CHAP, xvni] 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. Thompson12 divided a given square a2 into two parts

4rs+l '

such that each plus the square of the other is a square. Take s = r+l, Then the sum of fractions is a2 if 2r+l = a~1, whence r= (1 — a)/(2a).

J. Whitley13 took x2+y=(x+v)~, 2/2-He = (?/4-z)2, which give x} y in terms of v, z [Euler99 of Ch. XVI]. Take v = l—z. Then x+y^a- gives

J. Cunliffe14 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 1— 2x, 2—2x and 2x are to be squares . Take 2x = 4n2, n = s — 1 /2 . Then

gives s.

W. Wright and Winward15 took x and y as the numbers required in the last problem. Then 2x, 2y, x+y±(x*—y^ are to be squares. Set x+y-p, x—y = q. Then p±q and p±pq are to be squares. Take p+pg = n2. Then p-pg=D if l-g2= D = (l-r^)2, whence g = 2r/(r2+l). Set

Then p±g=D if (r2+l)(ra2dr2r) = D. Nowr2+l=D if r=(t;2-l)/(2z>). Take w=2, whence r = 3/4. Take wi=P/2. Then w2±2r = D if P2±6 = D. Set P2+6 = (3S-P)2, which gives P. Set E = J+2. Then P2~6=D if 4_) ----- |-9£4=D = (2+36£+3£2)2, whence i=47/6. B. Gompertz took x+y=pk2j l+x—y = lfp and by a long discussion obtained the preceding numerical answer.

"Jesuiticus" 16 imposed the further condition that x+y=D. Thus #-|-y=r2, 2x=p2, 27/ = g2, l-f£~- 7/ = m2, l—rc+y = n2, whence p2+^2 = 2r2, m2+n2=2. Take p=m, g=n, whence r=l. Then m2+n2 = 2 if m, n=(uz-~vi±2uv)l(u?+v*).

Several17 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. Euler18 made AJB±A, AB±B all squares. Set A—x/z, B=y/z] then xy±xz} xy^hyz are to be squares. Since a2+62±2a6= D, set

ag«2cd, yz=2ab.

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

1814, 229-30.

»76id., 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. 16 Ladies' Diary, 1839, 41-42, Quest. 1638. " Math. Quest. Educ. Times, 5, 1866, 60-1. 18 Novi Comm. Acad. Petrop., 19, 1774, 112; Comm. Arith., II, 53-63; Op. Om., (1), III, 338.