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



infinitude of solutions in this way, but not all. He stated that all solu- 

 tions are obtained from Brouncker's rule by setting r = a/e, whence 

 x = 2ae/(a 2 ne~), and choosing integers a, e such that a 2 ne 2 divides 

 2ae. 



Wallis 48 gave a long exposition of results which he implied are essentially 

 due to Brouncker. He gave a tentative method to solve na 2 + 1 = D. 

 For n = 7, take the square 3 2 just > 7; then 7 = 3 2 - 2, 7-2 2 = 6 2 - 8, 

 7.32 = 92 _ ig, whence we have a number 18 which is double the root 9; 

 hence 7-3 2 = (9 I) 2 1. In general, use the square c 2 just > n and 

 exceeding n by b. Employ no? = (ca) 2 ba 2 f or a = 1, 2, 3, -, until we 

 reach a value a of a for which ba 2 ^ 2ca, and then replace ca by (ca 1) + 1. 

 For each a ^ a, we thus have two values of na 2 . Presently we can make 

 a further reduction of ca 1 to ca 2, etc., etc. It is stated that we finally 

 reach an equation in which the number subtracted is unity and hence a 

 solution. Devices are suggested (pp. 465-74) to abbreviate the long calcu- 

 lations. 



Given (pp. 474-8) one solution, nr 2 + 1 = s 2 , set t = 2s; then the 

 values of x in the successive solutions of nx 2 + 1 = D arer, rt, r(t 2 1), 

 r(t 3 2), , while if ra, r(3 are any two consecutive terms, the next term is 

 r(tp - a). 



Wallis 49 explained in an example Brouncker's method of finding a funda- 

 mental solution. The example chosen was 13a 2 + 1 = D. Since 13 lies 

 between the squares 9 and 16, set 13a 2 + 1 = (3a + 6) 2 , whence 



4a 2 + 1 = Gab + b 2 , 2b > a > b. 



Hence set a = b + c, whence 2bc + 4c 2 + 1 = 36 2 , 2c > b > c. Set 

 b = c + d, c = d + e, d = e + f. Then e 2 + 1 = Qef + 4/ 2 , 7/ > e > 6f. 

 Hence set e = 6/ + g, f = g + h, g = h + i. Then 4hi + 3i 2 + 1 = 3/i 2 . 

 Thus h > i. Taking* h = 2i, we see that the last equation becomes 

 IK 2 + 1 = 12i 2 and holds for i = 1, whence h = 2, >, a= 180. It is 

 noted (pp. 492-3) that, since b, c, d, are decreasing integers, we finally 

 reach a term which divides the preceding, as in Euclid's process to find the 

 g.c.d., a process entirely analogous to the present one. If we had proposed 

 the example 13a 2 + 9 = D, we would get IW + 9 = 12i 2 , whence i = 3, 

 and similarly for any square in place of 1 or 9. But if k is not a square, 

 13a 2 + k = D is not always solvable, but when solvable the solution can 

 be found by the above method. 



As noted by H. J. S. Smith, 50 Brouncker's method is the same as that 

 given by Euler 65 - 72> 81 and really consists in the successive determination 



48 Commercium, 789, letter XVII to Brounckcr, Dec. 17, 1657; Oeuvres de Fermat, III, 



457-480. 



49 Commercium, 804, letter XIX to Brouncker, Jan. 30, 1658; Oeuvres de Fermat, III, 490- 



503. Cf. Wallis, Algebra, 1693, Ch. 98. 



* To proceed as would later writers, set h = i +j, whence - 4i 2 + 2y + 3j 2 = 1; then 

 i j -}- k t whence j 2 6jk 4fc 2 = 1, with unity as coefficient of a square term, so 

 that j = 1, k = is an evident solution. 



50 British Assoc. Report, 1861, 313; Coll. Math. Papers, 1, 193. Cf. Konen, 3 p. 39; Whitford, 4 



pp. 52-6; Wertheim. 42 



