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



J. Matteson 161 found four squares such that fifteen linear or quadratic 

 functions of the squares or their roots shall be squares. 



A. Martin and H. W. Draughon 162 found three integers such that the 

 square of the sum of any two less the square of the third is a square. 



A. Gerardin 163 treated x 2 -(y-z} 2 = a, y~-(x-z)~ = b, z 2 -(x-ijY~ = c. 

 Set y = z-\-u, x = z-\-u-\-w, z = w-\-h. Then c = /ir, a = rs, b = hs, where 



RATIONAL ORTHOGONAL SUBSTITUTIONS. 



L. Euler 164 stated that he had a general solution of the problem to find 

 16 integers arranged in a square such that the sum of the squares of the 

 numbers in each row or column or either diagonal are all equal, while the 

 sum of the products of corresponding numbers in any two rows or columns 

 is zero. The example given is the following: 



68 -29 41 -37 



-17 31 79 32 



59 28 -23 61 



-11 -77 8 49. 



Euler 165 treated orthogonal substitutions on n = 3, 4, 5 variables, i. e., 

 linear substitutions leaving unaltered the sum of the squares of the variables. 

 He expressed the coefficients in terms of trigonometric functions. For 

 n = 3, he noted the rational solution 



p 2 +9 2 r 2 s 2 2qr -\-2ps 2qs 2pr 



2qr 2ps p 2 q 2 +r 2 s 2 2pq+2rs 



2qs -\-2pr 2rs 2pq p 2 q 2 r 2 -f-s 2 , 



each entry being divided by p 2 -\-q 2 +r 2 +s 2 . For n 4 he gave two similar 

 rational solutions of which the second is 



ap+bq+cr+ds ar bs cp+dq as br+cq+dp aq bp+cs dr 



-aq+bp+cs dr as -\-br-\-cq-}- dp ar bs -\-cp- dq ap-\-bq cr ds 



ar+bs cp dq ap+bq cr-\-ds aq-\- bp+cs+dr as br cq-\-dp 



-as-\-br cq-\-dp aq bp -\-cs-\-dr ap -\-bq-\-cr-ds ar -\-bs-\-cp-\-dq, 



in which the sum of the products of corresponding numbers in any two 

 rows or columns is zero, while the sum of the squares of the numbers in 

 any row or column is a = (a 2 + b 2 + c 2 + d 2 ) (p 2 + q 2 + r 2 + s 2 ) . For his 164 former 

 problem, we require also that the sum of the squares of the numbers in 



161 Math. Quest. Educ. Times, 18, 1873, 35-7. Same in his Collection of Diophantine Prob- 

 lems with Solutions (ed., A. Martin), Washington, D. C., 1888, 22-4. 



102 Amer. Math. Monthly, 1, 1894, 361-2. 



63 Sphinx-Oedipe, 8, 1913, 30-1. 



164 Opera postuma, 1, 1862, 576-7, letter to Lagrange, Mar. 20, 1770. Quoted by Legendre, 

 ThSorie des nombres, 2, 1830, 144; Maser's German transl., II, 140. 



166 Novi Comm. Acad. Petrop., 15, 1770, 75; Comrn. Arith., I, 427-443. 



