corrupted. There is no doubt, however, about the first few words, "The given number must not be odd."

i.e. No number of the form 4n+ 3 [or 4n-1] can be the sum of two squares.

1

The text, however, of the latter half of the condition is, in Bachet's edition, in a hopeless state, and the point cannot be settled without a fresh consultation of the MSS.1 The true condition is given by Fermat thus. "The given number must not be odd, and the double of it increased by one, when divided by the greatest square which measures it, must not be divisible by a prime number of the form 4n-1." (Note upon v. 12; also in a letter to Roberval). There is, of course, room for any number of conjectures as to what may have been Diophantos' words'. There would seem to be no doubt that in Diophantos' condition there was something about "double the number" (i.e. a number of the form 4n), also about "greater by unity" and "a prime number." From our data, then, it would appear that, if Diophantos did not succeed in giving the complete sufficient and necessary condition stated by Fermat, he must at all events have made a close approximation to it.

1 Bachet's text has δεῖ δὴ τὸν διδόμενον μήτε περισσὸν εἶναι, μήτε ὁ διπλασίων αὐτοῦ ἡ μλ. μείζονα ἔχη μέρος δ. ἢ μετρεῖται ὑπὸ τοῦ σοῦ σοῦ

He also says that a Vatican Ms. reads μήτε ὁ διπλασίων αὐτοῦ ἀριθμὸν μονάδα α. μείζονα ἔχη μέρος τέταρτον, ἢ μετρεῖται ὑπὸ τοῦ πρώτου ἀριθμοῦ.

Neither does Xylander help us much. He frankly tells us that he cannot understand the passage. "Imitari statueram bonos grammaticos hoc loco, quorum (ut aiunt) est multa nescire. Ego verò nescio heic non multa, sed paene omnia. Quid enim (ut reliqua taceam) est μhre ò dirλaolwv avтoû ap μõ a, &c. quae causae huius poodɩopioμoû, quae processus? immo qui processus, quae operatio, quae solutio?"

Nesselmann discusses an attempt made by Schulz to correct the text, and himself suggests μήτε τὸν διπλασίονα αὐτοῦ ἀριθμὸν μονάδι μείζονα ἔχειν, ὃς μετ τρεῖται ὑπό τινος πρώτου ἀριθμοῦ. But this ignores μέρος τέταρτον and is not satisfactory.

Hankel, however (Gesch. d. Math. p. 169), says: "Ich zweifele nicht, dass die von den Msscr. arg entstellte Determination so zu lesen ist: Aeî ôn tòv didbμevov μήτε περισσὸν εἶναι, μήτε τὸν διπλασίονα αὐτοῦ ἀριθμὸν μονάδι ἃ μείζονα μετρεῖσθαι ὑπό του πρώτου ἀριθμοῦ, ὃς ἂν μονάδι & μείζων ἔχῃ μέρος τέταρτον.” Now this correction, which exactly gives Fermat's condition, seems a decidedly probable one. Here the words μépos тéтаρтov find a place; and, secondly, the repetition of μονάδι & μείζων might well confuse a copyist. του for τοῦ is of course natural enough; Nesselmann reads Twos for TOU.

H. D.

9

We thus see (a) that Diophantos certainly knew that no number of the form 4n+3 could be the sum of two squares, and (b) that he had, at least, advanced a considerable way towards the discovery of the true condition of this problem, as quoted above from Fermat.

(b) On numbers which are the sum of three squares.

In the problem v. 14 a condition is stated by Diophantos respecting the form of a number which added to three parts of unity makes each of them a square. If a be this number, clearly 3a + 1 must be divisible into three squares.

Respecting the number a Diophantos says "It must not be 2 or any multiple of 8 increased by 2.”

i.e. a number of the form 24n+7 cannot be the sum of three squares. Now the factor 3 of 24 is irrelevant here, for with respect to three this number is of the form 3m + 1, and this so far as 3 is concerned might be a square or the sum of two or three squares. Hence we may neglect the factor 3 in 24n.

