if this supposition is correct, we cannotidentify the two persons, and therefore cannot trust the state-ment of Abu'lfaraj. There is a further considerationthat thereign of Julian (361363) could certainly only have been theend of Diophantos' life, as we see by comparing Theon's date,above mentioned, to whom Diophantos is certainly anterior;he may indeed have been much earlier, because (1) Theonquotes him as a classic, and (2) the absence of quotations beforeTheon does not necessarily show that the two were nearlycontemporary, for of previous writers to Theon who would havebeen likely to quote Diophantos ?

(5) In the preface to his Algebra, published A.D. 1572,Rafael Bombclli gives the bare statement that Diophantos lived

^ "II uous donne ce Diophante, qu'il a connu et dont il ne fait d'ailleurepas grand cas, comme nO, non pas a, Alexandrie, ainsi que le matht-maticien,mais en Arable (AiocpavTos 6 'Apd^ios), et, d'autre part, conime prolessant AAthenes."


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14 DIOPIIANTOH OF ALEXANDRIA.in the reign of Antoninus Pins', giving no proof or evidence ofit. From the demonstrated incorrectness of certain other state-ments of Bombelli concerning Diophantos we may infer that weought not hastily to give credence to this ; on the other hand itis scarcely conceivable that he would have made the assertionwithout any ground whatever. The question accordingly arises,whether we can find any statement by an earlier writer, whichmight have been the origin of Bombelli 's assertion. M. Tannerythinks he has found the authority while engaged in anotherresearch into the evidence on which Peter Ramus ascribes toDiophantos a treatise on Harmonics '^ an assertion repeated byGessner and Fabricius^. As I cannot follow M. Tannery in hisconjecturesfor they are nothing better, but are rather con-jectures of the wildest kind, I will give the substance of hisremarks without much comment, to be taken for what they areworth. According to M. Tannery Ramus' source of informationwas a Greek manuscript on music ; this there is no reason todoubt; and in the edition of Antiquae musicae auctores byMeibomius we read, in the treatise by Bacchios 6


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HISTORICAL INTRODUCTION'. lobe less likely than the other ? I confess that it seems to mc hyfar the more likely of the two ; for the long ami short vowx-ls o,M must have been closely associated, as is proved by the factthat in ancient inscriptions^ we find O written for both O and ilindiscriminately, and in others H used for both sounds.) Tlicii,according to M. Tannery, Ramus probably took the name forAi6(f)avTo


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16 DIOPHANTOS OV ALEXANDRIA.sufficiently near in point of" time to Diophantos and the rest inorder to know their respective ages. Unfortuoately, however,that is far from certain, Bacchios' own date being very doubtful.He is generally supposed to have lived in the time of Constantinethe Great ; this is however questioned by M. Tannery whothinks that the epigram given by Meibomius, in which Bacchiosis associated with a certain Dionysios, refers to ConstantinePorphyrogenetes, who belongs to the sixth century. Next,grave doubts may be raised concerning the determination bymeans of the supposed chronological order; for the definitionsof rhythm given by Nikomachos and Diophantos (?) are verynearly alike, that of Diophantos being apparently a developmentof that of Nikomachos : kutu 8e NiKOfia^ov, '^povcou evTUKTO'iavvd


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HISTORICAL INTROnrcTION. 17open to no objection, it would seem best to accept it provisionally,as the least uusatistactory theory. We shall therefore be notimprobably right in placing Diophantos in the second half of thethird century of our era, making him thus a contemporary ofPappos, and anterior by a century to Theon of Alexandria andhis daughter Hypatia,

One thing is quite certain: that Diophantos lived in aperiod when the Greek mathematicians of great original powerhad been succeeded by a number of learned commentators, whoconfined their investigations within the limits already reached,without attempting to further the development of the science.To this general rule there are two most striking exceptions, indifferent branches of mathematics, Diophantos and Pappos.These two mathematicians, who would have been an ornamentto any age, were destined by fate to live and labour at a timewhen their work could not check the decay of mathematicallearning. There is scarcely a passage in any Greek writerwhere either of the two is so much as mentioned. The neglectof their works by their countrymen and contemporaries can beexplained only by the fact that they were not appreciated orunderstood. The reason why Diophantos was the earliest of theGreek mathematicians to be forgotten is also probably thereason why he was the last to be re-discovered after the Revivalof Learning. The oblivion, in fact, into which his writings andmethods fell is due to the circumstance that they were notunderstood. That being so, we are able to understand whythere is so much obscurity concerning his personality and thetime at which he lived. Indeed, Avhen we consider how littlehe was understood, and in consequence how little esteemed, wecan only congratulate ourselves that so much of his work hassurvived to the present day.


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CHAPTER II.THE WORKS OF DIOPHANTOS ; THEIR TITLES AND GENERAL

CONTENTS; THE PORTIONS OF THEM WHICH SURVIVE.

1. We know of three works of Diophantos, which bearthe following titles.

(1) Wpi6fxr]TtKci}v /Si/SXia ly.(2) Trep] TToXvyoovcov apidfioov.(3) TropiafMara.

With respect to tlie first title we may observe that themeaning of "dpid/jbrjTiKa' is slightly different from that assignedto it by more ancient writers. The ancients drew a markeddistinction between dpidfiijTiKT] and \ 0740- rt/c?;, both of whichwere concerned with numbers. Thus Plato in Gorgias 451 B*states that dpidfirjrtKy'] is concerned with the abstract propertiesof numbers, odd even, and so on, whereas XoytaTCKij deals withthe same odd and even, but in relation to one anotlier. Geminosalso gives us definitions of the two terms. According to himdpidfxrjTLKij deals with abstract properties of numbers, whileXoyiariKi] gives solutions of problems about concrete numbers.From Geminos we see that enunciations were in ancient timesconcrete in such problems. But in Diophantos the calculations

' 1 tIs fjie fpoiTo..!'(l SwAcpares, tL^ eariv rj dptOfirjTiKr] t^x*''?> cI'toim' S**avTip, tSairep dpri, 6ti twv 5id \6you tis t6 Kvpoi ixovauv. Kal et /xe iwavip-ono Twf TTipl tL ; etiroifi' Av, 6ti twv vtpl rb Apribv tc koI irtpiTTov Sj dp(Kdrepa Ti^yx'**'^' ^"'a- ' 5' av fpoiTO, Trjv 5^ XoyiariKriv rlva KoKds rix^rivilvoiix &v 6ti Kal ai>T7) iarl tCiv \6yifi t6 trdv Kvpovp-ivuv. Kal el IwavipoiTo 'HiTfpl tI ; etiToifJ.^ hv wainp o\ iv rc^ StJ^v


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HIS WORKS. 10take an abstract form, so that the distinction between XoyiaTiKijand apidfjiriTLKr] is lost. We thus have W.pid/xr}TiKd given asthe title of his work, whereas in earlier times the term couldonly properly have been applied to his treatise on PolygonalNumbers. This broader use by Diophantos of the term arith-metic is not without its importance.

