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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 number1. Hypsicles was also the writer of the sup- · plement 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 Alexandria1; 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 (pooedávnσe) 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. 470-3.

'Theo Alexandrinus in primum librum Ptolemaei Mathematicae Compositionis (on c. 1X.): see Dioph. 11. 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 or ἑταίρῳ) Διοφαντῳ συνοπτικώτατα προσεφώνησε. The MSS. read ἑτέρω, which is apparently a mistake for ετέρως or possibly for ἑταίρῳ. Tannery con. jcctures τῷ ἑταίρῳ, but this is very doubtful; if the article had been there, Διοφάντῳ τῷ ralpy would have been better. On the basis of èralpy Tarnery 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 233–247 and was Bishop there 248-265 A.D. Tannery conjectures then that Diophantus was a Christian and a pupil of Dionysius (Tanucry, "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'.

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The works on which the fame of Diophantus rests are:
(1) The Arithmetica (originally in thirteen Books).
(2) A tract On Polygonal Numbers.

Six Books of the former and part of the latter survive.
Allusions in the Arithmetica imply the existence of

(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 Mopiaσriká 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.

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

8x+1'2x+x+3 + 1 (x − 4 ) + 4 = x, or 3x=196 and x=65j.

This would substitute 108 for 14, 161 for 21, 251 for 33, 301 for 43, 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

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Hultsch, art. Diophantos in Pauly-Wissowa's Real-Encyclopädie.

♦ Iamblichus In Nicomachi arithm. introd. p. 127 (ed. Pistelli); Dioph. 11. p. 73.

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Dioph. II. p. 72 note.

remarks, such scholia would more naturally have been quoted as σχόλια and not by the separate title Μοριαστικά!. It may have been a separate work by Diophantus giving rules for reckoning with fractions; but I do not feel clear that the reference may not simply be to the definitions at the beginning of the Arithmetica.

With reference to the title of the Arithmetica, we may observe that the meaning of the word apieμnтiká here is slightly different from that assigned to it by more ancient writers. The ancients drew a marked distinction between αριθμητική and λογιστική, though both were concerned with numbers. Thus Plato states that apilμntik is concerned with the abstract properties of numbers (as odd and even, etc.), whereas λoyiσTIKń deals with the same odd and even, but in relation to one another'. Geminus also distinguishes the two terms. According to him ȧpılμntiký deals with numbers in themselves, distinguishing linear, plane and solid numbers, in fact all the forms of number, starting from the unit, and dealing with the generation of plane numbers, similar and dissimilar, and then with numbers of three dimensions, etc. XoyiσTIKŃ on the other hard deals, not with the abstract properties of numbers in themselves, but with numbers of concrete things (aioonτwv, sensible objects), whence it calls them by the names of the things measured, e.g. it calls some by the names unλirns and piaλirns. But in Diophantus the calculations take an abstract φιαλίτης. form (except in V. 30, where the question is to find the number of measures of wine at two given prices respectively), so that the distinction between λογιστική and αριθμητική is lost.

We find the Arithmetica quoted under slightly different titles. Thus the anonymous author of prolegomena to Nicomachus' Introductio Arithmetica speaks of Diophantus' "thirteen Books of Arithmetic." A scholium on Iamblichus refers to "the last theorem of the first Book of Diophantus' Elements of Arithmetic

1 Hultsch, loc. cit.

3 Gorgias, 451 B, C: τὰ μὲν ἄλλα καθάπερ ἡ ἀριθμητική ή λογιστικὴ ἔχει· περὶ τὸ αὐτὸ γὰρ ἐστι, τό τε ἄρτιον καὶ τὸ περιττόν· διαφέρει δὲ τοσοῦτον, ὅτι καὶ πρὸς αὐτὰ καὶ πρὸς ἄλληλα πῶς ἔχει πλήθους ἐπισκοπεῖ τὸ περιττὸν καὶ τὸ ἄρτιον ἡ λογιστική.

• Proclus, Comment. on Euclid 1., p. 39, 14-40, 7.

• Cf. Plato, Laws 819 B, C, on the advantage of combining amusement with instruction in arithmetical calculation, e.g. by distributing apples or garlands (μńλwv té TIVWV diaroμal kal otepávwv) and the use of different bowls of silver, gold, or brass etc. (pidλas ἅμα χρύσου καὶ χαλκοῦ καὶ ἀργύρου καὶ τοιούτων τινῶν ἄλλων κεραννύντες, οἱ δὲ ὅλας πως διαδιδόντες, ὅπερ εἶπον, εἰς παιδιὰν ἐναρμόττοντες τὰς τῶν ἀναγκαίων ἀριθμῶν χρήσεις). B Dioph. II. p. 73, 26.

