DIOPHANTOS OF ALEXANDRIA. 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. DIOPHANTOS OF ALEXANDRIA A STUDY IN THE lIISToUV OF GREEK ALGEBRA BY -ClJUilM T. L. HEATH, B.A. SCHOLAR OF TRINITY COI.LKCiK, CAMHRIlxiK. EDITED FOR THE SYNDJC&^^^^XH^ UNIVEPiSITY PliESS. -^-V3 ^-'-- 3^"\S>. ((UHIVEP.'ITY QTambriligc : AT THE UNIVERSITY PKK.SS. 1885 [All Ei'jhts A'.^.Tc;./.] CTambritigr : rniNTF.ri dy c. j. clay, m.a. and sox, AT THK rXIVERSITY PRKSS. PREFACE. The scope of the prosont book is sufficiently indicatod Ity the title and the Table of Contents. In the chapter on " Dioijhantos' notation and definitions" several suggestions are made, which I believe to be new, with regard to the origin and significance of the symbols employed by Diophantos. A few words may be necessary to explain the purp(».se of the Appendix. This is the result of the compression of a large book 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 of peculiar devices which is one of the most prominent of the characteristics of the Greek algebraist, but which cannot l)o brought under general rules and tabulated in the same way as the processes described in Chapter V. The Appendix, then, is a necessary part of the whole, in that there is much in Diophantos which could not be introduced elsewhere ; it must not, however, be considered as in any sense an alternative to the rest of the book: indeed, owing to its extremely con- densed form, I could hardly hope that, by itself, it would even be comprehensible to the mathematician. I will merely add that I have twice carefully worked out the .«;<.lution of H. D. ^ 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 details here. 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 a thorough revision of the text. I have, however, only cursorily inspected one MS. of my author, that in the Bodleian Library, which unfortunately contains no more than a small part of the first of the six Books. The best Mss, are in Paris and Rome, and I regret that I have had as yet no opportunity of consulting them. Though this would be a serious drawback were I editing the text, no collation of MSS. could afifect my exposition of Diophantos' methods, or the solutions of his problems, to any appreciable extent; and, further, it is more than doubtful, in view of the unsatisfactory results of the collation of three of the MSS, by three different scholars in the case of one, and that the most important, of the few ob- scure passages which need to be cleared up, whether the text in these places could ever be certainly settled. I should be ungrateful indeed if I did not gladly embrace this opportunity of acknowledging the encouragement which I have received from Mr J. W, L. Glaisher, Fellow and Tutor of Trinity College, to whose prospective interest in the work before it was begun, and unvarying kindness while it was proceeding, I can now thankfully look back as having been in a great degree the " moving cause " of the whole. And, finally, I wish to thank the Syndics of the University Press for their liberality in undertaking to publish the volume. T. L. HEATH. 11 May, 1885. 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 Latino redditum et Commeutariis explanatum Biusileae, 1575. Bachet, Diophanti Alexandrini Arithmeticoioim Libri sex, et de niuueri.s multaugulis 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 desseu Schrift liber die Polygon-zahlen. Aus dem Griecbi-scheu iibersetzt und 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 Diofanto Alessandrino 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 de Simon 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. >"»'' 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 Sanscrit of 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.'i I.. 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 ra of .Mohammed ben Musa. London, 1831. 1Ii:ii>er<;, Quacstiones Archimedeae. llauniae, 1879. 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 16—17 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 IS — 23 No trace of lost books to be restored from Arabia. Corruption must have taken place before 11th cent, and probably before 950 A.D 23— "iC, Poiisms lost before 10th cent. a.d. 2<) § 2. What portion of the Arithmetics is lost? The contents of the lost books. The Polygonal Numbers and Porism.i may have formed part of the complete ArithmcticK. Objections to this theory 2(>— 3."> Other views of the contents of thf lost Books .... 3J — 37 Conclusion 37 CHAPTER III. THE WlllTEKS UPON J»lolMIA.\ T« ),s. § 1. (heck 38-39 § 2. Arabian 39 — 12 § 3. European gencially 42— 5('> CONTENTS. CHAPTER IV. \OT.\TI(»N AND DEFINITIONS (»F DlOPH.\NTOS. VAC.KS § 1. Introduction ,57 § 2. Sign for the unknown quantity discubsed 57 — 67 § 3. Notation for powers of the unknown G7— 09 § i. Objection that Diophantos loses generality by the want of more algebraic symbols answered 69 Other questions of notation : operations, fractions, dc. . . 69—76 § 5. General remarks on the historical development of algebraic notation : three stages exhibited 76—80 § 6. Ou the influence of Diophantos' notation on his work . . 80—82 CHAPTER V. §1. §3. SI. diophantos' METHODS OF SOLUTION. General remarks. Criticism of the positions of Hankcl and Ncsselmann Diophantos' treatment of equations ..... (A) Determinate equations of different degrees. (1) Pure equations of different degrees, i.e. equations con taining 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 solved 2. Those which can be rationally solved only under 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 Transition ..... Mitiiod of limits .... Method of appro.\imation to limits . 88—114 88- -90 90- -93 93- -94 95- -98 95 95—98 99—105 105—107 107 108—111 111—112 112—113 113—114 114—115 115—117 117—120 CONTKNTS. CHAPTER VI. PAOEH 1. The PonsHis of Diophantos 121 I2.'i 2. Other theorems assumed or implied 12.>— 132 ('/) Numbers as the sum of two squares 127— 1:<0 (h) Numbers as the sum of three squares l:{0 — l:{l (c) Numbers as the sum of four squares 131— 1H2 §1. §2. §3. §4. §5. §6. §7. CHAPTER VII. HOW FAR WAS DIOPHANTOS ORIGINAL? Preliminary 133—134 Diophantos' algebra not derived from Arabia .... 134—135 Reference to Hypsikles 13.") — 130 The evidence of his language 13G— 138 Wallis' theory of Greek Algebra 138 Comparison of Diophantos with his Greek predecessors . . 139—142 Discussion in this connection of the Cattle-prohlem . . . 142 — 147 CHAPTER VIII. DIOPHANTOS AND THE EARLY ARABIAN ALGEBRAISTS. § 1. Preliminary § 2. Comparison of Diophantos with Mohammed ibn Musfi . . 149- § 3. Diophantos and Abu'1-Wafa § 4. An anonymous Arabic ms. 155- § 5. Abu Ja'far Mohammed ibn Alhusain §G. Alkarkhi 156- 14S -155 155 -156 156 APPENDIX. ABSTRACT OF DIOPHANTOS. A rithmetics. Book I . 16.S— 171 Bookn . 172—181 Bookm . 181-189 Book IV . 189-208 BookV . 209-224 Book VI . 225-237 Polygonal Numbers . 238—244 EREATUM. On p. 78, last line but one of note, for " Targalia" read "Tartaglia" DIOPHANTOS OF ALEXANDRIA. CHAPTER I. Historical Introduction. § 1. The doubts about l^iophantos begin, as has been remarked by Cossali^ with his very name. It cannot be posi- tively decided whether his name was Diophanfos or Diophan^es. The preponderance, however, of authority is in favour of the view that he was called Diophantos. (1) The title of the work which has come down to us under his name gives us no clue. It is Aiocpdvrov 'A\€^avBp€Q}avToi> e'^et rd(f)o<;, d /jiiya Oavfia, Kal T«iyyo<;, E/c 8e ydficov TrifMirrcp TralS' iirev^vaev eret. At ai r7]\vy€Tov BetXov Te«09, VjfXLav irarpo^, Tov 8e Kal 7; Kpvepo^ fierpov eXoov ^iotov. Iler'^o? 3' av iriavpecro'L Trapijyopecov eviavTOLonivuv, rlva \oyov ix^i vpbs dWijXa, fSo^af raOra fMT] dpOQi ytypaipivai rbv 'ArroWwi/iof. aCrrol di ravra SiaKaddpavres fypa\j/av wj Tjj/ iKoveiv TOV naTpos. ^yCo Si vartpov irtpUtreaov iTlpi^ /3t/JA/v i"r6 ' AiroWuvlov iKbfhotxivtfi, Kal TTtpUxovTi anobuiiu ijyiwi (?) irepl tov i/iroKfifj^vov. Kal fieydXtji i\J/vxaywy^Or)v iirl Ty irpofiXi^fxaTos ^T-qati. Tb fj.iv viro ' AwoWwi'lov iKbodh (oiKf KOiff, OKOirdv. Kal yap irtpKpiptrai, k. t. X." HISTORICAL INTRODUCTION. 5 on Regular Solids was probably written about 180 B.C. It was clearly a youthful productiou. Besides this we have another work of Hypsikles, of astronomical content, entitled in Greek dva^opLK6avTov. The words as found in the text cannot be right. Aiocfjdvrrjv TOV da-TpovofiiKov should (if the punctuation were right) be Aio(f>dvTr)v TOV d(TTpov6^ov, the former not being Greek. Ku.ster's conjecture '^ is that we should read vTrofivrjfia ek Aio(f)dvTov da-TpovofjLLKov Kavova' et? to, KwviKa ^ AttoWojvlov vTToiJ.vnp.a. If this is right the Diophantos here mentioned must have been an astronomer. In that case the person in question is not our Diophantos at all, for we have no ground whatever to imagine that he occupied himself with Astronomy. It is cer- tain that he was famous only as an arithmetician. Thus John of Jerusalem in his life of John of Damascus^ in speaking of some one's skill in Arithmetic compares him to Pythagoras and ' tiraTla rj O^wvos tov TtufUTpov Ovyarrip tov ' A\f^avSp^wi i.\oao darpovofiLKOv Kavova, etf ra koovlku AttoWcoviov vTro^vr,- fia'^. He then proceeds to remark that it shows that Hypatia wrote a Canon Astronomicus, so that she evidently was versed in Astronomy as well as Geometry (as shown by the Commen- tary on Apollonios), two of the three important branches of Mathematics. It is likely then, argues Bachet, that she was acquainted with the third. Arithmetic, and wrote a commentary on the AritJtmetics of Diophantos. But in the first place we know of no astronomical work after that of Claudius Ptolemy, and from the way in which 6 da-rpovo^iLKO'; Kavwv is mentioned it would be necessary to suppose that it had been universally known, and was still in common use at the time of Suidas, and yet was never mentioned by any one else whom we knjULUUi inexplicable hypothesis. ' 'ISipixoyivt) Tov larpov 6 affrpoXoyoi Ai6ai'T(X Eln-e /xovovi ^wfjs ivvia pLrjvas ^X^'"- KcLKeivos ycXdaai, Ti /jl(v 6 KpSvos ivvia. /xrjvwy, ^■qal, \^y€L, (TV voef Ta/xa 5i ci'inofxa. aoc Elwe Kal ^KTslvas fwvov Tjxj/aro' Kal AiO(pain-os 'AWov dve\iri^u)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 solo attactu non aegros modo, sed et ben(^ valentes, velut pestifero sidere afflntoa repente necaret. Itaque nisi Diopbantum nostrum Astrologiae iieritum fuissc negemus, 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 d© notre auteur..."; that is precisely what Bacbet did rmt read there. 10 DIOPHANTOS OF ALEXANDRIA. Next, the expression ek Aio^avrov has been objected to by Nesselmann as not being Greek. He maintains that the Greeks never speak of a book by the name of its author, and therefore we ought to have Atocfxivrov dpidfiijTiKa, if the reference were to Diophantos of the Arithmetics. M. Tannery, however, de- fends the use of the expression, on the ground that similar ones 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, and thinks that we should read et? At6oi, Ji'yxP°^°^ ^^ UdTririft ri^ 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^riv ; ilvoiix &v 6ti Kal ai>T7) iarl tCiv \6yifi t6 trdv Kvpovp-ivuv. Kal el IwavipoiTo 'H iTfpl tI ; etiToifJ.^ hv wainp o\ iv rc^ StJ^v Tl0V 7] XoyKTTlKrj. (.tOnjlUg, 451 B,C. HIS WORKS. 10 take an abstract form, so that the distinction between XoyiaTiKij and apidfjiriTLKr] is lost. We thus have W.pid/xr}TiKd given as the title of his work, whereas in earlier times the term could only properly have been applied to his treatise on Polygonal Numbers. This broader use by Diophantos of the term arith- metic is not without its importance. Having made this preliminary remark it is next necessary to observe that of these works which we have mentioned some have been lost, while probably the form of parts of others has suffered considerably by the ravages of time. The Arithmetics should, according to the title and a distinct statement in the introduction to it, contain thirteen Books. But all the six known MSS.^ contain only six books, with the sole variation that in the Vatican MS. 200 the same text, which in the rest forms six books, is divided into seveii. Not only do the MSS. practically agree in the external division of the work ; they agree also in an equally remarkable manner — at least all of them which have up to the present been collated — in the lacunae and the mistakes which occur in the text. So much is this the case that Bachet, the sole editor of the Greek text of Diophantos, asserts his belief that they are all copied from one original ^ This can, however, scarcely be said to be established, ^ The six mss. are : 1—3. 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 doctissimo viro ClauLlio Salmasio refcrente accepimus ; neque Vaticanus, quern vir suniniua lacobus Sirmondus mihi ex parte transcribendum curauit, quicquam amplius continent, quam sex hosce Arithmcticorum libros, et tractatum de iiumeris multangulis imperfectum. Sed et tarn infeUcitcr hi omnes codices inter ae consentiunt, vt ab vno fonte manasse et ab eodem exemplari dcscriptos fuisso non 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's name wrongly, giving it as Xilander. O 9 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 by Jacobus Sirmondus. It is possible therefore that the collation of the two remaining mss. in the Vatican might even now lead to important results respecting the settling of the text. The evidence of the existence in earlier times of all the thirteen books is very doubtful, some of it absolutely incorrect. Bachet says * that Joannes Regiomontanus asserts that he saw the thirteen books somewhere, and that Cardinal Perron, who had recently died, had often told him that he possessed a MS. containing the thirteen books complete, but, having lent it to a fellow-citizen, who died before returning it, had never re- covered it. Respecting this latter MS. mentioned by Bachet we have not sufficient data to lead us to a definite conclusion as to whether it really corresponded to the title, or, like the MSS. which we knoAv, only announced thirteen books. If it really corresponded to the title, it is remarkable how (in the words of Nesselmann) every possible unfortunate circumstance and even the " pestis " mentioned by Bachet seem to have conspired to rob posterity of at least a part of Diophantos' works. Respecting the statement that Regiomontanus asserts that he saw a MS. containing the thirteen books, it is clear that it is founded on a misunderstanding. Xylander states in two passages of his preface " that he found that Regiomontanus 1 "loannes tamen Regiomontanus tredecini Diophanti libros se alicubi vidisse 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 Diophanti codex codem fato nobis criperetur. Cum enim prccibus meis motus Cardi- nalis amplissimus, nullisque sumptibus pai-cens, apud heredes Gosselini codicem ilium diligenter exquiri mandassct, et quouis pretio redimi, nusquam repertus est." Ad lectorcm. ■•* "Inueni deinde tanquam exstantis in bibliothecis Italicis, sibique uisi mentionem a Regiomontano (cuius etiam nominis memoriam ueneror) factam." Xylander, Epistola nuncupatoria. "Sane tredeeim libri Arithmeticae Diophanti ab aliis perhibentur exstare in bibliotheca Vaticana; quos Regiomontanus illo uiderit." Ibid. HIS WORKS. 21 mentioned a MS. of Diophantos which he liad seen in an Italian library; and that others said that the thirteen books were extant in the Vatican Library, " which Regiomontanus saw." Now as regards the latter statement, Xylander was obviously wrongly informed ; for not one of the Vatican Mss. contains the thirteen books. It is necessary therefore to inquire to what passage or passages in Regiomontanus' writings Xylander refers. Nesselmann finds only one place which can be meant, an Oratio habita Patavii in praelectione Alfragani^ in which Regiomon- tanus remarks that " no one has yet translated from the Greek into Latin the thirteen books of Diophantosl" Upon this Nesselmann observes that, even if Regiomontanus saw a MS., it does not follow that it had the thirteen books, except on the title-page ; and the remarks which Regiomontanus makes upon 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 passage is interesting as being the first mention of Diophantos by a European writer; the date of the Speech was probably about 1462. The only other passage, which Nesselmann was acquaint- ed with and might have formed some foundation for Xylatider's conclusion, is one in which Regiomontamis (in the same Oratio) describes a journey which he made to Italy for the purpose of 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 turn propriis turn Ptolemaei, omnia cum demonstrationibus Geometricis et Addi- tionibus Joannis de Eegiomonte. Item Oratio introductoria in omnen scientias Mathematicas Joannis de Reijiomonte, Patavii habita, cum Alfraganum pnblice praelegeret. Ejusdem utilissima introductio in elementa Euclidis. Item Epis- tola Philippi Melanthonis nuncupatoria, ad Senatum Noribergensem. Omnia jam 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, are videlicet 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 quadratic equations, is not found in our Dioph. ; and even supposing that it was given in the MS. which liegiomontanus saw, this is not a point which would des4.•r^•o special mention. 22 DIOPHANTOS OF ALEXANDRIA. object of turning into Latin certain Greek mathematical works\ But Diopliantos is not mentioned by name, and Nesselmann accordingly thinks that it is a mere conjecture on the part of Cossali and Xylander, that among tlie Greek writers mentioned in this passage Diophantos was included ; and that we have no 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 passage in another place, which is later than the Oration at Padua, and shows to my mind most clearly that Regiomontanus never saw 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 been translated into Latin ; that in the proemium he defined the several 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 proemium he promises thirteen. If this book, a wonderful and difficult luork, could be found entire, I should like to translate it into Latin, for the knowledge of Greek I have lately acquired would suffice for thisV' &c. The date of this occurrence is stated 1 After the death of his teacher, Georg von Peurbach, he tells us he went to Eome &c. with Cardinal Bessaiion. "Quid igitur rehquum crat nisi ut orbitam viri clarissimi sectarer? coeptum felix tuum pro viribus exequerer? Duce itaquo patrono communi Romam profectus more meo Uteris exerceor, ubi scripta 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 place between 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 he delivered the Oration), Venice, Rome (a second time) and Vienna. 2 Given on p. 135 of Ch. Th. v. Murr's Memorabilia, Norimbergae, 1786, and partly 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 diiliniendo terminos huius artis ascendit ad cubum cubi, primura cnim uocat uumcrum, quern numeri uocant rem, secundum uocat potentiam, ubi uumeri dieunt censum, deinde cubum, deinde potentiam poteutiae, uocant numerum censum de ceusu, item cubum de ccusu ct taudom cubi. Ncscio tamen si oumes com- HIS WORKS. 2li in a note to be 1463. Here then we have a distinct contradicti-.u to 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's letter to Digby (15 August 1G57) in which he says: The nanu' of this author (Diophantos) " me donne I'occasion de vous faire souvenir de la promesse, qu'il vous a pleu me faire de recouvrer quelque manuscrit de c^t Autheur, qui contienne tous les treize livres, et de m'en faire part, s'il vous pent tomber en main." This is clearly no evidence that a complete Diophantos existed at 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, who lived in the time of the Byzantine Emperors Andronicus I. and II. in the first half of the 14th century, and wrote Scholia to the two first books of the Arithmetics, given in Latin in Xylander's translation of Diophantos, knew the work in the same form in which we have it, so far as the first two books are concerned. From these facts Nesselmann concludes that the corruptions and lacunae in the text, as we have it, are due to 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 original text as it has come down to us. We owe the recovery of some Greek mathematical works to the finding of Arabic translations of them, as for inststnce parts of Apollonios. Now we know binationes horum proseeutus fuerit. non enim reperiuntur nisi 6 eius libri qui nunc apud me sunt, in prohemio autem pollicetur se scripturum tredecim. Si liber hie qui reuera pulcerrimus est et diflicilimus, integer inueniretur [Doppel- mayr, inueHi'atur] curarem eum latiuum facere, ad hoc enim sufficereut mihi literae graecae quas in domo domini mei reuerendissimi didici. Curate et uos obsecro si apud uestros usquam inueniri possit liber ille integer, sunt enim in urbe uestra non nulli graecarum litterarum periti, quibus solent inter caetoros tuae facuitatis libros huiusmodi occurrere. Interim tamen, si suadebitis. Hex dictos libros traducere in latinum occipiam, quatenus latinitas hoc nouo et pretiosissimo munere non careat. " 1 "Egli e io, per arrichire il mondo di cosi fatta opera, ci dessimo i\ tradurlo e cinque libri {delU settc che sotio) tradutti ue abbiamo." Bombelli, pref. to Algebra. 24 DIOPHANTOS OF ALEXANDRIA. that Diophantos was translated into Arabic, or at least studied and commented upon in Arabia. Why then should we not be as fortunate in respect of Diophantos as with others ? In the 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 indicate that their author had deeply studied Diophantos, but are, many of them, directly taken from the Arithmetics with the change, occasionally, of some of the constants. The obliga- tions of Alkarkhi to Diophantos are discussed by Wopcke in his Notice sur le Fakhrl. In a marginal note to his MS. is a remark attributing the problems of section iv. and of section III. 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. Intervening between these two parts are twenty-five problems which are not found in our Diophantos. We might suppose then that we have here a lost Book of our author, and Wopcke says that he was so struck by the gloss in the MS, that he hoped he had dis- covered such a Book, but afterwards abandoned the idea for the reasons : (1) That the first twelve of the problems depend upon equations of the first or second degree which lead, with two exceptions, to irrational results, whereas such were not allowed by Diophantos. (2) The thirteen other problems which are indeterminate problems of the second degree are, some of them, quite unlike Diophantos ; others have remarks upon methods employed, and references to the author's commentaries, which we should not expect to find if the problems were taken from Diophantos. It does not seem possible, then, to identify any part of 1 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 (mauuscrit 1)52, supplement arabe de la bibliothequc Imperiale) pr^ced6 d'un m the Fakhrl as having formed a part of Diophantos' work now lost. Thus it seems probable to suppose that the form in which Alkarkhi found and studied Diophantos was not different from the present. This view is very strongly supported by the follow- ing evidence. Bachet has already noticed tliat the solution of Dioph. II. 19 is really only another solution of ii. 18, and does not agree with its own enunciation. Now in the Faklu^l we have a problem (iv. 40) with the same enunciation as Dioph. 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 always excepted). It is then sufficiently probable that il. 19 and 20 followed each other in the redaction of Diophantos known to Alkarkhi ; and the fact that he gives a non-Diophantine solution of II. 19 would show that he had observed that the enunciation and solution did not correspond, and therefore set himself to work out a solution of his own. In view of this evidence we may probably assume that Diophantos' work had already taken its present mutilated form when it came into the hands of the author of the Fakhrl. This work was written by Abu Bekr Mohammed ibu Alhasan Alkarkhi near the beginning of the 11th century of our era ; so that the cor- ruption of the text of Diophantos must have taken place before the 11th century. There is yet another Arabic work even earlier than this last, apparently lost, the discovery of which would be of the greatest historical interest and importance. It is a work upon Diophantos, consisting of a translation or a commentary by Mo- hammed Abu'1-Wafa, already mentioned incidentally. But it is doubtful whether the discovery of his work entire would enable us to restore any of the lost parts of Diophantos. There is no evidence to lead us to suppose so, but there is a piece of evidence noted by Wopcke* which may possibly lead to an opposite conclusion. Abu'1-Wafa does not satisfactorily deal with the possible division of any number whatever into four squares. Now the theorem of the possibility of such divi.siou 1 Journal Asiatique. Ciuqui^me s^rie, Tome v. p. 231. 2b DIOPHANTOS OF ALEXANDRIA. is assumed by Diophantos in several places, notably in iv. 31. We have then two alternatives. Either (1) the theorem was not distinctly enunciated by Diophantos at all, or (2) It was enunciated in a proposition of a lost Book. In either case Abu'1-Wafa cannot have seen the statement of the theorem ia Diophantos, and, if the latter alternative is right, we have an argument in favour of the view that the work had already been, mutilated before it reached the hands of Abu'1-Wafa. Now Abu'l-Wafa's date is 328—388 of the Hegira, or 940—988 of our Era. It would seem, therefore, clear that the parts of Diophantos' Arithmetics which are lost were lost at an early date, and that the present lacunae and imperfections in the text had their origin in all probability before the 10th century. It may be said also with the same amount of probability that the Porisms were lost before the 10th century a.d. We have perhaps an indication of this in the title of another work of 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 from the title of this work what its contents may have been. Are the " theorems " those which Diophantos assumes, referring for proofs of them to his Porisms ? This seems a not unlikely sup- position ; and, if it is correct, it would follow that the proofs of these propositions, which Diophantos must have himself given, in fact, the Porisms, were no longer in existence in the 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 evidence of the existence at any time subsequent to Diophantos himself of the Porisms. Of the treatise on Polygonal Numhers we possess only a fragment. It breaks off' in the middle of the 8th proposition. It is not however probable that much is wanting; practically the treatise seems to be nearly complete. § 2. The next (juestion which naturally suggests itself is : As we have apparently six books only of the Arithmetics out of thirteen, where may we suppose the lost matter to have been HIS WORKS. 