16

numbers by successive squares till a square lies between the products. Thus between 40 and 44, 90 and 99 no square lies, but between 160 and 176 there lies the square 169. Hence x2=169 will lie between the proposed limits. The method is very neatly used in the following instance. In IV. 34 the problem is 'to divide 1 into two parts, such that if 3 be added to the one part and 5 to the other, the product of the two sums shall be a square.' If one part be x-3, the other is 4-x. Then x (9) must be a square. Suppose it = 4x3: then x=g. But this will not suit the original assumption, since x must be greater than 3 (and less than 4). Now 5 is 4+1: hence what is wanted is to find a number y2 + 1 such that

9

y2+1

is > 3 and <4. For such a purpose y3 must be <2 and > 11. "I resolve these expressions into square fractions" says Diophantus and selects 128 and 8 between which lies the square 10 or 8. He

64

64

instead of 4x. Sometimes, indeed,

The solution is as Since a square + its

is the root of the

Diophantus solves a problem wholly or in part by (5) synthesis1. Thus IV. 31 is 'To find 4 squares, such that their sum added to the sum of their roots is a given number.' follows. "Let the given number be 12. root is a square, the root of which minus first-mentioned square, and since the four numbers added together= 12, which plus the four quarters (12+) is 13, it follows that the problem is to divide 13 into four squares. The roots of these minus each will be the roots of the four squares sought for. Now 13 is composed of two squares 4 and 9: each of which is composed of two squares, viz. 9, 28, 144 and 81. The roots of these, viz. 8, 8, 12 and g, minus each, are the roots of the four squares sought for, viz. 11, 76, 18, 18: and the four squares themselves are 121, 49, 361 and 168" Although it has been said above, and has been sufficiently shown by the foregoing examples, that Diophantus does not treat his problems generally and is usually content with finding any particular numbers which happen to satisfy the conditions of his problems,

1 Compare also III. 16, Iv. 32, v. 17, 23 etc.

25

yet it should be added that he does occasionally attempt (6) such general solutions as were possible to him. But these solutions are not often exhaustive because he had no symbol for a general coefficient. Thus in v. 21 'to find 3 numbers, such that each of them shall be a square minus 1 and their sum shall be a biquadrate (Svvaμodúvaμis)' he finds the 3 numbers in the form a* - 2x2, x2 + 2x and x2 - 2x, and adds 'the problem has been solved in general (dopíoтois) terms,' and at the end of IV. 37 (comp. also IV. 20) where a similar solution is given he remarks "A solution in general terms is such that the unknown in the expressions for the numbers sought may have any value you please." The problems IV. 20, 37, 39 and 41 are expressly problems for finding general expressions. He solves them by a 'tentative assumption.' For instance IV. 39 is 'To find two general expressions for numbers such that their product minus their sum is a given number.' The solution runs as follows: 'The given number is 8. The first number may be taken as x, the second as 3. Then 2x-3=8, and x = 51⁄2. Now 5 is 1, 11 is the given number plus the second: and 2 is the second minus 1. Hence at whatever value the second number be taken, if I add it to the given number and divide the sum by the second number minus 1, I get the first number. Suppose

the second number to be x+1: then

x+9

Ꮳ

is the first.' These

general solutions for two numbers are immediately afterwards (Iv. 21, 38, 40, 42) used in problems of a similar character for three numbers, of which two are first found in general terms and then the third by a determination of x in the usual manner. Sometimes, however (e.g. IV. 26 and frequently in the 6th book'), a problem after being solved by particular numbers (as 40, 27, 25) is solved generally (by 40x, 27x, 25x in Iv. 26). But though the defects in Diophantus' proofs are in general due to the limitation of his symbolism, it is not so always. Very frequently indeed Diophantus introduces into a solution (7) arbitrary conditions and determinations which are not in the

1 vi. 3, 4, 6, 7, 8, 9, 10, 11, 13, 15, 17. See Nesselmann, pp. 418–421. The problems of the vith Book deal almost

entirely with 'right-angled triangles', i.e. with sets of three numbers, such that x2+ y2=z2.

