Al-Khārizmi who lived in the first half of the 9th century, the absurdity of which view will appear. We must now consider in detail the (a) Internal evidence of the date of Diophantos. (1) It would be natural to hope to find, under this head, references to the works of earlier or contemporary mathematicians. Unfortunately there is only one such reference traceable in Diophantos' extant writings. It occurs in the fragment upon Polygonal Numbers, and is a reference to a definition given by a certain Hypsikles'. Thus, if we knew the date of Hypsikles, it would enable us to fix with certainty an upper limit, before which Diophantos could not have lived. It is particularly unfortunate that we cannot determine accurately at what time Hypsikles himself lived. Now to Hypsikles is attributed the work on Regular Solids which forms Books XIV. and XV. of the Greek text of Euclid's Elements. In the introduction to this work the author relates that his father knew a treatise of Apollonios only in an incorrect form, whereas he himself afterwards found it correctly worked out in another book of Apollonios, which was easily accessible anywhere in his time. From this we may with justice conclude that Hypsikles' father was an elder contemporary of Apollonios, and must have died before the corrected version of Apollonios' treatise was given to the world. Hypsikles' work itself is dedicated to a friend of his father's, Protarchos by name. Now Apollonios died about 200 B.C.; hence it follows that Hypsikles' treatise 1 Polyg. Numbers, prop. 8. “καὶ ἀπεδείχθη τὸ παρὰ Ὑψικλεῖ ἐν ὅρῳ λεγόμενον.” 2 “καί ποτε διελοῦντες (sc. Basileides of Tyre and Hypsikles' father) τὸ ὑπὸ ̓Απολλωνίου γραφὲν περὶ τῆς συγκρίσεως τοῦ δωδεκαέδρου καὶ τοῦ εἰκοσαέδρου τῶν εἰς τὴν αὐτὴν σφαίραν ἐγγραφομένων, τίνα λόγον ἔχει πρὸς ἄλληλα, ἔδοξαν ταῦτα μὴ ὀρθῶς γεγραφέναι τὸν ̓Απολλώνιον. αὐτοὶ δὲ ταῦτα διακαθάραντες ἔγραψαν ὡς ἦν ἀκούειν τοῦ πατρός. ἐγὼ δὲ ὕστερον περιέπεσον ἑτέρῳ βιβλίῳ ὑπὸ ̓Απολλωνίου ἐκδεδομένῳ, καὶ περιέχοντι ἀπόδειξιν ἡγιῶς (?) περὶ τοῦ ὑποκειμένου. καὶ μεγάλως ἐψυχαγωγήθην ἐπὶ τῇ προβλήματος ζητήσει. τὸ μὲν ὑπὸ ̓Απολλωνίου ἐκδοθὲν ἔοικε κοινῇ σκοπεῖν. καὶ γὰρ περιφέρεται, κ. τ. λ.” on Regular Solids was probably written about 180 B.C. It was clearly a youthful production. Besides this we have another work of Hypsikles, of astronomical content, entitled in Greek ἀναφορικός. Νow in this treatise we find for the first time the division of the circumference of a circle into 360 degrees, which Autolykos, an astronomer a short time anterior to Euclid, was not acquainted with, nor, apparently, Eratosthenes who died about 194 B.C. On the other hand Hypsikles used no trigonometrical methods: these latter are to some extent employed by the astronomer Hipparchos, who made observations at Rhodes between the years 161 and 126. Thus the discovery of trigonometrical methods about 150 agrees well with the conclusion arrived at on other grounds, that Hypsikles flourished about 180 B.C. We must not, however, omit to notice that Nesselmann, an authority always to be mentioned with respect, takes an entirely different view. He concludes that we may with a fair approach to certainty place Hypsikles about the year 200 of our era, but upon insufficient grounds. Of the two arguments used by Nesselmann in support of his view one is grounded upon the identification of an Isidoros whom Hypsikles mentions' as his instructor with the Isidoros of an article in Suidas: Ισίδωρος φιλόσοφος ὃς ἐφιλοσόφησε μὲν ὑπὸ τοῖς ἀδελφοῖς, Eйπeρ Tis aλλos, év μalýμaσiv: and, further, upon a conjecture of Fabricius about it. Assuming that the two persons called Isidoros in the two places are identical we have still to determine his date. The question to be answered is, what is the reference in ὑπὸ τοῖς ἀδελφοῖς ? Now Fabricius makes a conjecture, which seems hazardous, that the adeλpoi are the brothers M. Aurelius Antoninus and L. Aurelius Verus, who were joint-Emperors from 160 to 169 A.D. This date being assigned to Isidoros, it would follow that Hypsikles should be placed about A.D. 200. In the second place Nesselmann observes that according to Diophantos Hypsikles is the discoverer of a proposition respecting polygonal numbers which we find in a rather less perfect 1 Eucl. xv. 5. “ ἡ δὲ εὕρεσις, ὡς Ἰσίδωρος ὁ ἡμέτερος υφηγήσατο μέγας διδάσκαλος, ἔχει τὸν τρόπον τοῦτον.” form in Nikomachos and Theon of Smyrna; from this he argues that Hypsikles must have been later than both these mathematicians, adducing as further evidence that Theon (who is much given to quoting) does not quote him. Doubtless, as Theon lived under Hadrian, about 130 A.D., this would give a date for Hypsikles which would agree with that drawn from Fabricius' conjecture; but it is not possible to regard either piece of evidence as in any way trustworthy, even if it were not contradicted by the evidence before adduced on the other side. We may say then with certainty that Hypsikles, and therefore a fortiori Diophantos, cannot have written before 180 B.C. (2) The only other name mentioned in Diophantos' writings is that of a contemporary to whom they are dedicated. This name, however, is Dionysios, which is of so common occurrence that we cannot derive any help from it whatever. (3) Diophantos' work is so unique among the Greek treatises which we possess, that he cannot be said to recal