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



W. A. Whitworth 30 noted that the triangle with the altitude 12 and sides 

 13, 14, 15 is the only one in which the altitude and sides are consecutive 

 integers. 



G. Heppel 31 noted that there are 220 triangles with integral sides ^ 100 

 and integral areas, but repeated (39, 41, 50). He listed only 55 rational 

 scalene triangles with relatively prime sides. 



Worpitzky 32 gave without proof a formula equivalent to (1). 



R. Miiller 33 considered rational triangles whose sides are consecutive 

 integers x 1, x, x + 1. Since the area is to be rational, x 2 4 = 3y 2 , 

 whence x = 2u, y = 2v, u? 3v 2 = 1. Hence the triangles are (3, 4, 5), 

 (13, 14, 15), etc. 



A. Martin 34 noted that the triangle with the sides 2m 2 + 1, 2m 2 + 2, 

 4w 2 + 1 has a rational area. 



T. Pepin 35 gave a historical note on rational triangles. 



O. Schlomilch 36 gave the same method and results as Hart. 23 



C. A. Roberts 37 noted that, if u is a square and w the double of a 

 square, u -f- w, u + 2w, 2u + w are the sides of a triangle with rational 

 area (u + w} ^2uw and listed many triangles with sides < 500. The triangle 

 is special since one side equals one- third of the sum of the remaining two. 



S. Robins 38 tabulated rational triangles with a given base and a given 

 difference between the remaining two sides; also (pp. 262-3) rational 

 triangles with sides x, x -f- n, 2x n for given n's. 



H. F. Blichfeldt 39 derived (1) by use of Heron's formula for area. 



S. Robins 40 found rational triangles whose sides are consecutive integers 

 by taking x 2 and x + 2 as the segments of the base made by the per- 

 pendicular to the base. The altitude is (3x 2 3)*, which is made rational 

 by choice of x by means of convergents to the continued fraction for Vs. 



A. Martin 41 juxtaposed two right triangles in various ways to obtain 

 rational triangles. From Heron's formula for the area A of a triangle with 

 the sides x, y, z, 



( -. -Y.2 .. 7 ,2^2 _ ! T 2 7 ,2 _ A2 _ /1 T7/ . . A/7/7/1 2 if A W^ 



Q\Z x y ) x y a { 2 xy AqiP) , 11 a - - . 



c :/ 



Then 



30 Math. Quest. Educ. Times, 36, 1881, 42. 



31 Ibid., 39, 1883, 37-8. Cf. Martin. 59 



32 Zeitschr. Math. Naturw. Unterricht, 17, 1886, 256. 



33 Archiv Math. Phys., (2), 5, 1887, 111-2. 



34 Math. Magazine, 2, 1890, 6. 



36 Mem. Accad. Pont. Nuovi Lincei, 8, 1892, 85. 



36 Zeitschr. Math. Naturw. Unterricht, 24, 1893, 401-9. 



37 Math. Magazine, 2, 1893, 136. 



38 Amer. Math. Monthly, 1, 1894, 13-14, 402-3 (for base 9). 



39 Annals of Math., 11, 1896-7, 57-60. 

 "Amer. Math. Monthly, 5, 1898, 150-2. 

 fl Math. Magazine, 2, 1898, 221-236. 



