CHAP, xviii] QUADRATIC FUNCTIONS MADE SQUARES. 493 



took two perpendicular segments AC and CD; let CB be the altitude of 

 triangle ACD. Then AB and DB measure the required numbers. 

 T. Thompson 12 divided a given square a~ into two parts 



r 2 -2rs 2 s 2 +2r 2 s 

 4rs+l ' 4rs+l ; 



such that each plus the square of the other is a square. Take s = r+l. 

 Then the sum of fractions is a 2 if 2r+l = cr 1 , whence r = (1 a)/(2a). 



J. Whitley 13 took x 2 +y = (x-\-v}~, y~+x=(y+z) 2 , which give x, y in 

 terms of v, z [Euler 99 of Ch. XVI]. Take v = l-z. Then x+y = a- gives 

 g=(a-l)/(2o). 



J. Cunliffe 14 found two numbers whose sum increased or decreased by 

 their difference or difference of their squares give squares. He took x and 

 1 x as the numbers. Since either difference is l2x, 2 2x and 2x are 

 to be squares. Take 2x = 4n 2 , n = s 1/2. Then 



gives s. 



W. Wright and Winward 15 took x and y as the numbers required in the 

 last problem. Then 2x, 2y, x-\-y (x 2 y~) are to be squares. Set x-\-y = p, 

 xy = q. Then pq and ppq are to be squares. Take p+pq = n 2 . 

 Then p-pq=\3 if l-q*= D = (l-rg) 2 , whence g = 2r/(r 2 +l). Set " 



Then pg=D if 2 +l)(w 2 2r) = D. Now r 2 +l = D if r=(v 2 -l)/(2y). 

 Take v = 2, whence r = 3/4. Take m = P/2. Then ra 2 2r = D if P 2 6 = D . 

 Set P 2 +6 = (3#-P) 2 , which gives P. Set R = t+2. Then P 2 -6=D 

 if 4H ----- |-9Z 4 =n = (2+36Z+3i 2 ) 2 , whence = 47/6. B. Gompertz took 

 x+y = pk z , l+xy = l/p and by a long discussion obtained the preceding 

 numerical answer. 



"Jesuiticus" 16 imposed the further condition that x-\-y=[3. Thus 

 x+y=r*, 2x = p 2 , 2y = q*, l+xy = m?, lx+y = n z , whence p 2 +# 2 = 2r 2 , 

 rtf+n 2 = 2. Take p = m, q = n, whence r = 1. Then m 2 +n 2 = 2 if 



m, n = (w 2 V 2 db2wy)/(u 2 +y 2 ). 



Several 17 solved easily the problem to find two positive rational numbers 

 such that each and the sum s of their squares exceed their product by 

 squares, and the problem when s is replaced by Vs. 



FOUR QUADRATIC FUNCTIONS OF TWO UNKNOWNS MADE SQUARES. 



L. Euler 18 made ABA, ABB all squares. Set A=x/z, E = y\z\ 

 then xyxz, xyyz are to be squares. Since a 2 +6 2 2a6= D, set 

 xy = a 2 +b 2 = c 2 +d 2 , xz = 2cd, yz 



12 The Gentleman's Diary, or Math. Repository, No. 55, 1795. A. Davis' ed., London, 3, 



1814, 229-30. 



18 Ibid., No. 68, 1808, 36-7, Quest. 917. 

 "Ladies' Diary, 1810, p. 40, Quest. 1203; Leybourn's M. Quest. L. D., 4, 1817, 122-4. 



16 The Gentleman's Math. Companion, London, 3, No. 16, 1813, 421-4. 

 18 Ladies' Diary, 1839, 41-42, Quest. 1638. 



17 Math. Quest. Educ. Times, 5, 1866, 60-1. 



18 Novi Comm. Acad. Petrop., 19, 1774, 112; Comm. Arith., II, 53-63; Op. Orn., (1), III, 33S. 