Having made this preliminary remark it is next necessary toobserve that of these works which we have mentioned somehave been lost, while probably the form of parts of others hassuffered considerably by the ravages of time. The Arithmeticsshould, according to the title and a distinct statement in theintroduction to it, contain thirteen Books. But all the sixknown MSS.^ contain only six books, with the sole variationthat in the Vatican MS. 200 the same text, which in the restforms six books, is divided into seveii. Not only do the MSS.practically agree in the external division of the work ; theyagree also in an equally remarkable mannerat least all ofthem which have up to the present been collatedin the lacunaeand the mistakes which occur in the text. So much is thisthe case that Bachet, the sole editor of the Greek text ofDiophantos, asserts his belief that they are all copied from oneoriginal ^ This can, however, scarcely be said to be established,

^ The six mss. are

:

13. Vatican mss. No. 191, xiii. c, cbarta bombycina.No. 200, XIV. c, charta pergamena.No. 304, XV. c, charta.

4. MS. in Nat. Library at Paris, that used by Bachet for his text.5. MS. in Palatine Library, collated for Bachet by Claudius Salmasius.6. Xylander's ms. which belonged to Andreas Dudicius.

Colebrooke considers that 5 and 6 are probably identical.- "Etenim neque codex Eegius, cuius ope banc editionem adornavimus;

neque is quern prae manibus habuit Xilander; neque Palatinus, vt doctissimoviro ClauLlio Salmasio refcrente accepimus ; neque Vaticanus, quern vir suniniualacobus Sirmondus mihi ex parte transcribendum curauit, quicquam ampliuscontinent, quam sex hosce Arithmcticorum libros, et tractatum de iiumerismultangulis imperfectum. Sed et tarn infeUcitcr hi omnes codices inter aeconsentiunt, vt ab vno fonte manasse et ab eodem exemplari dcscriptos fuissonon dubitem. Itaque parum auxilij ab his subministratum nobis esse, veris-simu allirmare possum," Epintola ad Lectorem.

It will be seen that the learned Bachet spells here, as everywhere, Xylander'sname wrongly, giving it as Xilander.

O 9


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20 DIOPHANTOS OF ALEXANDRIA.for Bachet had no knowledge of two of the three Vatican MSS.and had only a few readings of the third, furnished to him byJacobus Sirmondus. It is possible therefore that the collationof the two remaining mss. in the Vatican might even now leadto important results respecting the settling of the text. Theevidence of the existence in earlier times of all the thirteenbooks is very doubtful, some of it absolutely incorrect. Bachetsays * that Joannes Regiomontanus asserts that he saw thethirteen books somewhere, and that Cardinal Perron, who hadrecently died, had often told him that he possessed a MS.containing the thirteen books complete, but, having lent itto a fellow-citizen, who died before returning it, had never re-covered it. Respecting this latter MS. mentioned by Bachetwe have not sufficient data to lead us to a definite conclusionas to whether it really corresponded to the title, or, like theMSS. which we knoAv, only announced thirteen books. If itreally corresponded to the title, it is remarkable how (in thewords of Nesselmann) every possible unfortunate circumstanceand even the " pestis " mentioned by Bachet seem to haveconspired to rob posterity of at least a part of Diophantos'works.

Respecting the statement that Regiomontanus asserts thathe saw a MS. containing the thirteen books, it is clear thatit is founded on a misunderstanding. Xylander states in twopassages of his preface " that he found that Regiomontanus

1 "loannes tamen Regiomontanus tredecini Diophanti libros se alicubividisse asseverat, et illustrissimus Cardinalis Perronius, quern nupei- ex-tinctum niagno Christianae et literariae Rcipublicae detrimeuto, conquerimur,mihi saepe testatus est, se codicem manuscriptum habuisse, qui tredeeim Dio-phanti libros integros contineret, quern cilm Gulielmo Gosselino conciui suo,qui in Diophantum Commentaiia meditabatur, perhumauiter more suo exhi-buisset, pauUo post accidit, ut Gossclinus peste correptus iuteriret, et Diophanticodex codem fato nobis criperetur. Cum enim prccibus meis motus Cardi-nalis amplissimus, nullisque sumptibus pai-cens, apud heredes Gosselini codicemilium diligenter exquiri mandassct, et quouis pretio redimi, nusquam repertusest." Ad lectorcm.

* "Inueni deinde tanquam exstantis in bibliothecis Italicis, sibique uisimentionem a Regiomontano (cuius etiam nominis memoriam ueneror) factam."Xylander, Epistola nuncupatoria.

"Sane tredeeim libri Arithmeticae Diophanti ab aliis perhibentur exstare inbibliotheca Vaticana; quos Regiomontanus illo uiderit." Ibid.


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HIS WORKS. 21mentioned a MS. of Diophantos which he liad seen in an Italianlibrary; and that others said that the thirteen books wereextant in the Vatican Library, " which Regiomontanus saw."Now as regards the latter statement, Xylander was obviouslywrongly informed ; for not one of the Vatican Mss. containsthe thirteen books. It is necessary therefore to inquire to whatpassage or passages in Regiomontanus' writings Xylander refers.Nesselmann finds only one place which can be meant, an Oratiohabita Patavii in praelectione Alfragani^ in which Regiomon-tanus remarks that " no one has yet translated from the Greekinto Latin the thirteen books of Diophantosl" Upon thisNesselmann observes that, even if Regiomontanus saw a MS.,it does not follow that it had the thirteen books, except onthe title-page ; and the remarks which Regiomontanus makesupon the contents show that he had not studied them thoroughly

;

but it is not usually easy to see, by a superficial examination,into how many sections a Ms. is divided. However,- this passageis interesting as being the first mention of Diophantos by aEuropean writer; the date of the Speech was probably about1462. The only other passage, which Nesselmann was acquaint-ed with and might have formed some foundation for Xylatider'sconclusion, is one in which Regiomontamis (in the same Oratio)describes a journey which he made to Italy for the purposeof learning Greek, with the particular (though not exclusive)

1 Printed in the work Eudimenta astronomica Alfrarfani. "Item Alba-tegnius astronomus peritissimus de motu stellarum, ex observationibus turnpropriis turn Ptolemaei, omnia cum demonstrationibus Geometricis et Addi-tionibus Joannis de Eegiomonte. Item Oratio introductoria in omnen scientiasMathematicas Joannis de Reijiomonte, Patavii habita, cum Alfraganum pnblicepraelegeret. Ejusdem utilissima introductio in elementa Euclidis. Item Epis-tola Philippi Melanthonis nuncupatoria, ad Senatum Noribergensem. Omniajam recens prelis publicata. Norimbergae anno 1537. 4to."

- The passage is: "Diofanti autem tredecim libros subtilissimos nemo osqne-hac ex Graecis Latinos fecit, in quibus flos ipse totius Arithmeticae latet, arevidelicet rei et census, quam hodie vocant Algebram Arabico nomine."

It does not follow from this, as Vossius maintains, that Kegiomontanus sup-posed Dioph. to be the inventor of algebra.

The "ars rei et census," which is the solution of determinate quadraticequations, is not found in our Dioph. ; and even supposing that it was given inthe MS. which liegiomontanus saw, this is not a point which would des4.r^ospecial mention.


