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§ 8. Discussion in this connection of the Cattle-problem.
DIOPHANTOS AND THE EARLY ARABIAN ALGEBRAISTS.
§ 1. Preliminary
§ 2. Comparison of Diophantos with Mohammed ibn Mūsā
§ 3. Diophantos and Abu'l-Wafā
§ 4. An anonymous Arabic мs.
DIOPHANTOS OF ALEXANDRIA.
§ 1. THE doubts about Diophantos begin, as has been remarked by Cossali', with his very name. It cannot be positively decided whether his name was Diophantos or Diophantes. 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 Διοφάντου Αλεξανδρέως Αριθμ ητικών βιβλία ιγ. Now Διοφάντου may be the Genitive of either Διόφαντος or Διοφάντης. We learn however from this title that he lived at Alexandria.
(2) In Suidas under the article 'Taría the name occurs in the Accusative and in old editions is given as Διοφάντην ; but Bachet' in the Preface to his edition of Diophantos assures us that two excellent Paris MSS. have ▲ιópavтov. Besides this, Suidas has a separate article Διόφαντος, ὄνομα κύριον. Moreover Fabricius mentions several persons of the same name Διόφαντος, but the name Διοφάντης nowhere occurs. It is on this ground probable that the correct form is Διόφαντος. We may compare it with "Expаvтos, but we cannot go so far as to say, with Bachet, that Acopávτns is not Greek; for we have the analogous forms Ἱεροφάντης, συκοφάντης.
1 "Su la desinenza del nome comincia la diversità tra gli scrittori" (p. 61). 2 "Ubi monendus es imprimis, in editis Suidae libris male haberi, els Acopávтηv, ut ex duobus probatissimis codicibus manu exaratis qui extant in Bibliotheca Regia, depraehendi, qui veram exhibent lectionem eis Acópavтov.”
(3) In the only quotation from Diophantos which we know Theon of Alexandria (fl. 365-390 A.D.) speaks of him as Διόφαντος.
On the other hand Abu'lfaraj, the Arabian historian, in his History of the Dynasties, is thought to be an authority for the form Diophantes, and certainly in his Latin translation of the two passages in which D. is mentioned by Abu'lfaraj, Pococke writes Diophantes. But, while in the first of the two passages in the original the vowel is doubtful, in the second the name is certainly Diophantos. Hence Abu'lfaraj is really an authority for the form Diophantos.
(5) of more modern writers, Rafael Bombelli in his Algebra, published 1572, writes in Italian " Diofante sponding to Διοφάντης. But Joannes Regiomontanus, Joachim Camerarius, James Peletarius, Xylander and Bachet all write Diophantus.
We may safely conclude, then, that Diophantos was the name of our author. Far more perplexing than the doubt as to his name is the question of the time at which he lived. no certainty can even now be said to have been reached on this point, it will be necessary to enumerate the indications which bear on the question. Before proceeding to consider in order the internal and external evidence, it will be well to give the only facts which are known of his personal history, and which can be gathered from an arithmetical epigram upon Diophantos. This epigram, the probable date of which it will be necessary to consider later along with the question of its authorship, is as follows :
Οὗτός του Διόφαντον ἔχει τάφος, ὦ μέγα θαῦμα,
Καὶ τάφος ἐκ τέχνης μέτρα βίοιο λέγει.
The solution of this epigram-problem gives 84 as the age at which Diophantos died. His boyhood lasted 14 years, his beard grew at 21, he married at 33; a son was born to him 5 years later and died at the age of 42, when his father was 80 years old. Diophantos' own death followed 4 years later at the age of 84. Diophantos having lived to so great an age, an approximate date is all that we can expect to find for the production of his works, as we have no means of judging at what time of life he would be likely to write his Arithmetics.
§ 2. The most important statements upon the date of Diophantos which we possess are the following:
(1) Abu'lfaraj, whom Cossali calls "the courageous compiler of a universal history from Adam to the 13th century,” in his History of the Dynasties before mentioned, places Diophantos, without giving any reason, under the Emperor Julian (361363 A.D.). This is the view which has been ordinarily held. It is that of Montucla.
(2) We find in the preface to Rafael Bombelli's Algebra, published 1572, a dogmatic statement that Diophantos lived under Antoninus Pius (138-161 A.D.). This view too has met with considerable favour, being adopted by Jacobus de Billy, Blancanus, Vossius, Heilbronner, and others.
Besides these views we may mention Bachet's conjecture, which identifies the Diophantos of the Arithmetics with an astrologer of the same name, who is ridiculed in an epigram attributed to Lucilius; whence Bachet concludes that he lived about the time of Nero (54-68) (not under Tiberius, as Nesselmann supposes Bachet to say). The three views here mentioned will be discussed later in detail, as they are all worthy of consideration. The same cannot be said of a number of other theories on the subject, of which I will quote only one as an example. Simon Stevin' places Diophantos later than the Arabian algebraist Mohammed ibn Mūsā
1 Les Oeuvres Mathém. de Sim. Stevin, augm. par Alb. Girard, Leyden, 1634. "Quant à Diophant, il semble qu'en son temps les inventions de Mahomet ayent seulement esté cognues, comme se peult colliger de ses six premiers livres."