problem. Of such "fudged" solutions, as a schoolboy would call them, two particular kinds are very frequent. Sometimes an unknown is assumed at a determinate value': as in I. 14 To find two numbers whose product is three times their sum,' where Diophantus, without a word of apology, takes the first number as, the second as 12. Sometimes a new condition is introduced, as in VI. 19, where, two numbers being sought such that the cube of one is greater by 2 than the square of the other, Diophantus takes the numbers as x-1 and x+1, thus introducing a condition that the difference between the two numbers shall be 2. A very remarkable case of the latter kind occurs in IV. 7 where the problem would be, in our symbolism, to find three numbers, a, b, c, so that a3 + c2 shall be a square, b2+c2 a cube. Diophantus begins his solution by taking b2+c2 = a3. Arbitrariness of this kind is of course different from the cases in which Diophantus merely takes a particular number, where any other would evidently do as well. In the latter, he is urged by the defects of his symbolism: in the former he is urged only by the want of a solution to a particular problem: the difference is one of kind and not of degree.

72. From the very brief survey of the Arithmetica, it will be obvious to the reader that it is a work of the utmost ingenuity but that it is deficient, sometimes pardonably, sometimes without excuse, in generalization. The book of Porismata, to which Diophantus sometimes refers, seems on the other hand to have been entirely devoted to the discussion of general properties of numbers. It is three times expressly quoted in the Arithmetica. These quotations, when expressed in modern symbols, are to the following effect. In v. 3 the porism2 is cited: 'If x + a = m2, y + a = n2, and xy + a= p2, then m=n+1': in v. 5: 'If three numbers x2, (x + 1)2, 4x2 + 4x + 4, be taken, the

1 Other examples in 1. 25, 26, 27, 28, II. 19, v. 7, 30, 31.

2 Nesselmann, pp. 441-443, shews that the conditions may be satisfied by numbers of other forms. Of the 2nd porism he says (p. 445) that more general expressions might be found for

the numbers but he will not trust himself to find them. Of the 3rd he says (pp. 445-446, after Fermat) that Vieta uses it in the last propositions of the 4th Book of his Zetetica, The 3rd porism is mutilated in the quotation.

product of any two + their sum, or + the remaining number, is a square in v. 19 'the difference between two cubes may be resolved into the sum of two cubes.' Of all these propositions he says ἔχομεν ἐν τοῖς πορίσμασιν, 'we find it in the Porisms'; but he cites also a great many similar propositions without expressly referring to the Porisms. These latter citations fall into two classes, the first of which contains mere identities, such as the algebraical equivalents of the theorems in Euclid II. For instance in Diophantus II. 31, 32, and iv. 17 it is stated, in effect, that x2 + y2 + 2xy is always a square (Eucl. II. 4): in II. 35, 36, III. 12, 14 and many more places it is stated that a +ab is always a square (Eucl. II. 5) etc1. The other

2

:

2

class contains general propositions concerning the resolution of numbers into the sum of two, three or four squares. For instance, in II. 8, 9 it is stated 'Every square number' (in II. 10 'every number which is the sum of two squares') 'may be resolved into the sum of two squares in an infinite number of ways' in v. 12 'A number of the form (4n+3) can never be resolved into two squares,' but 'every prime number of the form (4n+1) may be resolved into two squares': in v. 14 'A number of the form (8n +7) can never be resolved into three squares.' It will be seen that all these propositions are of the general form which ought to have been but is not adopted in the Arithmetica. We are therefore led to the conclusion that the Porismata, like the pamphlet on Polygonal Numbers, was a synthetic and not an analytic treatise. It is open, however, to anyone to maintain the contrary, since no proof of any porism is now extant.

With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards. A few scholiasts appear, such as Eutocius of

1 Nesselmann, pp. 446-450, cites 10 such identities, most of which are used more than once by Diophantus.

2 In Iv. 31, 32, v. 17 Fermat thought that Diophantus was using a proposition 'Every number whatever

can be resolved into four squares,' but Nesselmann (p. 460—1) inclines to the opinion that Diophantus did not know this proposition generally but was relying on the known properties of certain determinate numbers.

Askalon (cir. A.D. 550) who wrote on Archimedes, Asclepius of Tralles and his pupil John Philoponus (cir. A.D. 650) who wrote on Nicomachus, and the unknown commentators who have added lemmas to the arithmetical books of Euclid; but though there is evidence that the old mathematicians were still studied in Athens and Alexandria and elsewhere, no writer of genius appears and the history of arithmetic and algebra is continued henceforth by the Indians and Arabs.