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on Charmides 165 E also refers to the problem "called by Archimedes the Cattle-Problem." Krumbiegel, who discussed the arguments for and against the attribution to Archimedes, concluded apparently that, while the epigram can hardly have been written by Archimedes in its present form, it is possible, nay probable, that the problem was in substance originated by Archimedes'. Hultsch has a most attractive suggestion as to the occasion of it. It is known that Apollonius in his KUTÓKLOV had calculated an approximation to the value of closer than that of Archimedes, and he must therefore have worked out more difficult multiplications than those contained in the Measurement of a circle. Also the other work of Apollonius on the multiplication of large numbers, which is partly preserved in Pappus, was inspired by the Sand-reckoner of Archimedes; and, though we need not exactly regard the treatise of Apollonius as polemical, yet it did in fact constitute a criticism of the earlier book. That Archimedes should then reply with a problem involving such a manipulation of immense numbers as would be difficult even for Apollonius is not altogether outside the bounds of possibility. And there is an unmistakable vein of satire in the opening words of the epigram, "Compute the number of the oxen of the Sun, giving thy mind thereto, if thou hast a share of wisdom," in the transition from the first part to the second, where it is said that ability to solve the first part would entitle one to be regarded as "not unknowing nor unskilled in numbers, but still not yet to be counted among the wise," and again in the last lines. Hultsch concludes that in any case the problem is not much later than the time of Archimedes and dates from the beginning of the second century B.C. at the latest.

I have reproduced elsewhere, from Amthor, details regarding the solution of the problem, and I need do little more than state here its algebraical equivalent. Eight unknown quantities have to be found, namely, the numbers of bulls and cows respectively of each of four colours (I use large letters for the bulls and small letters for the cows). The first part of the problem connects the eight unknowns by seven simple equations; the second part adds two more conditions.

1 Zeitschrift für Math. u. Physik (Hist. litt. Abtheilung), xxv. (1880), p. 111 sq. Amthor added (p. 153 sq.) a discussion of the problem itself.

* Art. Archimedes in Pauly-Wissowa's Real-Encyclopadie, 11. 1, pp. 534, 535• The Works of Archimedes, pp. 319–326.

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where n is an integer. The solution given by the scholiast1 corresponds to n = 80.

The complete problem would not be unmanageable but for the condition (8). If for this were substituted the requirement that W+X shall be merely a product of two unequal factors ("Wurm's problem"), the solution in the least possible numbers is

W = 1217263415886, w= 846192410280,
X= 876035935422, x=574579625058,
Y = 487233469701, y=638688708099,

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But, if we include condition (8) and first of all find a solution satisfying the conditions (1) to (8), we have then, in order to satisfy condition (9), to solve the equation

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The value of W would be a number containing 206545 digits.

Such are the very few and scattered particulars which we possess of problems similar to those of Diophantus solved or propounded before his time. They show indeed that the kind of problem was not invented by him, but on the other hand they show little or no trace of anything like his characteristic algebraical methods. In the circumstances, and in default of discovery of fresh documents, the question how much of his work represents original contributions of his own to the subject must remain a matter of pure speculation. It is pretty obvious that one man could not have been the author of all the problems contained in the six Books. There are also inequalities in the work; some problems are very inferior in interest to the remainder, and some solutions may be assumed to be reproduced from other writers of less calibre, since they reveal none of the mastery of the subject which Diophantus possessed. Again, it seems probable that the problem v. 30, which is exceptionally in epigrammatic form, was taken from someone else. The Arithmetica was no doubt a collection, much in the same sense as Euclid's Elements were. And this may be one reason why so little trace remains of earlier labours in the same field. It is well known that Euclid's Elements so entirely superseded the works of the earlier writers of Elements (Hippocrates of Chios, Leon and Theudius) and of the great contributors to the body of the Elements, Theaetetus and Eudoxus, that those works have disappeared almost entirely. So no doubt would Diophantus' work supersede, and have the effect of consigning to oblivion, any earlier collections of problems of the same kind. But, if it was a compilation, we cannot doubt that it was a compilation in the best sense, therein resembling Euclid's Elements; it was a compilation by one who was a master of the subject, who took account of and assimilated all the best that had been written upon it, arranged the whole of the available material in due and progressive order, but also added much of his own, not only in the form of new problems but also (and even more) in the mode of treatment, the development of more general methods, and

so on.