27 placed in the treatise? Was it at tlie beginning, micUHe, or end? This question can only be decided when we have come to a conclusion about the probable contents of the lost p to. ofxoia dir6 twu 6/xoiwv, ?a)S &v ^»'ds(!) elSoj €pI eidei tjov -yiv-qTaf eav de ttws if OTror^pu} ivvirapxTJ^^), ^ ^v dfKporipois iveWei^f/r] (?) rivk etSr), de-qaei irpoaBelvaL to, Xeivovra etOT) if dfjLcpor^poii roh fxipeaii', ews Slp eKarepij) tQv fxepQiv rd ei5r] ivvirdpxovra. y^vrjTai. Kal TraXi;' a'^e- Xelv Ta 6p.oi.a diro rwc onoiuf, ?ws &v eKarepij) tCiv fxepQ)V if eTooi KaTa\(i 2 and < 3. From the use of this linear method in this one case in the Anthmetics, and commonly in the treatise on Polygonal Numbers, we see that even in the time of Diophantos the geometrical representation of numbers was thought to have the advantage 1 "Namentlich ist in dicser Hinsicht daa fuufte Buch stiefmutterlich von dcr Mutter Zeit behandelt woiden." p. 2GB. 3.2 DIOPHANTOS OF ALEXANDRIA. of greater clearness. It need scarcely be remarked how opposed this Greek method is to our modern ones, our tendency being the reverse, viz., to the representation of lines by numbers. The treatise 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 ; which remark would have been unnecessary but for a statement by Bossut to the effect: "II avoit dcrit treize livres d' arithmetiques, les six premiers (?) sont arrives jusqu'a nous : tons les autres sont perdus, si, ndanmoins, un septieme, qu'on trouve dans quelques(!) editions de Diophante, n'estpas de lui"; upon which Reimer has made a note : " This Book on Polygonal Numbers is an independent work and cannot possibly belong to the Collection of Diophantos' Arithmetics^" This statement is totally un- founded. With respect to Bossut's own remark, we have seen that it is almost certain that the Books we possess are not the first six Books ; again, the treatise on Polygonal Numbei's does not only occur in some, but in all of the editions of Diophantos from Xylander to Schulz ; and, lastly, Bossut is the only person who has ever questioned its genuineness. We mentioned above the Porisms of Diophantos. Our knowledge of them is derived from his own words ; in three places in the Arithmetics he refers to them in the words exo/j-ev iv Tot' uv avviarrjKe rd Trpaynara 0e(jif\lii)v, vTroaTTJffai Trjv iv tois dptOfioh tpvffiv T« Kal Swaniv. •* vvv 5^ iirl rds irpordans x^RV'^'^t^^"' '^- '''■ ^• * T^s irpay/xareiai avrQv kv TpiffKaldfKa fii^Xlois yiyivripiivr)s. HIS WORKS. ;i5 not only problems, but propositions on numbers, i.e. miglit include the Po7'isms and Polygonal Numbers as a part of the complete Arithmetics. These objections which I have made to Schulz's argument are, I think, enough to show that his objection to the view adopted by Nesselmaun has no weight. Schulz's own view as to the contents of the missing Books of Diophantos is that they contained new methods of solution in addition to those used in Books I. to vi., and that accordingly the lost portion came at the end of the existing six Books. In particular he thinks that Diophantos extended in the lost Books the method of solution by means of what he calls a double- equation {Bi7r\r] laorr]^ or in one word hi,Tr\ola6rr}<;). By means of this double-equation Diophantos shows how to find a value of the unknown, which will make two expressions containing it (linear or quadratic) simultaneously squares. Schulz accordingly thinks that he went on in the lost Books to show how to make three such expressions simultaneously squares, i.e. advanced to a triple-equation. This view, however, seems to have nothing to recommend it, inasmuch as, in the first place, we nowhere find the slightest hint in the extant Books of anything different or more advanced which is to come ; and, secondly, Diophantos' system and ideas seem so self-contained, and his methods to move always in the same well-defined circle that it seems certain that we come in our six Books to the limits of his art. There is yet another view of the probable contents of the lost Books, which must be mentioned, though we cannot believe that it is the riglit one. It is that of Bombelli, given by Cossali, to the effect that in the lost Books Diophantos went on to solve determinate equations of the third and fourth degree; Bombelli's reason for supposing this is that Diophantos gives so many problems the object of which is to make the sum of a square and any other number to be again a square number by finding a suitable value of the first square ; these methods, argues Bombelli, of Diophantos must have been given for the reason that the author intended to use them for the solution of the equation x*-\-px = q^. Now Bombelli had occupied himself 1 Cossali's words are (p. 75, 76):..."non tralascier " Id uero mihi accidit durum et uix superabile incommodum, quM mirificft deprauata omnia inueni, ctim neque problematum expositio interdum integra esset, ac passim numeri (iu quibus sita omnia esse in hoc arRumento, quia ignorat?) tarn problematum quam solutionum siue cxplicatiouum corruptissimi. Non pudebit me ingenue fateri, qualem me heic gesserim. Audacter, et summo cum feruore potius qusim alacritate auimi opus ipsum initio sum aggressus, laborque mihi omnis uoluptati fuit, tantus est meus rernm arithmoticarum amor, quin et gratiam magnam me apud omnes liberalium scientiarum amatores ac patronos initiirum, et praeclare de rep. litteraria nierituriim intclligebam, eamque rem mihi laudi (quam ii 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 milii addiderint, neque enim mihi novum aut insolons est aduersus librariorum incuriam certamen, et hac in re militaui, (ut Horatii nostri uerbis utar) non sine gloria, quod me non arroganter dicere, Dio, Plutarchu.^, Strabo, Stephanusque nostri testantur. Sed cum mox in ipsum pelagus nionstris scatons me cmsus abripuit: non dcspondi equidem animum, neque manus dedi, scd tamen saepius ad cram undo soluissem respeju, qujmi portum in quem csaet euadeudum cogitandu prospicerem, depracheudiquc non minus uerii quum ele- 48 DIOPHANTOS OF ALEXANDRIA. Next Xylander tells us how he came to get possession of a manu- script of Diophantos. 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 manuscript of Diophantos and informed him that it belonged to Andreas Dudicius whom Xylander describes as " Andreas Dudicius Sbardellatus, hoc tempore Imperatoris Ro- manorum apud Polonos orator." On his departure from Witten- berg Xylander wrote out and took with him the solution of a single problem of Diophantos, to amuse himself with on his journey. This he showed at Leipzig to Simon Simonius Lucen- sis, 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 Arithmetics 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 the Prince Ludwig, extolling the pursuit of arithmetical and alge- braical science and dwelling in enthusiastic anticipation on the influence which the Prince's patronage would have in help- ing and advancing the study of Arithmetic \ This Epistola ganter ea cecinisse Alcaeum, quae (si possum) Latino in hac quasi uotiua mea tabula scribam. Qui uela uentis uult dare, dum licet, Cautus futuri praeuideat modum Cursus. mare ingressus, marine Nauigct arbitrio necesse est. Sand, quod de Echeueide pisce fertur, eum nauim cui se adplicet remorari, poenh credibile fecit mihi mea cymba tot mendorum remoris retardata. Expediui tamen me ita, ut facilS omnes mediocri de his rebus iudicio praediti, iutellecturi sint incredibilem me laborem et aerumnas difticilimas superasso : pudore etiam stimulatum oneris quod ultro milii imposuissem, non perferendi. Paucula quae- dam non plane explicata, studio et certis de causis in alium locum reiecimus. Opus quidem ipsum ita absoluimus ut ueque eius nos pudere debeat, et Arith- meticae Logisticesque studiosi nobis se plurimum debere sint baud dubie professuri." ' "Hoc non modo tibi Princeps Illustrissimc, honorificum erit, atque glori- osum ; sed tc labores nostros approbantc, arithmcticae studium cum alibi, tum in tua Academia et Gymuasiis, excilabitur, confirmabitur, prouebetur, et ad THE WRITERS UPON DIOPIIANTOS. 49 Nuncnpatoria boars the date 14th August, 1574'. Xylandcr (lied on the 10th of February in the year following that of the publication, 1576. Some have stated that Xy lander published the Greek text of Diophantos as well as the Latin translation. There appears to be no foundation for the statement, which probably rests on a misunderstanding of certain passages in which Xylander refers to the Greek text. It is possible that he intended to publish the Greek original but was prevented by his death which so soon followed the appearance of his trans- lation. It is a sufficient proof, how^ever, that if such was his purpose it was never carried out, that Bachet asserts that he himself had never seen or found any one who had ever seen such an edition of the Greek text ^ Concerning the merits of Xylander and his translation of Diophantos much has been written, and chiefly by authors who were not weW acquainted with the subject, but whose very ignorance seems to have been their chief incitement to startling statements. Indeed very few persons at all seem to have studied the book itself: a fact which may be partly accounted for by its rarity. Nesselmaun, whose book appeared in 1842, tells us honestly that he has never been able to find a copy, but has been obliged to take all information on the subject at second hand from Cossali and Bachet '. Even Cossali, so far as he gives any opinion at all upon the merits of the book, seems to do no more than reproduce what Bachet had said before him. Nor does Schulz seem to have studied Xylander' s work : at least all his statements about it are vague and may very well have been gathered at second hand. Both he and perfectam eiu.s scientiam multi tuis auspiciis, nostro labore pcrducti, niognam hac re tuis in remp. beneficiis accessionem factam esse gratissima commemora- tione praedicabunt." 1 "Heidelberga. postrid. Eidus Sextiles cio lo lxxiv." - "An vero et Graece a Xilandro editus sit Diophantus, nondum certti com- perire potui. Videtur sanb in multis suorum Commcntariorum locis, de Graeco Diopbanto tanquam a se cdito, vcl mox edendo, verba facere. Sed banc cditi- onem, neque mihi vidisse, neque aliquem qui viderit hactcnus audivisse contigit." Bachet, Epist. ad Led. 3 There is not, I believe, a copy even in the British Museum, but I had the rare good fortune to find the book in the Library of Trinity College, Cambridge. H. D. * 50 DIOPHANTOS OF ALEXANDRIA. Nesselmann confine themselves to saying that it was not so worthless as many writers had stated it to be (Nesselmann on his part confessing his inability to form an opinion for the reason that he had never seen the book), and that it was well received among savants of the period, while its effect on the growth of the study of Algebra was remarkable \ On the other hand, the great majority of writers on the subject may be said to shout in chorus a very different cry. One instance will suffice to show the quality of the statements that have been generally made : to enumerate more would be waste of space. Dr Heinrich Suter in a History of Mathematical Sciences (Zurich 1873) says^ " This translation is very poor, as Xylander was very little versed in Mathematics." If Dr Hein- rich Suter had taken the trouble to read a few words of Xylander's preface, he could hardly have made so astounding a statement as that contained in the second clause of this sentence. This is only a specimen of the kind of statements which have been made about Xylander's book ; indeed I have been able to find no one who seems to have adequately studied Xylander except Bachet ; and Bachet's statements about the work of his predecessor and his own obligations to the same have been unhesitatingly accepted by the great majority of later writers. The result has been that Bachet has been uni- versally considered the only writer who has done anything considerable for Diophantos, while the labours of his prede- cessor have been ignored or despised. This view of the relative merits of the two authors is, in my vieAv, completely erroneous. From a careful study and comparison of the two editions I have come to the conclusion that honour has not been paid where honour was due. It would be tedious to give here in 1 Schulz. "Wie uuvollkomiucn Xylanders Ai-beit auch ausfiel, wie oft cr auch den rcchten Sinn verfehlte, und wic oft auch seine Aumerkuugeii den Laser, der sich Eathes eiholen will, im Stichc lasseu, so gut war dock die Aufnalimc, welche sein Uuch bei den Gelehitcn damaligcr Zcit fand; dcnn in der That giog den Matlicmatikcru durch die Erscheinung dieses Werkes ein neues Licbt auf, und es ist mir schr wabrscbcinlicb, dass er viel dazu beigetrageu hat, die allgemeiue Arithmetik zu ihrer nachnialigen Hohe zu erheben." - "Dicse Uebersetzung aber ist sebr schwach, da Xylander in Mathematik sebr wenig bewaudcrt war." THE WIUTEUS UPON DluPlIANToS. ol detail the particular facts which led me to this conchision. I will only say in this place that my suspicions were first aroused by reading Bachet's work alone, before I had seen tlie earlier one. From perusing Bachet I received the impression that his repeated emphatic and almost violent repudiation of obligation to Xylander, and his disparagement of that author suggested the very thing which he disclaimed, that he was under too great obligation to his predecessor to acknowledge it duly. 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 Anthmetico- 7-um libri sex, et de numeris nndtancjulis liber unus. JVtoic 2)riinuin Graece et Latine editi, atque absolidissimis Commentariis illitstrati. Auctore Claudio Gaspare Bacheto Mezinaco Sebusiano, V.G. Lidetiae Pansiorum, Surnptibus Hieronymi Drovart^, via Jacobaea, sub Scuto Solari. MDGXXI. (I should perhaps mention that we have a statement^ that in Carl von Montchall's Library there was a translation of Diophantos which the mathe- matician "Joseph Auria of Neapolis" made, but did not ap- parently publish, and which was entitled "Diophanti libri sex, cum scholiis graecis Maximi Planudae, atque liber de numeris polygonis, collati cum Vaticanis codicibus, et latine versi a Josepho Auria." Of this work we know nothing; neither Bachet nor Cossali mentions it. The date would presumably be about the same as that of Xylauder's translation, or a little later.) Bachet's Greek text is based, as he tells us, upon a MS. which he calls "codex Regius", now in the Bibliotheque Na- tionale at Paris; this MS. is his sole authority, except that Jacobus Sirmondus had part of a Vatican MS. transcribed for him. He professes to have produced a good Greek text, having spent incalculable labour upon its emendation, to have inserted 1 For "surnptibus Hierouymi Drovart" Nesselinann has "surnptibus Sebas- tiani Cramoisy, 1021 " which is found in some copies. The former (as given above) is taken from the title-page of the copy which I have used (from the Library of Trinity College, Cambridge). - Schulz, Vorr. xliii.: "Noch erwuhnen die Litteratorcn, dass eich in der Bibliothek eines Carl von Moutchall einc Bearbeitung des Diophantus von dem beruhmten Joseph Auria von Neapel (vermuthlich doch uur handschriftUoli) befuudeu habe, welche den Titel I'uhrte u. s. w." (see Text). 52 DIOPHANTOS OF ALEXANDRIA. 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. In spite however of Bachet's assurance I cannot help doubting the quality of his text in many places, though I have not seen the MS. which he used. He is careful to tell us what pre- vious works relating to the subject he had been able to con- sult. First he mentions Xylander (whom he invariably quotes as Xilander), who had translated the whole of Diophantos, and commented upon him throughout, "except that he scarcely touched a considerable pai't 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 (already mentioned) and the Zetetica of Vieta (in which the author treats in his own way a large number of Diophantos' 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 Diophantos assumes as known in many places, or sufficiently explained the causes of his opera- tions 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 work\ As regards bis Latin translation, he says that he gives us Diophantos 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, corrected ^ On the nature of some of Bacliet's proofs Nicholas Saunderson (formerly Lucasian Professor) remarks in Elements of Algebra, 1740, apropos of Dioph. III. 17. "M. Bachet indeed in the IGth and 17th props, of his second book of Porisms has given us demonstrations, 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 demonstrations he gives are only synthetical, and so abominably perplexed withal, that in each demonstration he makes uso of all the letters in the alphabet except I and 0, singly to represent the quantities he has there occasion for." THE WRITERS UPON DIuPllANTOS. 53 what was wrong and supplied 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 more fair 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 do not care'". 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 Montucla", "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 diffi- culty of the work, he would never have completed it, had it not been for the stubbornness which his malady generated iu him. As the first and only edition of the Greek text of Dio- phantos, 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 1G70. Diophanti Alexandrini Arithmeticorwni lihri sex, et de numeris multangidis liher itmis. Cum commentariis G. G. Bacheti V. G. et ohseruationibus D. P. de Fermat Senatoris Tolosani. Accessit Doctrinae Amdyticae inuentum nouum, collectum ex variis eiusdem D. de Fermat Epistolis. Tolusae, Excudehat Bernardus Bosc, ^ Regione CuUegii Societatis Jesu, MDGLXX. This edition was not pubhshed by Fermat himself, as certain writers imply ^ but by his son '■"Deinde Latinum damus tibi Diophantum ex Xilandri versione accura- tissime castigata, in qua duo potissimum nos praestitisse scias velim, nam et deprauata correximus, hiantesque passim lacunas repleuimus : et quae sub- obscure, vel ambigue fuerat interpretatus Xilander, dilucidius exposuimus; fateor tamen, inde tantam inductam esse mutationem, vt propemodum aequius sit ver- sioneni istam nobis quam Xilandro tribuere. Si quis autem potius ad eum \^t- tinere contendat, qu5d eius verba, quatenus Diophanto fraudi non erant, niordicus retinuimus, per me licet." 2 I. 323. ' So Dr Hcinrich Suter: "Diese Am(fahe witrde 1G70 ditrch Fernuit ernnt^rt, der sie mit seinen eigenen algebraischep Untersuchungen und Erfindungen ^asstattete," 54 DIOPHANTOS OF AT.KXAXDRIA. after his death. S. Fermat tells us in the preface that this publication of Fermat's notes to Diophantos was part of an attempt to collect together from his letters and elsewhere his contributions to mathematics. The "Doctrinae Analj'ticae In- uentum nouum" is a collection made by Jacobus de Billy from various letters which Fermat sent to him at different times. The notes upon Diophantos' 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 Diophantos, From their brevity they were obviously intended for the benefit of experts \ or even perhaps solely for Fermat's own, he being a man who preferred the pleasure which he had in the work itself to all considerations of the fame which might follow therefrom. Fermat never cared to publish his investiga- tions, 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 Diophantos 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 imperfec- tions have been already noticed by Nesselmannl Thus the reprinted edition of 1670 is untrustworthy as regards the text. ^ Lectori Beneuolo, p. iii. : "Doctis quibus tantum pauca sufficiunt, harum obseruationum auctor scribebat, vel potius ipse sibi scribens, his studiis exerceii malebat quam gloriari ; adco autem ille ab omni ostentationo alienus erat, vt nee lucubratioues suas ty]iie mandari curauerit, ct suonim qiiandoquc resjionsorum autographa nullo scruato exemplari pctentibus vitro miserit ; iiorunt scilicet ple- rique celeberrimorum huius saeculo Geomctrarum, quam libenter ille et quaut& bumanitate, sua iis inuenta patefecerit." 2 "Was dieser Abdruck an iiusserer Eleganz gewounen hat (denn die Ba- chet'sche Ausgcbe ist niit ausserst unangcnehmen, nanientlich Griechischeu Lettern gedruckt), das hat sie an inncrm Werthe in Bczug auf den Text ver- loren. Sie ist nicht bloss voller Diuckfchler in cinzelnen Worten und Zeichen (z. B. durchgehends ir statt "?>), 900) sondern audi ganze Zeilen sind ausgelassen Oder doppelt gedruckt, (z. B. iii. 12 cine Zeile doppelt, iv, 25 eine doppelt und THE WRITERS UPON DIOPHANTOS. 55 I omit here all mention of works which are not directly upon Diophantos (e.g. the so called "Translation" by Stevin and Alb. Girard). We have accordingly to pass from 1670 to 1810 before we find another extant work directly upon Diophantos. In 1810 was published an excellent translation (with additions) of the fragment upon Polygonal Numbers by Poselger : Dio- phantus von Alexandrien iiher die Polygonal-Zahlen. Uebersetzt mit Zusdtzen von F. Th. Poselger. Leipzig, 1810. Lastly, in 1822 Otto Schulz, professor in Berlin, published a very meritorious German translation with notes: Diupliantus von Alexandria arithmetische Aufgahen nebst dessen Schrift iiber die Pohjgon-Zahlen. Aus dem Griechischen ilbersetzt iind mit Anmerkungen begleitet von Otto Schulz, Professor am Berlinisch- Colnischen Gymnasium zum grauen Kloster. Berlin, 1822. In der Schlesingerschen Buck- und Musikhandlung. The former work of Poselger is with the consent of its author incorporated in Schulz's edition along with his own translation and notes upon the larger treatise, the Arithmetics. According to Nessel- mann Schulz was not a mathematician by profession: he pro- duced, however, a most excellent and painstaking edition, with notes chiefly upon the matter of Diophantos 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; so that nothing has been done for the Greek text since the original edition of Bachet (1621). I have now mentioned all the extant books which have been written directly upon Diophantos. Of books here omitted which are concerned with Diophantos indirectly, i.e. those which reproduce the substance of his solutions or solve his gleich hinterher eine ausgelassen, rv. 52 eine doppclt, v. 11 eine aup^'clnpsen, desgleichen v. 14, 2.5, 33, vi. 8, 13 und so weiter), die Zalileu Verstiimmcit, was aber das Aergste ist, die Bacbet'schen kritischen Zeicheu sind fast iiberall, die Klammer durcbgtingig weggefallen, so dass diese Ausgabe als Text des Diophant vcillig unbrauchbar geworden ist," p. 283. Accordingly Cantor errs when he says "Die beste Textamijabe ist die von Bachet de Meziriac mit Anmerkungen von Format. Toulouse, 1G70." (Getch. p. 31)0.) 56 DIOPHANTOS OF ALEXANDRIA. problems or the like of them by different methods a list has been given at the outset. As I have already mentioned a statement that Joseph Auria of Naples wrote circa 1580 a translation of Diophantos which was found (presumably in MS. form) in the library of one Carl von Montchall, it is necessary here to give the indications we have of lost works upon Dio- phantos. First, we find it asserted by Vossius (as some have understood him) that the Englishman John Pell wrote an un- published Commentary upon Diophantos. John Pell was at one time a professor of mathematics at Amsterdam and gave lectures there on Diophantos, but what Vossius says about his commentary may well be only a recommendation to undertake a commentary, rather than a historical assertion of its comple- tion. Secondly, Schulz states in his preface that he had lately found a note in Schmeisser's Orthodidaktih der Mathematik that Hofrath Kausler by command of the Russian Academy pre- pared an edition of Diophantos \ Of this nothing whatever is known; if ever written, this edition must have been only for private use at St Petersburg. I find a statement in the New American Cyclopaedia (New York, D. Appleton and Company), vol. VI. that "a complete translation of his (Diophantos') works into English was made by the late Miss Abigail Lousada, but has not been published." ^ The whole passage of Schmcisser is: "Die mechanische, geistlose Behand- lung der Algebra ist ins besondere von Herru Hofrath Kausler stark geriigt worden. In der Vorrede zu seiner Ausgabe des Vjlakerschen ExempcUmclis beginnt er so : ' Seit mehreren Jahren arbeitete ich fiir die Kussisch-Kaiserliche Akademie der Wissenschaften Diophants unsterbliches Werk iiber die Arithnietik aus, und fand darin einen solchen Schatz von den feinsten, scharfsinnigsten algebraischcn Auflosungen, dass mir die mechanische, geistlose Methode der neuen Algebra mit jedem Tage mehr ekelte u. s. w.' " (p. 33.) CHAPTER IV. NOTATION AND DEFINITIONS OF DIOPHANTOS. § 1. As it is my inteution, for the sake of brevity and perspicuity, to make use of the modern algebraical notation in giving my account of Diophantos' problems and general methods, it will be 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 the relations which would be represented in our time by algebraical equations, the extent to which he is able to manipulate unknown quantities, and so on. Apart, however, from the necessity of such a description for the proper and adequate comprehension of Diophantos, the general question of the historical develop- ment 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 Diophantos ; 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. § 2. First, as to the representation of an unknown quantity. The unknown quantity, Avhich Diophantos calls ttXj/^o? fiovdBoiu aXoyov i.e. "a number of units of which no account is given, or undefined " is denoted throughout (def. 2) by what is uni- versally printed in the editions as the Greek letter ? with an accent, thus ?', or in the form s°'. This symbol in verbal description he calls u aptOfxo'^, "the number" i.e. by inipli- 58 DIOPHANTOS OF ALEXANDRIA. cation, the number par excellence of the problem in question. (In the cases where the symbol is used to denote inflected forms, e.g. accusative singular or dative plural, the terminations which would have been added to the stem of the full word dpi6fi6hantos for apiOfwi in its ordiiKn-y sense, which I hold that I have proved. 3. Mr Gow's tliird objection is stated thus: "If s is for dp. then, by analogy, the full symbol should be s' (like 5", k'") and not j°." (a) I must first remark that I consider that arguments from analogy are inapplicable in this case. The fact is that there are some points in which all the five signs of which I have been speaking are undoubtedly analogous, and others in which some are not; therefore to argue from analogy here is futile, because it would be equally easy to establish by that means either of two opposite conclusions. I might, with the same justice as Mr Gow, argue backwards that, since there is undoubtedly one point in which s° and 5" are not analogous, namely the superposition in one case of terminations, in the other case of the second letter of the word, therefore the signs must be differently explained : a result which, so far as it goes, would favour my view, (b) Besides, even if we admit the force of Mr Gow's argument by analogy, is it true that s' (on the supposition that s is for dp) is analogous to 5" at all? I think not; for s does not corres- pond to 5, but (on my supposition) to 8v, and I only partially corresponds to v, inasmuch as t is the tliinl letter of the complete word in one case, in the other i; is the second letter, (c) As a matter of fact, however, I maintain that my suggestion does satisfy analogy in one, and (I think) the most important respect, namely that (as I have above explained) Diophantos proceeded on one and the same system in making his abbreviations, taking in each case the two first letters of the word, the only difference being that in one case only are the two letters contracted into one sign. Let us now enquire whether my theory will remove the difficulties stated by Mr Gow on p. 108 of his work. As reasons for doubting whether the symbol for dpid/xoi is really a final sigma, he states the following. "It must be remembered : (1) that it is only cursive Greek which has a final siijma, and that the cursive form did not come into use till the 8th or 9th century : (2) that inflexions are appended to Diophantus' symbol s' (e.g. s°", ss°S etc.), and that his other symbols (except f) are initial letters or syllables. The objection (1) might be disposed of by the fact that the Greeks had two uncial sigmas C and 1, one of which might have been used by Dioi)hantus, but I do not see my way to dismissing objection (2)." First, with regard to objection (1) Mr Gow rightly says that, supposing the sign were really s, it would be possible to dismiss this objection. On my tlieory, however, it is not necessai-y even to dismiss it : it does not exist. Secondly, my theory will dismiss objection (2). "Diophantus' other symbols (except /;>) are inititil letters or syllables." I answer "So is to." "Inflexions are appended to Diophantus' symbol s'." I answer "True; but the nature of the sign itself made this convenient," as I have above explained. NOTATION AND DKriNITIoNS uV 1 )li .pll VNToS. CyJ sake, subject to the remarks which 1 have here ma^le on the subject. § 3. Next, as regards the notatiou Avhich Diupliaiitus used to express the different powers of the unknown quantity, i.e. corresponding to x^, x^ and so on. The square of tlie unknown is called by Diophantos SviafiK: and denoted by the abbrevia- tion* B". Now tiie word Bvvafiiq ("power") is commonly used in Greek to express a square number. The first occurrence of the word in its technical sense is probably as early as the second half of the fifth century B.C. Eudemos uses it in quoting from Hippokrates (no doubt word for word) who lived about that time. The dilBference iu use between the words Bvvafj.i<; and T€Tpnycovol'IfANT(»s. 71 in tlie secoiul. As Diopliantos does nut go "beyond the sixtli power, the hist three phices in the tirst scale are left blank. An examination of these two scales will show also that the generation of the successive powers dirt'ers in the two systems. The Diophantine terms for them are based on the addition of exponents, the Arabic on their multipliaition \ Thus the "cube- cube" means in ])iophantos of, in the Italian and Arabic system x". The first method of generation may (says Cossali) be described as the method by which each power is represented by the product of the two lesser powers which are nearest to it, the method of viidtiplication ; the second the method of elevation, i.e. the method which forms by raising to the second or third power all powers which can be so formed, or the ^ih, Cth, 8th, 9th, &c. The intermediate powers which cannot be so formed are called in Italian Relati. Thus the fifth power is Relato 1", x' is Relato 2", a-'" is Censo di Relato 1", .r" is Relato 3", and so on. Wall is calls these powers supevsolida, reproduced by Montucla as sursolides. For Subtraction Diophantos uses a symbol. His full terra for Negation is Xei-\\r.<;, corresponding to inrap^i|re&)9 (njfietov yfr €X\nre>jt) with the top shortened. As Diophantos uses no distinct sign for +, it is clearly necessary, 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 x^ — ox' + Sx — 1, 1 This statement of Cossali's needs qualification however. There is at least one Arabian algebraist, Alkarkhi, the author of the Fakhri referred to above (pp. 24, 25), who uses the Diophantine system of powers of the unknown de- pending on the additioji of exponents. Alkarkhi, namely, expresses all powers of the unknown above the third by means of nUil, his term for tlie square, and ka% his term for the cube of the unknown, as follows. The fourth power is with hun mdl mdl, the fifth null Jca'b, the sixth ka'b ka'b, the seventh null mal ka'b, the eighth nial ka'b ka'h, the ninth ka'b ka'b ka'b, and bo on. » Dr Heinrich Suter however has the erroneous statement that Diopliantos would express j?