506 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xix 



Forg=(r 2 -s 2 )/(rs), 



i 4 = (m 2 qmn+n 2 ) 2 , 



if TW/TO = (<r+8)/(4g) . Or we may use 



-~ = -=-s, A =r 2 n 2 r 2 m 2 -\-2rsmn. B = s 2 m 2 s 2 n 2 -{-2rsmn. 

 (z+y) 2 z+y B 



Take B=(tn-sm} 2 to get ?w. Then A=rV if r = 4/s 2 /( 2 -3s 2 ). He 42 

 later used the same z, but took = 2wm(r 2 +s 2 ), a = (r 2 +s 2 )(m 2 n 2 ), whence 

 x 2 z 2 = a 2 . Set b = a rv, y = z+sv; then a 2 +s 2 = 6 2 +?/ 2 gives w in terms 

 of a, r, s, z. Finally, y 2 z 2 = D if a quartic in m is the square of (say) 

 m 2 mn (r 2 s 2 ) / (rs) + n 2 , whence 



m : n = r 4 +6r 2 s 2_j_ s 4 . 4 rs ( r 2_ s 2) > 



J. Cunliffe 43 obtained Calculator's 41 first result by the same method. 

 S. Ward 44 discussed Euler's 40 final condition. Set/=/ ; gr, h = h'k, 



which reduces to/' 4 //i' 2 =/' 4 -2. The latter is a square if / /2 = (Y 2 +2s 2 )/(2rs), 

 and r 2 +2s 2 = D if r = t 2 - 2, s = 2t. The value for/' 2 is a square if t(t 2 - 2) = D . 

 Taking t = 2, we get xfz = -41/9, y/z = 185/153. Or we may treat P=D 

 by setting q = mp and treating (mrp 4 l)(m 2 1) = D by the usual method 

 for quartics, one solution p = l being known. 



W. Lenhart 45 took the roots of the three squares to be 



O I O O t O 



x 2 -{-y 2 zr+wr 



- - j^ 



The square of either the first or the second exceeds unity by a square. 

 Hence it remains only to make the difference of their squares a square, 

 viz., (vx-\-wy)(vxwy)(vy-\-wx)(vywx) = [H. Take v = ty+x, w = txy, 

 whence vy+wx = t(vx wy). Then shall t(vx+wy}(vy wx) = D, which 

 holds if 



z 2 y 2 +2txy= D, y 2 x 2 +2xy/t= D. 



The second condition is satisfied if x 2y/t. Then the first becomes 

 4+3 2 = D = (2 pf) 2 , say, whence we get t and x = p 2 3, y = 2p, 

 y=(p 2 +l) 2 +8j w = 2(p z S)p. Or we may take x 2 y 2 -\-2txy = 

 whence x = p-+l, y = 2(p + ). Then F(y z -x-+2xy/C) = (ty-r) 2 if 



~ 6 *o [ 716 ^/ "" Tt/ 6 TC IJL>*L/ / 



Then 



r=a ^r- t= 



Dividing the values of x and y by d= (p 2 +l)/(8p) and those of v and w 



42 The Gentleman's Math. Companion, London, 4, No. 19, 1816, 628-31. 



ZWd., 5, No. 26, 1823, 262-4. 



44 J. R. Young's Algebra, Amer. ed., 1832, 339-341. 



Math. Miscellany, 2, 1839, 129-132; French transl., Sphinx-Oedipe, 8, 1913, 83-4. 