We must therefore credit Diophantos with the knowledge of the fact that no number of the form 8n +7 can be the sum of three squares.

This condition is true, but does not include all the numbers which cannot be the sum of three squares, for it is not true that all numbers which are not of the form 8n +7 are made up of three squares. Even Bachet remarked that the number a might not be of the form 32n +9, or a number of the form 96n+28 cannot be the sum of three squares.

Fermat gives the conditions to which a must be subject

thus:

Write down two geometrical series (common ratio of each 4), the first and second series beginning respectively with 1, 8,

1 4 16 64 256 1024 4096

8 32 128 512 2048 8192 32768

then a must not be (1) any number obtained by taking twice any term of the upper series and adding all the preceding terms,

or (2) the number found by adding to the numbers so obtained any multiple of the corresponding term of the second series.

Again there are other problems, e.g. v. 22, in which, though conditions are necessary for the possibility of solution, none are mentioned; but suitable assumptions are tacitly made, without rules by which they must be guided. It does not follow from the omission to state such rules that Diophantos was ignorant of even the minutest points connected with them; as however we have no definite statements, it is best to desist from speculation in cases of doubt.

(c) Composition of numbers as the sum of four squares.

Every number is either a square or the sum of two, three or four squares. This well-known theorem, enunciated by Fermat in his note to Diophantos IV. 31, shows at once that any number can be divided into four squares either integral or fractional, since any square number can be divided into two other squares, integral or fractional. We have now to look for indications in the Arithmetics as to how far Diophantos was acquainted with the properties of numbers as the sum of four squares. Unfortunately it is impossible to decide this question with anything like certainty. There are three problems [Iv. 31, 32 and v. 17] in which it is required to divide a number into four squares, and from the absence of mention of any condition to which the number must conform, considering that in both cases where a number is to be divided into three or two squares [v. 14 and 12] he does state a condition, we should probably be right in inferring that Diophantos was aware, at least empirically, if not scientifically, that any number could be divided into four squares. That he was able to prove the theorem scientifically it would be rash to assert, though it is not impossible. But we

may at least be certain that Diophantos came as near to the proof of it as did Bachet, who takes all the natural numbers up to 120 and finds by trial that all of them can actually be expressed as squares, or as the sum of two, three or four squares in whole numbers. So much we may be sure that Diophantos could do, and hence he might have empirically satisfied himself that in any case occurring in practice it is possible to divide any number into four squares, integral or fractional, even if he could not give a rigorous mathematical demonstration of the general theorem. Here again we must be content, at least in our present state of knowledge of Greek mathematics, to remain in doubt.

CHAPTER VII.

HOW FAR WAS DIOPHANTOS ORIGINAL?

§ 1. Of the many vexed questions relating to Diophantos none is more difficult to pronounce upon than that which we propose to discuss in the present chapter. Here, as in so many other cases, diametrically opposite views have been taken by authorities equally capable of judging as to the merits of the case. Thus Bachet calls Diophantos "optimum praeclarissimumque Logisticae parentem," though possibly he means no more by this than what he afterwards says, "that he was the first algebraist of whom we know." Cossali quotes "l' abate Andres" as the most thoroughgoing upholder of the originality of Diophantos. M. Tannery, however, whom we have before had occasion to mention, takes a completely opposite view, being entirely unwilling to credit Diophantos with being anything more than a learned compiler. Views intermediate between these extremes are those of Nicholas Saunderson, Cossali, Colebrooke and Nesselmann; and we shall find that, so far as we are able to judge from the data before us, Saunderson's estimate is singularly good. He says in his Elements of Algebra (1740), "Diophantos is the first writer on Algebra we meet with among the ancients; not that the invention of the art is particularly to be ascribed to him, for he has nowhere taught the fundamental rules and principles of Algebra; he treats it everywhere as an art already known, and seems to intend, not so much to teach, as to cultivate and improve it, by applying it to certain indeterminate problems concerning square and cube numbers, right-angled triangles, &c., which till that time seemed to have been either not at all considered, or at least not regularly treated of. These