196 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v 



Many 18 proved that if the sides and area be integers, the area is divisible 

 by 6. Take the sides to be the products of (1) by pqrs. Then the area is 

 pqrs(ps + qr)(pr qs). 



S. Tebay 19 tabulated 237 rational triangles arranged according to the 

 magnitude of the area, the greatest area being 46410 (cf. Martin 46 ). 



J. Wolstenholme 20 found a triangle whose sides and area are in arith- 

 metical progression. Take a b, a, a + b as the sides, a + 26 as the area. 

 Then 



q(3a 2 - 16) 



16 + 3a 2 



W. Ligowski 21 found a triangle whose sides a, 6, c, area F, and radii r 

 and p of circumscribed and inscribed circles, are all rational. He assumed 

 that s a = px, s b = py, s c = pz, where s is the semi-perimeter, and 

 readily proved that the sides are proportional to 



a = x(y* +1), b = y(x* +1), c = (x + y)(xy - 1), 

 whence 



P = xy-l, r = iO 2 + l)fe 2 +1), F = xy(x + y)(xy - 1). 

 W. Simerka 210 gave several methods of finding rational triangles and a 

 table of the 173 having sides ^ 100, showing also the area, tangents of the 

 half angles, and the coordinates of the vertices (cf. Scherrer 62a ). He proved 

 that the perimeter is always even. 



H. Rath 22 employed the segments a, /?, 7 of the sides determined by the 

 points of tangency of the inscribed circle. Then the sides are a + 0, 

 + 7, P + J an d the square of the area is aj3j(a + J3 + 7). The latter 

 is a rational square only for a = dj 2 , ft = dB, 7 = dC, where B and C are 

 any two positive relatively prime integers, and likewise for k and j t while 

 d/5 is the value of the fraction 



BC(B + C) 



k 2 - BCj z 



when reduced to its lowest terms. Each resulting set of rational numbers 

 , j8, 7 defines a rational triangle, the condition that the sum of any two 

 sides shall exceed the third being evidently satisfied. His final tables 

 show relatively prime integral sides, the triangles whose area is a multiple 

 of some side being listed separate from the others. He gave (p. 218) nine 

 rational triangles whose sides form an arithmetical progression, the common 

 difference being here given as a subscript : 



(3, 4, 5) lf (13, 14, 15) i, (15, 26, 37) n, (75, 86, 97) n, 

 (25, 38, 51) ia , (61,74,87)!,, (15, 28, 41) 13 . 



18 The Lady's and Gentleman's Diary, London, 1866, 61, Quest. 2044. 



"Elements of Mensuration, London and Cambridge, 1868, 113-5. Table reprinted by 

 G. B. Halsted, Metrical Geometry, 1881, 167-170. 



20 Math. Quest. Educ. Times, 13, 1870, 89-90. Same by D. S. Hart, 20, 1874, 56. 



21 Archiv Math. Phys., 46, 1866, 503-4. 

 * Ibid., 51, 1870, 196-240. 



22 Archiv Math. Phys., 56, 1874, 188-224. See the compact exposition by P. Bachmann, 



Niedere Zahlentheorie, 2, 1910, 440-1. Cf. Kommerell 270 of Ch. XXII. 



