CHAP. XIX] x-+y~, x~+z-, y~+z- ALL SQUARES. 499 



J. Cunliffe 9 set x-+y' i =(x+y a) 2 , z 2 +2 2 ==(.r-f z 6) 2 . From the re- 

 sulting two values of x we get 



2dy+a i -\ 



Then ?/ 2 +2 2 = D if 4dy i +4d(a 2 -2ad)?/H \-aW(a-d)-= D. Let it be 



the square of 2dy~-\- (a? 2ad)y ad(a d) . Then 



2ad(2ad - d 2 ) 2a(a - 6) (a 2 - 6 2 ) 



"Calculator" 10 first solved x zj ry 2 = a 2 , x"-{-z 2 = b 2 . Take b = rv a, 

 z = y sv, then a 2 y 2 = b- z 2 gives v = (2ra 2sy}/(r' 2 s-), whence b, z are 

 known. To satisfy x 2J ry 2 = a 2 take 



/ *> *)\ / 9 I 0\ / 9 9\ f C\ \ / O ON / O 9\ 



a = (r- s-)(m-+n-), y = (r* s-)(2mn), x= (r 2 s 2 )(m 2 n 2 ). 



Then 2 = (r 2 +s 2 ) -2mn 2rs(m 2 +n 2 ). Then 7/ 2 -J-z 2 becomes a quartic in m 

 which is equated to the square of m 2 ran(V 2 +s 2 )/(rs) n 2 , whence 

 77i : ?i = 4rs : r 2 +s 2 . Taking n = r 2 +s 2 , we have 



which equal the products of s 6 by Euler's 4 values for/=r/s. Cf. Euler. 6 

 S. Ward 11 took x 2 +?/ 2 = a 2 , z 2 +2 2 = (m-f n) 2 , y z +z* = (m-ri)*. Then 



Let 2a 2 = m 2 +16?i 2 . Then the first two expressions are squares and the 

 third becomes 3w 2 -12w 2 =D. Take m = np, 3p 2 -12=/ 2 (p-2) 2 . Thus 



.Q> 

 J" -3 



Set /= l+q. The quartic is the square of 8-2^+fg 2 if q= --16/3. Then 

 z = 240, ?/ = 44, 2 = 117, which appear to be the least numbers. 



W. Lenhart 12 took z = (p 2 -l)/(2p), y = 2qf(q*-l), z = l. Then 



if 



(p 2 -l) 2 (g 2 -l 



provided p 2 +g 2 = 5 = 1 +4. As usual, 



s + lor3. For s = 2, p = 11/5, 3 = 2/5. 



C. Gill 13 obtained Euler's 6 result by setting 



b = a cos A +z sin A, ?/ = 2 cos A a sin A 

 and c, a; to be the analogous functions of B. Then a?+z 2 = 



9 New Series Math. Repository (ed., Leybourn), London, 1, 1806, II, 39. Also in Math. 



Repository, 3, 1804, 5. 



10 The Gentleman's Math. Companion, London, 4, No. 19, 1816, 626-7. Same with altered 



lettering, S. Bills, The Mathematician, London, 3, 1850, 200-1. 



11 J. R. Young's Algebra, Amer. ed., 1832, 338-9. 



12 Math. Miscellany, 2, 1839, 132. Reproduced in Math. Magazine, 2, 1898, 215-6; Sphinx- 



Oedipe, 8, 1913, 84. 



13 Application of Angular Analysis . . . , N. Y., 1848; Reproduced in Math. Quest. Educ. 



Times, 17, 1872, 82-3. 