-5x^ + 8.r - 1 by k" d /A «"« »• ij ^ M° «'. which is exactly what h« would not do. 72 DIOPHANTOS OF ALEXANDKIA. Diophantos would write «" a s°"' ?; >//v S" e ^° d. With respect to this curious sign, given in the MSS. as ^ and described as an inverted truncated ■\\r, I must here observe tliat I do not believe it to be what it is represented as being. I do not believe that Diophantos used so fantastic a sign for minus as an inverted truncated -v/r. In the first place, an inverted -^^ seems too curious a sign, and too far-fetched. To one wlio was looking for a symbol to express mimis many others more natural and less fantastic than j/v must have suggested themselves. {Secondly, given that Diophantos used an inverted -x/r, why should he truncate it ? Surely that must have been unneces- sary ; Ave could hardly have expected it unless, without it, confusion was likely to arise; but ^ could hardly have been confused with anything. It seems to me that this very trunca- tion throws doubt on the symbol as we find it in the MS. Hence I believe that the conception of this symbol as an inverted truncated -^ is a mistake, and that the description of it as such is not Diophantos' description ; it appears to me to be an explanation by a scribe of a symbol which he did not under- stand*. It seems to me probable that the true explanation is the following : Diophantos proceeded in this case as in the others which we have discussed (the signs for apt^/xd?, ^vvaiJii<;, 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 all happens to come later in the alphabet than the corresponding first letter), so, in seeking an abbreviation for \et>|rt9 and cognate inflected forms developed from Xtir, he first took the initial letter of the word. The uncial" 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 1 I am not even sure that the description can be made to mean all that it is intended to mean. AXiWs scarcely seems to be sufliciently precise. Might it not be applied to ^ with any part cut off, and not only shortened at the top? * I adhere to the uncial form above for clearness' sake. If Diophantos used the "Majuskelcursive" form, the explanation will equally apply, the difference of form being for our purpose lU'Rlifiihle. NOTATION AND DEFINITIONS OF DIOl'llANTt )S. 7:{ letter of the stem Xnr is I, but Al is open to objectiuii \slnii so written. Hence Diophantos placed the I inside the A, thus, A. Of the possibility of this I entertain no doubt, because there are indubitable cases of combination, even in uncial writing, of two letters into one sign. I would refer in par- ticular to X, which is an uncial abbreviation for TAAANTON. Now this sign. A, is an inverted and truncated i/r (written in the uncial form, ^j; and we can, on this assumpti(tn, easily account for the explanation of the sign for minus which is given in the text. For Division it often happens that no symbol is necessary, i.e. in the cases where one number is to be divided by another which will divide it without a remainder. In other cases the division has to be expressed by a fraction, whether the divisor be an absolute number or contain the variable. Thus the case of Division comes under that of Fractions. To express nume- rical fractions Diophantos adopts a uniform system, which is also seen in other writers. The numerator he writes in the ordinary line like a number; then he places the denominator above the line to the right of the numerator, in the same place as we should write an exponent, usually placing a ciicumflex a-cent over the eud of it. Thus i| is represented by tf'^ , yJ^j^ is aP, -Win- is (v. 12) ^eTZT?""^", ^[\%' is (iv. 17) y.S'xicd^'^^'. Diophantos, however, often expresses fractions by simply putting iv fjLopiro or [lopiov between the numerator and the denominator, i.e. one number divided by another. Cf. IV, 29 pF-.^f ^ttS fiopiov HS^.fipixh, i.e. Vif/nV' ^^^'J "^'- -'^ /5-,^X ^^ p-opiw pK^.aKi, i.e. TyiC^o* There is a peculiarity in the way in which Diophan- tos expresses such complex fractions as ,.,,'• It will be best understood by giving a typical case. This jiarticular fraction Diopliantos writes thus, aoyXB^^^.u^, that is, it is as if he had written with our notation y^Y i. Instances of this - " - . fiH[)ii notation occur passim, cf. V. 2 T7r6i"'^.d^ is e(iuivalent to -p.f • Bachet reproduces Diophantos' notation by writing in the.se cases '/jY h ^^^^ iti h respectively. 74 DIOPHANTOS OF ALEXANDHIA. But there is another kind of fraction, besides the purely numerical one, which is continually occurring in the Arith- metics, 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 simply a power of the unknown, 9'. Concerning fractions of this kind Diophantos 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 ho- monymous with what are called dpiO/iot, or unknowns, are called after them, thus from dpi6fx6<; we name the fraction to dpiOfjioaTov [i.e. - from ;r], to Suvafioarov from BvvafiL<;, to Kv^oarov from Kv^ok iv fiopup B"6" a /x" "^ \ei\Jrei ., . . 60^-' + 2520 - .' ^, -'5 c>- c. - . ' sjc - - 5 s - r ^ 2j-' + ox- + ix + \ fiopiM o" a >i^ p /j.° a corresponding to ^ — ^^ . To connect the two sides of an equation Diophantos uses Avords (cro9 or i'ao9 iari, or the oblique cases of i.^ Ixfpt(j/ji6i S a I p.o.-^ inrop^is ^Pp.oI? (^ po.q Put for the numbers x + 10, 10 - x. Squaring we have .r2+20x+ 100, x2 + 100-20x. Thediff., 40x = 80. Dividing, x = 2. Result, greater is 12, less is 8. The comparison of these two forms under which the same operations appear is most interesting. It is indeed obvious that if we take the skeletons of work given in the MS, the similarity is most striking. It is true that the Greek notation for the equations is very much inferior to the modern, but on the other hand the words indicating the operations make the whole very little less concise than the modern work. The omission of these tabular skeletons supplied in the MS. is a very grave defect in Bachet's edition, and thanks are due to M. Rodet for his interesting quotations from the original source. The same writer quotes two other such tables, which, however, for brevity's sake, we omit here. Though in the Ms. the sign I is used 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 his earlier translation (1575) Xy lander does use a symbol to denote equalit3^ namely ||, two short vertical parallel lines, in his notes to Diophantos. Thus we find, for example (p. 7G) \Q -¥ 12 || \Q + 0^7 + 9, which we should ex- press by x' +12= x^ + Qx + ^d. § 5. Now that we have described in detail Diopliantos' method of expressing algebraical quantities and relations, we NOTATION AND DEFINITIONS OF DIOPIIANToS. 77 may remark on the general system which he uses that it is essentially different in its character from the modern notation. While in modern times signs and symbols have been developed which have no intrinsic relationship to the things which they syml)olise, but depend for their use upon convention, the case is quite different with Diophantos, where algebraic notation takes the form of mere abbreviation of words which are considered a.s pronounced or implied. This is partly proved by the symbols themselves, which in general consist of the first letter or letters of words (so written as to avoid confusion), the only possible exception being the supposed final sigma, 9, for dpiBfxof or the unknow^n quantit3^ Partly also it is proved by the fact that Diophantos uses the symbol and the complete word very often quite indifferently. Thus we find often in the same sentence 9 or 99 and dpt6fi6<;, dpiO/xol, S" and hvvaixi<;, ^ and \eiyjrei, and so on. The strongest proof, however, that Diophantos' algebraic notation >vas mere abbreviation is found in the fact that the abbreviations, which are his algebraical symbols, are used for the corresponding words even when those words have a quite different signification. So in particular the symbol 9 is used as an abbreviation for dpL6fi6<;, when the word is used, not in its technical Diophantine sense for the unknown, but in its ordinary meaning of a number, especially in enunciations ^\here dpi$fi6<; in its ordinary sense naturally occurs oftenest. Similarly ^ is not used only for Xelyjret but also for other inflexional forms of the stem of this word, e.g. for Xiiruv or \€Lyjra<; in ill. 3: Evpeh' Tpel<; dpiOfiov^ o7rco<; 6 diro tov avy/ceifievov iic rcov rpLiov JJ^ eKaarov nroifj rerpdycovoi'. Other indications are (1) the sepa- ration of the symbc.ls and coefficients by particles [cf. I. 4:J 99°"' dpa I] ; (2) the addition of terminations to the symbol to represent the different cases. Nesselmann gives a good instance in which many of these peculiarities are combined, 99°' dpa I fx° \ laot elalv 99°'* Td fiovdcrc Tt. I. ad Jin. In order to determine in what place, in respect of .systems of algebraic notation, Diophantos stands, Nesselmann observes that we can, as regards the form of exposition cf algebraic operations and equations, distinguish three historical stages of development, well marked and easily discernible. 1. Tho first ^ § 6. To the form of Diophantos' notation is due the fact that he is unable to introduce into his questions more than one unknown quantity. This limitation has made his procedure often very different from our modern work. In the first place he performs eliminations, which we should leave to be done in the course of the work, before he prepares to work out the problem, by expressing everything which occurs in such a way as to contain only one unknown. This is the case in the great majority of questions of the first Book, which are cases of the solution of determinate simultaneous equations of the first order with two, three, or four variables; all these Diophantos ex- presses in terms of one unknown, and then proceeds to find it from a simple equation. In cases where the relations between these variables are complicated, Diophantos shows extraordinary acuteness in the selection of an unknown quantity. Secondly, however, this limitation affects much of Diophantos' work in- juriously, for while he handles problems which are by nature indeterminate and woukl lead with our notation to an inde- terminate equation containing two or three unknowns, he is compelled by limitation of notation to assign to one or other of these arbitrarily-chosen numbers which have the effect of making the problem a determinate one. However it is but fair to say that Diophantos 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 wliatever, say such and such NOTATION AND DKFINITION'S (tF 1)F< )riIANT()S. Hi a one." Thus it can lianlly be said tliat there is (in LfiMieral) any loss of universality. We may say, then, that in general Diophantos is obliged to express all his unknowns in terms, or as functions, of one variable. There is something exce.ssively interesting in the clever devices by which he contrives .so to express them in terms of his single unknown, the simj»Ie ecpiation x = a. It may be observed that the greater proportion of the pro- blems in Book l. are such that more than one unknown quantify is sought. Now% when there are two unknowns and two condi- tions, both unknowns can be easily expressed in terms of one symbol. But when there are three or four quantities to be found this reduction is much more difficult, and Diophanto-s manifests peculiar adroitness in effecting it : the result being that it is only necessary to solve a simple equation with one unknown quantity. With regard to pure eciuations, .some have asserted that pure quadratics were tlie only form «>f ([uailratic ' Dof. 11: MerA 5^ Tavra iav airb irpofi\r)naT6s rii'ot yivijTai Crapiit ilitai To?s aiiToh fir} bixoirX-qdr) Si dirb iKaripwv tCiv fitpCiv, Strati i^xupfiy rd Ofioia aw6 Twv ofioiwu, iws av ifbs (?) eZSos ivl etdfi toov y^vrrrai. 4cu> bl Twt (V oTOTifHfi im- irdpxv^^) V ^v diJ. dn(l>oT^pois Tois ixipeaiv, ?wj dv inaTtpip tu.'v nfpwv to. ttorj iwwdpxovTa finfrai. Kal trdXiv d^eXfiv rd cp.oia dirb rwf ofioluv tm ay cKor^/x^ twi' fitpif In tlSot Kara\ui>0^. Bachet's text (lO-il). p. 10. 90 DIOPHANTOS OF ALEXANDRIA. solved in Diuplianto;;' : a statement entirely without foundation. We proceed to consider (2) Mixed quadratic equations. After the remarks in Def. 11 upon the reduction of pure equations until we have one term equal to another term, Diophantos adds*: "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 Arithmetics, as we possess the book, this promise is not fulfilled. The first 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. 23, "x = 4a; — 4, therefore cc = 2": vi. 7, "8-^x^ — lx = 7, hence a; = ^ " : vi. 9, " 630.^'^ — 73.r = 6, therefore x = f^": and, vi. 8, " G30j;*^ + 7Sx = G, and x is rational." These examples, though proving that somehow Diophantos had arrived at the result, are not a sufficient proof to satisfy us that he necessarily was 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 ti'ial. That, however, Diophantos' solutions of mixed quadratics were not merely empirical, is shown by instances in v. 3.1 In this problem he shows pretty plainly that his method was scientific, in that he indicates 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 : "x has to be so determined •^■'-GO , x'-iJO . 2 ,.n - that it must be > — - — and < -^ , i.e. .x - 00 > o.r, and o O x' - GO < S.r. Therefore x" = Hx + some number > GO, therefore x must be not less than II, and x^ <8x + 60, therefore a; must be not greater than 12." 1 Cf. Iloimer, translation of Bossut's Ge.^rh. d. Math. i. 55. Kliigel's Dictionary. Also Dr Hcinrich Siitcr's doubts in Oesch. d. Math. Zurich, 1873. * vaTfpov hi (TOL Sci^ofxev Kal ttois 8i'o iidJiv lawv iv\ KaraXfifdivTwv t6 towvtov \vtTai. DIOrilAXTOS' MKTIloDS UF SOLUTION. 1»1 Now by examining the roots of these two equations we rtnJ x> ^, and .'•<4 + V<(), or .v> lO-G'-ldi and x< Il'TITN. It is clear therefore that x inai/ be < 1 1 or > 12, and there- fore Diophantos' limits are not strictly accurate. As however it was doubtless his object to find integral limits, the limits 11 and 12 are those wdiich are obviously adapted for his purpose, and are a fortiori right. Later in the same problem he makes an auxiliary determination of x, which must be such that x^+m>1±c, a,-' + GO < Sir, which give x>\l+J(ji, a;<12 + J^\. Here Diophantos says x must be > 10, <21, wliioli again are clearly the nearest integral limits. The occurrence of these two examples which we have given of equations whose roots are irrational, and therefore could not be hit upon by trial, show's that in such cases Diophantos must have liad a method by which he approximated to these roots. Thus it may be taken for granted that Diophantos had a definite rule for the solution of mixed quadratic equations. We are further able to make out the formula or rule by which Diophantos solved such equations. Take, for example, the equation ax^ ■\- hx -\- c = 0. In our modern method of solution we divide by a and write the result originally in tlic form 7 /~T^ x= — y~ + A / —J . It docs not appear that Diophantos divided throughout by a. Rather he first multiplied by a so as to bring the equation into the form aV + ahx + ac = ; tlicn solving he found ax= — \h ±J\i}^ — ac, and regarded the result in the' f..rm ,^_^ ^ ± ^'l ^'-«^, Whether the inter- a mediate procedure was as we have described it is n(H certain ; but it is certain that he used the result in the form given. One remark however must be made upon the form of the root. ' Nessi'luiann, p. :ilO. Also IJo.li-t, Jountdl A^iniiijiir. .Iiiiivi.r, I'-T-. 92 DIOPHANTOS OF ALEXANDRIA. Diophantos takes no account of tlie existence of two roots, according to the sign taken before the radical. Diophantos ignores always the negative sign, and takes the positive one as giving the value of the root. Though this perhaps might not surprise us in cases where one of the roots obtained is nega- tive, yet neither does Diophantos use both roots when both arc positive in sign. In contrast to this Nessehnann points out that the Arabians (as typified by Mohammed ibn Milsa) and the older Italians do in this latter case recognise both roots. M. llodet, however, remarks upon this comparison between Diophantos and the Arabians, so unfavourable to the former, as follows (a) Diophantos did not write a text-book on Algebra, and in the cases where the equation arrived at gives two positive solutions one of them is excluded a priori, as for ex- ample in the case quoted by him, v. 13. Here the inequality 72.r > llx'^ + 17 would give a; < f f or else x<-^. But the other inequality to be satisfied is *l^x\% ov x> f^. As however -^j < /\j, the limits x<^\> fjj are impossible. Hence the roots of the equations corresponding to the negative sign of the radical must necessarily be rejected. (6) Mohammed ibn Mu^a, althougli recognising in theory two roots of the equation x^ + c = hx, in practice only uses one of the two, and, curiously enough, always takes the value cor- responding to the negative sign before the radical, whereas Diophantos uses the positive sign. But see Chapter viii. From the rule given in Def. 11 for compensating by addition any negative terms on either side of an equation and taking equals from equals (operations called by the Arabs aljahr and almulcahala) it is clear that as a preliminary to solution Diophantos so arranged his equation that all the terms were positive. Thus of the mixed quadratic equation we have three cases of which we may give instances : thus, -^,p + Jip' + m"+ mq, or in tliis case (•^)'' + 42, must be a scjuare in onh-r that the root may be rational, which in tliis case it is not. Case 2. Form mx^ = px + q. Diopliantos takes i u + V i i)^ + tun X = - f^ iii 1 . An example is IV. 45, where 2x-* > Gx + 1 S. Diophantos says : " To solve this take the square of half the co- efficient of X, i.e. 9, and the product of the ftbsolute term and the coefficient of x', i.e. od. Adding, we have 45, the square root^ of which is not^ < 7. Add half the coefficient of x and divide by the coefficient of x"^ ; whence x < 5." Here the form of the root is given completely; and the whole operation by which Diopbantos found it is revealed. Case 3. Form mx^ + 3' = P'' '• Dioplianto.s' root is — M— - — /. Cf. in V. 13 the equation already mentioned, 17.6-^+ 17 <72x. Diophantos says: "Multiply half the coeffi- cient of X into itself and we have 1296 : subtract the i)roduct of the coefficient of x^ and the absolute term, or 2 31. Achl half the coefficient of x, and the result is not > 07. Divide by the coefficient of x^, and x is not > f f ." Here again we have the complete solution given. (3) Cubic equation. There is no ground for supposing that Diophantos was acquainted with the solution of a cubic ct[uation. It is true there is one cubic e(piation which occurs in the Arithmetics, but it is only a very particular case. In vi. 19 the equation arises, a.'^ + 2^ + 3 = a;^ + 3a; - 3a;'^ - 1, and Diophantos says simply, "whence x is found to be 4." All that can be said of 1 For "coeflicient" Diophantos uses simply irX^t'os, luiinbcr: thus "number of apiOfiol " = coeff . of x. - Diophantos calls the "square root" irXei'pd or side. 2 7, though not accurate, is clearly the nearest integral limit which will servo the purpose. * As before, the nearest intajruJ limit. O-i DIOPHANTOS OF ALEXANDRIA. this is that if we write the equation in true Diophantine fashion, so that all terms are positive, x^ + oc = 4'X^+ 4. This equation being clearly equivalent to x{x^ + l) = 4(.r'^ + l), Diophantos probably detected the presence on both sides of the equation of a common factor. The result of dividing by it is a; = 4, which is Diophantos' solution. Of the two other roots X = ±J — 1 no account is taken, for reasons stated above. From this single example we have no means of judging how far Diophantos was acquainted with the solutions of equations of a degree higher than the second. I pass now to the second general division of equations. (B.) Indeterminate equations. As has been already stated, Diophantos does not in his Arithmetics, as we possess them, treat of indeterminate equa- tions of the first degree. Those examples in the First Book which would lead to such equations are, by the arbitrary assumption of one of the required numbers as if known, con- verted into determinate equations. It is possible that the treatment of indeterminate equations belonged to the missing portion which (we have reason to believe) has been lost between Books I. and II. But we cannot with certainty dispute the view that Diophantos never gave them at all. For (as Nessel- mann observes) as with indeterminate quadratic equations our object is to obtain a rational result, so in indeterminate simple equations we seek to find a result in ivhole numbers. But the exclusion of fractions as inadmissible results is entirely foreign to our author; indeed we do not find the slightest trace that he ever insisted on such a condition. We take therefore as our first division indeterminate equations of the second degree. I. Indeterminate equations of the second degree. The form in which these equations occur in Diophantos is universally this : one or two (and never more) functions of the uid/. This can be lationally solved accord- ing to Diophantos (a) When A is positive and a square, say a'. Thus oV -f C = _?/*, In this case ?/* is put = (ax ± in)* ; therefore a'x' + C = (ax ± m)^ a - m' 96 DIOPHANTOS OF ALEXANDRIA. (m and the doubtful sign being always assumed so as to give £c a positive value). {/3) When C is positive and a square number, say c*. Tlius Aa;' + r = y^. Here Diophantos puts ?/ = (mx ± c) ; therefore Ax^ + c^ = (mx ± cf, A — lit'- (7) When one solution is known, any number of other solutions can be found. This is enunciated in vi. 16 tlms, though only for the case in which C is negative: "when two numbers are given such that when one is multiplied by some square, and the other is s\ibtracted from the product, the result is a square number; another square also can be found, greater than the first taken square, which will have the same effect," It is curious that Diophantos does not give a general enunciation of this proposition, inasmuch as not only is it applicable to the cases ± Ax^ ±C^ = if, but to the general form Ax^ -\- Bx -\- G = y"'. In the Lemma at vi. 12 Diophantos does prove that the equation Ax^ + C = y can be solved when ^ + C is a square, i.e. in the particular case when the value x = l satisfies the equation. But he does not always bear this in mind, for in III. 12 the equation o2x^ + 12 = y^ is pronounced to be impos- sible of solution, although 52 -f 12 = G4, a square, and a rational solution is therefore possible. So, ill. 13, 2(j(jx^ — 10 = 3/* is said to be impossible, though a; = 1 satisfies it. / It is clear that, if a; = satisfies the ciiuation, (7 is a square, and therefore this case (7) includes the previous case (/?). It is interesting to observe that in VI. 15 Diophantos states that a rational solution of the equation Ax' -€' = 7/ is impossible unless A is the sum of two squares^. ' Nesselmann compares Lpj,'cmlro, Tlirorie des Xomhrrs, p. GO. DIOPHANTOS MI-:TH01)S Ol' SOLI TloN. !)7 Lastly, we must consider the Form Ax^ + Bx+C = y\ This equation can be reduced by means of a change of variable to the previous form, wanting the second term. Thus if we put x = z — ^ . , the transformation gives ^- + ^A - y ■ Diophantos, 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 aV+ Bx + C=y"; and Diophantos puts if= {ax + mf, whence Exx. II. 20, 21 &c. 2am -B' (yQ) When C is positive and a square, or the ccjuation is Ax^ + Bx -\- c' = y-; and Diophantos writes y' = {mx + c)^ whence X = —J Y • Exx. IV. 0, 10 &c. A - m' (7) When \B'^ — AG is positive and a square number. Diophantos never expressly enunciates the possibility of this case: but it occurs, as it were unawares, in iv. 33. In this problem 3« + 18 — a;^ is to be made a square, and the ec|uation ^x-\-\^ — a? = y'^ comes under the present form. To solve this Diophantos assumes 3a;+ 18 — j;* = 4x* which leads to the quadratic 3a; + 18 — ox^ = 0, and " the equation is not rational". Hence the assumption 4a/' will not do : "and we must seek a square [to replace 4] such that 18 times (this square + 1) + (f)"'' may be a square". Diophantos then solves this auxiliary eciuation 18 {x" + 1) + ;,' = if, finding x = IS. TIr-u he assumes 3x+l8-a,'=(18)V, which gives 325a;^ - 3a; - 18 = 0, whence Jc = -^^. H. D. 7 98 DIOPHANTOS OF ALEXANDRIA. It is interesting to observe that from this example of Dio- phantos we can obtain the reduction of this general case to the form At? + G^ = y^, wanting the middle term. Thus, assume with Diophantos that Ax^+Bx-^ C = m^x^, therefore by solution we have B ^ + and X is rational provided ~t- — AC+ Cnf is a square. This B'^ condition can be fulfilled if — - AC he a square by a previous case. Even if that is not the case, we have to solve (putting, for brevity, D for ~ — AG) the equation D + Cm' = f. Hence the reduction is effected, by the aid of Diophantos alone. (2) Double-equation. By the name "double-equation" Diophantos designates the problem of finding one value of the unknown quantity x which will make two functions of it simultaneously rational square numbers. The Greek term for the "double-equation" occurs variously as Bnr\ola-6T7](; or BtTrXi} caorij'i. We have then to solve the equations mx^ + ax + a= u'\ nx'-{- ^x + b = w') 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 x') is wanting. This is the simplest case, and wc shall accordingly take it first. y DIOPHANTOS MKTIKJDS OF SOLUTION. })0 1. Double equation of the first degree. Diophautos has one distinct method of solving the ociuations ax + a= II' ^x + b=w' taking slightly different forms according to the nature of the coefficients. (a) First method of solution of ^a; + b = wy This method depends upon the equation \ 2 ) '-^J-P Thus 100 DIOPHANTOS OF ALEXANDRIA. We have then to separate B:c + e into two factors ; let these factors be «, — + - , and \vc accordins^ly write p p 8x € u + v = 1- , p p /hx e Y , ^ SV 2hxfe \ fe V ,, s therefore —5- -\ {- + p] + { +p] =4 (ax + a). p p \p / \p / Now in order that this equation may reduce to a simple one, either (1) the coefficient of x^ must vanish or 8 = 0, therefore a = ^, or (2) the absolute term must vanish. Therefore ( ~ + i^ ) = 4a, or p' + 2e/ + e' = 4ap% i. e. / +2{a-h-2a) pi' + (a - 6)" = 0. Therefore {p^ — a + hf = 4a6, whence ah must be a square number. Therefore either both a and h are squares, in which case we may substitute for them c' and d\ p being then equal to c ±d, or the ratio a : 6 is the ratio of a square to a square. With respect to (1) we observe that on one condition it is not necessary that S should vanish, i.e. provided we can, before solving the equations, make the coefficients of x' in both equal by multiplying either equation or both by a square number, an operation which does not affect the problem, for a square multi- plied by a square is still a square. DIOPHANTOS MI-yniODS OF SOLUTION. 101 Thus if =^ or aii^ = ^m^, the coudition 8 = will bo jo il satisfied by multiplying the equations respectively by n' and i/r ; and thus we can also solve the equations like the equations •'■ + « = '^^l ,v + b = w') in an infinite number of ways. Again the equations under (2), ax+ 6^= ^i^ ^x + d' = w\ can be solved in two different ways, according as we write them in this form or in the form ^c'x + c'd' = 2U" obtained by multiplying them respectively by (T, c* in order that the absolute terms may be equal. We now give those of the possible cases which are found solved in Diophantos' own work. These are equations (1) of the form ant'x + a= ii^] a case which includes the more common one, when tiie co- efficients of X in both are equal. (2) of the form ^x + cP = 2uy solved in two different ways according as they are thus written, or in the alternative form, ad\c + c'd^ = u' ^c'x + c"d' = w 102 DIOPHANTOS OF ALEXANDRIA. General solution of Form (1), or, am an Multiplying respectively by if, nf, we have to solve the equations, am^n^x + an^ = il^\ amVic + hm^ = w'^) ' The difiFercnce = aif — hm^. Suppose this separated into two factors p, q. Put It' ± %d — p, n T w' = q, whence ..■^ ^ (P ±3)\ ^r- = (P-^)' , therefore am'^ifx + an^ = ( — 9- j , or a'm^n^x + hm'^= \—y^ Either equation will give the same value of x, and p^ + q^ arf + hmf i 2 X— — •xmSi^ since yq = arf — hnf. Any factors p, q may be chosen provided the value of x obtained is positive. Ex. from Diophantos. 65- G«=w') G5 - 24^ = to'} ' ,, . 2G() - 24.7- = w'-) therefore ... ^, . Oo — 24a; = w The difference = 195 = 15 . 13 say, therefore / 15 - 13 y ^ ^^^ _ ^^^^ 24a; = 64, or a; = §. C-^^) DIOPHANTOS MKTHODS OF SOLUTION. 1()3 General solution (first method) of Form (2), or, oc + c" = ii'^l ^x + d'' = 2o'y In order to solve by this method, we multiply by rf*, c* respectively and write (xd^x + c'dr = n") ^c'x + c\V = w") ' u being the greater. The diti: = {id' - /3r) x. Let the factors of this be px, 7, therefore ii^=(&i+lJ\\ Hence x is found from the equation This equation gives j9V + 2x (pq - 2a^) + q" - ^c'd' = 0, or, since pq = ad^ — y8c^ p'x' - 2x {ad' + /Sr ) + q' - ^c'd' = 0. In order that this may reduce to a simple equation, as Diophantos requires, the absolute term must vanish. Therefore q' - ^c\l' = 0, whence 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 this solution is _ 2 {ad:" + I3c') _ Sc'ff (gff + )9 c') Ex. from Diophantos. This method is only used in one particular case, IV. 45, w^here c' = d' as the equations originally stand, namely 8x+-i = u'\ (jx+4> = iv'] ' 104- DIOPHANTOS OF ALEXANDRIA. the difference is 2x and q is necessarily taken = 2>/i = 4, and the factors are ^ , 4, therefore 8a; + 4 = r^ + 2 j and rr = 11 2. General solution (second method) of Form (2), or Here the difference ={ol - ^) x + (c^ - (T) = hx -\- € say, for brevity. Let the factors oi dx + e\iQ p, h . Then, as before proved (p. 100), p must be equal to (c + d). Therefore the factors are «-/3 - J 7 , oj + c + a, c + ft, c + fZ and we have finally \c±d J kC ± d. two. (^^)^'^+4.f<;;/>-fo, 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 c + d. Ex. from Diophantos, III. 17 : to solve the equations 10a; + 9 = w'^) 5a; + 4 = 10^ } ' The difference is here 5x + 5, and Diophantos chooses as the factors 5, a; + l. This case therefore corresponds to the value c + d of jj. The solution is given by (i-y= lOx + !), whence x - 28. DIOPHANTOS METHODS OF SOLUTION. 