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22 DIOPHANTOS OF ALEXANDRIA.object of turning into Latin certain Greek mathematical works\But Diopliantos is not mentioned by name, and Nesselmannaccordingly thinks that it is a mere conjecture on the partof Cossali and Xylander, that among tlie Greek writers mentionedin this passage Diophantos was included ; and that we haveno ground for thinking, on the authority of these passages,that Regiomontanus saw the thirteen books in a complete form.But Nesselmann does not seem to have known of a passagein another place, which is later than the Oration at Padua,and shows to my mind most clearly that Regiomontanus neversaw the complete work. It is in a letter to Joannes de Blan-chinis^ in which Regiomontanus states that he found at Venice" Diofantus," a Greek arithmetician who had not yet beentranslated into Latin ; that in the proemium he defined theseveral powers up to the sixth, but whether he followed outall the combinations of these Regiomontanus does not know'^for not more than six books are found, though in the proemiumhe promises thirteen. If this book, a wonderful and difficultluork, could be found entire, I should like to translate it into Latin,for the knowledge of Greek I have lately acquired wouldsuffice for thisV' &c. The date of this occurrence is stated

1 After the death of his teacher, Georg von Peurbach, he tells us he wentto Eome &c. with Cardinal Bessaiion. "Quid igitur rehquum crat nisi utorbitam viri clarissimi sectarer? coeptum felix tuum pro viribus exequerer?Duce itaquo patrono communi Romam profectus more meo Uteris exerceor, ubiscripta plurima Graecorum clarissimorum ad literas suas disceudas me invitant,quo Latinitas in studiis praesertim Mathematicis locupletior redderetur."

Peurbach died 8 April, llGl, so that tlie journey must have taken placebetween 1-lGl and 1171, when he permanently took up his residence at Niim-berg. During this time he visited in order Eome, Ferrara, Padua (where hedelivered the Oration), Venice, Rome (a second time) and Vienna.

2 Given on p. 135 of Ch. Th. v. Murr's Memorabilia, Norimbergae, 1786, andpartly in Doppelmayr, Ilistorischc Nachricht von der Kiirnbergischen Mathe-vuiticis uml Kiimtlcrn, p. 5. Note y (Niiruberg, 1730).

3 The whole passage is" Hoc dico dominationi uestrae me reperisse nunc uenetiis Diofantum aritli-

meticum graecum nondum in latinum traductura. Hie in prohemio diiliniendoterminos huius artis ascendit ad cubum cubi, primura cnim uocat uumcrum,quern numeri uocant rem, secundum uocat potentiam, ubi uumeri dieuntcensum, deinde cubum, deinde potentiam poteutiae, uocant numerum censumde ceusu, item cubum de ccusu ct taudom cubi. Ncscio tamen si oumes com-


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HIS WORKS. 2liin a note to be 1463. Here then we have a distinct contradicti-.uto the statement that Regiomontanus speaks of having si-eu tliir-teen books ; so that Xylander's conchisions must be abandoned.No conclusion can be arrived at from the passage in F'ermat'sletter to Digby (15 August 1G57) in which he says: The nanu'of this author (Diophantos) " me donne I'occasion de vous fairesouvenir de la promesse, qu'il vous a pleu me faire de recouvrerquelque manuscrit de c^t Autheur, qui contienne tous les treizelivres, et de m'en faire part, s'il vous pent tomber en main."This is clearly no evidence that a complete Diophantos existedat the time.

Bombelli (1572) states the number of books to be seven\showing that the MS. he used was Vatican No. 200.

To go farther back still in time, Maximus Planudcs, wholived in the time of the Byzantine Emperors Andronicus I. andII. in the first half of the 14th century, and wrote Scholia tothe two first books of the Arithmetics, given in Latin inXylander's translation of Diophantos, knew the work in thesame form in which we have it, so far as the first two booksare concerned. From these facts Nesselmann concludes thatthe corruptions and lacunae in the text, as we have it, are dueto a period anterior to the 14th or even the 13th century.

There are yet other means by which lost portions of Diophan-tos might have been preserved, though not found in the originaltext as it has come down to us. We owe the recovery of someGreek mathematical works to the finding of Arabic translationsof them, as for inststnce parts of Apollonios. Now we knowbinationes horum proseeutus fuerit. non enim reperiuntur nisi 6 eius libri quinunc apud me sunt, in prohemio autem pollicetur se scripturum tredecim. Siliber hie qui reuera pulcerrimus est et diflicilimus, integer inueniretur [Doppel-mayr, inueHi'atur] curarem eum latiuum facere, ad hoc enim sufficereut mihiliterae graecae quas in domo domini mei reuerendissimi didici. Curate et uosobsecro si apud uestros usquam inueniri possit liber ille integer, sunt enim inurbe uestra non nulli graecarum litterarum periti, quibus solent inter caetorostuae facuitatis libros huiusmodi occurrere. Interim tamen, si suadebitis. Hexdictos libros traducere in latinum occipiam, quatenus latinitas hoc nouo etpretiosissimo munere non careat.

1 "Egli e io, per arrichire il mondo di cosi fatta opera, ci dessimo i\ tradurloe cinque libri {delU settc che sotio) tradutti ue abbiamo." Bombelli, pref. toAlgebra.


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24 DIOPHANTOS OF ALEXANDRIA.that Diophantos was translated into Arabic, or at least studiedand commented upon in Arabia. Why then should we notbe as fortunate in respect of Diophantos as with others ? Inthe second part of a work by Alkarkhi called the Fakhrl^(an algebraic treatise) is a collection of problems in deter-minate and indeterminate analysis which not only indicatethat their author had deeply studied Diophantos, but are,many of them, directly taken from the Arithmetics with thechange, occasionally, of some of the constants. The obliga-tions of Alkarkhi to Diophantos are discussed by Wopcke inhis Notice sur le Fakhrl. In a marginal note to his MS. is aremark attributing the problems of section iv. and of sectionIII. in part to Diophantos^. Now section IV. begins with pro-blems corresponding to the last 14 of Diophantos' Second Book,and ends with an exact reproduction of Book ill. Interveningbetween these two parts are twenty-five problems which are notfound in our Diophantos. We might suppose then that we havehere a lost Book of our author, and Wopcke says that he wasso struck by the gloss in the MS, that he hoped he had dis-covered such a Book, but afterwards abandoned the idea for thereasons : (1) That the first twelve of the problems depend uponequations of the first or second degree which lead, with twoexceptions, to irrational results, whereas such were not allowedby Diophantos. (2) The thirteen other problems which areindeterminate problems of the second degree are, some of them,quite unlike Diophantos ; others have remarks upon methodsemployed, and references to the author's commentaries, whichwe should not expect to find if the problems were taken fromDiophantos.

