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



let the angle A from which the median AD is drawn be acute. Let BC 

 denote the side whose mid point is D. Take any number, as 13, which 

 is a sum of two squares, 2 2 + 3 2 , and take DC = 2, AD = 3. Then 

 AB 2 + AC 2 = 2AD 2 + 2DC 2 = 2-13 = 5 2 + I 2 , since the double of a sum 

 of two squares is a sum of two squares. But 5 and 1 are not values of AB, 

 AC. Hence we divide 5 2 + I 2 into a sum of two other squares by Dio- 

 phantus II, 10, viz., (5 - N} 2 + (1 + 2N) 2 , whence N = 6/5, AB = 3f, 

 AC = 3|. Multiplying all by 5, we get AB = 19, AC = 17, BC = 20, 

 AD = 15. 



If A is obtuse, take DC = 3, AD = 2. We get the same values of AB 

 and AC as before, while BC = 30, AD = 10 (in place of the misprint 12). 



T. F. de Lagny 83 proved that in any parallelogram the sum of the squares 

 of the two diagonals equals the sum of the squares of the four sides and 

 noted the examples 9 2 + 13 2 = 2(5 2 + 10 2 ), 17 2 + 31 2 = 2(15 2 + 20 2 ). To 

 solve x 2 + y 2 = 2 (a 2 + 6 2 ) in integers, we may, for a = b, take y = 2a xb/c, 

 whence x = 4abc/(b 2 + c 2 ). Next, a special solution of 



z 2 + 2/ 2 = 2{a 2 + (a + b) 2 } 



is given by x = b, y b + 2a; to find the general solution, set c = 2a + 6, 

 x = cz,y = b=F zd/e; then z = ( 2bde =F 2ce 2 )/(d 2 + e 2 ). 



B. A. Gould 84 found a parallelogram with rational sides a, b and diagonals 

 x } y. The condition is x 2 + y 2 = 2(a 2 + b 2 ). Set a -\- b = s, a b = t, 

 whence x 2 + y 2 = t 2 + s 2 . A solution is fx = sd + te, fy = se td, if 

 f 2 = d 2 -f- e 2 . Wm. Lenhart called the sides a a' and diagonals 26, 26', 

 whence a 2 b 2 = 6' 2 a'\ which is satisfied if 



a, b = nn' mm'', b', a' = nm f mn'. 



J. Maurin 85 gave Gould's solution. 



E. Henet 86 noted that in the triangle with the sides x = pv + u, 

 y = pu v, z = u -\- v -+- p(u v), where u > v, p > 1, the median m z is 

 rational: 2m z = p(u -\- v) u + v. Also m y is rational if 



u : v = ( M - 2)(4p - /i) : (AI - 4)(2p + /*), 4 < M < 3 + p. 



M. A. Gruber 87 solved 2 (a 2 + 6 2 ) = c 2 + d 2 by setting b = a + p, 

 c = 2a + q, whence a follows rationally. W. F. King (pp. 320-2) proceeded 

 as had Gould. 84 



H. Schubert 88 discussed triangles with rational sides a, 6, c, and one or 

 more rational medians, that to side a being designated by t a . Since 



(2O 2 - (b - c) 2 = (b + c) 2 - a 2 = 4s(s - a), s = }(a + b + c), 

 the rationality of a implies that of x, where 



Z a i(6 c) = sx, t a + (& c) = (s a)/x. 



83 Hist. Acad. Roy. Sc. avec les M6m., ann<Se 1706, Paris, 1731, 319-333 (Hist., 83-99). 



84 Cambridge Miscellany, 1, 1843, 14. 



85 L'interme'diaire des math., 3, 1896, 210. 



86 Ibid., 240. 



87 Amer. Math. Monthly, 3, 1896, 219-221. 



88 Auslese Unterrichts- u. Vorlesungspraxis, Leipzig, 2, 1905, 68-92; same in Schubert, 47 

 3S-50. 



