502 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xix 



The preceding problem is evidently equivalent to that of finding a 

 rectangular parallelepiped whose edges and diagonals of faces are all 

 rational. If we add the condition that also a diagonal of the solid shall be 

 rational, we have a problem which H. Brocard 25 attempted to prove im- 

 possible by means of the terminal digits. P. Tannery 26 noted that the 

 proof is insufficient since it supposes that the numbers in question are 

 relatively prime in pairs. 



V. M. Spunar 27 noted that the last problem is impossible. 



A. Mukhopadhyay 28 proved it impossible [if the edges be relatively 

 prime integers]. The solutions of x 2 +?/ 2 =D are known to be x = 2k, 

 y = k 2 -!. Similarly, y = 21, z = 1 2 -1; z = 2m,x = m?-l. Then 



requires x = 2n, l 2 -\-l = n 2 1, whereas n 2 I 2 = 2 has no integral solutions. 



M. Rignaux 29 remarked that the problem is difficult and not yet solved. 

 He satisfied three of the conditions, but not the fourth. 



A. Transon 30 stated falsely that a tetrahedron with six integral edges 

 cannot have among its solid angles a tri-rectangular trieder, and stated 

 that one can find, in an infinitude of ways a tretahedron OABC with 

 integral values of the three edges meeting at 0, and of the areas of the four 

 faces, while the three face angles at are right angles. C. Chabanel 31 

 and C. Moreau 31 gave the solution 



OA=4xyz, OB = 2y(x 2 +y 2 -z 2 *), OC = 2x(x 2 +y 2 -z 2 ), 

 area ABC = 2xy(x 2 +y 2 -z 2 }(x 2 +y 2 +z 2 '). 



FOUR SQUARES WHOSE SUMS BY THREES ARE SQUARES. 



L. Euler 32 applied his 4 method to A=x, B, C and the following D, but 

 was led to a condition difficult to treat and abandoned that metrical. Next, 

 take A = y, B and C as in Euler, 4 and D= {2px-(p 2 -l}y}/(p 2 +l'). Then 



4/Y f 2 " 

 = x 2 +y 2 -2gxy, g= 



Since S = x 2 +y 2 , the condition S = y 2 +B 2 +C 2 +D 2 gives y 2 +D 2 -2gxy = Q. 

 Inserting the value of D, we get 



Take R = gp 2 +2p+g. Then p=-g, 4gxfy = 2(g*+l] or 4. Using the 



25 L'intermediaire des math., 2, 1895, 174-5. 



26 Ibid., 3, 1896, 227. 



27 Amer. Math. Monthly, 24, 1917, 393. 



28 Math. Quest. Educ. Times, 41, 1884, 60. 



29 L'interme'diaire des math., 26, 1919, 55-57. 



3n Nouv. Ann. Math., (2), 13, 1874, 64; correction, 200. 

 81 Ibid., 340-3. 



32 Novi Comm. Acad. Petrop., 17, 1772, 24; Comm. Arith., I, 467-72; Opera Omnia, (1), 

 III, 203. Second method reproduced by Martin, Math. Mag., 2, 1898, 217-8. 



