CHAP. XV] LINEAR FUNCTIONS MADE SQUARES. 449 



eliminate x and y, and choose z = uft so that 



ak bh 2 ck dh_ 

 ad cb ad cb 



In the " Repository solution of the problem to find three numbers the sum 

 and difference of any two of which are squares," 30 I5x 2x 2 =F2x*-\-5x*x* 

 are taken as the square roots of the sum and difference of the first and second 

 numbers, while ld=3x+Gx 2 T6o: 3 3x t= Fo; 5 are taken as the square roots 

 of the sum and difference of the first and third numbers. Hence the three 

 numbers are determined. Here x is any square. Taking x = 9, we get 

 numbers 4387539232, etc., of ten digits each. 



C. Hutton 31 noted that y+1 = D if y = 4x 2 -4x. Then %y+l = (2ax-l) 2 

 gives x. 



Euler 32 solved the problem to make z a 2 v, , z d 2 v squares, where 

 a 2 , , d 2 are four given squares, by investigating a quadrilateral the sines 

 of whose angles p, q, r, s are ax, , dx, where a, , d are given numbers. 

 Let A, - , D be their cosines. Since sin (p+g)+sin (r+s)=0, etc., we 

 get aB+bA+cD+dC = Q and two similar relations obtained by inter- 

 changing b, c, and B, C', or b, d and B, D. Hence we get the ratios of 

 A, -, D as cubic functions a, , 5 of a, , d. Thus A = ay, , 

 D = 8y. Then a 2 x 2 -\-a~y 2 = 1, b 2 x 2 -\-(3 2 y 2 = 1, and we find that x 2 = v/z, y 2 = 1/z, 

 where 



v=(a+b+c-\-d)(a-\-b c d)(b a+c d)(a-\-c b d), 



z = 4 (be ad) (ac bd) (ab cd) . 



Hence 



r 



a 



? = ~~T' 



Euler 33 required three numbers x, y, z such that the sum and difference 

 of any two are squares. Let x>y>z and set 



Then xy = (pq) 2 , xz = (rs) 2 . Also p 2 -(-g 2 = r 2 +s 2 if 



(4) p = ac+bd, q = ad bc, r = ad+bc, s = ac bd. 



Thus x = (a 2 + b 2 ) (c 2 + d 2 ) . It remains to make 



both squares. Their product is a square if 



Take d = a. Then a 2 = (?i 2 6 3 c*){(n 2 b c) . Take a = bc, and take 6 equal 

 to the numerator of the resulting fraction for b/c. Thus 



c = 2n 2 Tl, a = 



30 The Diarian Repository, or Math. Register . . . by a Society of Mathematicians, London < 



1774, 522-3. Cf. Euler. 28 



31 Miscellanea Math., London, 1775, 110. 



32 M<m. Acad. Sc. St. Petersb., 5, anno 1812, 1815 (1780), 73; Comm. Arith., II, 380-5. 

 3S Ibid., 6, 1813-4 (1780), 54; Comm. Arith., II, 392-5. Cf. Euler. 28 



30 