It does not seem possible, then, to identify any part of1 The book which I have made use of on this subject is: "Extrait dn Fakhrl,

traits d' Algl'bre par Abou liekr Mohammed ben Alhavan Alkarkhi (mauuscrit1)52, supplement arabe de la bibliothequc Imperiale) pr^ced6 d'un m


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HIS WORKS. 2.'>the Fakhrl as having formed a part of Diophantos' work nowlost. Thus it seems probable to suppose that the form in whichAlkarkhi found and studied Diophantos was not different fromthe present. This view is very strongly supported by the follow-ing evidence. Bachet has already noticed tliat the solutionof Dioph. II. 19 is really only another solution of ii. 18, anddoes not agree with its own enunciation. Now in the Faklu^lwe have a problem (iv. 40) with the same enunciation asDioph. II. 19, but a solution which is not in Diophantos' manner.It is remarkable to find this followed by a problem (iv, 41)which is the same as Dioph. ii. 20 (choice of constants alwaysexcepted). It is then sufficiently probable that il. 19 and20 followed each other in the redaction of Diophantos knownto Alkarkhi ; and the fact that he gives a non-Diophantinesolution of II. 19 would show that he had observed that theenunciation and solution did not correspond, and therefore sethimself to work out a solution of his own. In view of thisevidence we may probably assume that Diophantos' work hadalready taken its present mutilated form when it came intothe hands of the author of the Fakhrl. This work was writtenby Abu Bekr Mohammed ibu Alhasan Alkarkhi near thebeginning of the 11th century of our era ; so that the cor-ruption of the text of Diophantos must have taken place beforethe 11th century.

There is yet another Arabic work even earlier than thislast, apparently lost, the discovery of which would be of thegreatest historical interest and importance. It is a work uponDiophantos, consisting of a translation or a commentary by Mo-hammed Abu'1-Wafa, already mentioned incidentally. But itis doubtful whether the discovery of his work entire wouldenable us to restore any of the lost parts of Diophantos. Thereis no evidence to lead us to suppose so, but there is a pieceof evidence noted by Wopcke* which may possibly lead toan opposite conclusion. Abu'1-Wafa does not satisfactorily dealwith the possible division of any number whatever into foursquares. Now the theorem of the possibility of such divi.siou

1 Journal Asiatique. Ciuqui^me s^rie, Tome v. p. 231.


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2b DIOPHANTOS OF ALEXANDRIA.is assumed by Diophantos in several places, notably in iv. 31.We have then two alternatives. Either (1) the theorem wasnot distinctly enunciated by Diophantos at all, or (2) It wasenunciated in a proposition of a lost Book. In either caseAbu'1-Wafa cannot have seen the statement of the theorem iaDiophantos, and, if the latter alternative is right, we have anargument in favour of the view that the work had already been,mutilated before it reached the hands of Abu'1-Wafa. NowAbu'l-Wafa's date is 328388 of the Hegira, or 940988 ofour Era.

It would seem, therefore, clear that the parts of Diophantos'Arithmetics which are lost were lost at an early date, andthat the present lacunae and imperfections in the text hadtheir origin in all probability before the 10th century.

It may be said also with the same amount of probabilitythat the Porisms were lost before the 10th century a.d. Wehave perhaps an indication of this in the title of another workof Abu'1-Wafa, of which Wopcke's translation is " Demonstra-tions des thdoremes employes par Diophante dans son ouvrage,et de ceux employes par (Aboul-Wafa) lui-meme dans son com-mentaire." It is not possible to conclude with certainty fromthe title of this work what its contents may have been. Arethe " theorems " those which Diophantos assumes, referring forproofs of them to his Porisms ? This seems a not unlikely sup-position ; and, if it is correct, it would follow that the proofsof these propositions, which Diophantos must have himselfgiven, in fact, the Porisms, were no longer in existence inthe time of Abu'I-Wafa, or at least were lor him as good as lost.It must be admitted then that we have no historical evidenceof the existence at any time subsequent to Diophantos himselfof the Porisms.

Of the treatise on Polygonal Numhers we possess only afragment. It breaks off' in the middle of the 8th proposition.It is not however probable that much is wanting; practicallythe treatise seems to be nearly complete.

2. The next (juestion which naturally suggests itself isAs we have apparently six books only of the Arithmetics out ofthirteen, where may we suppose the lost matter to have been


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HIS WORKS. 27placed in the treatise? Was it at tlie beginning, micUHe, orend? This question can only be decided when we have cometo a conclusion about the probable contents of the lost p


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28 DIOPH.\^TOS OF ALEXANDRIA.a collection of problems which, with scarcely an exception, leadto indeterminate equations of the second degree, beginning withsimpler cases and advancing step by step to more complicatedquestions. These indeterminate or semideterminate problemsform the main feature of the collection. Now it is a great stepfrom determinate equations of the first degree to semideter-minate and indeterminate problems of the second; and we mustrecognise that there is here an enormous gap in the exposition.We ought surely to find here (1) determinate equations of thesecond degree and (2) indeterminate equations of the first.With regard to (2), it is quite true that we have no definitestatement in the work itself that they formed part of thewriter's plan; but that they were discussed here is an extremelyprobable supposition. With regard to (1) or determinatequadratic equations, on the other hand, we have certainevidence from the writer's own words, that the solution of theadfected or complete quadratic was given in the treatise as itoriginally stood ; for, in the first place, Diophantos promises adiscussion of them in the introductory definitions (def. 11)where he gives rules for the reduction of equations of thesecond degree to their simplest forms; secondly, he uses hismethod for their solution in the later Books, in some casessimply giving the result of the solution without working it out,in others giving the irrational part of the root in order to findan approximate value in integers, without writing down theactual root\ We find examples of pure quadratic equationsoninino Diophantus (!); agit duntaxat de eo problematum semidetenninatorumgenere, quae respiciimt quadrata, aut cubos numerorum, quae problemata utresolvantur, (juantitates radicales de industria sunt vitandae." Pref. to ana-litiche istituzioni.

^ These being tbe indications in the work itself, what are we to think of arecent writer of a History of Mathematics, who says: "Hieraus und aus demUmstand, dass Diophant nirgends die von ihm versprochene Theorie dcrAuflosung der quadratiscben Gleichungen gibt, schloss man, er habe dieselbenicht gekannt, und bat desshalb den Arabern stets den Ruhm dieser ErtinJuugzugctlieilt," and goes on to say that "nevertheless Nesselmann after a thoroughstudy of the work is convinced that D. knew the solution of the quadratic"?It is almost impossible to imagine that these remarks are serious. The writeris Dr Heinricli Suter, (Jcschichte d. Mathetmitischen WissemchaJ'ten. ZweiteAutliigf. Ziiricb, 1873.


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HIS Wol^KS. 29even in the first Book : a fact which shows that Diophantosregarded them as in reality simple equations, taking, as he does,the positive value of the root only. Indeed it would seem thatDiophantos adopted as his ground for the classification of theseequations, not the index of the highest power of the unknownquantity contained in it, but the number of terms left in itwhen it is reduced to its simplest form. His words are': "Ifthe same powers of the unknown occur on both sides but withdifferent coefficients we must take like from like until we haveone single expression equal to another. If there are on bothsides, or on either side, terms with negative coefficients, thedefects must be added on both sides, until there are the samepowers on both sides with positive coefficients, when we musttake like from like as before. We must contrive always, ifpossible, to reduce our equations so that they may contain onesingle term equated to one other. But afterwards we willexplain to you also hoiu, luhen two terms are left equal to a

- Diophantos' actual words (which I have trauslated freely) are: MtrA 5^Tavra eav d-rrb irpo^Xr^^iaTos tlvos -yh-qrai virap^ii eldeffi rots avroh jurj ofioTrXTjBfj5^ dirb eKar^puv twv fiepuiv, deriaa a.


