INTRODUCTION CHAPTER I DIOPHANTUS AND HIS WORKS THE divergences between writers on Diophantus used to begin, as Cossali said1, with the last syllable of his name. There is now, however, no longer any doubt that the name was Diophantos, not Diophantes2. The question of his date is more difficult. Abū'lfaraj, the Arabian historian, in his History of the Dynasties, places Diophantus under the Emperor Julian (A.D. 361-3), but without giving any authority; and it may be that the statement is due simply to a confusion of our Diophantus with a rhetorician of that name, mentioned in another article of Suidas, who lived in the time of Julian'. On the other hand, Rafael Bombelli in his Algebra, 1 Cossali, Origine, trasporto in Italia, primi progressi in essa dell' Algebra (Parma, 1797–9), I. p. 61: "Su la desinenza del nome comincia la diversità tra gli scrittori." 2 Greek authority is overwhelmingly in favour of Diophantos. The following is the evidence, which is collected in the second volume of Tannery's edition of Diophantus (henceforward to be quoted as "Dioph.," "Dioph. II. p. 36" indicating page 36 of Vol. II., while "Dioph. 11. 20" will mean proposition 20 of Book 11.): Suidas s.v. Trarla (Dioph. 11. p. 36), Theon of Alexandria, on Ptolemy's Syntaxis Book 1. c. 9 (Dioph. 11. p. 35), Anthology, Epigram on Diophantus (Ep. xiv. 126; Dioph. 11. p. 60), Anonymi prolegomena in Introductionem arithmeticam Nicomachi (Dioph. II. p. 73), Georgii Pachymerae paraphrasis (Dioph. II. p. 122), Scholia of Maximus Planudes (Dioph. 11. pp. 148, 177, 178 etc.), Scholium on Iamblichus In Nicomachi arithm. introd., ed. Pistelli, p. 127 (Dioph. 11. p. 72), a Scholium on Dioph. 11. 8 from the MS. “A” (Dioph. 1. p. 26ο), which is otherwise amusing (Η ψυχή σου, Διόφαντε, εἴη μετὰ τοῦ Σατανά ἕνεκα τῆς δυσκολίας τῶν τε ἄλλων σου θεωρημάτων καὶ δὴ καὶ τοῦ παρόντος θεωρή paros, "Your soul to perdition, Diophantus, for the difficulty of your problems in general and of this one in particular "); John of Jerusalem (10th c.) alone (Vita Ioannis Damasceni XI. Dioph. II. p. 36), if the reading of the MS. Parisinus 1559 is right, wrote, in the plural, ὡς Πυθαγόραι ἢ Διόφανται, where however Διόφανται is clearly a mistake for Διόφαντοι. 3 Λιβάνιος, σοφιστής ̓Αντιοχεύς, τῶν ἐπὶ Ἰουλιανοῦ τοῦ βασιλέως χρόνων, καὶ μέχρι Θεοδοσίου τοῦ πρεσβυτέρου· Φασγανίου πατρός, μαθητὴς Διοφάντου. H. D. I published in 1572, says dogmatically that Diophantus lived under Antoninus Pius (138–161 A.D.), but there is no confirmation of this date either. The positive evidence on the subject can be given very shortly. An upper limit is indicated by the fact that Diophantus, in his book on Polygonal Numbers, quotes from Hypsicles a definition of such a number'. Hypsicles was also the writer of the supplement to Euclid's Book XIII. on the Regular Solids known as Book XIV. of the Elements; hence Diophantus must have written later than, say, 150 B.C. A lower limit is furnished by the fact that Diophantus is quoted by Theon of Alexandria2; hence Diophantus wrote before, say, 350 A.D. There is a wide interval between 150 B.C. and 350 A.D., but fortunately the limits can be brought closer. We have a letter of Psellus (11th c.) in which Diophantus and Anatolius are mentioned as writers on the Egyptian method of reckoning. "Diophantus," says Psellus, "dealt with it more accurately, but the very learned Anatolius collected the most essential parts of the doctrine as stated by Diophantus in a different way (reading érépws) and in the most succinct form, dedicating (poσepávnσe) his work to Diophantus." It would appear, therefore, that Diophantus and Anatolius were contemporaries, and it is most likely that the former would be to the latter in the relation of master to pupil. Now Anatolius wrote about 278-9 A.D., and was Bishop of Laodicea about 280 A.D. We may therefore safely say that Diophantus flourished about 250 A.D. or not much later. This agrees well with the fact that he is not quoted by Nicomachus (about 100 A.D.), Theon of Smyrna (about 130 A.D.) or lamblichus (end of 3rd c.). 