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 oμorλŋ0ĥ 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 είδη), the defects must be added on both sides until the terms on both sides have none but positive coefficients (évvπáρɣovтa), 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 ax3- bx, ax1-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, ie, the complete quadratic ax2± bx ±c=0, excluding the case ax + bx + co. 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 Nesselmann, reasonably suppose the "Porisms" to have formed an introduction to the indeterminate and semi-determinate analysis of the second degree which forms the main subject of the Arithmetica, and to have been an integral part of the thirteen Books, intervening, probably, between Books I. and II.? Schulz, on the other hand, considered this improbable, and in recent years Hultsch' has definitely rejected the theory that Diophantus filled one or more Books of his Arithmetica exclusively with Porisms. Schulz's argument is, indeed, not conclusive. It is based on the consideration that "Diophantus expressly says that his work deals with arithmetical problems""; but what Diophantus actually says is "Knowing you, O Dionysius, to be anxious to learn the solution (or, perhaps, discovery,' eupeow) of problems in numbers, I have endeavoured, beginning from the foundations on which the study is built up, to expound (vπoσrĥoal to lay down) the nature and force subsisting in numbers," the last of which words would easily cover propositions in the theory of numbers, while "propositions," not "problems," is the word used at the end of the Preface, where he says, "let us now proceed to the propositions (πрoтáσeis)...... which have been treated in thirteen Books." = On reconsideration of the whole matter, I now agree in the view of Hultsch that the Porisms were not a separate portion of the Arithmetica or included in the Arithmetica at all. If they had been, I think the expression "we have it in the Porisms" would have been inappropriate. In the first place, the Greek mathematicians do not usually give references in such a form as this to propositions which they cite when they come from the same work as that in which they are cited; as a rule the propositions are quoted without any references at all. The references in this case would, on the assumption that the Porisms were a portion of the thirteen Books, more naturally have been to particular propositions of particular Books (cf. Eucl. XII. 2, "For it was proved 1 Hultsch, loc. cit. The whole passage of Schulz is as follows (pref. xxi): "Es ist daher nicht unwahrscheinlich, dass diese Porismen eine eigene Schrift unseres Diophantus waren, welche vorzüglich die Zusammensetzung der Zahlen aus gewissen Bestandtheilen zu ihrem Gegenstande hatten. Könnte man diese Schrift als einen Bestandtheil des grossen in dreizehn Büchern abgefassten arithmetischen Werkes ansehen, so wäre es sehr erklärbar, dass gerade dieser Theil, der den blossen Liebhaber weniger anzog, verloren ging. Da indess Diophantus ausdrücklich sagt, sein Werk behandele arithmetische Probleme, so hat wenigstens die letztere Annahme nur einen geringen Grad von Wahrscheinlichkeit." in the first theorem of the 10th Book that..."). But a still vaguer reference would have been enough, even if Diophantus had chosen to give any at all; if the propositions quoted had preceded those in which they are used, some expression like τοῦτο γὰρ προγέγραπται, “ for this has already been proved,” or δέδεικται γὰρ TOÛTO, "for this has been shown," would have sufficed, or, if the propositions occurred later, some expression like és é§ñs deiXOŃσETAI οι δειχθήσεται ύφ' ἡμῶν ὕστερον, " as will be proved in due course" or "later." The expression "we have it in the Porisms" (in the plural) would have been still more inappropriate if the "Porisms' had been, as Tannery supposes', not collected together as one or more Books of the Arithmetica, but scattered about in the work as corollaries to particular propositions'. And, as Hultsch says, it is hard, on Tannery's supposition, to explain why the three particular theorems quoted from "the Porisms" were lost, while a fair number of other additions survived, partly under the title Trópioμa (cf. I. 34, I. 38), partly as "lemmas to what follows," Anupa eis Tò ¿Es (cf. lemmas before IV. 34, 35, 36, V. 7, 8, VI. 12, 15). On the other hand, there is nothing improbable in the supposition that Diophantus was induced by the difficulty of his problems to give place in a separate work to the "porisms" necessary to their solution. The hypothesis that the Porisms formed part of the Arithmetica being thus given up, we can hardly hold any longer to Nesselmann's view of the contents of the lost Books and their place in the treatise; and I am now much more inclined to the opinion of Tannery that it is the last and the most difficult Books which are lost. Tannery's argument seems to me to be very attractive and to deserve quotation in full, as finally put in the preface to Vol. II. of his Diophantus. He replies first to the assumption that Diophantus could not have proceeded to problems more difficult than those of Book v. "But if the fifth or the sixth Book of the Arithmetica had been lost, who, pray, among us would have believed that such problems had ever been attempted by the Greeks? It would be the greatest error, in any case in which a 1 Dioph. II. p. xix. ' Thus Tannery holds (loc. cit.) that the solution of the complete quadratic was given in the form of corollaries to I. 27, 30; and he refers the three "porisms" quoted in v. 3, 5, 16 respectively to a second (lost) solution of III. 10, to III. 15, and to IV. 1, 2. Dioph. II. p. xx. thing cannot clearly be proved to have been unknown to all the ancients, to maintain that it could not have been known to some Greek mathematician. If we do not know to what lengths Archimedes brought the theory of numbers (to say nothing of other things), let us admit our ignorance. But, between the famous problem of the cattle and the most difficult of Diophantus' problems, is there not a sufficient gap to require seven Books to fill it? And, without attributing to the ancients what modern mathematicians have discovered, may not a number of the things attributed to the Indians and Arabs have been drawn from Greek sources? May not the same be said of a problem solved by Leonardo of Pisa, which is very similar to those of Diophantus but is not now to be found in the Arithmetica? In fact, it may fairly be said that, when Chasles made his reasonably probable restitution of the Porisms of Euclid, he, notwithstanding the fact that he had Pappus' lemmas to help him, undertook a more difficult task than he would have undertaken if he had attempted to fill up seven Diophantine Books with numerical problems which the Greeks may reasonably be supposed to have solved." On the assumption that the lost portion came at the end of the existing six Books, Schulz supposed that it contained new methods of solution in addition to those used in Books I. to VI., and in particular extended the method of solution by means of the double equation (διπλή ισότης or διπλοϊσότης). By means of the double equation Diophantus shows how to find a value of the unknown which will make two expressions (linear or quadratic) containing it simultaneously squares. Schulz then thinks that he went on, in the lost Books, to make three such expressions simultaneously squares, i.e. advanced to a triple equation. But this explanation does not in any case take us very far. Bombelli thought that Diophantus went on to solve determinate equations of the third and fourth degree'; this view, however, though natural at that date, when the solution of cubic and biquadratic equations filled so large a space in contemporary investigations and in Bombelli's own studies, has nothing to support it. Hultsch seems to find the key to the question in the fragment of the treatise on Polygonal Numbers and the developments to |