105 The other value c — (Z of jj 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 Diophantos, though he uses the prin- ciple in certain special cases of the single equation. Fermat was the first, apparently, to discover that this might always be done, if one value a of x were known, by substituting in the equations x+ a for .v. By this means it is possible to find a positive solution even if a is negative, by successive appli- cations of the principle. But nevertheless Diophantos 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 unsuit- ableness of the value found for x by the ordinary method), employed in iv. 45, gives a different way of solving a double- equation from that which has been explained, used only in a special case. (/S) Second method of solution of a double-equation of the first degree. Consider only the special case hx + if = u^, {h+f)x + n^ = iu\ Take these expressions, and ?^^ and write them in order uf magnitude, denoting them for convenience by A, B, C. A = {h+f)x + n\ B = hx + n\ G = n\ ,, „ A-B f , A-B=fx therefore F^ = ^ ^^^ 5- (7 = 1... Suppose now hx + if = {}j + nf, therefore hx = y~ + 2ny, therefore A-B = ^(if + ^nij), f or ^=(2/+«)' + ^(2/'+2ny), thus it is only necessary to make this expression a s(|uarc. 106 DIOPHANTOS OF ALEXANDRIA. Write therefore (l + {) / + 2'^ ({ + 1)2/ + '^' = (Vy - '')"' whence any number of values for y, and therefore for x, can be found, by varying p. Ex. The only example in the Arithmetics is in iv. 45. There is the additional condition in this case of a limit to the value of X. The double-equation 8a; + 4 = u^^ Qx + 4! = vf has to be solved in such a manner that x<1. A— B Here ^ — ^ ~ 3 > ^^^ ^ ^^ taken ' to be {y + 2)^ therefore ^-5=^^^, therefore ^ = 2^!+^^^ + ^^ + 4 1/ + 4 = ^|-V -|^ + 4 which must be made a square, or, multiplying by f , 3^/^ -I- 12t/ + 9 = a square, where y must be < 2. Diophantos assumes 32/^+123/ + 9 = (m2/-3r, 6w + 12 whence ?/ = -^ ., , and the value of m is then determined so that ?/ < 2. As we find only a special case in Diophantos solved by this method, it woidd be out of place to investigate the con- 1 Of course Diophantos uses the same variable .r where I have for clearness used y. Then, to express what I have called in later, he says: "I form a square from 3 minus some number of x's and .r becomes some number multiplied by 6 together with 12 and divided by the dillerence by which the square of the number exceeds three," DIOPHANTOS METHODS OF SOLUTION. 1()7 ditions under which more general cases might be solved in this manner \ 2. Double equation of the second degree, or the general form Ace' + Bx + C = 2i\ A'x'' + B'x + C' = w\ These equations are much less thoroughly treated in Diophan- tos than those of the first degree. Only such special instances occur as can be easily solved by the methods which we have described for those of the first degree. One separate case must be mentioned, which cannot be solved, from Diophantos' standpoint, by the preceding method, but which sometimes occurs and is solved by a peculiar method. The form of double-equation being ax' + ax = u'] (1), ^x'+bx = w'\ (2), Diophantos assumes whence from (1) a X = -2 m — a and by substitution in (2) a \~ ha , , + a must be a square, m —aj 7n —a a'^ + ha{m'-a) IS a square. Therefore we have to solve the equation abm^ + a (/9a — ah) = if, and this form can or cannot be solved by processes already given according to the nature of the coefficients^ , . OJ+bi • Bachet and after him Cossali proved the pos.Kibility of solving ^^^ j\ "V this method under two conditions. - Diophantos did not apparently observe that this form of e would, of course, be much more difficult ; for, putting i/ = nix, we find X = —, — r . and we have to solve or Ccm* + c(Bc- 2b C) m" + hc{hC-Bc) + A c' = u\ of which equation the above corresponding one is a very parti- cular case. § 3. Summari/ of the preceding investigation. We may sum up briefly the results of our investigation of Diophantos' methods of dealing with equations tiius. 1. Diophantos solves completely equations of the first degree, but takes pains beforehand to secure 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. H. D. 8 114 DIOPHANTOS OF ALEXANDRIA. 2. For determinate equations of the second degree Dio- phantos has a general method or rule of solution. He takes however in the Arithmetics no account of more than 07ie root, even when both roots are positive rational numbers. But his object is always to secure a solution in rational numbers, and therefore we need not be surprised at his ignoring one root of a quadratic, even though he knew of its existence. 3. No equations of a higher degree than the second are found in the book except a particular case of a cubic. 4. Indeterminate equations of the first degree are not treated in the work as we have it, and indeterminate equations of the second degree, e.g. Aaf + Bx -\-G = y', are only fully treated in the case where ^ or C vanishes, in the more general cases more imperfectly. 5. For " double-equations " of the second degree he has a definite method when the coefficient of x^ in both expressions vanishes ; this however is not of quite general application, and is supplemented in one or two cases by another artifice of particular application. Of more complicated cases we have only a few examples under conditions favourable for solution by his method. 6. Diophantos' treatment of indeterminate equations of higher degrees than the second depends upon the particular conditions of the problems, and his methods lack generality. 7. After all, more wonderful than his actual treatment of equations are the extraordinary 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, however, would have been out of place in this chapter, and can only be studied in the problems themselves. § 4. I shall, as I said before, not attempt to class as methods what Nesselmann has tried so to describe, e.g. "Solution by mere reflection," "solution in general expressions," of which there are few instances definitely described as such by Diophantos, and " arbitrary determinations and assumptions." It is clear that the most that can be done to formulate these " methods " is the DIOPHANTOS METHODS OF SOLUTION. 1 1 .') enumeration of a few instances. This is what Ncssclmann has done, and he himself regrets at the end of his chapter on "Methods of solution" that it must of necessity be so incomplete. To understand and appreciate these artifices of Diophantos it is necessary to read the problems themselves singly, and for these I refer to the abstract of them in the Appendix. As for the " Use of the right-angled triangle," all that can be said of a general character is that rational right-angled triangles (whose sides are all rational numbers) are alone used in Diophantos, and that accordingly the introduction of such a right-angled triangle is merely a convenient device to express the problem of finding two square numbers whose sum is also a square number. The general forms for the sides of a right-angled triangle are c^ -f h^, a^ — 1>\ 2ab, which clearly satisfy the condition {a' + by={a'-b'f + {2abY. The expression of the sides in this form Diophantos calls "form- ing a right-angled triangle from the numbers a and b." It is by this time unnecessary to observe that Diophantos does not use general numbers such as a, b but particular ones. " Forming a right-angled triangle from 7, 2 " means taking a right-angled triangle whose sides are 7'^ + 2', 7" — 2'^ 2 . 7 . 2, or o3, 4o, 28. § 5. Method of Limits. As Diophantos often has to find a series of numbers in ascending or descending order of magnitude : as also he does not admit negative solutions, it is often necessary for him to reject a solution which he has found by a straightforward method, in order to satisfy such conditions ; he is then very frequently obliged to find solutions of problems which lie within certain limits in order to replace the ones rejected. 1. A very simple case is the following: Required to find a value of X such that some power of it, x", shall lie between two assigned limits, given numbers. Let the given numbers bo a, b. Then Diophantos' method is : Multiply a and 6 both succes.sively by 2", 3", and so on until some (nf' power is seen which lies be- tween the two products. Thu.s suppose c" lith between up' and /*// ; 6—2 Il6 DIOPHANTOS OF ALEXANDRIA. then we can put « = - , in which case the condition is satisfied, P for(-| lies between a and 6. & Exx. In IV. 34 Diopbantos finds a square between f and 2 thus : he multiplies by a square, 64 ; thus we have the limits 80 and 128; 100 is clearly a square lying between these limits ; hence (lo)'^ or f| satisfies the condition of lying between | and 2. Here of course Diophantos might have multiplied by any other square, as 16, and the limits would then have become 20 and 82, between which there lies the square 25, and so we should have f§ again as the square required. In VI. 23 a sixth power (a " cube-cube ") is required which lies between 8 and 16. Now the sixth powers of the first natural numbers are 1, 64, 729, 409 6... Multiply 8 and 16 (as in rule) by 2° or 64 and we have as limits 512 and 1024, and 729 lies between them ; therefore "^^-^^ is a sixth power such as was required. To multiply by 729 in this case would not give us a result. 2. Other problems of finding values of x agreeably to certain limits cannot be reduced to a general rule. By giving, however, a few instances, we may give an idea of Diophantos' methods in general. Q Ex. 1, In IV, 26 it is necessary to find x so that , x' + x lies between x and x+l. The first condition gives 8 > a;'' + a;^ Diophantos accordingly assumes 8 = {x+lY = x' + x' + '^^ + Jj, which is >x'' + .c\ Thus x = ^ satisfies one condition. It also Q is seen to satisfy the second, or -5-— < a; -f- 1 : but Diophantos X + X '■ practically neglects this condition, though it turns out to be satisfied. The method is, therefore, hero imperfect. Ex. 2. Find a value of x such that x>y^x''-C)0) ox, x^ - 60 < 'Sx. DIOPIIANTOS' MKTIIODS ()F SOLrTIuX. II7 Hence, says Diophantos, rr is <(: 11 nor > 12. Wi- liave already spoken (pp. 00, 91) of the reasoning by which h.- arrives at this result (by taking only one root of the quadratic, and taking the nearest integral limits). It is also required that a? — 60 shall be a square. Assuming then a;^ - 60 = {x - mf, x = — ^ — , 2m which must be > 11 < 12, Avhence m' + 60 > 22m, m" + 60 < 2hn, and (says Diophantos) in must therefore lie between 10 and 21. Accordingly he writes x'' - 60 = (x - 20)^ and x = llh, which is a value of x satisfying the conditions. § 6. Method of Approximation to Limits. We come now to a very distinctive method called by Dio- phantos 7rapia6T7] 6. Take \^ or 6i and find what small fraction -^ added to it makes it a square : thus 6^ + -^ must be a square, or 26 4- -, X y is a square. Diophantos puts 26 + \_ = [^+l)\ or 26/ + 1 =(5^+ 1)', whence ?/ = 10 and „ = t.W. •»!" - = in,' •""' *'' *" i^>'> ~ ^ J y' '"" J- 118 DIOPHANTOS OF ALEXANDRIA. square = (f l)^ [The assumption of {oy + 1)^ is not arbitrary, for assume 26?/^ + 1 = {py + l)^ therefore y = ^ ^ ^ , and, since - should be a small proper fraction, therefore 5 is the most y suitable and the smallest possible value for ^^, 26 — p- being < Ip or p^ + 2jj + 1 > 27.] It is now necessary (says Diophantos) to divide 13 into two squares whose sides are each as near as possible to |^. Now the sides of the two squares of which 18 is by nature compounded are 3 and 2, and 3 is > fi by ^1 2 is < f^ by l^i ■ Now if 3 — -g^, 2 + ^ were taken as the sides of two squares their sum would be 2^^601 ^^^ 400 ' which is > 13. Accordingly Diophantos puts 3 - 9a;, 2 + ll.r, for the sides of the required squares, where x is therefore not exactly ^ but near it. Thus, assuming (3 -9a:)' + (2 + 11a;)' = 13, Diophantos obtains x = y^. Thus the sides of the required squares are \^\, f^. Ex. 2. Divide 10 into three squares such that each square is >3. Take ^o or 3^ and find what fraction of the form ^ added 9 to it will make it a square, i.e. make 30 + -^ a square or Wy'^ + 1, I 3 1 where - = - . X y DIOPHANTdS' METHODS OF SOLUTION. Ill) Diophautos writes 30/ + 1 = (r)y + 1)-, whence y = 2 and ^*- And 3^ + ^V = a square = '^' . [As before, if we assume 30//^ + 1 = {py + 1)*, 7 = , Z"' ^ , and since - must be a small proper fraction, 30 —;:>'' should < 2y; or p^+ 2jj + 1 > 81, and 5 is the smallest possible value of p. For this reason Diophantos chooses it.] We have now (says Diophantos) to make the sides of the required squares as near as may be to y. Now 10=9+l = 3^+(f)^+(|)^ and 3, f, 4 are the sides of three squares whose sum = 10. Bringing (3, f , 4) and y to a common denominator, wo have (f^, ^, M) and M. Now 3is>ffbyf§, f is 10. Diophantos accordingly assumes as the sides of the three required squares 3 - Zox, f + 37a-, I + 31a-, where x must therefore be not exactly ^'^, but near it. Solving (3 - rox)' + (f + 37a,f + (4 + 3U-)^ = 10, or 10-llG.t-+3555a;'=10, we have x = ^-^ ; the required sides are therefore found to be sw. w. sw> and the squares 'UM^^> VW^'. "Mi^- The two instances here given, though only instiinces, serve perfectly to illustrate the method of Diophantos. To have put them generally with the use of algebniical symbols, nistead <»f 120 DIOPHANTOS OF ALEXANDRIA. concrete numbers, would have rendered necessary the intro- duction of a large number of such symbols, and the number of conditions (e.g. that such and such an expression shall be a square) which it would have been necessary to express would have nullified all the advantages of this general treatment. As it only lies within my scope to explain what we actually find in Diophantos' work, I shall not here introduce certain investigations embodied by Poselger in his article " Beitrage zur TJnbestimmten Analysis," published in the Ahhandlungen der Koniglichen Akademie der Wissenschaften zu Berlin Aus dem Jahre 1832, Berlin, 1834. One section of this paper Poselger entitles "Annaherungs-methoden nach Diophantus," and obtains in it, upon Diophantos' principles \ a method of approximation to the value of a surd which will furnish the same results as the method by means of continued fractions, except that the approxi- mation by what he calls the " Diophantine method " is quicker than the method of continued fractions, so that it may serve to expedite the latter ^ ^ "Wenn wir den Weg des Diophantos verfolgen." 2 "Die Diopliantisehe Mctbode kann also dazii diencn, die Convergcnz der Partialbriiche des Kettenbruchs zu beschlcunigen." K^r.r^. CHAPTER VI. § 1. THE PORISMS OF DIOPIIANTOS. We have already spoken (in the Historical Introduction) of the Porisms of Diophantos as having probably foiined a distinct part of the work of our author. We also riiscussed the question as to whether the Porisms now lost formed an integral portion of the Arithmetics or whether it was a com- pletely separate treatise. What remains for us to do under the head of Diophantos' Porisms is to collect such references to them and such enunciations of definite porisms as are directly given by Diophantos. If we confine our list of Porisms to those given under that name by Diophantos, it docs not therefore follow that many other theorems enunciated, assumed or implied in the extant work, but not distinctly called Porisms, may not with equal propriety be supposed to have been actually pro- pounded in the Ponsm^. For distinctness, however, and in order to make our assumptions perfectly safe, it will be better to separate what are actually called porisms from other theorems implied and assumed in Diophantos' problems. First then with regard to the actual Porisms. I shall not attempt to discuss here the nature of the proposition which was called a porism, for such a discussion would be irrelevant to my purpose. The Porisms themselves too have been well enumerated and explained by Nesselmann in his tenth chapter; he has also given, mth few omissions, the chief of the other theorems assumed by Diophantos as known. Of necessity, therefore, in this section and the next I shall have to cover very much the same ground, anil shall acconliugly bo a.s brief us may be. 122 DIOPHANTOS OF ALEXANDRIA. Porism 1. The first porism enunciated by Diopbantos occurs in v. 3. He says " We bave from tbe Porisms tbat if eacb of two numbers and tbeir product wben severally added to tbe same number produce squares, the numbers are the squares of two consecutive numbers \" This theorem is not correctly enunciated, for two consecutive squares are not the only two numbers which will satisfy the condition. For suppose x + a = m\ y + a = n^, xy+a= p^. Now by help of the first two equations we find xy + a= m^n^ — a {m^ + n^ — \)+ a^ and this is equal to ]f. In order that m\^ — a {m^ + n^ — l)-\-d^ may be a square certain conditions must be satisfied. One sufficient condition is m^ + ?i" — 1 = 2inn, or m — n = + 1, and this is Diopbantos* condition. But we may also regard 2«,8 rrrn a {m? + 71" - 1) + a'' = p^ as an indeterminate equation in m of which we know one solution, namely m = n ± 1. Other solutions are then found by substituting z + {ii ±1) for m, whence we have the equation {re -a)z' + 2 {n' (n ±l)-a{n±l)}z + {ii' - a) {n ± If - a(?r— 1) + a'^ = p^, or {7i' - a) / + 2 {n' - a) (w ± 1) ^ + {n (n ± 1) - af = p\ which is easy to solve in Diopbantos' manner, the absolute term being a square. But in the problem V. 3 tlwee numbers are required such that each of them, and the product of each pair, severally added ^ Kal iirel ^xoM*" ^'' '''O'S irophixaffiv, on (of hvo dpiO/Jiol iKarepds re Kai 6 vir^ avTuiv /nerd rod avTov SoOivTos ttoltj Tfrpdycovov, ■yt-ybvaaiv dwb Svo reTpayuvuv rwv KOLTk Tb ^f ijs. THE PORISMS OF DIOPHANTOS. 123 to a given number produce squares. Thus, if the third number be z, three more conditions must be added, namely, z + «, zx-\-a, yz + a should be squares. The two last conditions are satisfied, if m + 1 = n, by putting z = 2{a; + y) — 1 = 4/m" + 4??i + 1 — 4a, when xz + a= {m {2m + 1) - Sa}", yz + a= {m {2m + 3) - (2a - l)f , and this means of satisfying the conditions may have affected the formulating of the Porism. V. 4 gives another case of the Porism with — a for + a. Porism 2. In V. 5 Diophantos says* , " We have in the Porisms that in addition to any two consecutive squares we can find another number which, being double of the sum of both and increased by 2, makes up three numbers, the product of any pair of which ^lus the sum of that pair or the third number produces a square," i.e. m\ m^ + 2??i + 1 , 4 ( 7?i' + m + l), are three numbers which satisfy the conditions. The same porism is assumed and made use of in the follow- ing problem, v. 6. Porism 3 occurs in v. 19. Unfortunately the text of the enunciation is corrupt, but there can be no doubt that the correct statement of the porism is " The difference of two cubes can be transformed into the sum of two cubes." Diophantos contents himself with the mere enunciation and does not pro- ceed to effect the actual transformation. Thus we do not know his method, or how far he was able to prove the porism as a perfectly general theorem. The theorems upon the trans- formation of sums and differences of cubes were investigated by Vieta, Bachet and Format. 1 Kal txoiJ^o irdXi;/ (v toZs wopifffMacriv on iraai 56o Ttrpaywvoif toTj (tori t6 iint irpooripov Kal SvUt. fxtiiuiv, t/xji apidfiodi TTOtei tSv 6 vwd 6iroiu}vo!}i> idvTe irpoffXdiir] avvan•« TeTpd-ywvov. 124 DIOPHANTOS OF ALEXANDRIA. Vieta gives three problems on the subject ^ (Zetetica iv.). 1. Given two cubes, to find in rational numbers two others whose sum equals the difference of the two given ones. As a solution of a^-I/ = x^ H- y^, he finds _ a{a'-^h') _ hj^a'j- If) ^- a' + h' ' y~ a' + b' ' 2. Given two cubes, to find in rational numbers two others, a +0 -X y, X- ^3_^, , y ^,_^, . 3. Given two cubes, to find in rational numbers two others, whose difference equals the difference of the given ones; a -u -X y, X- ^3_^^3 , y- ^^,_^^, . In 1 clearly x is negative if 2b^>a^', therefore, to secure a "rational" result, (v-j > 2. But for a "rational" result in 8 we must have exactly the opposite condition, t5 < 2. Fermat, who apparently was the first to notice this, remarked that in con- sequence the processes 1 and 3 exactly supplement each other, 1 Poselger {Berlin Abhandhuigcn, 1832) has obtained tlicse results. He gets, e.g. the first as follows: Assume two cubes {a-xf, (mx-j3f, which are to be taken so that their sum = a3 - /33. Now (a-x)^ = a^-3a-x + 3ax--x\ (fftx - /3)3 = - /33 + 3?»/3-x - 3»i''/3x2 + m^a?. If then G)' and 3(TO2/3-a) Sap' ^~ m3-l a3 + /33' (a-x)3 + (mx-/3)8 = a3-/3», a(a3-2^)l THB PORlSMS OP DIOPHANTOS. 125 SO that by employing them successively we can effect the trans- formation of 1, even when Process 2 is always possible, therefore by the suitable com- bination of processes the transformation of a sum of two cubes into a difference, or a difference of two cubes into a sum of two others, is always practicable. Besides the Po)nsins, there are many other propositions assumed or implied by Diophautos which are not definitely called porisms, though some of them are very similar to the porisms just described. § 2. Theorems assumed or implied by Diophantos. Of these Nesselmann rightly distinguishes two classes, the first being of the nature of identical formulae, the second theorems relating to the sums of two or more square num- bers, &c. 1. The first class do not require enumeration in detail. We may mention one or two examples, e.g. that the expressions C^-) - ah and a" (a -h 1)' + a' + (a + 1)" are squares, and that a {a^ — a) -\- a + (a^ — a) is always a cube. Again, Nesselmann thinks that Diophantos made use of the separation of a' — 1/ into factors in the solution of v. 8, in which he gives the result without clearly showing his mode of procedure in obtaining it ; though its separability into factors is nowhere expressly mentioned, and is not made use of in certain places where we should most naturally expect to find it, e.g. in iv. 12. 2. But ftxr more important than these identical formulae are the numerous propositions in the Theory of Numbers which we find stated or assumed as known in the Arithmetics. It is, in general, in explanation or extension of these that Fermat wrote his famous notes. So far as Diophantos is concerned it is extremely difficult, (jr rather impossible, to .say how far these 126 DIOPHANTOS OF ALEXANDRIA. propositions rested for him upon rigorous mathematical demon- stration, and how far, on the other hand, his knowledge of them was merely empirical and derived only from trial in particular cases, whereas he enunciates them or assumes them to hold in all possible cases. But the objection to assuming that Diophantos had a completely scientific system of investigating these propositions, as opposed to a merely empirical knowledge of them, on the ground that he does not prove them in the present treatise, would seem to apply equally to Fermat's o-svn theorems set forth in these notes, except in so far as we might be inclined to argue that Diophantos could not, in the period to which he belongs, have possessed such machinery of demon- stration as Format. Even supposing this to be true, we should be very careful in making assertions as to what the ancients could or could not prove, when we consider how much they did actually accomplish. And, secondly, as regards machinery of proof, we have seen that up to Fermat's time there had been very little advance upon Diophantos in the matter of notation. It will be best to enumerate here in order the principal propositions of this kind which we find in Diophantos, observing in each case any indication, which is perceptible, of the extent which we may suppose Diophantos' knowledge of the Theory of Numbers to have reached. It will be necessary and useful to refer to Fermat's notes occasionally. The question of the merits of Fermat's notes themselves this is not the place to inquire into. It is well known that he almost universally enunciates the theorems contained in these notes without proof, and gives as his reason for not inserting the proofs that his margin was too small, and so on. It is considered, however, that as his theorems are always true, he must necessarily have proved them rigorously. Concerning this statement I will only remark that in the note to v. 25 Format addresses himself to the solution of a problem which was " most difficult and had troubled him a long time," and says that he has at last found a general solution. Of this he gives a demonstration wliich is hopelessly wrong, and which vitiates the solution completely. THE PORISMS OF DIOPHANTOS. ] 27 (a) Theorems in Diophantos respecting the comjyosition of numbers as the sum of two squares. 1. Any square number can be resolved into two squares in any number of ivays, li. 8, 9. 2. Any number luhich is the sum of two squares can be resolved into two other squares in any number of ways, u. 10. N.B. It is implied throughout that the squares may be fractional, as well as integral. 3. If there are two whole numbers each of tuhich is the sum of tiuo squares, their product can be resolved into the sum of two squares in two ways, iii. 22. The object of ill. 22 is to find four rational right-angled triangles having the same hypotenuse. The method is this. Form two right-angled triangles from (a, b), (c, d) respectively, viz. a== + b\ a' - b\ 2ab, c' + d', c'-cr, 2cd. Multiplying all the sides of each by the hypotenuse of the other, we have two triangles having the same hypotenuse, {a' + h'){c'+d^), {a'-b'){c' + d?l 2ab{c' + d'), (a' + b') (c'+ cf), (a' + b') {& - d'), 2cd (a^ + b'). Two other triangles having the same hypotenuse are got by using the theorem enunciated, viz. (a' + ¥) (c' + d') = (ac ± bdf + (ad + be)', and the triangles are formed from ac ± bd, acl + be, being (a' + b') (c* + d'), 4abcd + (rr - b') (c' - d"), 2 (ac + bd) {ad - be), {a' + 6") {c' + d'), ^abcd - {a' - b') (c' - d'), 2 {ac - bd) {ad + be). In Diophantos' case a' + b' = V + 2"- = 5, d' + d'=2' + :i'=U; and the triangles are (65, 52, 39), (65, 60, 2o), (65, 63. 16), (65, c>i>, 33). 128 DIOPHANTOS OF ALEXANDRIA. [If certain relations hold between a, h, c, d this method fails. Diophantos has provided against them by taking two right- angled triangles viro iXax^o-Toov dptOfiwv (3, 4, 5), (5, 12, 13)]. Upon this problem Fermat remarks that (1) a prime number of the form ^n + 1 can only be the hypotenuse of a right-angled triangle in one way, the square of it in two ways, «&;c. (2) If a prime number made up of two squares be multiplied by another prime also made up of two squares, the product can be divided into two squares in two ways ; if the first is mul- tiplied by the square of the second, in three ways, &c. Now we observe that Diophantos has taken for the hypotenuse of the first two right-angled triangles the first tiuo prime numbers of the form 4n + 1, viz. 5 and 13, both of which numbers are the sum of two squares, and, in accordance with Format's remark, they can each be the hypotenuse of one single right-angled triangle only. It does not, of course, follow from this selection of 5 and 13 that Diophantos was acquainted with the theorem that every prime number of the form 4?i + 1 is the sum of two squares. But, when we remark that he multiplies 5 and 13 together and observes that the product can form the hypotenuse of a right-angled triangle in four ways, it is very hard to resist the conclusion that he was acquainted with the mathematical facts stated in Format's second remark on this problem. For clearly 65 is the smallest number which can be the hypotenuse of four rational right-angled triangles ; also Diophantos did not find out this fact simply by trijing all numbers up to G5 ; on the contrary he obtained it by multiplying together the first two prime numbers of the form 4?j-f- 1, in a perfectly scientific manner. This remarkable problem, then, serves to show pretty con- clusively that Diophantos bad considerable knowledge of the properties of numbers which arc the sum of two squares. 4. Still more remarkable is a condition of possibility of solution prefixed to the problem v. 12. The object of this problem is "to divide 1 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 very much THE PDKISMS (^F Dlnl'HANTOS. 1l>!» corrupted. There is no doubt, however, about the first few words, " The given number must not be odd." i.e. No number of the form 4/i-|- 3 [or 4« - 1] can be the sum of two squares. The text, however, of the latter half of the condition is, in Bachet's edition, in a hopeless state, and the point cannot be settled without a fresh consultation of the Mss.