the style or subject-matter of any other author, except, indeed, in the fragment on Polygonal Numbers; and even there the reference to Hypsikles is the only indication we can lay hold of. The epigram-problem, which forms the last question of the 5th book of Diophantos, has been used in a way which is rather curious, as a means of determining the date of the Arithmetics, by M. Paul Tannery'. The enunciation of this problem, which is different from all the rest in that (a) it is in the form of an epigram, (b) it introduces numbers in the concrete, as applied to things, instead of abstract numbers (with which alone all the other problems of Diophantos are concerned), is doubtless borrowed by him from some other source. It is a question about wine of two different qualities at the price respectively of 8 and 5 drachmae the xoûs. It appears also that it was wine of inferior quality as it was mixed by some one as drink for his servants. Now M. Tannery argues (a) that the numbers 8 and 5 were not hit upon to suit the metre, for, as these are the only numbers which occur in the epigram, and both are found in 1 Bulletin des Sciences mathematiques et astronomiques, 1879, p. 261, the same line in the compounds ὀκταδράχμους and πεντεδράχ μous, some other numerals would serve the purposes of metre equally well. (b) Neither were they taken in view of the solution of the problem, for each number of xóes which it was required to find are found to contain fractions. Hence (c) the basis on which the author composed his problem must have been the price of wines at the time. Now, says M. Tannery', it is evident that the prices mentioned for wines of poor quality are famine prices. But wine was not dear until after the time of the Antonines. Therefore the composer of the epigram, and hence Diophantos also, is later than the period of the Antonines. This argument, even if it is correct, does no more than give us a later date than we before arrived at as the upper limit. Nor can M. Tannery consistently assert that this determination necessarily brings us at all near to the date of Diophantos; for in another place he maintains that Diophantos was no original genius, but a learned mathematician who made a collection of problems previously known; thus, if so much had already been done in the domain which is represented for us exclusively by Diophantos, the composer of the epigram in question may well have lived a considerable time before Diophantos. It may be mentioned here, also, that one of the examples which M. Tannery quotes as an evidence that problems similar to, and even more difficult than, those of Diophantos were in vogue before his time, is the famous Problem of the Cattle, which has been commonly called by the name of Archimedes; and this very problem is fatal to the theory that arithmetical epigrams must necessarily be founded on fact. These considerations, however, though proving M. Tannery to be inconsistent, do not necessarily preclude the possibility that the inference he draws from the epigram-problem solved by Diophantos is correct, for (a) the date of the Cattle-problem itself is not known, and may be later even than Diophantos, (b) it does not follow that, if M. Tannery's conclusion cannot be proved to be necessarily right, it must therefore be wrong. 1 "Il est d'ailleurs facile de se rendre compte que ces prix n'ont pas été choisis en vue de la solution: on doit donc supposer qu'ils sont réels. Or ce sont évidemment, pour les vins de basse qualité, de prix de famine." On the vexed question as to how far Diophantos was original we shall have to speak later. We pass now to a consideration of the (b) External evidence as to the date of Diophantos. (1) We have first to consider the testimony of a passage of Suidas, which has been made much of by writers on the question of Diophantos, to an extent entirely disproportionate to its intrinsic importance. As however it does not bear solely upon the question of date, but upon another question also, it cannot be here passed over. The passage in question is Suidas' article "Taτía'. The words which concern us apparently stood in the earliest texts thus, ἔγραψεν ὑπόμνημα εἰς Διοφάντην τὸν ἀστρονομικόν. Κανόνα εἰς τὰ κωνικά· Απολλωνίου ὑπόμνημα. With respect to the reading Διοφάντην, we have already remarked that Bachet asserts that two good Paris MSS. have Διόφαντον. The words as found in the text cannot be right. Alopávтnv TÒν аσтρоνоμιкóv should (if the punctuation were right) be Διοφάντην τὸν ἀστρονόμον, the former not being Greek. Kuster's conjecture is that we should read ñóμvnμa eis Διοφάντου αστρονομικὸν κανόνα· εἰς τὰ κωνικὰ ̓Απολλωνίου Vπóμvημа. 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 certain 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 3 1 Υπατία ἡ Θέωνος τοῦ Γεωμέτρου θυγάτηρ τοῦ ̓Αλεξανδρέως φιλοσόφου καὶ αὐτὴ φιλόσοφος, καὶ πολλοῖς γνώριμος γυνὴ Ἰσιδώρου τοῦ φιλοσόφου ἤκμασεν ἐπὶ τῆς βασιλείας ̓Αρκαδίου· ἔγραψεν ὑπόμνημα εἰς Διοφάντην τὸν ἀστρονομικόν. Κανόνα εἰς τὰ κωνικά· ̓Απολλωνίου ὑπόμνημα. 2 Suidae Lexicon, Cantabrigiae, 1705. 3 Chapter XI. of the Life as given in Sancti patris nostri Joannis Damasceni, Monachi, et Presbyteri Hierosolymitani, Opera omnia quae exstant et ejus nomine circumferuntur. Tomus primus. Parisiis, 1712. 'Avaλoyías dè'ApiÐμNTIKÀS OÜTWS ἐξησκήκασιν εὐφυῶς, ὡς Πυθαγόραι ἢ Διόφαντοι, |