216 



HISTORY OF THE THEORY OF NUMBERS. 



[CHAP. V 



On the ratios of the sides to the radius of the inscribed circle see 

 Gerono, 150 Ch. XXIII. 



The following papers were not available for report : 



C. Klobassa, tjber Pythagoreische u. Heronische Zahlen, Progr., Troppau, 1908. 

 E. Haentzschel, Das Rationale in der algebraischen Geometric [an address], Unterrichtsblatter 

 Math. Naturw., 21, 1915, 1-5. 



RATIONAL QUADRILATERALS. 



A rational quadrilateral is one whose sides, diagonals and area are 

 expressed by rational numbers. 



Brahmegupta 1 ( 38) stated that " the legs of two right triangles multi- 

 plied reciprocally by the hypotenuses give the four sides of a trapezium." 

 Bhdscara 130 (born 1114) illustrated this construction of a rational quadri- 

 lateral by starting with the right triangles (3, 4, 5), (5, 12, 13). Multiplying 



the legs of the first by the hypotenuse of the 

 second, we get two opposite sides of the quad- 

 rilateral; multiplying the legs of the second by 

 the hypotenuse of the first, we get the two re- 

 maining sides of the quadrilateral. One diago- 

 nal is the sum 4 12 + 3 5 = 63 of the products 

 of the legs of one triangle by the corresponding 

 legs of the other. The other diagonal is 



4-5 + 3-12 = 56. 



As the Commentator Gane'sa (1545 A.D.) indicated (p. 81), the quadri- 

 lateral is formed by the juxtaposition of four right triangles obtained by 

 multiplying the sides of each given triangle by the perpendicular and base 

 of the other. Bhaseara noted that if we take the sides of the quadrilateral 

 in the new sequence 25, 39, 52, 60, one diagonal is still 56, but the other 

 is now the product 65 of the two hypotenuses. He noted ( 179-184, pp. 

 76-8) that the quadrilateral with the sides 40, 51, 68, 75 and diagonals 77, 

 85 has the area 3234. 



M. Chasles 131 made clear the true sense of Brahmegupta's theorem. 

 Let a, 6, c, d, e be integers, such as 3, 4, 5, 12, 13, for which a 2 + b 2 = c 2 , 

 c 2 -f d? = e 2 . Construct the quadrilateral A BCD with perpendicular diag- 

 onals AC, BD, crossing at / (see figure above), with 



AI = ac, CI = bd, BI = ad, DI = be. 

 Then 



AB = ae, BC = cd, CD = be, AD = c 2 . 



Hence the sides are rational and the quadrilateral is inscriptible in a circle, 

 since AI-CI = BI-DI; its diameter is ce/2. The area is 



!(ac + bd)(bc + ad). 



Lilavatf, $ 191-2; Colebrooke, 1 pp. 80-83. 



131 Aper^u historique, Bruxelles, 1837, Note 12, p. 440; ed. 2, Paris, 1875; ed. 3, Paris, 1889, 



p. 421. Cf. O. Terquem, Nouv. Ann. Math., 5, 1846, 636; H. G. Zeuthen, Bibliotheca 



Math., (3), 5, 1904, 108. 