^ The true con- dition is given by Fermat thus. " The given number mu.st not be odd, and the double of it increased by one, when, divided by the greatest square which measures it, mu^st not be divisible by a pnme number of the form 4h— 1." (Note upon v. 12; also in a letter to Roberval). There is, of course, room for any number of conjectures as to what may have been Diophantos' words'. There would seem to be no doubt that in Diophantos' condition there was something about "double the number" (i.e. a number of the form 4n), also about "greater by unity" and "a prime number." From our data, then, it would appear that, if Dio- phantos did not succeed in giving the complete sufficient and necessary condition stated by Fermat, he must at all events have made a close approximation to it. 1 Bachet's test has bel 8r] tov diS6iJLevoi> n-qn irepiaaov iXvai, firire 6 BiirXafflui' avTov q fi^a. fjceL^ova ^r) fxipos 5 . fj fxerpdrai viro rot a°". s°". He also says that a Vatican ms. reads /nTjre 6 diirXajlujv avrov api9iJ.ov fioudSa d. fiell^ova ^XV M^pos Tiraprov, fj neTpelrai viro tov irpwrov apiO/j-ov. Neither does Xylander help us much. He frankly tells us that he cannot understand the jmssage. ' ' Imitari statueram bonos grammaticos hoc loco, quorum (ut aiunt) est multa nescirc. Ego vcru noscio hoic non multa, scd pacnc omnia. Quid enim (ut reliqua taccam) est /xrjTe 6 onrXaalijsv avrov ap no a, &c. quae causae liuius irpocoi.opi.dfiov, quae processus ? immo qui processus, quae operatic, quae solutio?" * Nesselmann discusses an attempt made by Schulz to correct the text, and himself suggests nrfre rbv 8nr\affiova avrov apiOnov /xovdSi fitl^ova fx^iv, 6s fie- rpeirai vir6 rivoi irpurov api.6y.ov. But this ignores /i^pos riraprov and is not satisfactory. Haukel, however (Gesch.d. Math. p. 169), says: "Ich zweifele nicht, dass die von den Msscr. arg entstellte Determination so zu lesen ist: Sei Si) rbv 5iW/i*ror Urire ntpicabv ehai, /J-rire rov dnrXacrlova ai/Tou apiOnbv fjLovdSi a fitl^ova fitrptiadcu iino TOV irpil)Tov apid/iov, 8s aj' /louadi d ixd^uv IxV t'-^po^ TirapTov." Now this cor- rection, which exactly gives Fermat's condition, seems a decidedly probable one. Here the words p.ipos rirapTov find a place; and, secondly, the rept'tition of liovaSi d nd^uv might well confuse a copyist, tov for tov is of course natural enough ; Nesselmann reads nvos for tov. H. I). i» 130 DIOPHANTOS OF ALEXANDRIA. We thus see (a) that Diophantos certainly knew that no number of the form 4??. + 3 could be the sum of two squares, and (b) that he had, at least, advanced a considerable way to- wards the discovery of the true condition of this problem, as quoted above from Fermat. (6) On numbers luhich are the sum of three squares. In the problem v. 14 a condition is stated by Diophantos respecting the form of a number which added to three parts of unity makes each of them a square. If a be this number, clearly 3a + 1 must be divisible into three squares. Respecting the number a Diophantos says "It must not be 2 or any multiple of 8 increased by 2." i. e. a number of the form 24>i + 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 3w + 1, 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/i. We must therefore credit Diophantos with the knowledge of the fact that no number of the form 8n + 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 871 + 7 are made up of three squares. Even Bachet remarked that the number a might not be of the form 32?i -)- 9, or a number of the form 9G>i+ 28 cannot be the sum of three squares. Fermat gives the conditions to which a must be subject thus: Write down two geometrical series (common ratio of each 4), the hrst and second series beginning respectively with 1, 8, 14 16 C4 256 1024 4096 8 32 128 512 2048 8192 32768 then a must not bo (1) any number obtained by taking twice liny term of tlu' ii[)per scries ami adding all the preceding terms, THE PORISMS OF DIOPHAN'TOS. l.'H or (2) the number found by adding to the numbers so obtained any multiple of the correspondino: term of the second series. Thus (a) must not be, 8?i + 2.1 = 8" + 2, 32/1 + 2.4 + 1 = 32« + 9, 128n + 2.1G + 4<+ 1 =128n + 87, ol2n + 2.64 + 16 + 4 + 1 = 512n + 149, &c. Again there are other problems, e.g. v. 22, in which, though conditions are necessary for the possibility of solution, none are mentioned; but suitable assumptions are tacitly made, without rules by which they must be guided. It does not follow from the omission to state such rules that Diophantos was ignorant of even the minutest points connected with them ; as however we have no definite statements, it is best to desist from specula- tion in cases of doubt. (c) Goinposition of naniberti an the sum of four squares. Every number is either a square or the sum of two, three or four squares. This Avell-known theorem, enunciated by Format in his note to Diophantos iv. 31, 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 Arithmetics as to how far Diophantos was acquainted with the properties of numbers as the sum of four squares. Un- fortunately it is impossible to decide this question with any- thing like certainty. There are three problems [iv. 31, 32 and V. 17] 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 Sipiares [v. 14 and 12] he does state a condition, we should probably be right in inferring that Diophantos was aware, at least empirically, if not scientifically, that any number could be divided into four squares. That he was able to prove the theorem scientifically it would be rash to assert, though it i.s not impossible. But wc 9—2 132 DIOPHANTOS OF ALEXANDRIA. may at least be certain that Diophantos 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 ex- pressed as squares, or as the sum of two, three or four squares in whole numbers. So much we may be sure that Diophantos could do, and hence he might have empirically satisfied himself that in any case occurring in practice 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. Here again we must be content, at least in our present state of knowledge of Greek mathematics, to remain in doubt. CHAPTER VII. HOW FAR WAS DIOPHANTOS ORIGINAL ? § 1. Of the many vexed questions relating to Diophantos none is more difficult to pronounce upon than that which we propose to discuss in the present chapter. Here, as in so many other cases, diametrically opposite views have been taken by au- thorities equally capable of judging as to the merits of the ca.se. Thus Bachet calls Diophantos "optimum praeclarissiniumque Lo- gisticae parentem," though possibly he means no more by this than what he afterwards says, "that he was the first algebraist of whom we know." Cossali quotes "T abate Andres" as the most thoroughgoing upholder of the originality of Diophantos. M. Tannery, however, whom we have before had occasion to men- tion, takes a completely opposite view, being entirely unwilling to credit Diophantos with being anything more than a learned compiler. Views intermediate between these extremes are those of Nicholas Saunderson, Cossali, Colebrooke and Nessel- mann; and we shall find that, so far as we are able to judge from the data before us, Saunderson's estimate is singularly good. He says in his Elements of Algebra (1740), "Diophantos is the first writer on Algebra we meet with among the ancients ; not that the invention of the art is particularly to be ascribed to him, for he has nowhere taught the fundamental rules and principles of Algebra; he treats it everywhere as an art already known, and seems to intend, not so much to teach, as to culti- vate and improve it, by applying it to certain indeterminate problems concerning square and cube numbers, right-angled triangles, &c., which till that time seemed to have been either not at all considered, or at least not regularly treateil c»f. These 134 DIOPHANTOS OF ALEXANDRIA. problems are very curious and ratertaining; but j^et in the resolution of them there frequently occur difficulties, which nothing less than the nicest and moc^ refined Algebra, applied with the utmost skill and judgment, could ever surmount: and most certain it is that, in this way, no man ever extended the limits of analytic art further than Diophantos has done, or dis- covered greater penetration and judgment; whether we consider his wonderful sagacity and peculiar artifice in forming such proper positions as the nature of the questions under considera- tion required, or the more than ordinary subtilty of his reason- ing upon them. Every particular problem puts us upon a new way of thinking, and furnishes a fresh vein of analytical treasure, which, considering the vast variety there is of them, cannot but be very instructive to the mind in conducting itself through almost all difficulties of this kind, wherever they occur." § 2. We will now, without anticipating our results further, proceed to consider the arguments for and against Diophantos' originality. But first we may dispose of the supposition that Greek algebra may have been derived from Arabia. This is rendered inconceivable by what we know of the state of learning in Ai'abia at different periods. Algebra cannot have been developed in Arabia at the time when Diophantos wrote ; the claim of Mohammed ibn Musa to be considered the first important Arabian algebraist, if not actually the first, is ap- parently not disputed. On the other hand Rodet has shown that M(jhammed ibn Musii was largely indebted to Greece. There is moreover great dissimilarity between Greek and Indian algebra ; this would seem to indicate that the two were evolved independently. We may also here dispose of Bombelli's strange statement that he found that Diophantos very often quoted Indian authors \ We do not find in Diophantos, as we have him, a single reference to any Indian author whatever. There is therefore some difficulty in understanding Bombelli's positive statement. It is at first sight a tempting hypothesis to suppose that the "frequent quotations" occurred in parts of Diophantos' ^ "Ed in detta opera abbiamo ritrovato, ch' egli assai volte cita gli autori indiani, col cho rui lia fatto conoscere, che questa disciplina appo gl' indiani prima lu che agli arabi." HOW FAR WAS DIOPHANTOS ORIGINAL? 135 work contained only in the MS. which Bombclli used. But wo know that not a single Indian author is mentioned in that MS. We can only explain the remark by supposing that Bombelli confused the text and the scholia of Maximus Planudes ; for in the latter mention is made of an " Indian method of multiplica- tion." Such must be considered the meagre foundation for Bombelli's statement. There is not, then, mucli doubt that, if we are to find any writers on algebra earlier than Diophantos to whom he was indebted, we must seek for them among his own countrymen. § 3. Let us now consider the indications bearing upon the present question which are to be found in Diophantos' own work. Distinct allusions to previous writers there are none with the sole exception of the two references to Hypsikles which occur in the fragment on Polygonal Numbers. These references, how- ever, are of little or no importance as affecting the question of Diophantos' originality; for, so far as they show anything, they show that Diophantos was far in advance of Hypsikles in his treatment of polygonal numbers. And, so far as we can judge of the progress which had been made in their theoretical treat- ment by writers anterior to Diophantos from what we know of such arithmeticians as Nikomachos and Theon of Smyrna, we must conclude that (even if we assume that the missing part of Diophantos' tract on this subject was insignificant as compared with the portion which has survived) Diophantos made a great step in advance of his predecessors. His method of dealing with polygonal numbers is new ; and we look in vain among his precursors for equally general propositions with regard to such numbers or for equally scientific proofs of known pro- perties. Not that previous arithmeticians were uuaccjuaiuted with Diophantos' propositions as applied to particular jwlygonal numbers, and even as applicable generally ; but of their general application they convinced themselves only empirically, and by the successive evolution of higher and higher orders of sucli numbers. We may here remark, with respect to the term "arithmetic" which Diophantos applies to his whole work, that he is making a new use of the term. According to the previously ucc.-j»t.-< for Heron no more than a matter of arithmetical calcnlation. He solved such e, u ^eive, pArp-qffov, povTlb' eiricrTrjcras, el /xer^x^'J ffO(pir]s, iroffffij dp' iv ireStois HiKeXrjs ttot' (^6ffKeTo vrjffov Qpivad-qs, rerpaxv (rri^ea Saffffafi^v-q Xpo^V" oXXaaaovra' t6 iih XevKoio yaXaKTos, Kvav4(i> 8' irepov xpw/toTt Xa/iirdfievov, dXXoye /xiv ^avObf, t6 di iroiKlXoy. 'Ev 5e eKaartf) aricpet iv TaipoLS irda'r)% eh vofiov ipxop^vris. Savdorpix^^y dyeXijs W/urrT(f> fi^pei rjoi Kai <\T dpiOp-bv, Xwpis 5' oB drjXeiai Saai Kari xpoidv iKacTai, oiiK d'idpb Ke Xiyoi, oi)5' dpi0p.Qv dSa^i, oil pirjv iribye (, iravra fiirpa. ^PX^o Kv5i6o}v inKri(p6po%' taOi Tf iravTUii KCKpifi^fot rairrxi uinrvLO% cp ffocply. now FAR WAS DIOl'HANTOS OHKMNAL? U.', date of the epigram. If the " Cattle-problem " of which we arc now speaking were really due to Archimedes, we should, sup- posing M. 'i'aunery's theory to hold good, scarcely have found the result in such glaring contradiction to what cannot but have been the facts of the case, Nesselmann further argues in ftivour of his view by pointing out (1) that the problem is clearly at an end, when it is said that he who solves the problem must be not unskilled in numbers, i.e. where I have shown the division ; and the addition of two new conditions with the preface "And yet he could not pretend to proficiency in wise calculations" unless he could solve the rest, shows the marks of the interpolator on the face of it, and, moreover, of a clumsy interpolator who could neither solve the complete problem itself, nor even conceal his patchwork. (2) The lan- guage and versification are against the authenticity. (3) The Scholiast's solution does not, as it claims, satisfy the whole problem, but only the first part. (4) The impossibility of solution with the Greek numeral notation and the absurdly large numbers show that the author, or authors, could not have seen what the effect of the many heterogeneous conditions would be. Nesselmann draws the conclusion above stated ; and we may safely assume, as he says, that this ej)igram is from the historical point of view worthless, and could not, even if it were shown to be earlier than the date of Diophantos, be held to prove anything against his originality. M. Tannery takes the opposite view and uses the epigram for the express purpose of proving his assumption that Dio- phantos was not an original writer. M. Tannery takes a passage attributed to Geminos in which he is describing the distinction between XoytariKi] and apLOfirjTiKt'j. XoyiariKi] according to Geminos dewpet to /zef KXijOki' vtt \\pxiM^ov., the Arabic text of which, together with a Persian commentary by Roshan Ali, was printed at Calcutta (IHTi, Svo), the following explanation is given : ' The side (of the e3 of the supposition as to the origin of the rules used by Diophan- tos, which was mentioned above (pp. 140, 141), and we may pro- perly conclude that algebra, as we find it in Diophantos, was the result of a continuous development which extended from the time of Euclid to that of Heron and of Diophantos, and was independent of external influences. I now pass to the consideration of the actual rules which Mohammed ibn Mfisii gives for the solution of the complete quadratic, as compared with those of Diophantos. We remarked above (p. 91) that Diophantos would appear, when solving the equation ax' -\- bx = c, to have first multiplied by a throughout, so as to make the first term a square, and that he would, with b our notation, have given the root in the form — — . Mohammed ibn Musa, however, first divides by a throughout : " The solution is the same when two squares or three, or more or less, be specified ; you reduce them to one single square and in the same proportion you reduce also the roots and simple numbers which are connected therewith \" This discrepancy between the Greek and the Arabian algebraist is not a very striking or important one; but it is worth while to observe that Mohammed ibn Musa's rule is not the early Indian one ; for Brahmagupta (born 598) sometimes multiplies throughout by a like Diophantos, sometimes by 4a, which was also the regular practice of (^'rldhara, who thus obtained the root in the form . This rule of Crldhara's is quoted and followed 2a ' by Bhaskara. Another apparent discrepancy between Moham- med ibn Musa and Diophantos lies in the fact that Diophantos never shows any sign, in his book as we have it, of recognising two roots of a quadratic, even where both roots are positive and real, and not only when one of them is negative: a negative or irrational value he would, of course, not recognise ; unless an equation has a real positive root it is for Diophantos "impossible." Negative and irrational roots appear to be tacitly ])ut aside by ' Rosen, The Alr) for dpi0/x6s. The second of these objections asserted that it is of very rare occurrence, and is not found in the >iss. of Nikomachos and Pappos, where it might most naturally be expected. In reply to this, I pointed out that it was not in the least necessary for my theory that it should occur anywhere except in Diophantos ; and accordingly I did not raise the question whether the symbol was found in mss. so rarely as Mr Gow appears to suppose. Since then I have thought that it would be interesting to inquire into this point a little further, without, however, going too far afield. ■\;\liile reading Heiberg's Quaestiones Archimeikae in connection with the Cattle- problem discussed in chapter viii. it occurred to me that the symbol for apiOnos would be likely to be found, if anywhere, in the mss. of the De areiiae mimero Ubclhts of Archimedes, which Heiberg gives at the end of the book, and that, if it did so occur, Heiberg's textual criticisms would i^lace the matter beyond doubt, without the necessity of actually collating the mss. My expectation proved to be fully justified ; for it is quite clear that the symbol occun-cd in the MSS. of this work of Archimedes rather frequently, and that its form had given rise to exactly the same confusion and doubt as in the case of Diophantos. I will here give references to the places where it undoubtedly occurred. See the following pages in Heiberg's book, p. 172. p. 174. Heiberg reads dpL6fwv, with the remark "/cat omnes." But the similarity of the signs for api6fi6s and Kai is well known, and it could hardly be anything else than this similarity which could cause such a difference of readings. p. 187. Heiberg's remark " apiOfxiov om. codd. Bas. E ; excidit ante s (Kai)" speaks for itself. Also on the same page "dpiOnwu] si FBC." p. 188. js three times for dpiBixuv. p. 191. Here there is a confusion between 5" (six) and dpiOfxis, where Heiberg remarks, "Error ortus est ex compendio illo uevhi dpiOfios, de quo dixi ad I, 3." p. 192. iXaTTuv and dpid/xos given as alternative readings, with the obser- vation, " Confusa sunt compendia." Thus it is clear that the symbol in question occurs tolerably often in the mss. of another arithmetical treatise, and that the only one which I have investigated in this connection : a fact whicli certainly does not support Mr Gow's statement that it is veiy rarely found. APPENDIX. ABSTRACT OF THE ARITHMETICS AND THE TRACT ON POLYGONAL NUMBERS. H. D. u J DIOPHANTOS. ARITHMETICS. BOOK I. Introduction addressed to Dionjjsios. Definitions. 1. "Square" and "side," "cube," "square-square," etc. 2. " Power." Notation 8", k", 88", Sk", kk", /a", c^. 3. Corresponding fractions, the reciprocals of the foi-mer ; names used corresponding to the " nimibers." 4. "]S'umber"x"Number"= square. Square x square = "squarc- squai-e," &c. 5. " Number " x corresponding fraction = unit (/xoi-ds). 6. "Species" not changed by multiplication with monads. * I Reciprocal x reciprocal - reciprocal scpiare, etc. 9. Minus multiplied by minus gives j^lus. Notation for minus, /p. 10. Division. Remark on familiarity with processes. 11. Simplification of equations. Frohlenis. 1. Divide a given number into two having a given difference. Given nund)er 100, given difference 10. Lesser number required x. Therefore 2.«+ 40-100, X - 30. The required numbers an- 70, 30. 2. To divide a given number into two having a given nitio. Given number GO, given ratio ."3 : 1. Two numbers x, 2>x. Therefore ./j- ir». Till" numbers arc 45, 15. 11—2 164 DIOPHANTOS OF ALEXANDRIA, 3. To divide a given number into two having a given ratio and diffei-ence '. Given number 80 ; ratio 3:1; diflference 4. Smaller number x. Therefore the larger is 3x + 4, x = 1 9. The numbers are 61, 19. 4. Find two numbers in a given ratio, their difference also being given. Given ratio 5:1. Diffei'ence 20. Numbers 5x, x. Therefore x-6, and the numbers are 25, 5. 5. To divide a given number into two such tliat the sum of given fractions (not the same) of each is a given number. Necessary condition. The latter given number must lie between the numbers arising when the given fractions are taken of the first given number. First given number 100, given fractions ;^ and -, given sura 30. Second part 5a-. Therefore first part = 3 (30 - x). Therefore 90 + 2a; = 100, x = 5. The required parts are 75, 25. 6. To divide a given number into two parts, such that a given fraction of one exceeds a given fraction of the other by a given difierence. 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 ; fractions - and -respectively : excess 20. 4 G Second part Gx Therefore 10.v+80=100, x = 2, and the parts are 88, 12. 7. From the same (rcqiiired) number to take away two given numbers, so that the remainders are in a given ratio. Given numbers 100, 20; ratio 3:1. X required number. Therefore a;- 100 : a- 20 = 1 : 3, .«;^ 140. 8. To two given numbers to add tlie same (required) number, so that the sums ai-e in a given ratio. 1 By thiH Diojihantos means " such that one is so many times the other ylm a given number." ARITHMETICS. BOOK I. 165 Cotulilion. This ratio must bo less tlmu that of the greater given number to tlie smaller. Given numbers 100, 20, given ratio 3:1. X required number. Therefore 3a; + 60 = a: +100, and x=20, 9. From two given numbers to subtract the same (required) one so that the two remainders are in a given ratio. Condition. This ratio must be greater than that uf the greater given number to the smallei'. Given numbers 20, 100, ratio 6 : 1. X required number. Therefore 1 20 - 6x- = 100 - X, and a; - 4. 10. Given two numbers, to add the same (required) number to the smaller, and subtract it from the lai-ger, so that the sum in the first case may have to the difference in the second a given ratio. Given numbers 20, 100, given ratio 4:1. X required number. Therefore 20 + x- : 100-cc = 4 : 1, and a; = 76. 11. Of two given numbers to add the first to, and subtract the second from, the same (required) number, so that the numbers which arise may have a given ratio. Given numbers 20, 100 respectively, ratio 3:1. X required number. Therefore 3a; - 300 = a; + 20, and x = 160. 12. To divide a given number twice into two parts, such that the fii-st of the first pair may have to the first of the second a given ratio, and also the second of the first pnir to the second of the second another given ratio. Given number 100, ratio of greater of fii-st parts to less of second 2:1, ratio of greater of second parts to less of first 3:1. X smaller of second parts. The parts then are , J"" . 1 'int^ ^^^ ~ ^■" [ • Therefore 300 - 5x- = 1 00, x - 40, 100- 2a; j X J and the parts are (80, 20), (60, 40). 13. To divide a given number thrice into two parts, such that one of the first parts and one of the second pai-ts, the other of the second 166 DIOPUANTOS OF ALEXANDRIA. parts and one of the third parts, the other of the tliird parts and the remaining one of the first parts, are respectively in given ratios. Given number 100, ratio of gi-eater of first parts to less of second 3 : 1, of greater of second to less of third 2:1, and of greater of third to less of first 4:1. X smaller of third parts. Therefore greater of second = 2x, less of second = 100 - 2x, greater of first = 300 - Gx. Therefore less of first = Gx- 200. Hence greater of third = 2ix - 800. Therefore 25^-800 = 100, a; =36, and the respective divisions are (84, 16), (72, 28), (64, 36). li. To find two numbers such that their product has to their sum a given ratio. [One is arbitrarily assumed subject to the] Condition. The assumed value of one of the two must be greater than the numerator of the ratio [the denominator being 1]. Ratio 3:1. x one number, the other 12 ( > 3). Therefore 12a; = 3a; +30, a; = 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, ratio being then 2:1, and the second 50 from the first, ratio being then 3:1. cc 4- 30 the second. Therefore the first = 2x - 30, and a; + 80 : 2a;- 80=3 : 1. Therefore x = 64, and the numbers arc 98, 94. 16. To find three numbers such that the sums of each pair are given numbers. Condition. Half the sum of all must be greater than any one singly. Let (1) + (2) = 20, (2) + (3) = 30, (3) + (l) = 40. x the sum of the three. Therefore the numbers are X - 30, X - 40, X - 20. Hence the sum x = 3x' - 90. Therefore a; = 45, and the numbers ai-e 15, 5, 25. 17. To find four numbers such that the sums of all sets of three are given. Condition. One third of the sum of all must be greater than any one singly. Sums of threes 22, 24, 27, 20. J ARITHMETICS. HOOK I. 107 X the sum of all four. Therefore the mimhera uro .T-22, a; -24, a; -27, a; -20. Therefore 4x- - 93 = x, .r = 31 , and the numbers are 9, 7, 4, 11. 18. To find three numbers such that the sum of any puir exceeds the third by a given number. Given excesses 20, 30, 40. 2x sum of all, x = 45. The numbers arc 25, 35, 30. 19. [A different solution of the foregoing problem.] 20. To find four numbers such that the sum of any three exceeds the fourth by a given number. Condition. Half the sum of the four given dilterences must be greater than any one of them. Given differences 20, 30, 40, 50. 2a; the sum of the four required numbers. Tlierefoi-c the numbers are a;- 10, a;- 15, x-20, x-2b. Therefore 4x- 70 = 2.f, and .^•= 35. Therefore the numbers are 25, 20, 15, 10. 21. [Another solution of the foregoing.] 22. To divide a given number into three, such that the sum of each extreme and the mean has to the remaining extreme a given ratio. Given number 100 i (1) + (2) = 3 . (3), (2) + (3) 4.(1). X the third. Hence the sum of first and second = 3a-. There- fore 4a; =100. X = 25, and the sum of the first two = 75. y the first '. Therefore (2) + (3) = 4v/. Therefore by = 100, y = 20. The required parts are 20, 55, 25. 23. 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 the same given fraction of the greatest, but the least exceeds the same given fraction of the middle number by a given number. 1 As already remarked on pp. 80, 81, Diophantos docs not uso a second syllable for the uuknowu, but uses dpiO/xos for the second oiHiratiou as well ob for the lirtit. 168 TJlOPHANTOS OF ALEXANDRIA. Condition. The middle number must exceed the least by such a fraction of the greatest, that if its denomiuator be multiplied into the excess of the middle number over the least, the result is gi-eater than the middle number. Greatest exceeds middle by ^ of least, middle exceeds least by o of greatest, least exceeds ^ of middle by 10. aj + 10 the least. Therefore middle = 3aj, greatest = 6a; -30. Therefore x- 12^, and the numbers are 45, 37^, 22|^. 24. [Another solution of the foregoing.] 25. 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. 5 Assume the second to be a number divisible by 4, say 4. 3a; the first, and x = '2. The numbers are 6, 4, 5. 26. 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, - , -, ^ being the other fractions. Assume the second to be a multiple of 4, say 4. 3a; the first. The second after giving and taking becomes a; + 3. Therefore first after giving x to second and receiving of fourth = a; + 3. Therefore fourth =18- 6a;. And fourth after giving 3 - a; to first and receiving r^ of third = a; + 3. Therefore third = 30a; - 60. Lastly the third after giving 6.x- - 1 2 to fourth and receiving 1 from second = a; + 3. Therefore 50 24a; - 47 - a; + 3, a; - ^o • ARITHMETICS. ]J0OK I. 100 m, I- *i 1 1^0 , 120 111 Therefore the numbers are ^ , 4, -^ * ~ok ' or, multiplying by the common denominator, 150, 92, 120, 114. 27. To find three numbers such that, if each receives a given fraction of the sum of the other two, the results are all equal. The fractions being , -r, _ , the sum of the second and third is assiimed to be 3, and x put for the first. The numbers are, after multiplying by a common denominator, 13, 17, 19. 28. To find four numbers such that, if each receives a given fraction of the sum of the remaining three, the four results are eqiial. The e;iven fractions being 77 , 7 , ^ , ;r , we a^suvie the sum of ° ° 3 4 6 the last three numbers to be 3. Putting X for the first, Diophantos finds in like manner that numbers are 47, 77, 92, 101. 29. Given two numbers, to find a third which, when multiplied by each successively, makes one product a square and the other the side of that square. Given numbers 200, 5. X required number, 200a; = (oxY, a; = 8. 30. To find two numbers ivhose sum and whose product are given. Condition. The square of ludf the sum 7nust exceed tlie product by a square number, Iuti Se tovto irXaaixaTiKov '. Given sum 20, product 9G. 2x the difierence of the required numbers. Therefore numbers are 10 + x, 10 -x. Hence 100-aj'=96. Therefore x ^ 2, and the difierence = 4. The required numbei-s are 12, 8. 1 There has been much controversy as to the moaninR of this diflicult phrase. Xylancler, the author of the Schoha, Bachct, Cossali, Schulz, NcbhoI- mann, all discuss it. As I do not profess here to bo commcntinR on the text I shall uot criticise their respective views, hut ouly remark tliat I think it is best to take TrXaff/xariKov in a passive sense. "And this condition can (easily) be formed," i.e. can be investigated (and shown to bo tnu), or 05 " 11. To find two square numbers diflWing by a t/iven ntimber. Given difference GO. Side of one number x, side of the other x phis any number whose square < GO, say 3. Therefore (x- + 3)^' - a;' = 60, and the required .squares are 72], 132 j. » Diophantos' words are: "I form the s(iuare from any number of apiBnol minus as many units as are contained in the side of U)." The prt-eftution im- plied throuRhout in the choice of m is that we must assume it so tli-ii tli. r. suli may be rational in Diophantos' sense, i.e. rational and positive. 174 DIOPHANTOS OF ALEXANDRIA. 12. To add such a number to each of two given numbers that the results shall both be squares. (1) Given numbers 2, 3, required number x. x + 2) Therefore ^M^^^^st each be squares. x+ o) ^ This is called a double-equation. To solve it, take the difference between them, and resolve it into tioo factors^ : in this case say 4 and -. . Then take either (a) the square of half the difference bettveen these factors and equate it to the smaller expression, or (b) the sqriare of half the sum and equate it to tlie larger. 