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



pyramid. For example, if the sides of the triangular base are 13, 14, 15, 

 take cos n = 65/97, sin /* = 72/97; then the altitude is h = 9, lateral edge 

 97/8, and volume 252. 



Schubert 151 discussed rational spherical triangles, i. e., having rational 

 values for the tangents of half of each side and angle. 



R. Giintsche 152 made use of F. Bessell's 153 relations between the face and 

 trihedral angles and reduced the problem of the rational tetrahedron to 

 a diophantine equation quadratic in q and quadratic in r with coefficients 

 involving an arbitrary parameter p. Euler's 144 process of Ch. XXII is 

 used to find solutions q, r rational in p, so that the six edges, the surface 

 areas and volume are expressed rationally in p. 



Giintsche 154 considered tetrahedra whose edges, surface areas and 

 volume are all rational and having all faces congruent. He reduced the 

 problem to the solution of 



W(W + <A + 6 - 1)(^0 - t - e - 1) = h\ 



but did not solve it in general. But seven particular sets of solutions in- 

 volving an arbitrary parameter are found. 155 The tetrahedra of Hoppe 149 

 are all of the type here considered. 

 . E. Haentzschel 156 wrote Giintsche's cubic function in the form 



A 3 (0 3 - e) - - 4i 2 2 - t(e z - 0) 



and reduced it to Weierstrass' normal form 4H(s e,-) by the substitution 



= 4(8 + 2 /3) 

 3 - 



obtaining e! = - 2 /3; e 2 ,e z = =F 3 /4 + 2 /6 =F 0/4. By use of Weierstrass' 

 ^-function, he solved 411 (s e t ) = y 2 . The case = 7/3 is treated in detail. 



* 0. Schulz 157 treated rational tetrahedra. 



For special tetrahedra, see papers 30-31 of Ch. XIX. 



161 Auslese . . . Unterrichts- und Vorlesungspraxis, 3, 1906, 202-250. 

 152 Sitzungsber. Berlin Math. Gesell., 6, 1907, 2-16. 



163 Archiv Math. Phys., 65, 1880, 363-372, on spherical triangles with rational values for the 



sines and cosines of the angles and sides. Cf. M. Bambas, (3), 26, 1918, 195-6. 



164 Sitzungsber. Berlin Math. Gesell., 6, 1907, 38-53. 



166 He gave two such sets in Archiv Math. Phys., (3), 11, 1907, 371. 



iM Sitzungsber. Berlin Math. Gesell., 12, 1913, 101-8. Continued, 17, 1918, 37-9. 



167 Ueber Tetraeder mit rationalen Masszahlen der Kantenlangen und des Volumens, Halle, 



1914, 292 pp. Cf. Haentzschel. 144 



