226 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



Leonardo Pisano, 3 in his Liber Quadratorum of 1225, proved (1) and 

 used it to solve x 2 + y 2 = a 2 + & 2 , given a solution of c 2 + d 2 = e 2 : 



x (ac + bd)/e, y = (ad bc)/e. 



This solution was reproduced without proof by Lucas Paciuolo and Cardan 

 hi their arithmetics (full titles on p. 6 and p. 8 of Vol. I). 

 F. Vieta 4 noted that X 2 = F 2 + G 2 , Z 2 = B 2 + D 2 imply 



(!') (XZ) 2 = (BG DF) 2 + (BF =F G) 2 . 



If B and D are the hypotenuses, M, N and MD/B, ND/B the pairs of 

 legs of two similar right triangles, a third right triangle with the legs 



(BM d= ZW)/ and (BN =F DM)/5 has the hypotenuse BMD 2 . In 

 the special case F = B,G = D, (10 becomes (X 2 ) 2 = (2J5D) 2 + (5 2 - D 2 ) 2 ; 

 the right triangle with the sides 2BD, D 2 B 2 , X 2 is called the triangle of 

 double angle. Using the latter and the given triangle (B, D, X), and 

 applying the same rule, we obtain the triangle (3BD 2 B 3 , D 3 3B 2 D, X 3 ) 

 of triple angle, etc. [equivalent to De Moivre's formulas for cos na, sin na 

 in terms of cos a, sin a]. 



Vieta, 5 to express Z 2 = B 2 + D 2 as the sum of two new squares, employed 

 a second right triangle (F, G, X) to obtain (10, whence [cf. L. Pisano 3 ]] 



(BGDF\ 2 

 * ~\ X / 



He noted that the method of Diophantus II, 10 consists in denoting the 

 sides of the required squares by A + B, SA/R D. Thus 



2SRD - 2R 2 B 2SRD + B(S 2 - R 2 ) 



S 2 + R 2 ' + ~ S 2 + R 2 



Hence from (B, D, Z} and the triangle (2SR, S 2 - R 2 , S 2 + R 2 ), formed 

 from S, R, construct a third triangle by (10 and reduce the sides in the 

 ratio R 2 + S 2 . 



G. Xylander, 6 in his comment on Diophantus V, 12, stated incorrectly 

 that a must be the double of a prime. 



C. G. Bachet 7 remarked that 10 is the double of a prime, while 

 2-10 + 1 = 21 is neither a square nor the sum of two integral squares, 

 and expressed his belief that 21 is not the sum of two rational squares. 

 While Diophantus seemed to infer that the double of the even number a, 

 increased by unity, should be a prime, this would exclude 22, 58, 62, for 

 which 2-22 + 1 =45 = 36 + 9, etc., whereas 45, 117 are not primes. 



8 Tre Scritti inediti, 1854, 66-70, 74-5; Scritti L. Pisano, 2, 1862, 256. Review by O. 



Terquem, Annali Sc. Mat. Fis., 7, 1856, 138; Nouv. Ann. Math., 15, 1856, Bull. Bibl. 



Hist., 61. Cf. Woepcke, Jour, de Math., 20, 1855, 57; A. Genocchi, Annali Sc. Mat. 



Fis., 6, 1855, 241-4; M. Chasles, Jour, de Math., 2, 1837, 42-9, who gave a geometrical 



proof. 

 4 Ad Logisticem Speciosam Notae Priores, Props. 46-48; Opera Math"., 1646, 34. French 



transl. by F. Ritter, Bull. Bibl. Storia Sc. Mat., 1, 1868, 267-9. 

 6 Zetetica, 1591, IV, 2, 3; Opera Math., 1646, 62-3. 



6 Diophanti Alex. Rerum Arith. Libri sex, Basel, 1575, 129, 1. 9. 



7 Diophanti Alex. Arith., 1621, 301-4. 