225 In this case (a) the square of half the diflference = -^ . m r o 225 - 97 Iheretorc x+ ^= yrr > ^^^ ^ — wr, o4 d4 ... ^. 225 289 Avhilc the squares are -rr- , -ttt • (2) In order to avoid a double-equation, First find a number which added to 2 gives a square, say x^ — 2. Therefore, since the same number added to 3 gives a square, x' + 1= square = {x- 4y say, the absolute term (in this case 4) being so chosen that the solution may give x^>2. 15 Therefore x=-^ , 97 and the required number is ^ , as before. 13. Fro77i two given numbers to take the same (required) number so tluit both the remainders are squares. Given numbers 9, 21. Assuming 9 - x^ as the required number we satisfy one condi- tion, and it remains that 12 + a;" = a square. Assume as the side of this square x minus some number whose square > 12, say 4. 1 Wc must, as usual, choose suitable factors, i.e. such as will give a "ra- tioual" result. This must always be premised. ARITHMETICS. HOOK II. 175 Therefore (x - 4)^ = 1 2 + x\ 1 ^ = 2' and the required number is 8|. 14. Fro7n the same {required) number to subtract sui-cessiv:/;/ two given munbers so that the remainders may both be squares. 6, 7 the given numbers. Tlien (1) let cc be required number. Tlierefore cb - G~| , , ^ \ are both squares. The difference = 1, which is the product of 2 and .^ ; and, l)y the rule for solving a double-equation, 121 (2) To avoid a double-equation seek a number which exceeds a square by 6, i. e. let a;" + G be the required number. Therefore also x- - 1 : square = {x- 2)' say. Hence ^ ~ I ' and the number required = tr • 15. To divide a given number into tioo ])arts, and tojind a square number tohich when added to either of the two parts gives a square number. Given number 20. Take two numbci-s the sum of whose squares < 20, say 2, 3. Add x to each and square. "We then have x^ + 4a; + 41 X' + Gx + 'd) ' and if 4.x- + 4]^ 6a; + 9j are respectively subtracted the remaindei-s are the same s<|uiire. Let then x- be the square required, and therefore 4a; + 4"| Gx + Oj the required parts of 20. Then 10X-+13-20, and X ^ iV, . 176 DIOPHANTOS OF ALEXANDRIA. (68 132\ iTi ' ~i7r ) ' 49 and the required square rrrrr . 16. To divide a given number into two parts and find a square which exceeds either part by a square. Given number 20. Take (x + rtif for the required square, where m' < 20, i.e. let (x + 2Y be the required sqi;are (say). This leaves a square if either 4a; + 4) . , ^ ^ , ^ ^ „ ?■ 1*5 subtracted, or 2.r + 3J Let these be the parts of 20, and x=-7, . b /76 44\ Therefore the parts required are (-w, /. ) > .625 and the required square is -^^ . ob 17. Find two numbers in a given ratio such that cither together with an assigned square produces a square. Assigned square 9, ratio 3:1. If we take a square whose side is mx + 3 and subtract 9 from it, the remainder will be one of the numbers required. Take e. g. (x + 3)^ - 9 = cc^ + 6a; for the smaller number. Therefore 3a;* + 18a; = the larger number, and 3x* + 18a; + 9 must be made a square - (2.x-- 3)" say. Therefore x = 30, and the required numbers are 1080, 3240. 18. 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 equal. First gives to second - of itself + 6, second to third ^ of itself i> b + 7, third to first = of itself + 8. Assume that the first two are 5x, Gx [equivalent to one con- ,.,. -, , ,. , ,, , ^ - 90 108 105 ditionj, aiul we find the numbei"s to be - , - , -=- . ARITHMETICS. BOOK H. 177 19. Divide a number into throe parts .sati-sfyinfj tlio conditions of tlie preceding problem. Given number 80. First gives to second p of itself + G ic, and results are equal. [Diophantos a.ssumes 5x, 12 for tlie first two nunilM-rs, and his ,^ . 170 228 217 , result IS -y^- , — , — ; but the solution does not cor- respond to the question.] (See p. 2.5.) 20. To find three squares such that the difference of the great- est and the second is to the diffei-ence of the .second and the lea.st in a given ratio. Given ratio 3:1. Assume the least square ^- .r^, the middle = .r* -I- 2x -f 1. Therefore the greatest = x° + 8x + 4 = square = {x + SYsay. Therefore x ^ '■ , and the squares are 30|, 12^, 6;^. 21. To find two numbers such that the square of either added to the other number is a square. X, 2a; -1- 1 are assumed, which by their form satisfy one con- dition. The other condition gives 4x- + 5x + \ ^ square = (2.c - 2)" say. 3 Therefore ,-« = --, 3 19 and the numbers are :^ , ^^ . Id lo 22. To find two numbers such that the square of eitlier utitntu the other number is a square. a; -f- 1 , 2.r -f- 1 are assumed, satisfying one condition. Therefore 4.7;^ -i- 3x - square - 9,r* say. 3 Therefore -^ = ? > and the numbers are ^ . c • 5 23. To find two numbers such that the sum of the square of either and the sum of both is a square. Assume x, x+l for the numbers. These satisfy one condi- tion. H. D. 12 178 DIOPnANTOS OF ALEXANDRIA. Also a;* + ix + 2 must be a square = {x - 2)* say. Therefore x = -, . 4 1 5 Hence the numbers are t , ^ . 4 4 24. To find two numbers such that the clifTerence of the square of either and the sum of both is a square. Assume x + 1, x for the numbers, and we must have a;^ - 2a; - 1 a square = {x- 3)- say. Therefore x=^, and the nvimbei's are 3|, 2^. 25. To find two numbers such that the sum of either and the square of their sura is a square. Since 03" + 3x^, x^ + 8x' are squares, let the numbers be 3x-, Sx- and their sum x. Therefore 1 Icc^ = a; and a; = yy . 3 8 Therefore the numbers are y^ , ynT • 26. To find two nunibers such that the difference of the square of the sum of both and either number is a square. If we subtract 7, 12 from 16 we get squares. Assume then 12a;', 7x^ for the numbers, 1 6a;" = square of sum. 4 Therefore Idoif = ix, x = ^1) f 192 112 and the numbers are ^„^ , ^r^^ . obi obi 27. To find two numbers such that the sum of either and their product is a square, and the sum of the sides of the two squan s so arising equal to a given number, 6 suppose. Since x (4a; - 1) + a; = square, let a;, 4a; - 1 be the numbers. Therefore 4a;* + 3a; - 1 is a square, whose side is 6 - 2a;. 37 Therefore x = --, in 1 37 121 and the numbers are - , . ^1 til 28. To find two numbers sndi that tlu; difference of their pro- duct and either is a square, and the sum of the sides of the two squares so arising equal to a number, 5. ARITIIMKTICS. ROOK II. 17!» Assume 4a; + 1, x for tlie nuinheis, wliicli tlicit.'foie satisfy on*- condition. Also 4.V* - 3x- - 1 - {') - 2, 16 23 37 and the numbers are n , -q- , -q • 35. To find three numbers such that, if the square of any one be added to the sum of all, the result is a square. /7>i-n\ ^ ^^^^ .^ ^ square. Take a number soparublo into ARITHMETICS. BOOK 111. IM two factors (m, u) in three ways, say 12, whicli is tlio pro duct of (1, 12), (2, 6), (3, 4). The values then of — ^ — are 5.\, 2, - . Let now o\x, 2x, ^x be the numbers. Their sum is 12.r'. Therefore So; = 1 2.t*, a: = !^ , and the numbei-s are - , ^ , q • O ij and the numbers are 9, 328, 73. i^^)'- 186 DIOPIIANTOS OF ALEXANDRIA. 17. To find three numbers such that the product of any two added to the sum of those two gives a square. Leimna. The squares of two consecutive numbers liave this property. Let 4, 9 be two of the numbers, x the third. Therefore 10a; + 9) , , , , I must both be squares, 5a; + 4 j ^ and tlie difference = 5a; + 5 = 5 (a; + 1). Therefore by Book ii., 10a: + 9 and a; =28, and (4, 9, 28) is a solution. 18. \_Another solution of the foo'egoinff problem.] Assume the first to be x, the second 3. Therefore 4a; + 3 = square = 2-5 say, whence a; = 5^, and 5^, 3 satisfy one condition. Let the third be a;, 5^ and 3 being the first two. Therefore 4a; + 3 ) , , ., , -, _, > must both be squares, C^a; + 5iJ ^ bid, since the copfficients in one expression are both greater than those in the other, but neither of the ratios of corre- sponding ones is that of a square to a square, our method will not solve them. Hence (to replace 51, 3) we must find two numbers such that their product + their sum = square, and the ratio of the numbers each increased by 1 is the ratio of a square to a square. Let them be y and 4?/ + 3, which satisfy the latter con- dition ; and so that product + sum = square we must have 4y^ + 81/ + 3 - square = (2^ - 3)'', say. 3 Therefore y = -- 3 Assume now ^tt, 45, a; for the numbers. Therefore oja; + 4M 13x 3 \- ave both squares. To" "^ 10) or 130a; +1051 , . 130X+ 30J -- 1-^1-^"--^' ARITHMETICS. BOOK III. 187 the ditlerence = 75, -wliicli has two factors 3 ami 2"), 7 and X - :^A gives a solution, 3 7 the numbers being , 4J, . 19. To fniil three numbers siu-li tliat the iiroduct of any two exceeds the sum of those two by a square. Put first = X, second any number, and we fall into the same difficulty as in the preceding. We have to find two numbers such that their product minus their sum = square, and when each is diminished by one they have the ratio of squares. 4?/+l, y+l satisfy the latter condition, and it/- - 1 - square -{2i/ — 2)' say. 5 Therefore !/ = :>• c 13 28 Assume then as the numbers -^- , ~, x. b b Therefore 2}^x-3U 5 , _ ^ /• are both squares, 8 ''*'' ~ 8 ) or lOx-U] . ^. 10a;-2Gj "'"^ 'l"'^'"^^' the difterence = 12 = 2 .G, and x =^ 3 is a solution. 13 The numbers are -^ , 3h, 3. o 20. To find two numbers such that their product added to both or to cither gives a square. Assume x, ix - 1 , since x{ix- 1) + x = ix^ = square. Therefore also 4a,-' + 3.x- - 1 ) , ^, , > arc both squares, 4x^ + ix-l) the diflference = x = 4a; , 7 , 4 and X = ,j^ gives a solution. 21. To find two numbers such that the product exceeds tho sum of both, and also cither severally, V)y a stjuare. Assume x + 1, ix, since 4a; (x- + 1 ) - 4a; = scpiare. 188 DIOPIIANTOS OF ALEXANDRIA. Therefore also 4a;* + 3.« - 1 ~) , , 2 'j- are both squares, the difference = ix = 4x. 1. Therefore a;=l], and (2|, 5) is a solution. 22. To find four numbers such that, if we take the square of the sum ± any one singly, all the resulting numbers are squares. Since in a rational right-angled triangle square on hypotenuse = squares on sides, square on hypotenuse =*= twice product of sides = square. Therefore we must find a square which will admit of division into two squares in four ways. Take the right-angled triangles (3, 4, 5), (5, 12, 13). Multiply the sides of the first by the hypotenuse of the second and vice versa. Therefore we have the triangles (39, 52, 65), (25, 60, 65). Thus 65^ is split up into two squares in two ways. Also G5 = 7' + 4' = 8^-hP. Therefore 65^ = {T - ^J + 4 . 7^ 4" = (8^ - 1')^ + 4 . 8M* . = 33'' + 56^= 63* +16', which gives two more ways. Thus 65* is split into two squares in four ways. Assume now as the sum of the numbers 65a;, first number = 2 .39 . 52a;* = 4056a;*^ second „ =2 25 . 60x* = 3000a;* third „ =2.33. 56a;* = 3696a;* i fourth „ =2.16.63.«*=2016a;*j 65 Therefore 12768x-* = 65a; and x=i21Q9>' and the numbers arc found, viz. 17136600 12675000 15615600 8517600 163021824' 163021824' 163021824' 163021824" 23. To divide a given number into two j)arts, and to find a sijuare which exceeds either of the parts by a scpiare. Let the given number be 10, and the square x" + 1x + 1. Put one of the parts 2.'b+1, the other 4a;. Therefore the conditions are satisfied if fix- -I- 1 = 10. and the sum = 12768x* Tlierefore ARITHMETICS. HOOK IV. 189 3 X- and the parts are 6, 4, the square 6]. 24. To divide a given number into two parts, and to find ii square which added to either of the parts produces a square. Given number 20. Let the square be x^ + 2x+\. Tliis is a square if we add 2a; + 3 or 4a; + 8. Therefore, if these ai'e the parts, the conditions are .satisfied when 6a; + 1 1 = 20, or a; ~ H. Therefore the numbers into which 20 is divided an- (6, 14) and the required square is 6^. BOOK IV. 1. To divide a given number into two cubes, such that the sum of their sides is a given number. Given number 370, sum of sides 10. Sides of cubes 5 + x,5-x. Therefore 30x' 4 250 = 370, x = 2, and the cubes are 7^, 3^. 2. To find two numbers whose difference is given, and also the difference of their cubes. Difference 6. Difference of cubes 504. Let the numbcra be- a; + 3, a; - 3. Therefore 1 8a;* + 54 = 504. Tlierefore a;' = 25, a; = 5, and the sides of the cubes are 8, 2. 3. A number multiplied into a square and its side makes the latter product a cube of which the foi-mer product is the side; to find the square. Let the square be a;". Therefore its side is x, and let tho Q number be - . Hence the products are 8a;, 8, and {Sxy = rr,, . . 1 1 Therefore a; = ^ , ^ = 7- 190 DIOPnANTOS OF ALEXANDRIA. 4. To add the same number to a square and its side and make them the same, [i.e. make the first product a square of which the second product is side]'. Square cc^, whose side is x. Let the number added to x' be such as to make a square, say 3a;". Therefore Zx" + x = side of 4x° = 2x and x = ^. The square is 3 and the number is - . 5. To add the same number to a square and its side and make them the opposite. Square x^, the number ix° - x. Hence 5a;^ -x = side of 4a;" = 2.r, and x--=. 6. To add the same square number to a square and a cube and make them the same. Let the cube be o;^ and the square any square number of x^'s,, say Ox--. Add to the square 16a;^ (The 16 is arrived at by taking two factors of 9, say 1 and 9, subtracting them, halving the remainder and squaring.) 1 o Therefore x^ + 16.«" = cube = S.c'' suppose and a; = — . Whence the numbers are known. 7. Add to a cube and a square the same square and make them the opposite. [Call the cube (1), first square (2), and the added square (3)]. Now suppose (2) + (3) = (1) [since (2) + (3) = a cube\. Now a' + y^2ah is a square. Suppose then {\) =a- + 1/, (3) = 2ah. But (3) must be a square. Therefoi-e 2ab must be a square ; hence we put « = 1, & = 2. Tlius suppose (1) = 5x^, (3) = 4,r-, (2) = x^. Now (1) is a cube. Therefore a? - 5, and (1)- 125, (2) = 25, (3) = 100. ' 111 this aud the following enunciations I have kept closely to the Greek, partly for the purpose of showing Diophautos' mode of expression, and partly for the brevity gained thereby. "To make them the same" means in the case of -1 what I have put in brackets; "to make them the opposite" means to make the first product a side of which the second product is the square. ARITHMETICS. BOOK IV. 191 8. [Another solution of the foregoing.] Since (2) + (3) = (l), a cube, and (1) + (3) = .square, I havn to find two squares whose .sum + one of thoni - a S(iuan', and whose sum = (l). Let the fii-st square be ar*, the second 4. Therefore 2.r + 4 = a square = (2^; - 2)' say. Therefore x = i, and the squares are IG, 4. Assume now (2) = ix\ (3) = IGx'. Therefore 20a;' = a cube, and x = 20, thus (8000, 1600, 6400) is a solution. 9. To add the same number to a cidje and its side, and make them the same. Added number x, cube 8x-^, say. Therefore second sum = 3a;, and this must be the side of cube Sx^ + x, or 8a;^ + a; = 27x'. Therefore 19x' = a;. But 19 is not a square. Hence we must find a square to replace it. Kow the side 3a; comes fx-om the assumed 2j;. Hence we must find two consecutive numbers whose cubes differ by a square. Let them he y, y+\. Therefore 3v/" + 3v/ + 1 =square = (l — 2?/)* say, and y = l. Thus instead of 2 and 3 we must take 7 and 8. Assuming now added number = x, side of cul)e = Ix, side of new cube — 8a;, we find 343.<;' + x = 512x''\ Therefore a;^ = ^J^, a; = l. (343 7 1 \ 2197'T3'13;^'^'"^"^^""- 10. To add the same number to a cube and its sid'! and i/uikc them the opjwsite. Suppose tlie cube 8x-^, its side 2x, the number 27x-' - 2a;. Therefore 35x' - 2a; = side of cube 27j;', therefore 3r)x*-5 = 0. This gives no rational value. Now 35 = 27 + 8, 5 = 3 + 2. '^ Therefore we must find two numbers the sum of wlioso cubes bears to the sum of the numbers tlie ratio of a square to a square. Let sum of sides = anything, 2 say, and side of first cul>c = :. Therefore 8 - 12^3 + 6s' = twice a square. Therefore i-(jz + 3z^ = a .square = (2 - 45)* sjiy, and : = on- 192 DIOPHANTOS OF ALEXANDRIA. of the sides = ^ ^ , and the other side = j^ . Take for them 5 and 8. Assuming now as the cube \'25x^, and as the number 5 1 2^' - 5x, we get 637a;^ -5x = 8x, and a; = - , , /125 5 267\ . 11. To find two cubes whose sum equals the sum of their sides. Let the sides be 2x, Sx. This gives 35a;' = 5x. This equation gives no rational result. Finding as in the preceding problem an equation to replace it, 637a;' = 1 Sx, a; = = , 1.1 1 125 512 and the cubes are ^^ , -^^^ . 12. To find two cubes whose difference equals the difference of their sides. Assume as sides 2x, 3x. This gives Idx^ = x. Irrational; and their difference we have to find two cubes such that difference of sides = ratio of squares. Let them be (z + 1)*, z\ Therefore Sz' + 3;^ + 1 = square = (1 - 2z)- say. Therefore z = 7. Now assume as sides 7a;, Sx. Therefore 169a;* = a;, and .'c=t^. 1 o Therefore the two cubes (i^> a 13. To find two numbers such that the cube of the greater + the less = the cube of the less + the greater. Assume 2a;, 3a;. Therefore 27a;' + 2a; = Sa;' + 3a;. Therefore 19.x' = a;, which gives an irrational result. Hence, as in 12th problem, we must assume 7a;, 8a;, 7 8 and the numbers are as there v:^ , ^g • 1 4. To find two numbers such that either, or their sum, or their difference increased by 1 gives a square. Take unity from any square for the first number ; let it be, say, 9a;" + 6.r. ARITHMETICS. BOOK IV. 19:^ But the second + 1 =a square. Therefore wc must Cnd a squar.- such that the square found + 9x* + 6x = a square. Taku factors of 9j;* + 6.r, nz, (9x+6, x). Square of half dif ference -16xf + 24a; + 9. Therefore, if we put the second number IG.c" + 24.C + 8, threo conditions are satisfied, and the remaining condition gives difference + 1 = square. Hence 7a;* + 1 Sx + 9 = square = (3 - 3x-)- say. Therefore x = 18, and (3024, 5624) is a solution. 15. To find three square numbers such that their sum equals the sum of their differences. 8umo{diSerences=A-B+B-C+A-C = 2{A-C) = A+B+C, by the question. Let least (C) = 1, greatest = a;' + 2.x- + 1. Therefore sum of the three squares = 2x^ + ix = x^ + 2a; + 2 + the middle one. Therefore the middle one (JB) = a;' + 2a; - 2. This is a 9 square, = (x - 4)" say. Therefore a; = - , and the squares are (s^, ~nr ? 1 ) '^'' (196, 121, 25). 16. To find three numbers such that the sum of any two multi- plied by the third is a given number. Let (fii-st + second) . third = 35, (second + third) . first = 27, and (third + first) . second = 32, and let the third = x. 3.3 Therefore first + second = Assume first = 1*^ — , second X 250 Therefore ^^^ If + 10 = 27~ + 25 = 32 These equatians are inconsistent, but if 25 - 10 u-ere t^ual to 32 - 27 or 5 they would be right. Therefore we have t<» divide 35 into two parts differing by 5, i.e. 15 and 20. Thus first number = — , second = — . Therefore . +15-27, r*^ X * II. 1). X--5, and (3, 4. 5) is a solution. 13 194 DIOPHANTOS OF ALEXANDRIA. 17. To fiiid three numbers whose sum is a square, and such th<(t the sum 0/ the square of each and the succeeding number is a square. Let the middle number be 4a;. Therefore I must find what square + ^x gives a square. Take two numbers whose product is ix, say 2ic and 2. Therefore {x-\y is the square. Thus the first number = £c- 1. Again 16x-* + third = square. Therefore third = a square - 1 6a;* = (4a; + Vf—YQtx? say, = 8a;+ 1 . Now the three together = square, therefore 13a;=square=1697/- say. Therefore x=\ 3^. Hence the numbers are ISy^'-l, 52/, 104/ +1. Lastly, (third)* + first = square. Therefore 10816/ + 221/ = a square or 10816/ + 221= a 220 55 square = (104y/ +1)° say. Therefore y = -"- = — , , /36621 157300 317304\ . , . 18. To find three numbers whose sum is a square, and such that the difference of the square of any one and the succeeding number is a square. The solution is exactly similar to the last, the numbers being in this case 13/ +1, 52/, 104/ -1. The resulting equation is 10816/ - 221 = square = (104^ - 1)-, whence 2/ =-^^-^, , /170989 640692 1270568\ . ^^^ (T08I6-' T0816 ' 10816 ; ^^ " ^^^"*^°'^- 19. To find two numbers such that the cube of the first + thr second - a cube, the square of the second + the first = a square. Let the first be x, the second 8-a;'', therefore a;''-16.r''+64 + a' = a square = (x* + 8)* say, whence 32a;^ = x. This gives an irrational result since 32 is not a square. Now 32 = 4. 8. Therefore we must put in our assumptions 4 . 64 insteail. Then the second number is 64 -a;^, and we get, as an equation for x, 256a;» = 1 . Therefore x = — , lb 1 262143 and tlu. numbers are jg, ^^gg . ARITHMETICS. UOOK IV. 10.", 20. To jlud three numbers imfffinite/i/^ s?/<7( (/uU the ]>ro t^ > -i « > if. 22. Find three numbers which are proportional and such that the difference of any two is a square. Assume a; to be the least, x + 4 the midiUe, x + 1 3 the greatoist. therefore if 13 were a square we shoidd have an indefi- nite solution satisfying three of the conditions. We muHt 1 I.e. in general expressions. 2 Product of second and third + 1 = (a: + 2) (ix + 4) + 1 - i-c' ♦ l'^-^ + 9. ^hidi i« a square. 1.1—2 196 DIOPHANTOS OF ALEXANDRIA. therefore replace 13 l)y a squai-e wliich is tlie sura of two squares. Thus if \vc assume x, x + 9, x+ 25, three conditions are satisfied, and the fourth gives x (x + 25) = {x + 9)", there- fore x = -;^ , I . ,81 144 256\ . . ^. I i IT , „- . r,- IS a solution. /«i 144 ZDt)\ [j ' T ' 7') 23. To find three numhers such that the sum of their solid content^ and any one of them is a square. Let the product be x' + 2x, and the first number 1, the second £C*+ 2x ix + 9 ; therefore the third = :, — -^ . This cannot be divided out generally unless x^: 4:X=2x : 9 or a;" : 2.^'=4a; : 9, and it could be done if 4 were half of 9. Now ix comes from (jx - 2x, and 9 from 3^, therefore we have to find a number m to replace 3 such that 2»i-2 = -^, therefore m^ = 4w - 4 or m = 2. ^ We put therefore for the second number 2x + 4, and the third then becomes Ix. Therefore also [third condition] 5 x^ + 2x + \x - square = ix^ say, whence x= , . (. 34 1\\ . solution. 24. To find three numbers such that tJie difference of their solid content and any one of them is a square. Fii'st x, solid content x^.+ x; therefore the product of second and third = 03 + 1 ; let the second = 1 . Therefore the two remaining conditions give a i both squares [Double equation.] Difference = cc = ^ . 2a;, aay ; therefore (.t + :^)' = vX-*+k-1, x- .™.(^;,i, 8 -— ) is a .solution. I.e. the continued product of all three. Observe the solution of a mixed quadratic. X- 27' 26 27 136 ARITHMETICS. UooK IV. l!l7 25. Divide a givou number into two jiarts wljose prcKluct i.s a cube mltius its side. Given number 6. First part x ; therefore second = 6 - x, and 6a; - a;* = a cube minus its side = {^x-lf- (2x - 1 ) say, so that 8x^ - 1 2.t' + ix = Gx - x'. This wouUl reduce to a simple equation if the coefficient of x were the same on both sides. To make tliis so, since G is lixcd, we must put m for 2 in our assumption, where 3m - )n = G, or m = 3. Therefore, altering the assumption, (3x•-l)^-(3x•-l)=6x•-.t•^ , 26 whence and the parts are 2G. To divide a rjiven number into three parts sii.<7i that their con- firmed 2)roduct equals a cube lohose skle is the sum oj' their differencen. Given number 4. Let the product be 8a;' : now the sum of differences = twice difference between third and firat; therefore difference between third and first parts = x. Let the first be a multiple of a;, say 2a;. Therefore the third = 3a;. Hence the second = r. ^, ii"d, if the second had lain between o the first and third, the problem would have been solved. Now the second came from dividing 8 by 2 . 3,- so that we have S to find two consecutive numbers such that ^, . , ^ their product lies between them. Assume m, m + 1 ; therefore -, 7/t + m lies between m and m + 1. g Therefore —5 + 1 > ?« + L m + m Therefore m* + m + 8 > 7?i' + 2«t' + m, or 8 > m" i in\ 1\' Take (IN ' "i + ., ) ) which is > m"" + m, and p.|u:itc it t< Therefore m + ;^ - 2, and m = \^ 27 cube. 198 DIOPHANTOS OF ALEXANDRIA. Hence we assume for the numbers 5 9 8 3^' 5'^'' 3^' or (25x, 27a;, -iOx), multiplying throughout by 15. Therefore the sum = d2x= 4, and •'k = oo j and f -^ , , „ j are the three parts required. o [N.B. The condition —^ x* = 1 4,x-, x = yoi 1 o ^"^ (13-' res) i« ^ soi^^ti^"- 28. To find two numbers such that the difference between the product and either is a cube. Let the first be 8a;, the second a;^ + 1 (since 8a;'+8a; - 8a;= cube) ; also 8if^ + 8a; - a;* - 1 must be a cube, which is "im- possible " [for to get rid of the third power and the abso- lute term we can only put this equal to (2a;- 1)^ which gives an " irrational " result]. Assume then the first = 8a; + 1, the second = a;* (since 8a;^ + a;^ - x^ = cube). Therefore 8a;' + a;" - 8.r - 1 = a cube = {2x - If say. 14 Therefore x = y^ , wi ^ 125 196 and the nunilx'rs are „- , . Id 169 29. To find two numbers such that their product =t their sum - a cube. Let the first cube be G4, the second 8. Therefore twice the sum of the numbers = 64 - 8 = 56, and the sum of the numbers = 28, but thi-ii- product + their sum = 64. Therefore their product - 36. ARITHMETICS. BOOK IV. 199 Therefore we have to tind two numbers whose sum -• 28, and whose product = 36. Assuming 14 +a;, 14 -x for these numbers, 196 -x':::^ 36 and x* = 160, and if 160 were a a square we could solve it rationally. Now 160 arises from 1 4'' -36, and 14= J. 28 --.56 2 4 = T (diflference of cubes) ; 36 = ^ sum of cubes. Therefore we have to find two cubes sucli that ( - of their difiereuce ) - ^^ their sum = a square. Let the sides of these cubes be s + 1 , s - 1 . 1 3 1 Therefore - of their difference = r> ~' + ;j> '^^^^ the square of this 9,3,1 =r ^2^'^4- Hence ( - . differer cc ) ^ .^ • sum = t ~ + ;> =' + a~~) v"^ "*" "')• Therefore dz' + Gz' + 1 - 4^' - 1 2s = a square = (3;:' + 1 - 6c)» say, 9 whence 32s^ ■= 36s*, and ~ ^ o • Therefore sides of cubes are 17 1 , , ^ 4913 1 -, g, and the cubes .^,,-- 4913 Now put product of numbers + their sum = -,- product - sum = ^y^ therefore their sum 2456 ~ 512 ' their product 2457 ~ 512 ■ Then let the first number = 1228 X + half sum = x + — - - , second 1228 = 512 -"• -ru r 1507984 Therefore -j^-^^-^-^ -x* '•*" ""- 512- Therefore 2621 4 4x •- 250000. 200 DIOPHANTOS OF ALEXANDRIA. „ 500 Hence x = ^—- , 512 , /1728 728\ . ""'"^ V^12 ' 512 j '' ^ '"^"*'°''- 30. To find two numbers such that their product ± their sum = a cube [same problem as the foregoing]. Every square divided into two parts, one of tohich is its side, makes the 2)roduct of these ])arts + their sum a cube. [i.e. x(x^ - x) + x' — x + x= a cube.] Let the square be x^ ; the parts are x, x^ — x, and fi-om the second part of the condition x^ -x^ -X- = x^ - 2x^ = a cube = [7,] say. Therefore 5 *^ = 2a;'', x = -^ , o < , /16 144\ . and ( -^ , -jq 1 IS a solution. 31. 7'o find four square numbers such that their sum + the su?n of their sides - a yiven number. Given number 1 2. Now x* + a; + j = a square. Therefore the sum of four squares + the sum of their sides + 1 = 13. Thus we have to divide 13 into 4 squares, and if from eacli of their sides we subtract ^ we shall have the sides of the required squares. 10 . n G4 36 144 81 Now l3 = 44-9 = --f25+-25+05' and the sides of the required squares are 11 ^ 19 13 10' lO' 10' 10" 32. To find four squares such that their sum ininus the sum of their sides equals a given number. Given number 4, Then similarly f side of first - - j + ... = 5. -,.,..,-. 9 IG G4 36 and 5 is divided into ^-^ , ^rz , ^^ . ?,- > 2y 25 25 2;) and the sides of the squares arc ( . - 13 21 17^ 10 ' 10 ' lOy ARITHMETICS. BOOK IV. 201 33. To divide unity into two parts snch that, if given numbers be added to each, the j)roduct of the resulting expressions may be a square. Let 3, 5 be the numbers to be added, aud let the parts be , 1 -xj Therefore (a; + 3) (6 - a;) = 18 + 3x — a;' - a square = ■l.c* say. Hence 18 + 3a; = 5a;*; but 5 comes from a S([uare+1, and the roots cannot be rational unless (this square +l)18 + [7jj =a square. Put (m- +l)18 + f^j =a square, or 72/?r + 81 = a square = (8ni + 9)* say. Therefore «i = 1 8. Hence we must put (x + 3) (6 - a;) = 18 + 3a; - a;' - 3•24a;^ Therefore 325x'' - 3a; - 18 = 0. 78 6 Therefore sohition. 31. [Another solution of the foregoing.] Suppose the first a; - 3, the second i -x; therefore a; (9 - a;) = square = 4a;* say, 9 and 5a;- = 9a;, whence a;=p, but I cannot take 2 from 9 - , and X must be > 3 < 4. 9 Now the value of a; comes from - — r . Therefore, since a square + 1 a; > 3, this S(piare + 1 < 3, therefore the square < 2. It is 5 also > -. . 4 Therefore I mustfind a square between -and 2, or ^^ and — . And -TT-r- or ^ will satisfy the conditions. 64 16 Put now x(9-x)- j^, .c. 202 DIOPIIANTOS OF ALEXANDRIA. Therefore U4 IT 21 20^ /21 20\ . , ^. and ( .,- , . 1 IS a solution, 35. To divide a given number into three j)arts such that the pro- duct of the first and second, with the third added or subtracted, may be a square. Given number G, tlie third part x, the second any niimher less than G, say 2, Therefore the first — ^ - x. Hence 8 - 2x' ± a; = a square. \^Doid)le-equation.^ And it cannot be solved by our method since the ratio of tlie coefficients of X is not a ratio of squares. Therefore we must find a number y to replace 2, such that ?/ + 1 , r = a squai-e = 4 say. 2/-1 5 Therefore 2/ + 1 = -iy — 4, and y = q • 5 13 Put now the second part = - , therefore the first - ~ —x. o o G5 5 Therefore -^ -^x^x^^s. square. Thus „„ o . }• are both squares, or „„^ ~ i^ .^l- are both squares : difference =195 = 15. 13. 2G0-24:x) Hence (^^^ ~ Y = 65 - 24a;, and 24a; = 64, a; = | . /5 5 8\ Therefore the parts arc ( ^ , ^ , ^ j . 