CHAPTER IV. 



RATIONAL RIGHT TRIANGLES. 

 METHODS OF SOLVING x 2 + y 2 = z 2 IN INTEGERS. 



According to Proclus, 1 Pythagoras represented the smaller leg by 

 x = 2a + 1, the larger leg by y = 2a 2 + 2a, and the hypotenuse by 

 z = y + 1. Plato 1 took the difference z y to be 2 (instead of 1) and 

 obtained 2 x = 2a, y = a 2 I, z = a 2 + 1. 



The Hindus Baudhayana and Apastamba, 3 about the fifth century B.C., 

 obtained independently 4 (?) of the Greeks the solutions (3, 4, 5), (5, 12, 13), 

 (7, 24, 25), which are cases of the rule of Pythagoras, and (8, 15, 17), 

 (12, 35, 37), cases of the rule of Plato. 



Euclid 5 gave the set of solutions 



af3 7 , X/3 2 - 7 2 ), X/3 2 + 7 2 ), 

 as well as (II, 5; X, 30) the related set 



' ifvfv* O \ I iv iv I O \ * * v *v ) 



t \ / i ft \ J 



Marcus Junius Nipsus, 6 at least a century before Diophantus, gave 

 two rules to find right triangles with integral sides, one leg being given. 

 Expressed algebraically, his rules give, as solutions of z 2 y 2 = x 2 , 



z = |x 2 + 1, y = ^x 2 1, for x even, 



formulas equivalent to those of Pythagoras and Plato, respectively. 



Diophantus 7 took a given value (in fact, 4) for z and required that 

 z 2 x 2 shall be a square of the form (mx z} 2 . Thus 



9w? /w 2 - 1 \ 



lil,^ I lid -L \ 



Here m is any rational number; replacing it by m/n, and taking z = m 2 + n 2 , 

 we get 



(1) x = 2mn, y = m 2 n 2 , z = m 2 + n 2 . 



1 Proclus Diadochus, primum Euclidis elem. libr. comm. (5th cent.), ed. by G. Friedlein, 



Leipzig, 1873, 428. Elements d' Euclide avec les Comm. de Proclus, 1533, 111; Latin 

 trans, by F. Barocius, 1560, 269. M. Cantor, Geschichte Math., ed. 3, I, 1907, 185-7, 

 224. G. J. Allman, Greek Geometry from Thales to Euclid, 1889, 34. 



2 Cited by Heron of Alexandria, Geometric, p. 57; Boethius (6th cent.), Geometric, lib. 2. 



3 Sulbasutra, publ. by A. Btirk with German transl., Zeitschrift der deutschen morgenland- 



ischen Gesell., 55, 1901, 327-91, 543-91. 



4 Biirk. 3 H. G. Zeuthen, Bibliotheca Math., (3), 5, 1904, 105-7. M. Cantor, Geschichte 



Math., ed. 3, I, 1907, 636-45; 96 for 3 2 + 4 2 = 5 2 in Egypt. 



6 Elementa, X, 28, 29, lemma 1; Opera, ed. by J. L. Heiberg, 3, 1886, 80. M. Cantor, Ges- 



chichte Math., ed. 3, I, 1907, 270-1, 482. 



6 Cf. J. B. Biot, Jour, des Savants, 1849, 250-1; Comptes Rendus Paris, 28, 1849, 576-81 

 (Sphinx-Oedipe, 4, 1909, 47-8). M. Cantor, Die romischen Agrim . . . Feldmess., 

 1875, 103, 112, 165. C. Henry, BuU. Bibl. Storia Sc. Mat. Fis., 20, 1887, 401-2. 



7 Arith., II, 8; Opera, ed. by P. Tannery, 1, 1893, 90; T. L. Heath, 1910, 145. 



165 