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so DIOPHANTOS OF ALEXANDRIA.third, such a question is solved." That is to say, "reduce whenpossible the quadratic to one of the forms x = a, or x^ = b. Iwill give later a method of solution of the complete equationx^ax= b." Now this promised solution of the completequadratic equation is nowhere to be found in the Arithmeticsas we have them, though in the second and following Booksthere are obvious cases of its employment. We have to decide,then, where it might naturally have come; and the answer isthat the suitable place is between the first and second Books.

But besides the entire loss of an essential portion of Dio-phautos' work there is much confusion in the text even of thatportion which remains. Thus clearly problems 6, 7, 18, 19 ofthe second Book, which contain determinate problems of thefirst degree, belong in reality to Book I, Again, as already re-marked above, the problem enunciated in ii. 19 is not solved atall, but the solution attached to it is a mere " dXKco^" of ii. 18.Moreover, problems 15 of Book il. recall problems alreadysolved in i. Thus il. l = l. 34: ii. 2 = 1. 37: ii. 3 is similarto I. 33 : II. 4 = I. 35 : li. 5 = I. 36. The problem i. 29 seemsalso out of place in its present position. In the second Book anew type of problem is taken up at il. 20, and examples of itare continued through the third Book. There is no sign of amarked division between Books ii. and ill. In fact, expressedin modern notation, the last two problems of li. and the firstof III. are the solutions of the following sets of equations

II. 35. x''+[x + y + z) = a^y^+{x + y + z)=h'':^ + [x -ir y -\- z) = c"

II. 36. x^{x + y-\-z)= a-y--{x + y + z) = lAz" -{x+y + z)=c- ]

III. 1. (x + y + z) -.!' = a"{x + y + z)~y' = U'{x + y + z)-2' = c'

These follow perfectly naturally upon each other; andtherefore it is quite likely that our division between the two


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HIS WORKS. 31Books was not the original one. In fact tlie frequent occur-rence of more definite divisions in tlie middle of the Books,coupled with the variation in the Vatican Ms. which divides oursix Books into seven, seems to show that the work may havebeen divided into even a larger number of Books originally.Besides the displacements of problems which have probably takenplace there are many single problems which have been muchcorrupted, notably the fifth Book, which has, as Nesselmannexpresses it\ been "treated by Mother Time in a very step-motherly fashion". It is probable, for instance, that betweenV. 21 and 22 three problems have been lost. In several othercases the solutions are confused or incomplete. How the im-perfections of the text were introduced into it we can only con-jecture. Nesselmann thinks they cannot be due merely to thecarelessness of a copyist, but are rather due, at least in part, tothe ignorance and inexpertness of one who wished to improveupon the original. The view, which was put forward byBachet, that our six Books are a redaction or selection madefrom the complete thirteen by a later hand, seems certainlyuntenable.

The treatise on Polygonal Numbers is in its subject related 'to the Arithmetics, but the mode of treatment is completelydifferent. It is not an analytical work, but a synthetic onethe author enunciates propositions and then gives their proofsin fact the treatise is quite in the manner of Books vil.X. ofEuclid's elements, the method of representing numbers bygeometrical lines being used, which Cossali has called linearArithmetic. This method of representation is only once used inthe Arithmetics proper, namely in the proposition v. 13, whereit is used to prove that if a; + 7/=l, and a; and y have to be sodetermined that aj + 2, ?/ + 6 are both squares, we have to dividethe number 9 into two squares of which one must be > 2 and< 3. From the use of this linear method in this one case in theAnthmetics, and commonly in the treatise on Polygonal Numbers,we see that even in the time of Diophantos the geometricalrepresentation of numbers was thought to have the advantage

1 "Namentlich ist in dicser Hinsicht daa fuufte Buch stiefmutterlich von dcrMutter Zeit behandelt woiden." p. 2GB.


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3.2 DIOPHANTOS OF ALEXANDRIA.of greater clearness. It need scarcely be remarked how opposedthis Greek method is to our modern ones, our tendency beingthe reverse, viz., to the representation of lines by numbers. Thetreatise on Polygonal Numbers is often, and probably rightly,held to be one of the thirteen original Books of the Arithmetics.There is absolutely no reason to doubt its genuineness ; whichremark would have been unnecessary but for a statement byBossut to the effect: "II avoit dcrit treize livres d' arithmetiques,les six premiers (?) sont arrives jusqu'a nous : tons les autressont perdus, si, ndanmoins, un septieme, qu'on trouve dansquelques(!) editions de Diophante, n'estpas de lui"; upon whichReimer has made a note : " This Book on Polygonal Numbers isan independent work and cannot possibly belong to the Collectionof Diophantos' Arithmetics^" This statement is totally un-founded. With respect to Bossut's own remark, we have seenthat it is almost certain that the Books we possess are not thefirst six Books ; again, the treatise on Polygonal Numbei's doesnot only occur in some, but in all of the editions of Diophantosfrom Xylander to Schulz ; and, lastly, Bossut is the only personwho has ever questioned its genuineness.We mentioned above the Porisms of Diophantos. Ourknowledge of them is derived from his own words ; in threeplaces in the Arithmetics he refers to them in the words exo/j-eviv Tot


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Ills WORKS. 3?integral part, now lost, of the original thirteen books ? If thissupposition is correct the Po7'isms also must have intervened be-tween Books I. and ll., where we have already said that probablyDiophantos treated of indeterminate problems of the firstdegree and of the solution of the complete quadratic. Themethod of the Ponsms was probably synthetic, like the Poly-gonal iVwrnfters, not (like the six Books of the Anthmetics)analytical ; this however forms no sufficient reason for refusingto include all three treatises under the single title of thirteenBooks of Arithmetics. These suppositions would account easilyfor the contents of the lost Books ; they would also, with theadditional evidence of the division of our text of the Arithmeticsinto seven books by the Vatican MS., show that the lost portionprobably does not bear such a large proportion to the whole asmight be imagined. This view is adopted by Colebrooke \ andafter him by Nesselmann, who, in support of his hypothesisthat the Arithmetics, the Porisms and the treatise on PolygonalNumbers formed only one complete work under the generaltitle of dptd/jLTjTLKa, points out the very significant fact that wenever find mention of more than one work of Diophantos, andthat the very use of the Plural Neuter term, dpid/xrjTiKa, wouldseem to imply that it was a collection of different treatises onarithmetical subjects and of different content. Nesselmann, how-ever, does not seem to have noticed an objection previously urged

^ Algebra of the Hindus, Note M. p. lxi."In truth the division of manuscript books is very uncertain: and it is by

no means improbable that the remains of Diophantus, as we possess tlicni, maybe less incomplete and constitute a larger portion of the thirteen books an-nounced by him (Def. 11) than is commonly reckoned. His treatise on polygonnumbers, which is surmised to be one (and that the last of the thirteen), follows,as it seems, the six (or seven) books in the exemi)lar8 of the work, as if thepreceding portion were complete. It is itself imperfect: but the manner isessentially different from that of the foregoing books: and the solution ofproblems by equations is no longer the object, but rather the demonstrationof propositions. There appears no gi-ouud, beyond bare surmise, to presume,that the author, in the rest of the tracts relative to numbers which fulfilledhis promise of thirteen books, resumed the Algebraic manner: or in short,that the Algebraic part of his performance is at all mutilated in the copiesextant, which are considered to be all transcripts of a single imperfectexemplar."