36. To find two numbers such that the first with a ce^-tain fraction of the second is to the remainder of the second, and the second with the same fraction of the first is to the remainder of the first, each in given ratios. Let the first with the fraction of the second = 3 times the remainder of the second, and the second with the same fraction of tlie first = 5 times the remainder of the first. Let the second = x + 1, and let the part of it received by the first be 1. Therefore the first - 3.c - 1 [for 3.f - 1 + 1 - 3.c]. ARITHMETICS. 1U)0K IV. 203 Also fii-st + second - ix, uiul first + second = sum of tlie numbers after interchange, therefore J^I$±^^S^l_ ^ q remainder of tiret 2 Therefore the remainder of the first = ^x, and hence the second receives from the first 3.0 - 1 - ^ x- = r a; - 1 . o o Hence l^^^^, = r, tlierefore r,x^ + tx~\ = 3.c - 1 , 6x-\ x+\ 3 3 and x = ^ . 8 12 Therefore the first number = -;; , and the second = -''- : and 1 is < t rr- of the second. Multiply by 7 and the numbers are 8, 12; and the fraction is ^ ; but 8 is not divisible by 12, so multijily by 3, and (24, 36) is a solution. 37. To find two numbers indefinitely such their product + their sum = a given number. Given number 8. Assume the first to be x, the second 3, Therefore 3a; + a; + 3 = given number = 8. Therefore x-'j, and the numbers are ( . , 3 j . 5 8 — 3 Now - arises from -. — !r . Therefore we may put mx + n for 4 3+1 "^ ^ 1 1 1 1 1 z' 8 ~ ("*•*-' + '" ) the second number, and tiie nrst = r- . mx + n+l 38. To find three numbers such that (the product + the sum) of any two equals a given number. Condition. Each number must be 1 less than some squai-o. Let product + sum of first and second = 8, of second and tbinl - 15, of third and first = 24. Thi-n , — , = the first: let the second = x- 1. second + 1 Therefore ^-"•^- first --1. Similarly third — -1. X X •'-■ 204 DIOPHANTOS OF ALEXANDRIA. Therefore (^^ - 1 V^ - l) + ^ - 2 = 24, and ^.^ - 1 = 24, 12 therefore aj = -. , o , /33 7 68\ . , ^. and I _ , ^ , 1 IS a solution, 39. To find two numbei-s indefinitely such that their product exceeds their sum by a given number. Let the first number be x, the second 3. Therefore product - sum = 3.V - a; - 3 = 2a; - 3 = 8 (say). Therefore x = -- . Thus the first = — , tlie second = 3. But — = — :r . x + d Hence, putting the second = a; + 1 , the first = — — . 40. To find three numbers such that the product of any two exceeds their sum by a given number. Condition. Each of the given numbers must be 1 less than some square. Let them be 8, 15, 24. Therefore first number = :; — =- = , say. Therefore second - 1 a; 9 IG the first = - + 1 , the second = a; + 1 , and the third = — + 1 . X X Therefore ('^ + 1 V^ + l) - ^ - 2 = 24. Orl4i-ll24, Ij', X- 5 md ( q-s ,,.,,_) is a solution. /57 17 9Z\ . (l2' 5'12;^^^ 41. To find two numbers indefinitely whose product has to their sum a given ratio. Let the ratio be 3 : li the first number x, the second 5. 15 Therefore 5x = 3 (5 + x), and x = — . r,ut -^ = r~~:^ , and, putting x for 5, the indefinite solution is: first = -^_ , second =x. X — o ARITUMETICS. BOOK IV. 205 42. To find three numbers such that for any two their product bears to theii- sum a given ratio. _ first and second multiplied . , , , Let 5 — 7 , — = 6, and let the other ratios be nrst + st^cond 4 and 5, the second number .r. Therefore first = — , third X — 3 4a; ~ x-i' 3x ix . 3x 4a; \ , ^ , Also -. j = 5 { —5+ r) or 12x- x-Z a;-4 \a;-3 a;- 4/ 35.0-' - 1 20.r. \a; — o x — -±/ 120 Therefore x = -^, , , /360 120 480\ . . , . ""n^' 23' 28;"''^'''"^^'^""- 43. To find three numbers such that the product of any two h«s to the sum of the three a given ratio. Let the ratios be 3, 4, 5. First seek three numbers such that the product of any two has to an arbitrary number (say 5) the given ratio. Of these, let the product of the first and the second =15. 15 Therefore if x = the second, the first = ■ — . X But the second multiplied by the third = 20. 20 20 15 Therefore the third - — , and "^—^^ — = 25. x or Therefore 25.v'=20.15. And, if 20 . 15 were a square, what is required would be done. Now 15 = 3.5 and 20 = 4 . 5, and 15 is made up of the ratio 3 ; 1 and the arhilrary number 5. 12»r Therefore we must find a number m such that — = ratio of a square to a square. Thus 1 2/?r . 5»i = GOm' - square = 900/»', say. Tliereforo m = 15. Let then the sum of the three =15, and the product of the first and sccon and x ^ G. X X 206 DIOPUANTOS OF ALEXANDRIA. 45 Therefore the first number =- , the third = 10, 47 and the sum of the three = 23| =— . Now, if this loere 15, the j)rohlevi loould he solved. Put therefore 15a;° for the sum of the tlirce, and for the numbers l^x, Qx, \0x. 47 Therefore 2^x = 15a;-, and x=---, /705 282 470\ whence fgQ-,3Q,3QJ is a sohition. 44. To find three numbers such that the product of their sum and the first is a triangular number, that of their sn7n and the secoml a square, and that of their su7h and the third a cube. Let the sum be x', and the first —, , the second -, , the third — ^ , X X .r which will satisfy the three conditions. 1 8 But the sum =—3 = .r* or 18 = x\ X Therefore loe must rejilace IS by a fourth iiower. But 18 = sum of a triangular number, a square and a cube; let the fourth power be x^, which must be made up in the same way, and let the square be x* -1o? + 1. Therefore the triangular number + the cube = 2a;*— 1; let the cube be 8, therefore the triangular number = 2a;^ - 9. But 8 limes a triangular number + 1 = a square. Therefore IG.x'^ - 71 = a square = (4a;- 1)^ say; therefore x = 9, and the triangular number = 153, the square =6400 and the cube = 8. Assume tlien as the first number -^-, as the second — „ , x' ' a' as the third -3 . Therefore — „- = a;* and x = 9. a; ,,,, /153 6400 8\ . , ^. ARITHMETICS. BOOK IV. 207 45. To find three numbers such that the dijj'crence of the greatest and the middle has to tlie difference of the middle and the least a given ratio, and also the sinn of any pair is a square. Ratio 3. Since middle number + lea.st = a square, let them = l. Therefore middle > 2 ; let it be x + 2, so that least = 2 - x. Therefore the interval of the greatest and the middle = 6a-, whence the greatest = Tx + 2. Therefore ' > are both squares [Donble equation] : take two numbers Avhose product = 2x, say - and 4, and pro- ceed by the rule. Therefore x= 112, biit I cannot take \\2 from 2; therefore x must be found to be < 2, so that 6.f+4 6m + 18, and 18 . 2 + 3- = 4.5 ; therefore we may put in-'- + ^. Thus we have 3;:- + 12^ + 9 = (3 - bz)'. Hence s = yi , -'^'i^l the side of the middle square ■--■ ... aii«l the square itselr - . Turning to the original problem, wo i>ut y^j'^ ^•'-" + ^• Therefore x = , ' , which is < 2. <26 208 DIOPHANTOS OF ALEXANDRIA. Hence the greatest of the required numbers = 7x + 2 - - 11007 726 ' 2817 and the second of them = as + 2 = - , lab 87 and the thii'd = 2-x = ^^z . 46. To find three numbers such that the difference of the squares of the f/reatcst and the middle numbers has to the difference of the middle and the least a given ratio, and the sums of all 2}airs are severally squares. Ratio 3. Let greatest + middle = the square 1 (Sx^. Therefore greatest is > 8a;^, say 8a;" + 2, Hence middle = Bx* - 2, and greatest + middle > greatest + least, therefore great- est + least < 1 6a;* > Sx^ = 9a;-, say; therefore the least number = a;® - 2. Now difference of squares of greatest and middle = 64a;*, and difference of middle and least = 7a;", but 64 ^ 21. Now 64 comes from 32 . 2, so that I must find a number m 21 such that 32m = 21. Therefore ««. = ^ . Assume now that the gi'eatest of the numbers sought 21 ,21 „ 21 = 8a;* + -^ , the middle = 8a; - — , the least = a; - — . [Therefore difference of squares of greatest and middle = 21a;* = 3. 7a;*.] The only condition left is 21 21 8a;* - p + a;* - -gT, ^ a square 9.1; — = a square = {S.r - 6)" say. r,., r 597 Therefore x = yr:^ . 5/6 /3069000 2633544 138681\ . , ^. "^'"^^ (331776' 331776 ' 331776;^^''^ ^°^^^^^'^"- AIUTILMETICS. BOOK V. BOOK V. 1. To find three numbers in c. p. such that each exceeds a given number by a square. Given number 12. Find a square which exceeds 12 by a square [by ii. 11], say 42^. Let the first number be 42^, the third x^, so that the middle one = 6i.r. a;2_ 12") Therefore „. ,_> are both squai-es : their diflference therefore as usual we find the value of x, viz. — - , A^, 2346 i 130321\ . ^^ (^21,^^, -10816 j-^ solution. 2. To find three numbers in g. p. such that each together icith the same given number equals a square. Given number 20. Take a square whicli exceeds 20 by a square, say 36, so that IG + 20 - 3G = a square. Put then one of the extremes 16, the other x*, so that the middle term = \x. ^ + 20 "i Therefore , ^^> are both squares : their difference \x + 20j = y? -ix^x{x- 4), whence we have 4a; + 20 = 4, which gives an irrational result, but the 4^1(16), and we should have in i)lace of 4 some number > 20. Therefore to replace 16 we must find some square > 4 . 20, and such that with the addition of 20 it becomes a square. Now 81>80; therefore, putting for the nijuired square {m + 9)-, (»4 + 9)' + 20 = square -{m-U)' .siiy. Therefore m = .i, and the square = (9A)'' OOj. H.D. " " 1* 210 DIOPIIANTOS OF ALEXANDRIA. Assuming now for the numbers 90^, O^x, a?, we have, ^^ > are both squares : and the difference =a;(a;-9i), 9^a;+20j whence we derive x = -r— - , 152 ' /nm 389.1 1681 \ . ^^ V ^^' T52 ' 23I04J ^^ ^ '°^^^^^"^- 3. Givoi one numhei; to find three others such that any one of them or the product of any two, when added to the given number, pro- duces a square. Given number 5. Porism. If of two numbers each and their pi'oduct together with the same number make squares, the two numbers arise from two consecutive squares. Assume then {x + 3)-, {x + 4)-, and put for the first number a;^+6cc + 4, and for the second a;° + 8x+ll, and let the third equal twice their sum minus 1, or ix^ + 28a; + 29. Therefore 4a;' + 28a; + 34 = a square = (2x - 6)^ say. Hence a; = ^r^; , 26 /2861 7645 20336\ . , ,. and I -;r=-;r , -sv^/T . Wr,,^ IS a solution. V676 ' 676 ' 676 / 4. Given one number, to find three others such that each, and the product of any two exceed the given number by some square. Given number 6. Take two consecutive squares x", a;* + 2a; + 1, add 6 to each, and let the first number = a;* + 6, the second number = a;^ + 2a; + 7, the third being equal to twice the sum of first and second mhiics 1, or 4a;^ + 4a; + 25. Therefore third minus 6 =4a;' + 4a; + 19 = square =(2a; -6)* say. 17 Therefore a; = — : , /4993 6729 22660\ . *^^ (-784' T84' -78r)-^-'^<^l"*^°^^- [Observe in this problem the assumption of the Porism numbered (1) above (pp. 122, 123).] 5. To find three sqiiares such thai the procbict of any tivo, added to the sum of those two, or to the remaining one, gives a square. Porism. If any two consecutive squares be taken, and a third number which exceeds twice their sum by 2, tliese three ARITHMETICS. BOOK V. 211 numbers have the property of the numbers recpiired by the problem. Assume as the first a;^ + '2a; + 1 , and as the second x" + 4x + 4. Therefore the third = Ax^ + 1 2x- + 1 2. Hence x- + 3.v + 3 = a square -{x- 'df say, and a; = ^ . m c /25 64 196\ . iherefore ( ^ > 'K ■> ~q~ ) ^^ '^ solution. C. To find three numbers such that each exceeds 2 by a square, and the product of any two minus both, or minus the remaining one = a square. Add 2 to numbers found as in 5th problem. Let the first be x^ + 2, the second x^ +'2x+ 3, the third ■^if + -ix + 6, and all the conditions are satisfied, except 4x^ + 4a; + 6 - 2 = a square = 4 (a; - 2)* say. 3 Therefore x = -^ , o /59 114 246\ . ^^^ (,25' 25' 25 J '' ^' '°^"^^""- 7. To find two numbers such that the sum of their product and the squares of both is a square. [^Lemma to the following j)roblem.^ First number x, second any number {m), say 1. 3 Hence a;' + a; + 1 = a square = {x — 2)- say, and -c = r . Therefore K , 1] is a sokitiou, or (3, 5). 8. To find three right-angled triangles ' whose areas are equal. First find two numbers such that their product + sum of their squares = a square, i.e. 3, 5, as in the preceding problem [15 + 3' + 5^'= 7']. Now form three right-angled triangles from (7, 3), (7, 5), (7, 3 + 5), respectively, i,e. the triangles (7« + 3S 7*- 3', 2.7.3), Ac. ^ I.e. rational right-angled triangles. Ax 6. Thus if I divide 13 into squares whose difference < 1, this condition is satistied. 13 Take -^ = 6|, and I wish to add to 6i a small fraction which will make it a square, or, multiplying by 4, I wish to make ;, + 2G a square, or 26x* + 1 = a square = {5x+\ )' Siiy, whence x = 10. 214 DIOPHANTOS OF ALEXANDRIA. Therefore to make 2G into a square I must add y^, or to make 6^ into a square I must add 400^ 1 13 /51\' ^^^ 4o-o-'-2=Uo;- Tlierefore I must divide 1 3 into two squares such tluit their sides 51 Tnay be as nearly as possible equal to ^ . [TrapicronjTos dywyrj, above described, pp. 117 — 120.] Now 13 = 2^ + 3^ Therefore I seek two numbers such that 51 9 3 minus the first = ^ , or the first = — , and 2 plus 51 11 the second = ^ , so that the second = — . I write accordingly (11a +2)°, (3 - 9a;)° for the required squares substituting x for -— . Therefore the sum = 202a;- - lO-i; + 13 = 13. 5 ,^, ., 257 258 Hence x = y^ , and the sides are y^. ' Tni » and, subtracting 6 from the squares of each, we find as the pai-ts of unity / 4843 ^358- V10201 ' 10201, 13. To divide unity into two jyarts such that, if we add given numbers to each, the results are both squares. Let the numbers be 2, G, and let them be represented in the figure. Suppose DA = 2, AB = l, BE=G, G a point in A£ so chosen that DG, GE may both be squares. Now Tlierefore I have to divide 9 into tioo squares such that one of them lies between 2 and 3. Let the hitter square be x^. Therefore the second square = 9 - a;', wliere x-" > 2 < 3. Take two squares, one > 2, tlie other < 3, [the former square ARITHMETICS. BOOK V. 21') being the smaller], say ^ ■ , -^^ . Therefore, if we can make x" lie between these, what wtis required is done We must have ^^^,<\l Hence, in making 9 - x^ a square, we must find 17 19 67)1 ^?Tl 17 19 >12"I2- Thus 72w>17m'' + 17, and 36*- 17. 17 = 1007 which' is '^ Sl°, hence m is :}► — . Similarly ?« is -j; tt; • Let m = 3i Therefore 9 -x' = h - '-x\ , and x = =Ti • 53 TT » 7056 ,^, ^ „, /1438 1371\ Hence ar ^ ^^^^ , and the segments of 1 are (^^gog ' 2809J " 14. To divide unity into three pm-ts such that, if we add the same number to the three parts severally/, the results are all squares. Comlition. Given number must not be 2 [Condition remarked upon above, pp. 130, 131.] Given number 3. Thus 10 is to be divided into three squares such that each > 3. Take ^ of 10, or 3^, and find x so that ^^7,+ 3J may be a square, or 30x° + 1 = a square = {6x +\f say. Therefore a; = 2, 1 121 and 36 '*' ^^ " "36" " ^ s^^''^^"^- Therefore we have to divide 10 into three squares each near to — . , [7rapio-OT7;T09 aywyr;']. 00 1 I.e. the integral part of the root is ^31. The limits taken arc .1 fortiori limits as explained on p. 93, n. 3 and 4. Strictly speaking, wc could only say, taking integral limits, that x/Iu07<32, but this limit is not narrow enough to secure a correct result in the work which follows. 21 G DIOPHANTOS OF ALEXANDRIA. 9 16 Now 10 = 3^ + 1' = the sum of the three squares 9, ^, ^ . 3 4 11 Comparing the sides 3, -, p with -^, or (multiplying by 30) 90, 18, 24 with 55, we must make each side approach 55. Put therefore for the sides 3-35CC, 31a; + g, S7x+-^ [35 = 90-55, 31 = 55-24, 37 = 55-18], we have the sum of the squares = 35550;^- 116rK+ 10 = 10. Therefore x = ^rv^- , 3555 and this solves the problem. 15. To divide unity into three parts such that, if three given numbers be added, each to one of the parts, tlie results are all squares. Given numbers 2, 3, 4. Then I have to divide 10 into three squares such that the first > 2, the second > 3, the third > 4. Let us add - unity to each, and find three squares whose sum is 10, the first lying between 2, 2^, the second between 3, 3i, and the third between 4, 4|. Divide 10 into two squares, one of which lies between 2 and 2^. Then this square minus 1 will give one of the parts of unity. Next divide the other square into two, one lying between 3, 3J ; this gives the second part, and therefore the third. 16. To divide a given number into three parts such that the sums of all pairs are squares. Number 10. Then since the greatest + the middle jmrt = a square, &c., the sum of any pair is a square < 10, but twice the sum of the three = 20, There/ore 20 is to he divided into three squares each of lohich < 10. Now 20=16 + 4. Therefore we must divide 16 into two squares, one of which lies between 6 and 10; we then have three squares each of which is < 10, and whose sum = 20, and by subtracting each of these squares from 10 we get the parts required; [16 must be divided into the two squares by v. 13.] ARITHMETICS. ROOK V. 217 17. To divide a given number into four parts such that the sum of any three is a square. Number 10. Then three times the sum = the sum of four squai'es. Hence 30 must be divided into four squares, eacli of wliich < 10. If we use the method of Tra^icro'-nj? and make each near 7|, and then subtract each square found from 10, wc have the required i)arts. But, observing that 30-1G + 9 + 4+1, I take i, 9 and divide 17 into two squares each of which < 10 > 7. Then sub- tract each of the four squares from 10 and we have the required parts. 18. To find three numbers S2ich that, if we add any one of them to the cube of their sum, the result is a cube. Let the sum be x, the numbers 7x^, 26a;', G3x^. Hence, for the last condition, 9Ga;^ = x. But 9G is not a square. There- fore it must be replaced. Now it arises from 7 + 2G + G3. Therefore I have to find three numbers, each 1 less than a cube, whose sum is a square. Let the sides of the cubes be wi+ 1, 2-m, 2, whence the numbers are m^ + 3'w' + 3m, 7- 12ni+ Gnr-m'', 7, aud the sum = ^m" - dm + 14 = a square = (3?« - 4)-. 2 Therefore 7n = -r-. , 15 1538 18577 ^ and the numbers are qq^t- > qqtF ' '• Therefore, putting the sum - x, and the numbers of the problem 3375^' 3375 '"'''*'' 15 we find X = ^ : therefore, «fec. 54 1 9. To find tloree numbers such t/uif, if we subtract any one of them from t/ie cube of the sum, the result is a cube. Let the sum be x, the numbex"8 - x', " x', . x'. rr. r 4877 a / Therefore ^r^;-- x" ^ x ; |( n .io \ 4877 but irz^TT-, ^ 3 - the sum of three cubes. 1728 218 DIOPHANTOS OF ALEXANDRIA. Therefore we must find three cubes, each < 1, such that (3 -their sum) = a square =2,^ say. Tlierefore we liavetofind three 3 162 cuhes whose sum is -=^, or we have to divide 162 into three cubes. But 162 = 125 + 64 - 27. Now (Porism) the difference of two cuhes can be transformed into the sum, of two cuhes. Having then found the three cubes we start again, 2 and x = 1\x^, so that x = -^, which, with the three cubes, determines the result. 20. To find three numbers such that, if we subtract the cube of tlieir sum from any one of them, the result is a cube. Sum =x, and let the numbers be 2x', 9a;^, 28a;^. Therefore 39 x^ = 1, and we must replace 39, which = sum of three cubes + 3. Therefore we must find three cubes whose sum + 3 = a square. Let their sides be m, 3 -m, and any number, say 1. Therefore 9m^ + 31 - 277/2- = square = (3m - 7)" say, so that 6 , , .. ^.1 , 6 9 m- - , and the sides of the cubes are ■= , - , 1. Starting again, let the sum be x, and the numbers 341 3 854 3 250 3 Wb^' 125*' 125*' 25 5 so that 1445a;' =^ 125, ^''=289' ^ " 17 ^ thus the numbers are known. 21. To find three numbers, whose sum is a square, such that the cube of their sum added to any one of them gives a square. Let the sum be x^, the numbers 303', Bx", IS.'c". Therefore 26a;* = 1 ; and, if 26 were a fourth power, this would give the result. To replace it by a fourth power, wc must find three squares whose sum diminished by 3 = a fourth power, or thi-ee numbers such that each increased by 1 - a square, and the sum of the three - a fourth power. Let these he ARITHMETICS. ROOK V. 219 m* - 2ni^, m^ + 2m, m^ - 2m [sura ^ m*] ; then if we put m anything, say 3, the numbers are 63, 15, 3. Thus, putting for the sum x^, and for the numbers 3j;', 15x', GSx", a; = 5 , and the problem is solved. 22. To find three numbei-s whose sum equals a square, and such that tlie cube of the sum exceeds any one of them by a square. [Incomplete in the text.] 23. To divide a given fraction into three parts, such that each exceeds the cube of the sum by a square. Given fraction - . Therefore each = — + a square. Therefore 3 the sum of the three = sum of three squares + — . 13 Therefore we have simply to divide — into three squares. 24. To find three squares such that tJieir contimced product added to any one of them gives a square. Let the "solid content" = x", and we want three squares such that each increased by 1 gives a square. They can be got from right-angled triangles by dividing the square of one of the sides about the right angle by the square of the other. Let the squares then be 9 , _25 , ^ 16 '^' 144*' 225 '*^- 14400 Therefore the solid content = x". This = x'. Olo'iUU 120 Therefore " oq ** ~ ^ ' 120 . but _g, IS not a square. Thus we must find three right-angled triangle.s such that, if 6's are their bases, ;/s arc tlieir p<'q)endicular8, p H/9^6,6, 6^ == square, or assuming one tnangle arbitrarily 3»,6, (3, 4, 5), we have to make l2pj)J>J'j ^ square, or 220 DIOPHANTOS OF ALEXANDRIA. a square. " This is easy" (Diophantos ') and the triangles (3, 4, 5), (8, 15, 17), (9, 40, 41), satisfy the condition, and 03 = -^ ; 25 256 .1 .1 /25 256 9\ the squares then are ( -7- , -5, , Vr / • 25. To find three squares such that their continued product exceeds any one of them hy a square. Let the " solid content " = a;*, and let the numbers be got fx'om right-angled triangles, being namely 16 ^ 2 _64 , 25^^' 169 '^' 289^' m r 4.5.8 , , Therefore — r^ a; = 1, and the first side ought to be a square. As before, find three triangles, assuming one (3, 4, 5) such that hjiji^2^^pj>^= Q. square^, [letters denoting hypotenuses and bases], or such that 20 v^-^ — a square. [For the rest the text is in a very unsatisfactory state.] 1 Diophantos does not give the work here, but merely the results. Moreover there is a mistake in the text of (5, 12, 13) for (H, 15, 17), and the problem is not finished. Schulz works out this part of the problem thus : Find two right-angled triangles whoso areas are in the ratio vi : 1. Let the Bides of the first be formed from {2m + 1, m - 1), and of the second from (m -f- 2, 7K-1), BO that two sides of the first are Am" -2m -2, ^m- + (im and the area =6m'» + 9Hi3-9,/i2-6m. Two sides of the second are 2m'^ + 2m-i, G?;! + 3, and m times the area = Cm'* + 9m' - 9m- - Om. Now jmt e.g. jft = 3, therefore the first k-ianglc is formed from 7, 2, viz. (28, 4,5, ,'53) ; second from 5, '2, viz. (20, 21, 29). 2 Cossali remarks: "Construct the triangles (/, h,p) [i = ipotenusa], [-b ' —b ' ir=^^j' /ib^' + iip^ bAbp-p(b^-ip^) p Abp + b {b^ - ip-) ,,\ ^""^ [^b—' b ' b ^^)' ARITHMETICS. HOOK V. 221 26. To find three squares such that each exceeds tlicir continued product by a square. Let the "solid content " = a;', and the squares have to be found by means of the same triangles as before. We put 25x*, 625a;-, 1 4784a;- for them, ic. [Text again corrupt.] 27. To find three squares such that the product of any two increased by 1 is a square. Product of fii-st and second + 1 = a square, and the third is a square. Therefore solid content + each = a square ; and the pnjblem reduces to the 24th above. 28. To find three squares such that the product of any two diminished by 1 is a square. [Same as 25th problem.] or the solid content of the three hypotenuses has to that of the three perpen- diculars the ratio of a square to a square. It is in his note on this imperfect problem that Fermat makes the error which I referred to above. He says on the problem of finding tico triiuujks such that the products of hij2)otcnuse and one t)(l +2x) a square, whereas, the quantity to be made a square is (1 + 6x + Ox' + 2x^1 { - 1 - 2x). The solution is thus incurably wTong.] Fermat seems afterwards to have discovered that his solution did not help to solve this particular problem of Diophantos, but docs not seem to have seen that the solution is inconsistent with his own problem itself. 222 DIOPHANTOS OF ALEXANDRIA. 29. To find three squares such that unity diminished by the product of any two = a square, [Same as (26).] 30. Given a number, find three squares such that the sum of any two together with the given number jyroduces a square. Given number 15. Let one of the required squares =9. Therefore I must find two other squares, such that each + 24 = a square, and their sum + 15 = a square. Take two pairs of numbers whose product = 24, and let them be the sides of a right-angled triangle' which contain 4 the right angle, say - , Gx ; let the side of one square be 2 half the difierence, or - - 3a;. X 3 Again, take other factoi'S - , 8x, and half the difierence 3 = 4:X = side of the other square, say. 2x / 3 \^ /2 \" Therefore (^ — 4ccj +( — 3a; j +15 = a square, or -f + 25a;^ - 9 = a square = 25a;^ say. g Therefore x = ^ , and the problem is solved. 31. Given a number, to find three squares such that the sum of any pair exceeds the given number by a square. Given number 13. Let one of the squares be 25. Therefore we must seek two more such that each + 12 = a square, and (sum of both) - 13 ^ a square. Divide 12 into products (3a;, -) and (4a;, -J, and let the squares /3 2\" / 3 \" Therefore (^ a; - - j + ( 2a; - .^ j - 13 = a square, or ^ + 61 a;" - 25 = a square = -| say. a;" * '^ XT Therefore x = 2, and the problem is solved. 1 I. c. corrcspoudiug factors in the two pairs, in this case G.r, Sj. be AllITHMETICS. BOOK V. 223 32. To find three squares such that the sum of thdr squares is a mare. Let one be x", the second 4, the third 9. Therefore a;* + 97 = a square = (x' - 10)* say. 3 Therefore ^'' = 2o> ^"* *^--^ ^^ ^^ot a square and must be replaced. Hence I have to find p*, q* and m such that "^~ ^ ^^ = a 2m square. Let ;r = s-, q' = \, and //t = ;:' + 4. Therefore m''-;/-5' = («* + 4)*-s--16 = 8c'. Hence we must , 8«' 4;:^ ^ave, ggT^g = ^ square, or - — - = a square. Put s' + 4 = (;s+l)«say. Therefore s = ^ , and the squares are i^" = t , q^ = 4, and 25 m= -Y, or, taking 4 times each, ^r = 9, (7' = IG, 7?i =: 25. Starting again, put the first square = : Q (x" - GO) < g {x^ - 60). Therefore a;' > 5a; + 60 < 8x + 60. Hence x" = 5a; + a number > 60, or x is' -^ 11. Also a;- -f: 8a; + 60. Therefore a; is :|» 12, so that x must lie between 11 and 12. But a;'-60=(a;-«^)^ therefore a;= —^ — , which > 11, < 12, whence vf + 60 > 22m < 24??t. From these we find, m is not > 21, and not < 19. Hence we put x' - 60 = {x - 20)', and a; = 11^. Thus a;^-132i, a;^- 60 = 721, and 72^ has to be divided into two numbers such that - of the first + Q of the second = 11 A. Let the first = z. o Therefore |.^(72i-.) = lH, or and 5. 79 "' ~ 1 o 79 Therefore the number of xocs at five-drachmas = j^ - 59 eight „ -j2- [At the end of Book v. Bachet adds 45 Greek arithmetical epi- grams collected by Salmasius, which however have nothing to do with Diophantos.J 1 See pp. 'JO, *J1 for uu uxijlaualiuu of thusu liiuils. 225 BOOK VI. 1. To find a rational right-anfjkd triamjle such that the hypote- nuse exceeds each side by a cube. Suppose a triangle formed from tlie two numbei-s x, 3. Therefore hypotenuse =.x--+ 9, perpendicular = G.r, base=x'-9. Therefore by the question x^ + 9 - (x* - 9) should l)e a culx?, or 18 should be a cube, which it is not. Now 18 = 2. 3*, therefore we must replace the number 3 by m, where 2»r = a cube ; i. e. m = 2, Thus, forming the triangle from .r, 2, viz. (x" + 4, \x, x^ - 4), we must have a;" - 4a; + 4 a cube. Therefore {x - 2)- = a cube, or x-2 - a cube - 8 say. Hence x =10, and the triangle is (40, 96, 104). 2. To find a right-angled triangle such that the sum of the hypotenuse and either side is a cube. Form a triangle as before from two numbers, and one of them must be a number twice whose square - a cube, i.e. 2. Therefore, forming a triangle from x, 2, or (x' + 4, 4x, 4 - x*) we must have a;* + 4x + 4 a cube, and x^ < 4. ■ " 27 Hence a; + 2 = a cube, which must be < 4 > 2 -^ — say. o Therefore ^ "^ "s" ' . , . /135 352 377\ and the triangle is (^_ , -^ - , — j . 3. To find a right-angled triangle such that the sum of the area and a given number is a square. Let 5 be the given number, (3a;, 4a;, o.^) the triangle. Therefore 6 1* + 5 = square = 9x' say. 5 Hence 3x-^ = 5, and ^ is not a square ratio. Hence I must find a triangle and a iuiml)er such that the difference of the square of the number and the area of the triangle has to 5 a square ratio, L e. - ^ of a square. .. .. 13 226 DIOPHANTOS OF ALEXANDRIA. Form a triangle from x, - : then the area = a;* — 5, and let the ° ' «' a;' 1 2.5 ^. _ . 101 1 - number =x-i , so that i . 5 -\ — j- = - of a square, or, 4 . 2o + — 15- = a square = (10 + - j 24 Whence cc = -^ . o 24 5 The triangle must therefore bo formed from -^ j ni > and the number is -7:7; . oU Put now for the original triangle (Jix, 2>x, bx), where (hj^b) is 24 5 pbx' 170569 , and we have the solution. 4. To find a right-angled triangle such that its area exceeds a given number by a square. Number 6, triangle {3x, ix, 5x). Therefore Gx^ - 6 = square = 4aj* say. Hence, as before, we must find a triangle and a number such that the area of the triangle - (number)^ = -^ of a square. Form the triangle from ?», — . ° 7)1 1 fi 1 Therefore its area = ju' :,, and let the number he m--^. — . 1)1, z m Hence G (G s \ m. or, 36;>i" - GO = a square = (6»i - 2)*. Therefore vi = .. , and the triangle must be formed from (-^, -A, the number being ^ . 