(αριθμητικής στοιχειώσεως)." A scholium on one of the epigrams in Metrodorus' collection similarly speaks of the "Elements of Diophantus","

None of the MSS. which we possess contain more than the first six Books of the Arithmetica, the only variation being that some few divide the six Books into seven', while one or two give the fragment on Polygonal Numbers with the number VIII. The idea that Regiomontanus saw, or said he saw, a MS. containing the thirteen Books complete is due to a misapprehension. There is no doubt that the missing Books were lost at a very early date. Tannery' suggests that Hypatia's commentary extended only to the first six Books, and that she left untouched the remaining seven, which accordingly were first forgotten and then lost; he compares the case of Apollonius' Conics, the first four Books of which were preserved by Eutocius, who wrote a commentary on them, while the rest, which he did not include in his commentary, were lost so far as the Greek text is concerned. While, however, three of the last four Books of the Conics have fortunately reached us through the Arabic, there is no sign that even the Arabians ever possessed the missing Books of Diophantus. second part of an algebraic treatise called the Fakhri by Abū Bekr Muḥ. b. al-Hasan al-Karkhi (d. about 1029) is a collection of problems in determinate and indeterminate analysis which not only show that their author had deeply studied Diophantus, but in many cases are taken direct from the Arithmetica, with the change, occasionally, of some of the constants. In the fourth section of this work, which begins and ends with problems corresponding to problems in Diophantus Books II. and III. respectively, are 25 problems not found in Diophantus; but the differences from Diophantus in essential features (e.g. several of the problems lead to equations giving irrational results, which are always avoided by Diophantus), as well as other internal evidence, exclude the hypothesis that we have here a lost Book of Diophantus. Nor is there any sign that more of the work than we possess was known

1 Dioph. II. p. 73, 17; Iamblichus (ed. Pistelli), p. 132, 12. Dioph. 11. p. 62, 25.

Thus the

3 e.g. Vaticanus gr. 200, Scorialensis -1-15, and the Broscius MS. in the University Library of Cracow; the two last divide the first Book into two, the second beginning immediately after the explanation of the sign for minus (Dioph. 1. p. 14, 1).

Dioph. II. p. xvii, xviii.

See F..Woepcke, Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed ben Alhaçan Alkarkhī (manuscrit 952, supplément arabe de la bibliothèque Impériale), Paris, 1853.

to Abu'l Wafa al-Būzjānī (940-998 A.D.), who wrote a "commentary (tafsir) on the algebra of Diophantus" as well as a "Book of proofs of the propositions used by Diophantus in his work..." These facts again point to the conclusion that the lost Books were lost before the 10th c.

Tannery's suggestion that Hypatia's commentary was limited to the six Books, and the parallel of Eutocius' commentary on Apollonius' Conics, imply that it is the last seven Books, and the most difficult, which are lost. This view is in strong contrast to that which had previously found most acceptance among competent authorities. The latter view was most clearly put, and most ably supported, by Nesselmann', though Colebrooke' had already put forward a conjecture to the same effect; and historians of mathematics such as Hankel, Moritz Cantor, and Günther have accepted Nesselmann's conclusions, which, stated in his own words, are as follows: (1) that much less of Diophantus is wanting than would naturally be supposed on the basis of the numerical proportion of 6 to 13; (2) that the missing portion is not to be looked for at the end but in the middle of the work, and indeed mostly between the first and second Books. Nesselmann's general argument is that, if we carefully read the last four Books, from the third to the sixth, we find that Diophantus moves in a rigidly defined and limited circle of methods and artifices, and that any attempts which he makes to free himself are futile; "as often as he gives the impression that he wishes to spring over the magic circle drawn round him, he is invariably thrown back by an invisible hand on the old domain already known; we see, similarly, in half-darkness, behind the clever artifices which he seeks to use in order to free himself, the chains which fetter his genius, we hear their rattling, whenever, in dealing with difficulties only too freely imposed upon himself, he knows of no other means of extricating himself except to cut through the knot instead of untying it." Moreover, the sixth Book forms a natural conclusion to the whole, in that it consists of exemplifications of methods explained and used in the preceding Books. The subject is the finding of rightangled triangles in rational numbers such that the sides and area satisfy given conditions, the geometrical property of the right-angled triangle being introduced as a fresh condition additional to the purely arithmetical conditions which have to be satisfied in the 1 Algebra der Griechen, pp. 264–273. Algebra of the Hindus, Note M, p. lxi.

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