H. D. 3


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34 DIOPllANTOS OF ALEXANDRIA.against the theory that the three treatises formed only one work,by Schulz, to the effect that Diophantos expressly says that hiswork treats of arithmetical problems^. This statement itselfdoes not seem to me to be quite accurate, and I cannot thinkthat it is at all a valid objection to Nesselmann's view. Thepassage to which Schulz refers must evidently be the openingwords of the dedication by the author to Dionysios. Diophantosbegins thus: "Knowing that you are anxious to become ac-quainted with the solution [or ' discovery,' eupecri?] of problemsin numbers, I set myself to systematise the method, beginningfrom the foundations on which the science is built, the pre-liminary determination of the nature and properties in numbers^."Now these "foundations" may surely well mean more than isgiven in the eleven definitions with which the treatise begins,and why should not the "properties of numbers" refer to thePorisms and the treatise on Polygonal Numbers .? But there isanother passage which might seem to countenance Schulz'sobjection, where (Def. 11) Diophantos says "let us now proceedto the propositions'... which we will deal with in thirteen Books\"The word used here is not problem {Trpo/SXTj/xa) but proposition(TrporaaL^;), although Bachet translates both words by the sameLatin word " quaestio," inaccurately. Now the word irporaaL^;does not only apply to the analytical solution of a problem : itapplies equally to the synthetic method. Thus the use of theword here might very well imply that the work was to contain

1 Schulz remarks on the Porisms (pref. xxi.): "Es ist daher nicht uuwahr-Bcheinlich dass diese Porismeu eine eigene Schrift uuseres Diophautus wareu,welche vorziiglich die Zusammensetzung dcr Zahlen aus gew-issen Bestaud-theilen zu ihrem Gegeustando hattc. Kunnte man diesc Schrift gar als eineBestandtheil des grossen in dreizehn Biichern abgefassten arithmetischenWerkes anseheu, so wiire es sehr erkliirbar, dass gerade dieser Theil, der denblossen Liebhaber weniger anzog, verloren ging. Da indess Diophantus aus-driickiich sagt, sein Werk behandele arithmetische Probleme, so hat weuigstensdie letztere Annahme nur einen geringen Grad von Wahrscheinhchkeit."

* Diophantos' own words are: Tiju tvptcnv twv iv roh apid/ioTs Trpo^XijfidTuiv,TifU(l)TaT^ fiOL AiovOffie, -yivilKTKtiiv ae cnrovdalus ^xovra naOuv, opyavwcrai r^c /j^dodoviweipdOrji', dp^ofKifOS d(f>' uv avviarrjKe rd Trpaynara 0e(jif\lii)v, vTroaTTJffai Trjv iv toisdptOfioh tpvffiv T Kal Swaniv.

* vvv 5^ iirl rds irpordans x^RV'^'^t^^"' '^- ''' ^* T^s irpay/xareiai avrQv kv TpiffKaldfKa fii^Xlois yiyivripiivr)s.


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HIS WORKS. ;i5not only problems, but propositions on numbers, i.e. miglitinclude the Po7'isms and Polygonal Numbers as a part of thecomplete Arithmetics. These objections which I have madeto Schulz's argument are, I think, enough to show that hisobjection to the view adopted by Nesselmaun has no weight.Schulz's own view as to the contents of the missing Books ofDiophantos is that they contained new methods of solution inaddition to those used in Books I. to vi., and that accordinglythe lost portion came at the end of the existing six Books. Inparticular he thinks that Diophantos extended in the lost Booksthe method of solution by means of what he calls a double-equation {Bi7r\r] laorr]^ or in one word hi,Tr\ola6rr}


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36 DIOPHAXTOS OF ALEXANDRIA.much, almost during his whole life, with the then new methodsof solution of equations of the third and fourth degree ; and, forthe solution of the latter, the usual method of his time led tothe making an expression of the form Ax^+ Bx + C a. square,where the coefficients involved a second unknown quantity.Nesselmann accordingly thinks it is no matter for surprise thatin Diophantos' entirely independent investigations Bombellishould have seen, or fancied he saw, his own favourite idea.This solution of the equation of the fourth degree presupposesthat of the cubic with the second term wanting ; hence Bombelliwould naturally, in accordance with his view, imagine Diophantosto have given the solution of this cubic. It is possible also thathe may have been influenced by the actual occurrence in theextant Books [vi. 19] of a cubic equation, namely the equationx^ + x = 4x^ + 4, of which Diophantos at once writes down thesolution a; = 4, without explanation. It is obvious, however,that no conclusion can be drawn from this, which is a veryeasy particular case, and which Diophantos probably solved^ bysimply dividing out by the factor x^+l. There are strongobjections to Bombelli's view. (1) Diophantos himself states(Def. XI.) that the solution of the problems is the object in itselfof the work. (2) If he used the method to lead up to thesolution of equations of higher degrees, he certainly has not goneto work the shortest way. In support of the view it has beenasked "What, on any other assumption, is the object of definingin Def ll. all powers of the unknown quantity up to the sixth ?rapitici, si avanzasse egli a sciogliere 1' equazionc x*+px-q, parendogli, dienei libri riinastici, con proporsi di trovar via via numeri quadrati, cammini unastrada a qucU' intento. Egli e di fatto procedendo sn queste tracce di Diofanto,che Vieta deprime 1' esposta equazione di giado quarto ad una di secondo.Siccome pen"!* cio non si effettua che mediante una cubica mancante di secondotermine; cosi il pcnsiero sorto in auimo a Bombelli iniporterebbc, che Diofantonei libri perduti costituito avesse la regola di sciogliere questa sorta di equa-zione cubicbe prima d' innoltrarsi alio scioglimento di quella equazione di quartogrado."

' This is certainly a simpler explanation than Bachet's, who derives thesolution from the proportion ar* : .x-=x : 1.Therefore x' + x : x- + l = .r : 1.Therefore x^+ x : 4x^ + i = x : i.

But the equation being .r''-)-.r= 4j-'-' + 4, it follows that x-i.


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HIS WORKS. 37Surely Diophantos must have meant to use them." The answerto which is that he has occasion to use them in the work, butreduces all the equations which contain these higher powers byhis regular and uniform method of analysis.

In conclusion, I may repeat that the most probable view isthat adopted by Nesselmann, that the works which we knowunder the three titles formed part of one arithmetical work,which was, according to the author's own words, to consist ofthirteen Books. The proportion of the lost parts to tlie wholeis probably less than it might be supposed to be. The Ponsnisform the part, the loss of which is most to be regretted, forfrom the references to them it is clear that they containedpropositions in the Theory of Numbers most wonderful for thetime.


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CHAPTER III.THE WRITERS UPON DIOPHANTOS.

