against the theory that the three treatises formed only one work, by Schulz, to the effect that Diophantos expressly says that his work treats of arithmetical problems'. This statement itself does not seem to me to be quite accurate, and I cannot think that it is at all a valid objection to Nesselmann's view. The passage to which Schulz refers must evidently be the opening words of the dedication by the author to Dionysios. Diophantos begins thus: "Knowing that you are anxious to become acquainted with the solution [or 'discovery,' evpeσis] of problems in numbers, I set myself to systematise the method, beginning from the foundations on which the science is built, the preliminary determination of the nature and properties in numbers"." Now these "foundations" may surely well mean more than is given in the eleven definitions with which the treatise begins, and why should not the "properties of numbers" refer to the Porisms and the treatise on Polygonal Numbers? But there is another passage which might seem to countenance Schulz's objection, where (Def. 11) Diophantos says "let us now proceed to the propositions... which we will deal with in thirteen Books"." The word used here is not problem (Tpóßλnua) but proposition (πρóτασis), although Bachet translates both words by the same Latin word "quaestio," inaccurately. Now the word πрóτασis does not only apply to the analytical solution of a problem: it applies equally to the synthetic method. Thus the use of the word here might very well imply that the work was to contain 1 Schulz remarks on the Porisms (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 hatte. Könnte man diese Schrift gar als eine 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." * Diophantos own words are: Τὴν εὕρεσιν τῶν ἐν τοῖς ἀριθμοῖς προβλημάτων, τιμιώτατέ μοι Διονύσιε, γινώσκων σε σπουδαίως ἔχοντα μαθεῖν, ὀργανῶσαι τὴν μέθοδον ἐπειράθην, ἀρξάμενος ἀφ' ὧν συνέστηκε τὰ πράγματα θεμελίων, ὑποστῆσαι τὴν ἐν τοῖς ἀριθμοῖς φύσιν τε καὶ δύναμιν. 3 νῦν δὲ ἐπὶ τὰς προτάσεις χωρήσωμεν, κ. τ. λ. 4 τῆς πραγματείας αὐτῶν ἐν τρισκαίδεκα βιβλίοις γεγενημένης. not only problems, but propositions on numbers, i.e. might include the Porisms and Polygonal Numbers as a part of the complete Arithmetics. These objections which I have made to Schulz's argument are, I think, enough to show that his objection to the view adopted by Nesselmann has no weight. Schulz's own view as to the contents of the missing Books of Diophantos is that they contained new methods of solution in addition to those used in Books I. to VI., and that accordingly the lost portion came at the end of the existing six Books. In particular he thinks that Diophantos extended in the lost Books the method of solution by means of what he calls a doubleequation (διπλῆ ἰσότης or in one word διπλοϊσότης). By means of this double-equation Diophantos shows how to find a value of the unknown, which will make two expressions containing it (linear or quadratic) simultaneously squares. Schulz accordingly thinks that he went on in the lost Books to show how to make three such expressions simultaneously squares, i.e. advanced to a triple-equation. This view, however, seems to have nothing to recommend it, inasmuch as, in the first place, we nowhere find the slightest hint in the extant Books of anything different or more advanced which is to come; and, secondly, Diophantos' system and ideas seem so self-contained, and his methods to move always in the same well-defined circle that it seems certain that we come in our six Books to the limits of his art. There is yet another view of the probable contents of the lost Books, which must be mentioned, though we cannot believe that it is the right one. It is that of Bombelli, given by Cossali, to the effect that in the lost Books Diophantos went on to solve determinate equations of the third and fourth degree; Bombelli's reason for supposing this is that Diophantos gives so many problems the object of which is to make the sum of a square and any other number to be again a square number by finding a suitable value of the first square; these methods, argues Bombelli, of Diophantos must have been given for the reason that the author intended to use them for the solution of the equation x*+px=q'. Now Bombelli had occupied himself 1 Cossali's words are (p. 75, 76):..."non tralascierò di notare l' opinione, di cui fu tentato Bombelli, che nelli sei libri cioè dal tempo, di tutto distruggitore, much, almost during his whole life, with the then new methods of solution of equations of the third and fourth degree; and, for the solution of the latter, the usual method of his time led to the making an