It is perhaps desirable to add a few words on the previous history of the theory of polygonal numbers. The theory certainly goes back to Pythagoras and the earliest Pythagoreans. The triangle came first, being obtained by first taking 1, then adding

2 to it, then 3 to the sum; each successive number would be represented by the proper number of dots, and, when each number was represented by that number of dots arranged symmetrically under the row representing the preceding number, the triangular form would be apparent to the eye, thus:


Next came the Pythagorean discovery of the fact that a similar successive addition of odd numbers produced successive square numbers, the odd numbers being on that account called gnomons, and again the process was shown by dots arranged to represent squares. The accompanying figure shows the successive squares and gnomons.

Following triangles and squares came the figured numbers in which the "gnomons," or the numbers added to make one number of a given form into the next larger of the same form, were numbers in arithmetical progression starting from 1, but with common difference 3, 4, 5, etc., instead of 1, 2. Thus, if the common difference is 3, so that the successive numbers added to I are 4, 7, 10, etc., the number is a pentagonal number, if the common difference is 4 and the gnomons 5, 9, 13, etc., the number is a hexagonal number, and so on. Hence the law that the common difference of the gnomons in the case of a n-gon is


Perhaps these facts had already been arrived at by Philippus of Opus (4th c. B.C.), who is said to have written a work on polygonal numbers'. Next Speusippus, nephew and successor of Plato, wrote on Pythagorean Numbers, and a fragment of his book survives', in which linear numbers, polygonal numbers, triangles and pyramids are spoken of: a fact which leaves no room for doubt as to the Pythagorean origin of all these conceptions'.

Hypsicles, who wrote about 170 B.C., is twice mentioned by Diophantus as the author of a "definition" of a polygonal number,

1 Bioypapo, Vitarum scriptores Graeci minores, ed. Westermann, 1845, P. 448. Theologumena arithmeticae (ed. Ast), 1817, pp. 61, 61; the passage is translated with notes by Tannery, Pour l'histoire de la science hellène, pp. 386–390.

Cantor, Geschichte der Mathematik, 13, p. 249

which is even quoted verbatim'. The definition does not mention any polygonal number beyond the pentagonal; but indeed this was unnecessary: the facts about the triangle, the square and the pentagon were sufficient to enable Hypsicles to pass to a general conclusion. The definition amounts to saying that the nth a-gon (I counting as the first) is

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Theon of Smyrna, Nicomachus and Iamblichus all devote some space to polygonal numbers. The first two, who flourished about 100 A.D., were earlier than Diophantus, and are accordingly of interest here. Besides a description of the successive polygonal numbers, Theon gives the theorem that two successive triangular numbers added together give a square. That is,

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The fact is of course clear if we divide a square

into two triangles as in the figure.

Nicomachus gave various rules for transforming triangles into squares, squares into pentagons, etc.

1. If we put two consecutive triangles together we get a square (as in Theon's theorem).

2. A pentagon is obtained from a square by adding to it a triangle the side of which is 1 less than that of the square; similarly a hexagon from a pentagon by adding a triangle the side of which is less than that of the pentagon; and so on.

In fact,

} n (2 + (n − 1) (a − 2)} + } (n − 1) n = } n [2 + (n − 1) {(a + 1) − 2}]. Next Nicomachus sets out the first triangles, squares, pentagons, hexagons and heptagons in a diagram thus:

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• Expositio rerum mathematicarum ad legendum Platonem utilium, ed. Hiller,

pp. 31-40.

• Introductio arithmetica, ed. Hoche, 11. 8-12, pp. 87-99.

• In Nicomachi arithmeticam introd., ed. Pistelli, pp. 58-61, 68-72.

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