1. In this chapter I purpose to give a sketch of what hasbeen done directly, and (where it is of sufficient importance)indirectly, for Diophantes, enumerating and describing briefly(so far as possible) the works which have been written on thesubject. We turn first, naturally, to Diophantos' own country-men ; and we find that, if we except the doubtful " commentaryof Hypatia," spoken of above, there is only one Greek, who haswritten anything at all on Diophantos, namely the monk Maxi-mus Planudes, to whom are attributed the scholia attached toBooks I. and ii. in some MSS., which are printed in Latin inXylander's translation of Diophantos. The date of these scholiais the first half of the 14th century, and they represent all thatwe know to have been done for Diophantos by his own country-men. How different his fate would have been, had he lived alittle earlier, when the scientific spirit of the Greeks was stillactive, what an enormous impression his work would then havecreated, we may judge by comparing the effect which it hadwith that of a far less important work, that of Nikomachos.Considering then that up to the time of Maximus Planudesnothing was written about Diophantos (beyond a single quota-tion by Theon of Alexandria, before mentioned, and an occa-sional mention of the name) by any Greek, one is simplyastounded at finding in Bossut's history a remark like thefollowing : " L'auteur a eu parmi les anciens une foule d'inter-prfetes (!), dont les ouvrages sont la plupart (!) perdus. Nousregrettons, dans ce nombre, le commentaire de la cdlebreHipathia (sic)." Comment is unnecessary. With respect to


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40 DIOPHANTOS OF ALEXANDRIA.Alexandria. He wrote Kitab Sina'at al-jabr," i.e. "the book ofthe art of algebra."

(h) p. 283, Among the works of Abii'1-Wafa is mentioned"An interpretation' (tafsir) of the book of Diophantos aboutalgebra."

(c) On the same page the title of another work of Abu'l-Wafa is given as " Demonstrations of the theorems employedby Diophantos in his work, and of those employed by (Abu'l-Wafa) himself in his commentary " (the word is as beforetafsir).

{d) p. 295, On Kosta ibn Luka of Ba'lbek it is mentionedthat one of his books is tafsir on three-and-a-half divisions(Makalat) of the book of Diophantos on " questions of numbers."We have thus in the Fihrist mentions of three separateworks upon Diophantos, which must accordingly have beenwritten previously to the year 990 of our era. ConcerningAbu'1-Wafa the evidence of his having studied and commentedupon Diophantos is conclusive, not only because his other workswhich have survived show unmistakeable signs of the influenceof Diophantos, but because the proximity of date of the Fihristto that of Abu'1-Wafa makes all mistake impossible. As I havesaid the Fihrist was written circa 990 A.D. and the date ofAbu'1-Wafa is 328388 a.h. or 940998 A.D. He was anative of Buzjan, a small town between Herat and Nishapur inKhorasan, and was evidently, from what is known of his worksone of the most celebrated astronomers and geometers of histime^. Of later notices on this subject we may mention those

' There is a little doubt as to the exact meaning of tafsirwhether it meansa translation or a commentary. The word is usually applied to the literal exe-gesis of the Koran ; how much it means in the present case may perhaps beascertainable from the fact that Abu'1-Wafa also wrote a tafsir of the Algebra ofMohammed ibn Musa al-Khruizml. It certainly, according to the usual sense,means a commentary not a mere translatione.g. at p. 249 al-Nadim clearlydistinguishes translators of Aristotle from the mufassirln or makers of tafsir, i.e.commentators.

For this information I am indebted to the kindness of Professor RobertsonSmith.

- Wcipcke, Journal Asiatiqne, Ft'vrier-Mars, 1855, p. 244 foil.Abu'l-Wafa's full name is Mohammed ibn Mohammed ibn Yahya ibn Ismailibn Al'abbfis Abu'1-Wafa Al-Iiuzjani.


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THE WRITERS UPON DIOPHANTOS. 41in the Tarlkh Ilokoma (Hajji-Khalifa, No. 2204), by tho ImamMohammed ibn. 'Abd al-Karim al-Shahrastani wlio died A.ii. 548or A.D. 1153\ Of course this work is not so trustworthy anauthority as the Fihrist, which is about 160 years earlier, andthe author of the Tarlkh HoJcoma stands to the Fihrist in therelation of a compiler to the original source. In the TarikhHokoma we are told {a) that Abu'1-Wafa " wrote a commen-tary on the work of Diophantos concerning Algebra," (6) that' Diophantos, the Greek of Alexandria, conspicuous, perfect,famous in his time, wrote a famous work on the art ofAlgebra, which has gone over into Arabic," i. e. been trans-lated. We must obviously connect these two notices. Lastlythe same work mentions (c) another work of Abu'1-Wafa,namely ' Proofs for the propositions given in his book byDiophantos."

A later writer still, the author of the History of the Dynas-ties, Abu'lfaraj, mentions, among celebrated men who lived inthe time of Julian, Diophantos, with the addition that " Hisbook^..ou Algebra is celebrated," and again in another placehe says upon Abu'1-Wafa, " He commented upon the work ofDiophantos on Algebra."

The notices from al-Shahrastani and Abu'lfaraj are, as I have^ The work Bibliotheca arabico-hispana Escurialensis op. et studio Mich.

Casiri, Matriti, 1760, gives many important notices about mathematiciansfrom the Ta'rikh Hokoma, which Casiri denotes by the title Bibliotheca philo-sophorum.

Cossali mentions the Ta'rikh Hokoma as having been written about a.d. 119^!by an anonymous person: "II hbro piti antico, che ci fornisca tratti relativiall' origine dell' analisi tra gli arabi e la Bihlioteca arabica de' jilosoji, scrittacirca 1' anno 1198 da anonimo egiziano" (Cossali, i. p. 174). There is howevernow apparently no doubt that the author was al-Shahrastani, as I have said inthe text.

2 After the word "book" in the text comes a word Ab-kismet which is un-intelligible. PocoQke, the Latin translator, simply puts A. B. for it: "cuius liberA. 13. quern Algebram vocat, Celebris est." The word or words are apparentlya corruption of something ; Nesselmann conjectures that the original word wasan Arabic translation of the Greek title, Arithmeticsa supposition which, iftrue, would give admirable sense. The passage would then mark the Arabianperception of the discrepancy (according to the accepted meaning of termn)between the title and the subject, which is obviously rather algebra than arith-metic in the strict sense.


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42 DIOPHANTOS OF ALEXANDRIA.said, for obvious reasons not so trustworthy as those in theFihrist. They are, however, interesting as showing that Dio-phantos continued to be kno^vn and recognised for a consider-able period after his work found its way to Arabia, and wascommented upon, though they add nothing to our informationas to what was done for Diophantos in Arabia. It is clear thatthe work of Abu'1-Wafa was the most considerable that waswritten in Arabia upon Diophantos directly ; about the obliga-tions to Diophantos of other Arabian writers, as indirectlyshown by similarity of matter or method, without direct refer-ence, I shall have to speak later.