5. To find a riglit-anglcd tri;nigle such that a given number exceeds the area by a S(juare. Number 10, triangle (3x', ix, 5x). Therefore lO-Gx-'-a square, ARITHMETICS. BOOK VI. 227 and a triangle and a number must be found sudi tli:it (nund.fr)* + area of triangle = -- of a square. Form a triangle frum m, - , and let the number be - + 5m. Dt m Therefore 260«i' + 100 = a square, or 65»i* + 25 ^- a square = (8ni + 5)^ say. Therefore m - 80. The rest is obvious. 6. To find a right-angled triangle such that th'' sum of the area and one side* about the right angle is a given number. Given number 7. Triangle {3x, 4x, 5x), therefore 6x"+3£c=7. ■^j +6.7 not being a square, is not possible. Hence we must siibstitute for (3, 4, 5) a right-angled triangle , , /onesideX* ^ . , sucli that ( — ^ j + I times the area = a square. Let one side be x, the other 1. 7 1 Therefore ^x + j = a. square, or lix + 1 = a square) Also, since the triangle is rational, x'+l = a squarei ' Now the difference — x^ — lix = x(x~ 14). Therefore, putting 24 7^ = 14a; + 1, we have x- -=-. Therefore the triangle is /24 25\ (-^ , 1, -;^ ), or we may make it (24, 7, 25). Going back, we take as the triangle (24.c, 7x, 25x). Therefore 84a;" + lx-l, and x^ - . 4 . / 7 25 Hence the triangle is ( 6, t , . 7. To find a right-angled triangle sucii that its area exceeds ouo of its sides by a given number. 1 N.B. For brevity and distinctness I slmll in future call llic flidcs about the ri«ht angle simply "sides," and not apply the term to the hyiwUjnuMC. which will always be called "hypotenuse." 1 .;— 2 228 DIOPHANTOS OF ALEXANDRIA. Given number 7. Therefore, as before, we have to find a right-angled triangle such that ( ~ ) + ^ times area = a square, i.e. the triangle (7, 24, 25). Let the triangle of the problem be (7a;, 24a;, 25a;). Therefore 84a;^ - 7a; = 7, and x= ^. 8. To find a right-angled triangle such that the sum of its area and both sides = a given number. Number 6. Again I have to find a right-angled triangle such /sum of sidesX' that ( ~ J -f times area = a square. Let «i, 1 1 .1 -1 ..1 p fm+l\- „ m" 7m 1 1 be the sides; therefore ( -— j +3m = ^ +-^ + - = a square, and m" -f- 1 = a square. Therefore vi' + 1 im + 1) , , , , V are both squares, m- + 1 J ^ ' and the difierence = 2«i . 7. 45 Therefore "'' ^ 28 ' (45 53\ Assume now for the triangle of the problem (45a;, 28a;, 53a;). Therefore G30a;' + 73a; = 6, and X is rational, 9. To find a right-angled triangle such that its area exceeds the sum of both sides by a given number. Number 6. As before we find subsidiary triangle (28, 45, 53). Therefore, taking for the required triangle (28a;, 45a;, 53a;), we find 6S0x- - 73a; = 6, and x= ^ . 65 10. To find a right-angled triangle such that the sian of its area, hypotenuse, and one side is a given number. Given number 4. Assuming hx, px, bx, ,-, - + hx ^-bx=i, and in order that this equation may have a rational solu- tion I must find a triangle such that /hypotenuse -f one sideV -f- 4 times area = a square. I AlUl'lI MIOTICS. BOOK VI. 229 ^[ake a right- angled triangle from ;«, m+ 1. Therefore /hyiJOtenuse + one sicle\ * /2«t* + 2//i + 1 + '2m + 1\* V 2 ; =v 2 ) = Hi* + im' + Gin' + 4//t + 1 and 4 times area = im {/a + 1) (2m + 1) which = Sm^ + 12/«- + im. Therefore m* + I2iu^ + ISm- + Sm + 1 ^ a square = (7/4- + Gm - 1)^ say. Hence m=,, 4 and the triangle must be formed from ( , - j, or (5, 9). Thus we must assume for the triangle of the problem the similar triangle {2Sx, 45.f, ~^3.c), and G30.C* + 81a-= 4. 4 Therefore x = — — . lOo 11. To find a right-angled triangle such that its area exceeds the sum of the hypotenuse and one side by a given number. Number 4. As before, Vjy means of the triangle (28, 45, 53) we get G30a;" — 81.'; = 4. Therefore x = ^ . 6 12. To find a right-angled triangle such that the difference of its sides is a square, and also the greater alone is a square, and, thirdly, its area -1- the less side - a square. Let the triangle be formed from two numbei-s, the gi-eater side being twice their product. Hence I must find two numbers such Uiat twice their product is a square and also exceeds the difference of their squares by a square. This is true for any two numbers of which the gi-eater - twice the less. Form then the triangle from x, 2x, and two conditions are fulfilled. The third condition gives Gx* + 3x* - a wjuarc, or 6x^ -I- 3 = a square. Therefore we must seek a number such that six times its square with 3 produces a square, i.e. 1, and an infinite number of others. Hence the triangle required is formed from 1, 2. Lemma. Given two numbers whose sum is a s(iuai-o, an infinite number of squares can be found which by multiplication with one of 230 DIOPHANTOS OF ALEXANDRIA. the given ones and the addition of the other to this product give squares. Given numbers 3, G. Let x" + 2a; + 1 be the square required, which will satisfy 3 (x^ + 2a; + 1) + 6 = a square, or 2>x- + G.r + 9 = a square. This indeterminate equation has an infinite number of solutions. 1 3. To find a right-angled triangle such that the sum of its area and either of its sides = a square. Let the triangle be (5a;, 12a;, 13x). Therefore 30.«' + 1 2a; = a square = 36a;^ say. Therefore 6a; = 12, and a; = 2. But SOx^ + 5x is not a square when x = 2. Therefoi'e I must find a square m^a;^ to replace 36a;" such that the value 12 —5 — :ryr of X IS veol and satisfies 30a;- + 5a; = a square, m - 30 rru- • 1 1 .-. .• 60m^+2520 This gives by substitution -, — z^^ ,, . . = a square. *' ^ m* - 60m + 1)00 ^ Therefore 60??r + 2520=a square. If then [by Lemma'] we had 60 m' + 2520 equal to a square, the equation could he solved. Now 60 arises from 5, 12, i.e. from the product of the sides of (5, 12, 13); 2520 is the continued product of the area, the greater side and the difference of the sides [30. 12.1235]. Hence we must find a subsidiary triangle such that the pro- duct of the sides + the continued product of greater side, difierence of sides and area - a square. Or, if we make the greater side a square, we must have [dividing by it], less side + product of difference of sides and area = a square. Therefore we must, given two numbers (area and less side), find some square such that if we multiply it by the area and add the less side, the result is a square. 2'his is done hy the Lemmas^ and the auxiliary triangle is (3, 4, 5). 1 Diophantos has expressed this rather curtly. If (h p b) bo the triangle (b>p), we have to make hp + ^bp . (b -p) b a Kquarc, or if b is a square, 2' + i ^'P U'-p) must be a square. ARITHMETICS. I50<)K VI. liSl Thus, if the original triangle is {Zx, ix', 5x), we have Gx-* + 4a;) , . 4 Let !»= — a —„ be the solution of the first equation. 9G 12 Therefore the second gives — ; — q-^; — 7. — Trr. + . " ^ = a S(iuarc. " 7Ji'-12//i' + 30 m'-G ' Hence 12 m^ + 24 = a square, and we must find a square such that twelve times it + 24 = a square [as in Lemma], Therefore m^ = 25, and a; = Tj-jT . ^. . , . , . , . /12 16 20\ Therefore the triangle required isLq, jq, -.q)- 14. To find a right-angled triangle such that its area exceeds either side by a square. The triangle found as before to l)c similar to (3, 1, 0), i.e. (3a;, ix, 5x). Therefore 6.r - 4a; = square ^ m" {< G). 4 Hence x = Q-m" 96 12 '''''^ (6 - my 6 - wr a square, or 24 + 12»t- a stjuaiv Let m = 1 say. Therefore 4 X ~ ^ , /12 16 A and the triangle i« v g . 5 ' - / • Or, putting m = z+l, we tiiid 3r + 6c+9 a square, ami 13 22 .1 z^^, 3+1^0 , SO that X- is rational. This relation can be satisfied in an infinite number of ways it b- pin a «,uaro, and also ;; + i />i). ., ,._ Therefore wo liave to find a triant^le such that Krcatcr side ^^luore. difference of sides = s(iuare, less side + area = square. Form the triangle from (u, h), therefore greater side =2.1,. which ib a Hqimro. if a -26, difference of sides =16^-36^= a square, less side + area -3&> + 0fc»= a square. 232 DIOPHANTOS OF ALEXANDRIA. 15. To find a right-angled triangle such that its area exceeds either the liypotenuse or one side by a square. Let the triangle be (3a;, ix, 5x). Therefore 6a;* -5a;) , ^, V are both squai-es. \)X — ox\ 3 Making the latter a square, we find x = ;, Cm? < 6). Therefore from the first ,^ ^, - t, ; = a square, or (G - m)- - m 15wt* - 36 = a square. This equation we cannot solve, since 15 is not the sum of two squares. Now 15??i^=the product of a squai-e less than the ai-ea, the hy})0teuuse, and one side ; 36 = the continued product of the area, one side, and the difference between the hypote- nuse and that side. Hence we must find a right-angled triangle and a square such that tlie square is < 6, a^id the continued pi'oduct of the square, the hyj)otenuse of the triangle, and one side of it exceeds the continued product of the area, the said side and the difference hetioeen the hypotenuse and that side by a square. [Lacuna and coiTuption in text']. Foi*m the triangle from two "similar plane numbers" [numbers of the form ah, oir], say 4, 1. This will satisfy the con- ditions, and let the square be 36. (< area.) The triangle is then (8*, \bx, 17a;). Therefore GO.r- - 8a; = 360;^^ say. 1 Thus x = y^, and the triangle ^^ (3' ^' y ) 1 Schulz works out the subsidiary part of this problem thus, or rather only proves the result given by Diophantos that the triangle must be formed from two "similar plane numbers'' a, aU- [i.e. a. 1 and ah. h.] ; and hyp. h = a-h^^-a-. greater side ij = a-b* - «-, less side k = 2a-b'\ area /= ^ kg. Now h-k = a^b* - 2a-b'^ + a" = {ab"^ - ay, ft square ; and hkz'^ - Jcfih - ft) is a square i{ z-=k (h - k) k, for, if we then divide by the square h - k and twice by the square kk, we get 2 (k-(i)^ia\ which is a square. AUlTilMETlCS. I'.uolv VI. 233 16. Given two numbers, if some square be multiplictl by one of them, and the other be subtracted, the result being u square, then another square can be found greater than the tii-st square wliich han the same property. [Leitwia to the following problem.] Numbei-s 3, 11, side of square 5, so that 3. 25 -11= 64 = a square. Let the required square be (.« + 5)*. Therefore 3 (a; + 5)- - 1 1 = 3.v" + 30x + 64 = a square = (8 - 2.r)» say. Hence x = 62. The side of the square = 67, and the square it.self = 4489. 17. 2h find a riyht-amjled triamjle such that the sum vf the area ami either the hypotenuse or one side = a square. We must first seek a triangle {h, k, (j) and a square s' such that hkz- - ka {h -k) = a, squai-e, and z' > «, the area. Let the triangle be formed from 4, 1, and the square be 36, but, the triangle being (8, 15, 17), the square is not > area. Therefore we must find another square to replace 36 by the Lemma in the preceding. But hk = 136, ka {h - i) = 480 . 9 = 4320. Thus 36 . 136 - 4320 = a square, and we want to find a larger square {m') than 36 such that 136?«' - 4320 - a square. Putting m = z+ 6, (s- + 122 + 36) 136 - 4320 = square, or, 136^' + 16322 + 576 = a square = (»z - 24)* say. This equation has any number of solutions, of which one gives 676 for the value of {z + 6)' [putting n = 16]. Hence, putting for the triangle (8x, 15.i-, 17x), we get 60x* + 8a; = 676x-*, Therefore ^ ^ 77 " 18. To find a right-anfied triangle such that the Hue hiscctiw/ nti acute angle is rational. Let the bisector (A D) = 5a; and one .section of tlu- has,- ( Itli) .ij, so that the perpendicular \x. 234 DIOPIIANTOS OF ALEXANDRIA. Let the whole base be some multiple of 3, say 3. Then CD = 3-3x. But, since AD bisects the i BAC, the hypotenuse = - (3 - 3a;), therefoi-e the hypotenuse = 4 - 4a;. Hence IGa;' - 32a; + 16 = 16a;= + 9, and a; = ^ . Multiplying throughout by 32, the perpendicular = 28, the base = 96, the hypotenuse = 100, the bisector =: 35. 19. To find a right-angled triangle such that the sum of its area and hypotenuse = a square, and its perimeter = a cube. Let the area = x, the hypotenuse = some square minus x, say 16 -a;; the product of the sides = 2x. Therefore, if one of the sides be 2, the other is x, and the perimeter = 18, which is not a cube. Therefore we must find a square which by the addition of 2 becomes a cube. Let the side of the square be {x+ 1), and the side of the cube (a:-l). Thei-efore a;^ - 3a;^ + 3a; - 1 = a;^ + 2a; + 3, from which a; = 4. Hence the side of the square is 5, and of the cube 3. Again, assuming area = x, hypotenuse = 25 - a?, we find that the perimeter = a cube (sides of triangle being x, 2). But (hypotenuse)'' = sum of squares of sides. Therefore of - 50a; + 625 = a;- + 4, 621 and x=-^. 20. To find a right-angled triangle such that the sum of its area and hg2>otenuse = a cube, and the perimeter = a square. Area x, hypotenuse some cube mimis x, sides x, 2. Therefore we have to find a cube which by the addition of 2 becomes a square. Let the side of the cube = m-1. (3 \* ^m + lj say. <¥)'• ' Put then the area a;, the sides x and 2, the hypotenuse ^y^ -x. (4913 \* —rrr xj = a;^ + 4 gives a;. 21 Thus wi = -J-, and the cube ■■ 4913 AlUTIIMETICS. ROOK VI. 235 21. To find a right-angled triancjle mch that thf sum of its area and one side is a square and its perimeter is a cube. Make a riglit-angk'il triangle from .r, x + 1. Therefore the i)erpendicular -2x+\, the base = 2x* + 2x, the hypotenuse = 2x* + 2x* + 1 . First, Ax- + Ga; + 2 = a cube, or (4a; ■(- 2) {x + 1 ) = a cube. If wo divide all tlie sides by x + 1 we have to make 4x + 2 a cube. Secondly, area + perpendicular = a square. ^, - 2x^ + 3xVx 2x+l Ihereiore — ; r-r^ — + -^ = a square. (x + 1)* X + 1 ^ 2x' + 5x* + 4x+l „ , Hence ^39" i " ^ 2x+ 1 =a square. But 4x+ 2 = a cube. Therefore we must find a cuU- which is double of a square. 3 Tlierefore 2x + 1 = 4, x = - , and .1 . ■ 1 • /8 15 17\ the triangle IS (^g, ^ , -j 22. To find a right-angled triangle such that tlie sum of its area and one side is a cube, while its perimeter is a square. Proceeding as before, we have to make 4x + 2 a squarej 2x + 1 a cube / ' Therefore the cube = 8, the square = IG, »-• = .,, and the triani , . /16 63 65\ 23. To fiml a right-angled triangle such that its perimeter is a square, and the sum of its perimet^ and area is a cube. Form a right-angled triangle from x, 1. Therefore the sides are 2.7;, x-*- 1, and the hypotenuse x* + 1. Hence 2x* + 2x should be a square, and x' -I- 2x* -f x a cul>c. It is easy to make 2x'' + 2x a square : let it ^ 7«V. 2 Therefore x- ^"' ^ , and from the second condition m* -2 8 8 _2^ {m' - 2)' "^ {m' - 2)' "^ m' - 2 must be a cube, i.e. 7-^ — rr-3 = a culto. (w -J) 236 DIOPHANTOS OF ALEXANDRIA. Therefox'e 2m* = a cube, or 2m = a cube = 8 say. 2 1 I Thus ??i = 4, .X' = r7 = - . ^^^ *" ^ 7a • 14 7 49 But foi* one side of the triangle we have to subtract 1 from this, which is impossible. Therefore I must find another value of a; > 1 : so that m" > 2 < 4. And I must find a cube such that \ of the square of it > 2 < 4. Let it be n^, so that ?i" > 8 < 1 G, This is satisfied by , 729 3 27 -"=G4''^==T- 97 729 512 Therefore m=^^, nr = . _ _ , x = ^^^ , and the square of this 16 25G 21/ > 1. Thus the triangle is known. 24. 7'o Jiiid a right-angled triangle such that its 2)erimeter is a cube and the sum of its perimeter and area = a square. (1) We must first see how, given two numbers, a triangle may be formed whose perimeter = one of the numbers, and whose area - the other. Let 12, 7 be the numbers, 12 being the perimeter, 7 the area. Therefore the product of the sides = 14 = - . 14.u Thus the hypotenuse = 12 — ; — 1 4x'. Therefore from the right-angled triangle 1 24 1 172 + 4 + 19Ga;'' - 336x - — = -^ + 196a;^ a;- X X or, 172 - 336a;- ^- = 0. ' X This equation gives no rational solution, unless 86"- 24. 336 IS a square. But 172 :^ (perimeter)- + 4 times area, 24 . 336 = 8 times area multiplied by (perimeter)". (2) Let now the area = x, the perimeter = any number which is both a square and a cube, say 64. Therefore ( — a ] - 8 . 64" . a; must be a square, or, 4a;' - 2 4 5 7 6a; + 4 1 9 4 3 4 ^ a square. AUITMMETICS. I500K VI. li.ST Therefore x' - GlU.c + 1048570 is a square.) Also X + 04 is a square./ To solve this double equation, multiply the second equation by such a square as will make the absolute t<.'rm the same as in the first. Then, taking the difference and factors, itc, the equations are solved. [In the text we find i$i(Tw(r$o} aoi ol dpiO/JLoi, which, besides being ungrammatical, would seem to be wrong, in that dpiOfjiOL is used in an unprecedented manner for /loraoe?.] 25. To find a right-angled triangle such that the square of its hypotenuse = the sum of a square and its side, i'odicct of the greatest and the middle + tlie square of the least = a square whose side is (greatest + twice middle number). Let the numbei-s be AB, BG, BD (in fig.) we have to prove 8 {AB){BG) + {BDy- = [AB + 2BGy. E A B..D...G Now AB = BG+GI). Therefore SAB . BG - 8 (BG' + BG . GD) = iAB . BG + iBG' + 4BG . GD. and iBG . GD +DB'^ AB' [for AB=BG + GD, DB = BG-GD\ and we have to seek how AB" + iAB. BG -\- iBG^ can be made a square. Take AE^ BG. Therefore iAB . BG = iAB . AE. This together witli ABG' or iAE' makes iBE.EA, and this together with AB' = [BE+EA)- = (AB + 2BGy. 3. If there are any numbers in A. p. the difference of the greatest and the least > the common difference in the ratio of the number of terms dimiuiahcd by 1. POLYGONAL NUMBERS. 239 Let AB, BG, BD, BE... he in a. p. B.A..G..D.. E Therefore we must have, difference of AB, BE^ (difference of AB, BG) X (number of terms- 1). AG, GD, DE arc all equal. Therefore EA = AG >i (number of the terms AG, GD, DE) ^ AG x (number of term.s in series- 1). Therefore (kc. 4. If there are any numbers in a.p. {greatest + least) x number of terms = double the sum of all. [2s = 7i{l + a).] Let the numbers be ^, 2^, C, D, E, F. (A +F) X the number of them shall be twice the sum. A.B.C.D.E.F H.L.M.K...G The number of terms is either even or odd ; and let their number be the number of units in IIG. First, let the number be even. Divide IIG into two equal parts at A'. Now the difference of i^, Z) = the difference of C, A. Therefore F+ A =C + I),h\it F + A = {F+ A) HL. Hence C + D = {F+A)LM, E+B = {F+A)MK. Therefore A + B + ... = (F+A) UK. And {F + A) IIG ^twice (A +B +...). 5. Secondly, let the number of terms be odd, A, B, C, D, E, and let there be as many units in FU as there are terms, «J:c. A.B.C.D.E F.G.K.II 6. If titer e are a series of numbers beginning loilh 1 and increas- ing in A. p., then the sicm of all x eight times the common difference + the square of {common difference - 2) = a square, whose side dimin- ished by 2^ the common difference multijilied by a number, which increased by 1 is double of the number of terms. [Let the a.p. be 1, \ + a, ... 1 + n - 1 . a. Therefore we have to prove s.8a + {a-2y = {a{2n-\) + 2y, i, e. 8as = 4a V - 4 (a - 2) na, or 2» - an' - (o - 2) « = n (2 + n- la)]. 240 DIOPHANTOS OF ALEXANDRIA. Proof. Let AB, GD, EZ be numbers in A. p. starting from 1. A.K..N...B G D E.L Z H.M X—T Let HT contain as many units as there are terms including \. Difference between EZ and 1 = (difference between -4 5 and 1) X a number 1 less than IIT [Prop. 3]. Put AK, EL, HM each equal to unity. Therefore LZ=MT.KB. Take KN = 2 and inquire whether the sum of all x eight times KB + square on NB makes a square whose side diminished by 2 = KB X sura of HT, TM. Sum of all = I product {ZE + EL) .IIT=\ {LZ + 2EL) HT, and LZ= AIT . KB from above. Therefore the sum = \ (KB . iVT . TH+ 2TH), or, bisecting MT at X, the sum = KB . TH . TX+ HT. Thus we inquire lohether KB. TH. TX. SKB + 8KB . HT + square on KB is a square. Now SHT . TX . KB' = iHT . TM . KB', and SKB . HT = AHM. KB + i (HT + TM) KB. Therefore toe must see lohether i.HT. TM. KB' + iHM. KB + 4 (HT + TM) KB + NB"- is a square. But 4/7.1/ . KB = 2KB . NK, and 2KB.NK+NB-=KB- + KN% and again /JA'^ = HM' . BK\ and HM\BK"- + UlT . TM . BK'= {HT+ TMf BK\ Hence our expression becomes {HT+ TMf Bid + 4 {HT + TM) KB + A'iV^^ A.K..N ...B R H.M A'— T or, putting {HT + TM) BK= NR, NR' + iNR + KN' and 4.NR ^ 2NR . NK. Therefore the given expression is a square whose side is RK, and RK -2 = NR, which is KB {HT + TM), and HT+ TM+ 1 = twice the number of terms. Thus th(! proposition is proved. 7. Let POLYGONAL NUMBERS. HT+TM^A, KB=B. K 1j 241 Therefore square on .1 x square on B = square ou G, where G = {HT+TM)KB. Let DE = A, EZ =^ B, in a straight line. Complete squares DT, EL, and complete TZ. Then DE : EZ^DT : TZ, and TE ■ EK=TZ : EL. Therefore TZ is a mean proportional between the two squares. Hence the product of the squares = the square of TZ, and DT^ {IIT -h TMf, ZK = square on KB. Thus the product (HT + T2If. KB' = NB^. 8. If there are any number of terms heginning from 1 in a. p. the svm is a jiolygonal number, for it has as many angles as the common difference increased by 2 contains units, ami its side = the number of terms inclibding the term 1. The numbers being represented in the figure, (sum of series multiplied by ^KB) + NB- - RK\ O.A.K..N...B RG- II . M A'- -D /•; . L- -T Therefore, taking another unit AO, KO - 2, KN -- 2, and OB, BK, BN are in arithmetical progression, so that S.OB.BK + BN' = {OB + 2BKy, [Prop. 2], and OB + 2BK- OK - ZKB an.l 3+12.2, or 3 is one less than the double of the common difference of OB, BK, BN. Now as the sum of the terms of the j)rogressiou, including unity, It; H. D. 242 DIOPHANTOS OF ALEXANDRIA. is subject to the same laws as Oi? ', while OB is any number and OB always a polygonal (the first term being AO [1] and AB the term next after it) whose side is 2, it follows that the sum of all terms in the progression is a polygonal equiangular to OB, and having as many angles as there are units in the number which exceeds by OK, or 2, the difference KB, and the side of it is HT which = number of terms, including 1. And thus is demonstrated what is said in Hypsikles^ definition. If there are any numbers increasing from unity by equal intervals, when the interval is 1, the sum of all is a tri- angular number : wlien 2, a square: when 3, aj)entagon and so on. And the number of angles = 2 + common difference, the side = number of terms including 1. So that, since we have triangles when the diffei'ence = 1, the sides of them will be the greatest term in each case, and the product of the greatest term and the greatest term increased by 1 - twice the triangle. And, since OB is a polygonal and has as many angles as units, and when multiplied by 8 times (itself - 2) and increased after multiplication by the square of (itself — 4) [i.e. NB-] it becomes a square, the definition of polygonal numbers will be : Every polygonal multiplied 8 times into (number of angles — 2) + square of (number of angles — 4) = a squax'e. The Hypsiklean definition being proved, it remains to show how, given the sides, we may find the numbers. Now having the side HT and the number of angles we know also KB, therefore we have {IIT + TM) KB = NR. Hence KR is given [NK^1\ * This result Nesselmann exhibits thus. Take the aiithmetical progression 1, 6 + 1, 2& + l...(K-l)t + l. If s is the sum, Qsh + (l) - 2)'^=[h (2k - 1) + 2p, If now we take the three terms 6-2, h, h + 2, also in a. p., 8b(?;+2) + (?i-2)' = [(6 + 2) + 26]' = (3!> + 2)2, Now 6 + 2 is the sum of the first two terms of first series; and 3 = 2.2-1, therefore 3 corresponds to 2h - 1. Hence s and h + 2 are subject to the same law. POLYGONAL NUMBERS. 243 Therefore we know also the square of KR. Subtracting from it the square of NB, we have tlie remaining term which = number x '^KB. Similarly given the number we can find the side. 9. Rule. To Jhid the number from the side. Take the side, double it, subtract 1, and multiply the remamder by (number of angles - 2). Add 2 to the product, and from the square of the number subtract the square of (number of angles — 4). Dividing the remainder by 8 times (number of angles - 2), we find the required polygonal. To filed the side from the numher. Multiply it l>y 8 times (number of angles - 2), add to the product the square of (number of angles - 4). We thus get a square. Subtract 2 from the side of this square and divide remainder by (number of angles - 2). Add 1 to quotient and half the result is the side required. 10. [A fragment.] Given a numher, to find in how viany loays it can he a polygcmal. Let AB be the given number, BG the number of angles, and in BG take GD = 2, GU - 4. A . T B E..D..G K Z H Therefore, since the polygonal AB has BG angles, %AB . BD + BE- = a square = ZIP say. Take in AB the length AT=\. Therefore MB . BI)= iAT . BD + i (AB + TB) BD. Take DK=i{AB+TB), and for AAT.BD put 2BD . DE. Therefore ZIP = KD . BD + 1BD .DE + BE*, but 2BD . DE + BE' = BD' + DE\ Hence ZU ' ^ KD . BD + BD* + DE\ and KD . BD + BD^ - KB . BD. Thus Zir=KB.BD+DE\ and, since DK = 4 {AB + TB), DK> 4 J T > 4, and half 4 - DG, GK>GD. 244 DIOPHANTOS OF ALEXANDRIA. Therefore, if DK is bisected at L, L will fall between G and K, and the sqiiare on LB = LD' + KB . BD. A . T B E..D..G L K Z H N M Therefore ZE' = BU - LD' + DE\ or ZH' + DL' = BU- + DE\ and LD"-~DE' = LB'~ZH\ Again since ED = DG and DG is produced to L, EL.LG + GD'=DL\ Therefore DL' - DG' = DL' - DE' = EL . LG. Hence EL . LG = LB' ~ ZIP. Put ZM = BL {BL being > ZII). Therefore ZM' - ZH' = EL . LG ; but DK is bisected in L, so that DL = 2 (AB + BT) ; and DG = 2 AT. Therefore GL = iB T, and BT-^^GL, but also AT {ov l) = ^^6-'(or 4). Therefore AB = \ EL, but TB also = \ GL. 4 4 Hence AB.TB=^EL. LG, or EL.LG=1(JAB.BT. Thus UAB.BT = MZ' - ZII ' = 21 H ' + 2ZH . II M. Therefore IIM is eve^i. Let it be bisected in JV [Here the fragment ends.] INDEX [The references are to pages.] Ab-kismet, 41 u. Abu'lfaraj, 2, 3, 12, 13, 41 Abu'l-Waffi Al-Biizjfmi, 13, 25—20, 40—42, 148, 155, 157 Abu Ja'far Mohammed ibn AUiusain, 156 Addition, how expressed by Diophantos, 69 ; Bombelli's sign for, 45 ; Vieta's, 78 «. Algebraic notation, three stages of, 77 —SO aljabr, 40, 92, 149—150, 158 Alkarkhi, 24—25, 71 ?;., 156—159 Al-Kharizmi, see Mohammed ibn Mfisfi almuktibahi, 92, 149—150, 158 Al-Nadim, 39, 40 ii. Al-Shahrastani, 41 Alsirfij, 24 u., 159 avaipopiKos of Hj-psikles, 5 6x)pl(TTu%, iv doplarcf), 140 ApoUonios, 4, 8, 9, 23 Approximations, 117—120, 147 Apukius, 15 Arabian scale of powers compared with that of Diophantos, 70—71, 150— 151 Arabic translations, Ac, 23, 24, 25, 39—42, 148—159 Archimedes, 7, 142, 143, 144, 146, 147 Aristoxenos, 14, 15 Arithmetic and Geometry, 31, 141— 142 'ApiOfiriTLKo. of Diophantos, 33 and pas- sim apidfiijriKri and XoyiaTiK-f), IH, 136, 145 dpidfioi, 6 ; Diophantos' technical use of the word, 57, 150; his sj-mbol for it, 57- 66, 137—138, 160 apidfiOffTov, 74 Ars rei et census, 21 h. Auria, Joseph, 51, 56 Autolykos, 5 Bacchios 6 y^pwv, 14, 15, 16 Bachet, 49 — 53 and passim "Back-reckoning," H5 — 86, 114; ex- amples of, 110, 111, and in the ap- pendix passim Bhaskara, 153 Billy, Jacobus de, 3, 54 Blancauus, 3 Bombelli, 13, 14, 15, 23, 35, 36. 42— 45, 52, 134—135; his algebraic no- tation, 45, 68 Bossut, 32, 38, 90 n., 138—139 n. Brahmagupta, 153 Brassinne, 221 n. Camcrarius, Joachim, 2, 42 Cantor, 55 h., 58, 59. 67, 141 n., 151. 152, 156, 157 Cardan, 43, 46, 70 Casiri, 41 n. Cattle-problem, the, 7, 142—117 Censo, 70 Coefficient, 93 «. Colebrooke, 12, 19 n., 33, 133. 136. 137 n. Cosa, 45, 70 Cossali, 1, 3, 10, 12, 31, 36, 41 n., 43 n.. 49, 51, 70, 71. 107 n.. 133. 136. HO. 169 ;i.. 220 ;i. Tridhara, 153 Cubes : transformation of a Bom of two cube-s into the difference of two others, and vice rer$ii, 123—125 Cubic equation. 30, 93—91. 114 246 Data of Euclid, 140 Dedication to Dionysios, 136 Definitions of Diophantos, 28, 29, 57, 67, 7-4, 137, 138, 163 Determinate equations : see contents ; reduction of, 29, 149—150 Diagonal numbers, 142 Didymos, 14, 15, 16 Digby, 23 Dioi^hautos, see contents s , . - J 35, 98 Division, how represented by Diophan- tos, 73 Double-equations of the first and second degrees, 98 — 107 ; of higher degrees, 112—113 Svva/jLis and the sign for it, 58 n., 62, 63, 66 7i., 67, 68, 140, 151 ; dvfa/xis and Terpaywvoi, 67 — 68 dwa/xoduva/jus and the sign for it, 67 — 68 dvvainoOvvafJ.oa'Toi', 74 dvva/jMKv^os and the sign for it, 65 ii., 67—68 Swa/JLOKV^offTov, 74 dwafioarov, 74 er5os = power, 29 7i. Elements of EucUd, 4, 5, 142, 158 Epanthema of Thymaridas, 140 Epigrams, 2, 6, 7, 9, 142—147, 223 Equality, Diophantos' expression of, 75—76 ; Xylander's sign for, 76 Equations, classes of, see contents; reduction of determinate equations, 29, 149-150 Eratosthenes, 5 Euclid, Elements, 4, 5, 142, 158; Data, 140 Eudemos, 67 Eunapios, 13 Fabricius, 1, 5, 14 Fakhn, the, 24—25, 71 n., 156—159 Fermat, 13, 23, 53, 54, 68, 123, 124, 125, 126, 128, 129, 130, 131, 221 n. Fihrist, the, 39, 40, 41, 42 Fractious, representation of, 73 — 75 Gardthausen, 60, 64 Geminos, 18, 145—146 Geometry and algebra, 140 — 141, 151 —153, 156, 158 Geometry and arithmetic, 31, 141 — 142 Girard, Albert, 3 n., 55 Gow on Diophantos, 64 — 66 7i., 137 n., 160 Hankel, 83—85, 129 n. Harmonics of Diophantos, 14 ; of Pto- lemy, 15 Harriot, 78 n. Heiberg, 146—147, 160 Heilbronner, 3 Herakleides Ponticus, 16 Heron of Alexandria, 141, 153 Hipparchos, 5, 141 Hippokrates, 67 History of the Dynasties, see Abu'Ifaraj Holzmann, Wilhelm, see Xylander Hultsch, 146 n. Hypatia, 1, 8, 9, 10, 11, 17, 38, 39 n. Hypsikles, 4, 5, 6, 135, 242 Z for tffos, 75 lambHchos, 78, 79, 140 Identical formulae, 125 Indeterminate equations, 94 — 113, 144, 146, 147, 157, 158, 159 Irrationality, Diophantos' view of, 82 Isidoros, 5 Italian scale of powers, 70, 71 jabr, 40, 92, 149—150, 158 jiclr, 150 John of Damascus, 8 John of Jerusalem, 8 ka'b, 71 n., 157, 158 Kitab AljUtrist, 39 Kliigel, 11, 90 n., 144 Kostfi ibn Luk:l, 40 KvfioKvpos and the sign for it, 67—68 KvjBos and the sign for it, 58 n., 62, 63, 66h., 67— 68 Kuster, 8 INDEX. 247 Lato, 70 Lehmann, 60 Xfr^ty, and the symbol for it, 66 ;i., TI- TS, 137, 163 Xei^tj iirl \e'i\pLv ■jroWairXaffiaffOuaa TTOtet virap^LV, 13T n. \i6((>avTos or Aew^aj'Tos, 14 Lessing, 142, 143, 144, 146 h. Limits, jnethod of, 86, 8T, 115— IIT ; approximation to, IIT — 120 \(yyi.