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 api@unтixá 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 apiμntin is concerned with the abstract properties of numbers (as odd and even, etc.), whereas XoyIOTIKń deals with the same odd and even, but in relation to one another. Geminus also distinguishes the two terms. According to him ȧpioμnτin 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 hand 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 paxirns. But in Diophantus the calculations take an abstract 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. 1 Gorgias, 451 Β, C: τὰ μὲν ἄλλα καθάπερ ἡ ἀριθμητικὴ ἡ λογιστικὴ ἔχει· περὶ τὸ αὐτὸ γάρ 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, eg. by distributing apples or garlands (λ Té TWD Baroual kal oreødrwy) and the use of different bowls of silver, gold, or brass etc. (øvdλas 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). Dioph. 11. p. 73, 26. (αριθμητικῆς 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 seven3, 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. Thus the 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 (eg. 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. 72, 17; Iamblichus (ed. Pistelli), p. 132, 12. 2 Dioph. 11. p. 62, 25. 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. 11. p. xvii, xviii. * See F. Woepcke, Extrait du Fakhrī, traité d'Algèbre par Abou Bekr Mohammed ben Alhaçan Alkarkhí (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. problems of the earlier Books. But, assuming that Diophantus' resources are at an end in the sixth Book, Nesselmann has to suggest possible topics which would have formed approximately adequate material for the equivalent of seven Books of the Arithmetica. The first step is to consider what is actually wanting which we should expect to find, either as foreshadowed by the author himself or as necessary for the elucidation or completion of the whole subject. Now the first Book contains problems leading. to determinate equations of the first degree; the remainder of the work is a collection of problems which, with few exceptions, lead to indeterminate equations of the second degree, beginning with simpler cases and advancing step by step to more complicated questions. There would have been room therefore for problems involving (1) determinate equations of the second degree and (2) indeterminate equations of the first. There is indeed nothing to show that (2) formed part of the writer's plan; but on the other hand the writer's own words in Def. 11 at the beginning of the work promise a discussion of the solution of the complete or adfected quadratic, and it is clear that he employed his method of solution in the later Books, where in some cases he simply states the solution without working it out, while in others, where the roots are "irrational," he gives approximations which indicate that he was in possession of a scientific method. Pure quadratics Diophantus regarded as simple equations, taking no account of the negative root. Indeed it would seem that he adopted as his ground for the classification of quadratics, not the index of the highest power of the unknown quantity contained in it, but the number of terms left in it when reduced to its simplest form. His words are1: "If the same powers of the unknown occur on both sides, but with different coefficients (μn óμomλnon dé), we must take like from like until we have one single expression equal to another. If there are on both sides, or on either side, any terms with negative coefficients (ἐν ἐλλείψεσί τινα εἴδη), the defects must be added on both sides until the terms on both sides have none but positive coefficients (évvπáρxovтa), when we must again take like from like until there remains one term on each side. This should be the object aimed at in framing the hypotheses of propositions, that is to say, to reduce the equations, if possible, until one term is left equated to one term. But afterwards I will 1 Dioph. 1. Def. 11, p. 14. show you also how, when two terms are left equal to one term, such an equation is solved." That is to say, reduce the quadratic, if possible, to one of the forms ax2= bx, ax2= c, or bx = c; I will show later how to solve the equation when three terms are left of which any two are equal to the third, i.e. the complete quadratic axbx + c = 0, excluding the case ax + bx + c = 0. The exclusion of the latter case is natural, since it is of the essence of the work to find rational and positive solutions. Nesselmann might have added that Diophantus' requirement that the equation, as finally stated, shall contain only positive terms, of which two are equated to the third, suggests that his solution would deal separately with the three possible cases (just as Euclid makes separate cases of the equations in his propositions VI. 28, 29), so that the exposition might occupy some little space. The suitable place for it would be between the first and second Books. There is no evidence tending to confirm Nesselmann's further argument that the six Books may originally have been divided into even more than seven Books. He argues from the fact that there are often better natural divisions in the middle of the Books (eg. at II. 19) than between them as they now stand; thus there is no sign of a marked division between Books I. and II. and between Books II. and III., the first five problems of Book II. and the first four of Book III. recalling similar problems in the preceding Books respectively. But the latter circumstances are better explained, as Tannery explains them, by the supposition that the first problems of Books II. and III. are interpolated from some ancient commentary. Next Nesselmann points out that there are a number of imperfections in the text, Book V. especially having been "treated by Mother Time in a very stepmotherly fashion"; thus it seems probable that at V. 19 three problems have dropped out altogether. Still he is far from accounting for seven whole Books; he has therefore to press into the service the lost "Porisms" and the tract on Polygonal Numbers. If the phrase which, as we have said, occurs three times in Book v., "We have it in the Porisms that ...," indicates that the "Porisms" were a definite collection of propositions concerning the properties of certain numbers, their divisibility into a certain. number of squares, and so on, it is possible that it was from the same collection that Diophantus took the numerous other propositions which he assumes, either explicitly enunciating them, or implicitly taking them for granted. May we not then, says |