346 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



special remark that Qx 2 + 3 = D has an infinitude of solutions, since it 

 has one solution x = 1. 



Diophantus solved Ax 2 + Bx + C = y 2 only in the following cases. 



(a) If A is a square, a 2 , set y = ax + m, whence x is found rationally; 

 examples in II, 20, 21, 23, 24, 33, III, 9, 16, 18, IV, 15, 21, V, 3, 4, 18, 20. 



(b) If C = c 2 , set y = mx + c; examples in II, 17, IV, 9, 10, 12, 14, 45. 



(c) In IV, 33, 18 + 3x x 2 is to be made a square, say ra 2 x 2 , where 

 (m 2 + 1) 18 + (f) 2 = CU. Then, multiplying by 4, 72m 2 + 81 = D, say 

 (8m + 9) 2 , whence m = 18, 18 + 3x - 325z 2 = 0, x = 6/25. In general, 

 as remarked by Nesselmann 11 (pp. 333-4), the corresponding condition that 

 the root x of Ax 2 + Bx + C = m 2 x 2 be rational is IB 2 - AC + Cm? = D, 

 and, as in (b), can be satisfied if IB 2 AC is a square. 



While H. Hankel 24 believed that Diophantus was influenced by Indian 

 sources, M. Cantor 25 took the opposite view except as to integral solutions. 

 P. Tannery 26 went to the extreme of believing that the Greeks influenced 

 the Indians also in the question of integral solutions, while even the cyclic 

 method [next explained] is only a variation of the Greek method of solv- 

 ing t 2 Du 2 = 1, since from the Greek method of deriving from one ap- 

 proximation to VD a closer approximation it is easy to pass to the Indian 

 method. 



E. B. Crowell 27 compared the work of Diophantus with that of Brahme- 

 gupta, 28 and the first solution by Brouncker 13 with that of Bhascara. 30 



Brahmegupta 28 (born 598 A.D.) gave a rule to find x so that Cx 2 + 1 

 shall be a square. Assume any " least root" L and add to CL 2 such an 

 " additive" number A that the sum is a square G 2 ; call G the "greatest 

 root" [L and G are values of x, y satisfying Cx 2 + A = y 2 ~\. Write L, G, 

 A twice. By cross multiplication, we obtain a least root LG + GL, while 

 CLL + GG is a greatest root, for additive A A ; dividing these new roots 

 by A, we get roots for additive unity. For details, see Bhascara. 30 



For example (67), let C = 9?. Take L = 1, A = 8, whence G = 10. 

 Then 2LG = 20, 92L 2 + G 2 = 192 are least and greatest roots for additive 

 64. Dividing them by 8, we get 5/2 and 24 as roots for additive unity. 

 By composition of the last pair with itself, we get other roots 120 and 1151 

 for additive unity. 



By composition of the roots for additive unity with the roots for additive 

 A, we get roots for additive A (68, p. 364). For example ( 77, p. 368), 

 from 3-30 2 + 900 = 60 2 , 3 -I 2 + 1 = 2 2 , we get the least root 



30-2 + 1-60 = 120 

 and greatest root 3-30-1 + 60-2 == 210 for 3-120 2 + 900 = = 210 2 . 



24 Zur Geschichte dcr Math, in Alterthum und Mittelalter, 1874, 204. 



25 Vorles. iiber Geschichte Math., 1, 1880, 533; ed. 2, 556; ed. 3, 596. 

 28 Me"m. Soc. Sc. Phys. Nat. Bordeaux, (2), 4, 1882, 325. 



27 M. Elphinstone's History of India, ed. 9, 1905, 142, Note 16 (ed., Crowell). 



28 Brahme-sphut'a-sidd'hanta, Ch. 18 (algebra), 65-66. Algebra, with arith. and mensura- 



tion, from the Sanscrit of Brahmegupta and Bhascara, transl. by H. T. Colebrookc, 1817, 

 p. 363. Cf. Simon. 3 "'' 