3. I now pass to the writers on Diophantos in Europe.From the time of Maximus Planudes to a period as late asabout 1570 Diophantos remained practically a sealed book, andhad to be rediscovered even after attention had been invited toit by Regiomontanus, who, as was said above, was the firstEuropean to mention it as extant. We have seen (pp. 21, 22)that Regiomontanus referred to Diophantos in the Oration atPadua, about 1462, and how in a very interesting letter toJoannes de Blanchinis he speaks of finding a MS. of Diophantos atVenice, of the pleasure he would have in translating it if he couldonly find a copy containing the whole of the thirteen books, andhis readiness to translate even the incomplete work in six books,in case it were desired. But it does not appear that he everbegan the work ; it seems, however, very extraordinary that theinterest which Regiomontanus took in Diophantos and tried toarouse in others should not have incited some of his Germancountrymen to follow his leading, at least as early as 1537,when we know that his Oration at Padua was published. Hardto account for as the fact may appear, it was left for an Italian,Bombelli, to rediscover Diophantos about 1570; though thementions by Regiomontanus may be said at last to have bornetheir fruit, in that about the same time Xylander was en-couraged by them to persevere in his intention of investigatingDiophantos. Nevertheless between the time of Regiomontanusand that of Rafael Bombelli Diophantos was once more for-gotten, or rather unknown, for in the interval we find twomentions of tlie name, (a) b} Joachim Camerarius in a letter


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THE WRITERS UPON DIOPHANTOS. 43published 1556\ in which he mentions that there is a MS. ofDiophantos in the Vatican, which he is anxious to see, (6) byJames Peletarius^ who merely mentions the name. Of theimportant mathematicians who preceded Bombelli, Fra LucaPacioli towards the end of the loth century. Cardan and Tar-taglia in the 16th, not one so much as mentions Diophantos^

The first Italian to whom Diophantos seems to have beenknown, and who was the first to discover a MS. in the VaticanLibrary, and to conceive the idea of publishing the work, wasRafael Bombelli. Bachet falls into an anachronism when hesays that Bombelli began his work upon Diophantos after theappearance of Xylander's translation*, which was published in1575. The Algebra of Bombelli appeared in 1572, and in the

^ De Graecis Latinisque tiumerorum notis et praeterea Saracenis sett Indicts,etc. etc., studio Joachimi Camerarii, Papeberg, 1556.

In a letter to Zasius : " Venit mihi in nientem eorum quae et de bac et aliisliberalibus artibus dicta fuere, in eo convivio cujus in tuis aedibus me et Peuce-rum nostrum participes esse, suavissima tua invitatio voluit. Cum autem deautoribus Logistices verba fierent, et a me Diophantus Graecus nominaretur, quiextaret in Bibliotbeca Vaticana, ostendebatur turn spes quaedam, posse nobiscopiam libri illius. Ibi ego cupiditate videndi incensus, fortasse audacius nontamen iiifeliciter, te quasi procuratorem constitui negotii gerendi, mandatevoluntario, cum quidem et tu libenter susciperes quod imponebatur, et fidessolenni festivitate firmaretur, de illo tuo et poculo elegante ct vino optimo.Neque tu igitur oblivisceris ejus rei, cujus explicationem tua benignitas tibicommisit, neque ego non meminisse potero, non modo excelleutis \-irtuti8 ctsapientiae, sed singularis comitatis et incredibilis suavitatis tuae."

- Arithmeticae practicae methodus facilis, per Gemmani Frisium, etc. Hueaccedunt Jacobi Peletarii annotationes, Coloniae, 1571. (But pref. of Peletariusbears date 1558.) P. 72, Nota Peletarii: "Algebra autem dicta videtur a GebroArabe ut vox ipsa sonat ; hujus artis si non inventore, saltern excultore. Aliitribuunt Diophanto cuidam Graeco."

'' Cossali I. p. 59, "Cosa pero, che reca la somma maraviglia si 6, che largoin Italia non si spandesse la cognizionc del codice di Diofanto : che in fioreessendovi lo studio della greca lingua, non veuisse da qualche dotto a comanvantaggio tradotta; che per 1' opposto niuna menzione ne faccia Fra Luca versoil fine del secolo xv, e niuna Cardano, e Tartaglia intorno la metA del secoloXVI ; che nelle biblioteche rinianesse sepolto, ed andassc dimenticato per modo,che poco prima degli anni 70 del secolo xvi si riguardasse per una scoperta1' averlo rinvonuto nella Vaticana liiblioteca."

"Non longo post Xilandrum interuallo llaphael Bombellius Bononiensis,Graecum e Vaticana Bibliotheca Diophanti codicem nactus, omnes priorumquattuor librorum quaestiones, et 6 libro quinto nonuuUas, probk-matibus uiubiuseruit, in Algebra sua quam Italico sermono conwcripsit."


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44 DIOPHANTOS OF ALEXANDRIA.preface to this work * the author tells us that he had recentlydiscovered a Greek book on Algebra in the Vatican Library,written by a certain Diofantes, an Alexandrine Greek authorwho lived in the time of Antoninus Pius ; that, thinking highlyof the contents of this work, he and Antonio Maria Pazzi de-termined to translate it ; that they actually translated fivebooks out of the seven into wliich the MS. was divided ; butthat, before the whole was finished, they were called away fromit by other labours. The date of these occurrences must be afew years before 1572. Though Bombelli did not carry out hisplan of publishing Diophantos in a translation, he has neverthe-less taken all the problems of Diophantos' first four Books andsome of those of the fifth, and embodied them in his Algebra,interspersing them with his own problems. Though he hastaken no pains to distinguish Diophantos' problems from hisown, he has in the case of Diophantos' work adhered prettyclosely to the original, so that Bachet admits his obligations toBombelli, whose reproduction of the problems of Diophantos hemaintains that he found in many points better than Xylander'stranslation'^ It may be interesting to mention a few points of

1 This book Nesselmann tells ns that he has never seen, but takes his infor-mation about it from Cossali. I was fortunate enough to find a copy of itpublished in 1579 (not the original edition) in the British Museum, the titlebeing UAlgebra, opera d'l Rafael BomheUi da Bolorjiia diiiisa in tre Libri InBologna, Per Giovanni Rossi. MDLXXIX. I have thus been able to verify thequotations from the preface. The whole passage is :

"Questi anni passati, essendosi ritrouato una opera greca di questa disciplinanella libraria di Nostro Signore in Vaticano, composta da un certo DiofanteAlessandrino Autor Greco, il quale fCl a tempo di Antonin Pio, e havendo melafatta vedere Messer Antonio Maria Pazzi Reggiano publico lettore delle Matema-tiche in Roma, e giu dicatolo con lui Autore assai intelligente de numeri (an-corche non tratti de numeri irrationali, ma solo in lui si vcde vn perfetto ordinedi opcrare) egli, ed io, per arrichirc il mondo di cosi fatta opera, ci dessimo atradurlo, e cinque libri (delli sette che souo) tradutti ne habbiamo ; lo restautenon haueudo potuto finire per gli trauagli aueuuti all' uno, e all' altro, e in dettaopera habbiamo ritrouato, ch' egli assai volte cita gli Autori Indiani, col che miha fatto conoscere, che questa disciplina appo gl' indiani prima ih, che a gli Arabi."

The parts of this quotation which refer to the personality of Diophantos, theform Diofante, &c., have already been commented upon ; the last clauses weshall have occasion to mention again.

'^ Continuation of quotation in note 4, p. 43"Sed suas Diophanteis quaestionibus ita immiecuit, ut has ab illis distiu-


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46 DIOPHANTOS OF ALEXANDRIA.then found that mention had been made of his work by Regio-montanus as being extant in an Italian Library and havingbeen seen by him. But, as the book had not been edited, hetried to reconcile himself to the want of it by making himselfacquainted with the works on Arithmetic which were actuallyknown and in use, and he apologises for what he considers tohave been a disgrace to him\ With the help of books only hestu

138-161 circa_diophantos of alexandria_00

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