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



F. Vieta 75 generalized the method of Diophantus III, 12 [13J. Let A 

 be the second number. Then the first is (B 2 a)/A, and the third is 

 (D- a)/A. Hence must 



B 2 -a D 2 -a 



We can make B 2 a=F 2 , D 2 a = G~ in an infinitude of ways. Then 

 F-G 2 +aA 2 is to be a square, say (FG-HA}\ Hence A = 2HFG/(H--a). 



C. G. Bachet 76 , who doubted that Diophantus had a general solution, 

 used the canon : Subtract the given number from each of two squares and 

 divide the remainders by the difference of the roots of these squares; 

 then the quotients and the difference of the roots are three numbers giving 

 a solution. For a = 6, take N+3 and 2N-\-3 as the roots of the squares; 

 then N, N+6+3/N and 4N+12+3/N give a solution. 



De Sluse 77 took an arbitrary square 6 2 and set d = b 2 a, xy = x 2 +2xb-\-d, 

 whence xy + a = (x -f- b) 2 . Similarly, we can set z = xc~(e~ + 2bc/e + dfx, whence 

 xz-\-a=(xc/e+b) 2 . Let yz-\-a be the square of (cx+cfy/e+b+d/x. Thus 



26 2 c dc 2 _b 2 c 2 2dc 

 H r rH --- 



e- e 2 



When 6 2 is replaced by d-\-a, this reduces to 2 = cje, so that the required 

 numbers are x, x+2b-{-d/x, 4#+4&+d/#. For a negative, a= A, call the 

 numbers x, y = x+A/x, z = xb 2 /c 2 -\-Alx. Then xy A=x 2 , xz A=x 2 tf/c 2 , 

 yz-A = (xb/c+AfxY- if fe/c = 2. 



N. Saunderson 78 (blind from infancy) gave the solution 



r~ a s 2 -a 



x = - -, y = - -, z = r s or 2x+2y (r s), 

 r s r s 



where r and s exceed Va and r>s. For a = 1, a solution is 



x, y = a-x+2a, z = px+2l3, a-j8 = 



V. Ricatti 79 treated the problem. 



L. Euler 79a set xy+a = p 2 , z = x+y2p, whence 

 yz-{-a = (yp} 2 . For a = 12, p = 4, then x = y = 2, z = l2. For a = 12, p = 5, 

 then x = l, 2/ = 13, z = 4 or 24. In art. 231, he noted that for a = l the 

 general solution is 



x=(p 2 -l)lz, y = (q 2 -l)/z, z={(p 2 -l}(ql)-r 2 }f(2r). 

 Euler 80 treated AB-l=p 2 , AC-l = q 2 , BC-l=r\ Thus 



2 = l(r 2 +l), I=(p 2 +l}(q 2 +l)=m 2 +n 2 , m = pql, n = p^q, 

 A 2 B 2 C 2 =(mr+ri) 2 +(nr-m} 2 . 



"Zetetica, 1591, V, 7[8], Francisci Vietae Opera matheraatica, ed. Francisci A Schooten, 

 Lugd. Bat., 1640, 78. 



76 Diophanti Alex., 1021, 149, 215. 



77 Renati Francisci Slueii, Mesolabum, acccssit pars altera de analysi et miscellanea, Leodii 



Eburonum, 1008, 177-8. 



78 The Elements of Algebra, 2, 1740, 390-5. 



79 Institutiones analyticae a Vincentio Riccato, Bononiae, I, 1765, 64. 



790 Algebra, 2, 1770, art. 232 (end of art. 233); 2, 1774, p. 305 (pp. 310-1); Opera Omnia, 

 (1), 1, 405 (408). 



80 Posth. paper, Comm. Arith., II, 577-9; Opera postuma, 1, 1802, 129-131. 



