CHAP, xix] SQUARES PLUS LINEAR FUNCTIONS MADE SQUARES. 525 



Diophantus, II, 34, made x~y, y*z, z~x squares. In IV, 18, these 

 and x+y+z are made squares. 



T. Strong 129 made x 2 y, x 2 z, y 2 x, y 2 z all squares. Take 



x 2 y = (x ay) 2 , x 2 z = (x bz) 2 , y 2 x = (ycx) 2 . 



Hence x, y, z are rational functions of a, b, c. Equate the resulting expres- 

 sion for if z to (e 1/fr) 2 . We get b rationally in terms of e, a, c. For 

 a = l t c = e = 2, we get a; = 5/4, # = 3/2, 2 = 14/9. 



Ricatti 130 found three numbers such that if the square of each be added 

 to the remaining two the sums are squares. He used the numbers x, 2x, 1 . 



R. Adrain 131 took 



and solved the resulting system of three linear equations for x, y, z. 



To make s+z 2 , s+y 2 , s-\-z 2 squares, where s = x-{-y-\-z, "A.B.L." 132 

 equated them to (x+v) 2 , (y+t) 2 , (z+k) 2 and solved algebraically the result- 

 ing linear equations. "Epsilon" took #+2 = 1/4. Then 



gives x, and x+l+y 2 = D if %p = q 2 -}-2pq-2qy, which gives y. W. Wright 

 took (v l)r, (x l)r and (y l)ras the numbers, and r 2 as their sum, whence 

 r = v4-x+y 3- The conditions become v 2 2v+2= n = (p v) 2 , etc., which 

 determine v, x, y. 



H. J. Anderson 133 found n numbers whose sum s exceeds the square of 

 each by a square. Express s = x 2 +y 2 as a sum of two squares x' 2 -\-y' 2 , 

 x" 2 +y" 2 , , in n ways (Euler, Algebra, II, 219) by taking x' = a f y b'x, 

 y' = a'x+b'y, x" = a"y-b"x, y" = a"x+b"y, , where 



,_ 2mn m 2 -n 2 ,, _ 2pq ,,,_P 2 -g 2 



a ;rr ~, o . , ci 



Take x } x', x", - - - as the required numbers. Their sum s is of the form 

 Ax+By. Thus s = x*+y z if 4By-4y 2 +A 2 = D. Forn = 4, C. Farquhar 

 used the numbers w, wx, wy, wz. Set <r = I+x+y+z. Then wv w 2 = D 

 = x 2 w z gives <T. Then take 



which determines z. 



J. R. Young 134 found three squares x\ and a number a such that x 2 ia 

 are all squares. Take z- = ra-+n-, a = 2ra t -n;, mf = r--s?, n i = 2r i -s i , whence 

 Xi=*r*+8*. It remains to make the values 4r;S;(r; s 2 ) of a equal. Take 

 ri = rz = r 3 = r. Thus s t -(r 2 si) are to be equal. The values for i = l and 2 

 are equal if r 2 = Si+SiS 2 +S2. Thus 4r 2 3sl is to be a square. Hence take 



129 Amer. Jour. Sc. and Arts (ed., Silliman), 1, 1818, 426-7. 



130 Institutiones analyticae a Vincentio Riccato, Bononiae, 1, 1765, 64. 



131 The Math. Correspondent, New York, 2, 1807, 13-14. 



132 The Gentleman's Math. Companion, London, 5, No. 25, 1822, 125-30. 



133 Math. Diary, New York, 1, 1825, 151-4. 



134 Algebra, 1816. S. Ward's Amer. ed., 1832, 346-7. A like discussion for two squares 



had been given by J. Cunliffe, New Series of Math. Repository (ed., T. Leybourn), 

 1, 1806, I, 221-2. 