1 Dioph. I. p. 470-2. 2 Theo Alexandrinus in primum librum Ptolemaei Mathematicae Compositionis (on c. 1x.) : see Dioph. II. p. 35, καθ ̓ ὁ καὶ Διόφαντός φησι· τῆς γὰρ μονάδος ἀμεταθέτου οὔσης καὶ ἑστώσης πάντοτε, τὸ πολλαπλασιαζόμενον εἶδος ἐπ ̓ αὐτὴν αὐτὸ τὸ εἶδος ἔσται κ.τ.έ. 3 * Dioph. II. p. 38-9: περὶ δὲ τῆς αἰγυπτιακῆς μεθόδου ταύτης Διόφαντος μὲν διέλαβεν ἀκριβέστερον, ὁ δὲ λογιώτατος ̓Ανατόλιος τὰ συνεκτικώτατα μέρη τῆς κατ' ἐκεῖνον ἐπιστήμης ἀπολεξάμενος ἑτέρω (? ἑτέρως οι εταίρῳ) Διοφάντῳ συνοπτικώτατα προσεφώνησε. The MSS. read érépw, which is apparently a mistake for érépws or possibly for éralow. Tannery conjectures τῷ ἑταίρῳ, but this is very doubtful; if the article had been there, Διοφάντῳ τῷ éralpy would have been better. On the basis of eralpy Tannery builds the further hypothesis that the Dionysius to whom the Arithmetica is dedicated is none other than Dionysius who was at the head of the Catechist school at Alexandria 232-247 and was Bishop there 248-265 A.D. Tannery conjectures then that Diophantus was a Christian and a pupil of Dionysius (Tannery, "Sur la religion des derniers mathématiciens de l'antiquité," Extrait des Annales de Philosophie Chrétienne, 1896, p. 13 sqq.). It is however difficult to establish this (Hultsch, art. "Diophantos aus Alexandreia" in PaulyWissowa's Real-Encyclopädie der classischen Altertumswissenchaften). The only personal particulars about Diophantus which are known are those contained in the epigram-problem relating to him in the Anthology'. The solution gives 84 as the age at which he died. His boyhood lasted 14 years, his beard grew at 21, he married at 33; a son was born to him five years later and died, at the age of 42, when his father was 80 years old. Diophantus' own death followed four years later. It is clear that the epigram was written, not long after his death, by an intimate personal friend with knowledge of and taste for the science which Diophantus made his life-work". The works on which the fame of Diophantus rests are: Six Books of the former and part of the latter survive. (3) A collection of propositions under the title of Porisms; in three propositions (3, 5 and 16) of Book v. Diophantus quotes as known certain propositions in the Theory of Numbers, prefixing to the statement of them the words "We have it in the Porisms that......” (ἔχομεν ἐν τοῖς Πορίσμασιν ὅτι κ.τ.έ.). A scholium on a passage of Iamblichus where he quotes a dictum of certain Pythagoreans about the unit being the dividing line (μelópiov) between number and aliquot parts, says "thus Diophantus in the Moriastica......for he describes as 'parts' the progression without limit in the direction of less than the unit." Tannery thinks the MopiaσTIKά may be ancient scholia (now lost) on Diophantus I. Def. 3 sqq.; but in that case why should Diophantus be supposed to be speaking? And, as Hultsch 1 Anthology, Ep. XIV. 126; Dioph. 11. pp. 60-1. 1 The epigram actually says that his boyhood lasted of his life; his beard grew after more; after more he married, and his son was born five years later; the son lived to half his father's age, and the father died four years after his son. Cantor (Gesch. d. Math. 13, p. 465) quotes a suggestion of Heinrich Weber that a better solution is obtained if we assume that the son died at the time when his father's age was double his, not at an age equal to half the age at which his father died. In that case Zx+12x+7x+5+}(x−4)+4=x, or 3x=196 and x=65§. This would substitute 10 for 14, 161 for 21, 25 for 33, 30 for 42, 611 for 80, and 65 for 84 above. I do not see any advantage in this solution. On the contrary, I think the fractional results are an objection to it, and it is to be observed that the scholiast has the solution 84, derived from the equation Jx+x+4x+5+Zx+4=x. * Hultsch, art. Diophantos in Pauly-Wissowa's Real-Encyclopädie. • Iamblichus In Nicomachi arithm, introd. p. 127 (ed. Pistelli); Dioph. II. p. 72. 5 Dioph. II. p. 72 note. |
