CHAP. IV] SOLUTION OF x"-\-y 2 = z 2 . 167 



[These correspond to the rules of Pythagoras and Plato.] Leonardo 14 

 obtained rational solutions of x 2 + y 2 = a 2 by a method quite different 

 from that of Diophantus; starting with any known rational triangle for 

 which o? + /3 2 = y 2 , he took x = aa/y, y = a/3/7- 



F. Vieta 15 (1540-1613) used the method of Leonardo, last cited, and 

 that of Diophantus. 



M. Stifel 16 called a -b a diametral number if a 2 + 6 2 = c 2 and stated in- 

 correctly that a b is a diametral number if and only if a/b belongs to one of 

 the series li, 2f, 3f, and if, 2r|, Syf, , and hence in effect that 

 a : b = 2n 2 + 2n : 2n + 1 or a : b = 4n 2 + 8n + 3 : 4n + 4 [cf. Meyer 46 ], 

 which correspond to the solutions of a 2 + b 2 = c 2 by Pythagoras and Plato. 

 These diametral numbers are not those defined by Theon of Smyrna 2 of 

 Ch. XII. 



The Japanese manuscript of Matsunago 17 of the first half of the eigh- 

 teenth century contains three proofs of (1). 



T. Fantet de Lagny 18 replaced m by d + n in (1) and obtained 



x = 2n(d + n), y = d(d + 2ri), z = x + d 2 = y-\- 2n 2 . 



Taking d = 1 or n = 1, we obtain the rule of Pythagoras or that of Plato. 



C. A. Koerbero 19 proved that the sides of any rational right triangle are 

 proportional to the numbers (1). 



L. Euler 20 expressed the hypotenuse c as b + anfm. By a 2 -\- b 2 = c 2 , 

 b: a = m? n 2 : 2mn. Hence a, b, c are proportional to the numbers (1) 

 with m > n > 0. 



Euler 21 noted that the sum of the squares of x + 1/x and y + 1/y is a 

 square if 



= _ + _ + + = n ^ 



a; + p 



the latter being true if (p 2 l)z = 4p. 



J. P. Griison 22 noted that n + 1 and n generate a triangle [of Pythagoras' 

 type] whose larger leg y = 2n 2 -f 2n and hypotenuse y + 1 generate a 

 new triangle whose least side is a square [2y + 1 = (2w + I) 2 ]. 



L. Poinsot 23 noted that every set of integral solutions of z 2 y 2 = x 2 

 is given by z = (p + <?)/2, y = (p <?)/2, where a; 2 has been expressed 

 in every way as a product of two integers p and q, both odd and relatively 

 prime or both even, but with no common factor > 2. 



14 Liber Abbaci, Ch. 15 (Scritti L. Pisano, Rome, 1, 1857). 

 15 Franciscus Vieta, Zetetica, 1591, IV, 1; Opera Math., 1646, 62. 



16 Arith. Integra, Niirnberg, 1544, f. 14v-f. 15v. Copied by loseppo Vnicorno, De 1'Arith. 



Universale, Venetia, 1598, 62. 



17 Y. Mikami, Abh. Gesch. Math. Wiss., 30, 1912, 229. Report by K. Yanagihara, TShoku 



Math. Jour., 6, 1914-5, 120-3; continued, 9, 1916, 80-7 (by use of progressions). 



18 Hist. Acad. Sc. Paris, 1729, 318. 



19 Nova trianguli rectanguli analysis, Halae Magd., 1738, 8. 

 20 Comm. Acad. Petrop., 10, 1738, 125; Comm. Arith., 1, 1849, 24. 



21 Opusc. anal., 1, 1783, 329; Comm. Arith., II, 46. 



22 Enthullte Zaubereyen u. Geheimnisse der Arith., Berlin, 1796, 104-6. 



23 Comptes Rendus Paris, 28, 1849, 581-3; also p. 579 by J. B. Biot. 