expression of the form Ax2+ B + C a square, where the coefficients involved a second unknown quantity. Nesselmann accordingly thinks it is no matter for surprise that in Diophantos' entirely independent investigations Bombelli should have seen, or fancied he saw, his own favourite idea. This solution of the equation of the fourth degree presupposes that of the cubic with the second term wanting; hence Bombelli would naturally, in accordance with his view, imagine Diophantos to have given the solution of this cubic. It is possible also that he may have been influenced by the actual occurrence in the extant Books [VI. 19] of a cubic equation, namely the equation x3 + x = 4x2 + 4, of which Diophantos at once writes down the solution = 4, without explanation. It is obvious, however, that no conclusion can be drawn from this, which is a very easy particular case, and which Diophantos probably solved' by simply dividing out by the factor +1. There are strong objections to Bombelli's view. (1) Diophantos himself states (Def. XI.) that the solution of the problems is the object in itself of the work. (2) If he used the method to lead up to the solution of equations of higher degrees, he certainly has not gone to work the shortest way. In support of the view it has been asked "What, on any other assumption, is the object of defining in Def. II. all powers of the unknown quantity up to the sixth? rapitici, si avanzasse egli a sciogliere l'equazione x1+px=q, parendogli, che nei libri rimastici, con proporsi di trovar via via numeri quadrati, cammini una strada a quell' intento. Egli è di fatto procedendo su queste tracce di Diofanto, che Vieta deprime l'esposta equazione di grado quarto ad una di secondo. Siccome però ciò non si effettua che mediante una cubica mancante di secondo termine; cosi il pensiero sorto in animo a Bombelli importerebbe, che Diofanto nei libri perduti costituito avesse la regola di sciogliere questa sorta di equazione cubiche prima d' innoltrarsi allo scioglimento di quella equazione di quarto grado." 1 This is certainly a simpler explanation than Bachet's, who derives the solution from the proportion x3 : x2=x: 1. But the equation being x3+x=4x2+4, it follows that x=4. Surely Diophantos must have meant to use them." The answer to which is that he has occasion to use them in the work, but reduces all the equations which contain these higher powers by his regular and uniform method of analysis. In conclusion, I may repeat that the most probable view is that adopted by Nesselmann, that the works which we know under the three titles formed part of one arithmetical work, which was, according to the author's own words, to consist of thirteen Books. The proportion of the lost parts to the whole is probably less than it might be supposed to be. The Porisms form the part, the loss of which is most to be regretted, for from the references to them it is clear that they contained propositions in the Theory of Numbers most wonderful for the time. CHAPTER III. THE WRITERS UPON DIOPHANTOS. men. § 1. In this chapter I purpose to give a sketch of what has been done directly, and (where it is of sufficient importance) indirectly, for Diophantos, enumerating and describing briefly (so far as possible) the works which have been written on the subject. We turn first, naturally, to Diophantos' own countrymen; and we find that, if we except the doubtful " commentary of Hypatia," spoken of above, there is only one Greek, who has written anything at all on Diophantos, namely the monk Maximus Planudes, to whom are attributed the scholia attached to Books I. and II. in some MSS., which are printed in Latin in Xylander's translation of Diophantos. The date of these scholia is the first half of the 14th century, and they represent all that we know to have been done for Diophantos by his own countryHow different his fate would have been, had he lived a little earlier, when the scientific spirit of the Greeks was still active, what an enormous impression his work would then have created, we may judge by comparing the effect which it had with that of a far less important work, that of Nikomachos. Considering then that up to the time of Maximus Planudes nothing was written about Diophantos (beyond a single quotation by Theon of Alexandria, before mentioned, and an occasional mention of the name) by any Greek, one is simply astounded at finding in Bossut's history a remark like the following: "L'auteur a eu parmi les anciens une foule d'interprètes (!), dont les ouvrages sont la plupart (!) perdus. Nous regrettons, dans ce nombre, le commentaire de la célèbre Hipathia (sic)." Comment is unnecessary. With respect to |