¿23 
THE APPLICATION OF ALGEBRA 
From whence, and yy + a — u x y — uu, the value of y 
( = V 
a — u 
) is likewise given, 
2 
PROBLEM LXXII 
The sum [a), the sum of the squares (b), and the sum o f 
the cubes (c), of any four numbers in geometrical propor 
tion being given ; to find the numbers. 
Let half the sum of the two means be x, and half their 
difference y ; also lçt half the sum of the two extremes 
be z, and half their difference v, and then the numbers 
themselves will be expressed thus, 2 — v, x— r 4 y, 
2 4 v : whence, by the conditions of the problem, we 
have 
z — v 4 x — y 4- x 4y + * 4 v = a, 
— ¿J 1 4 x — y | l 4 x 4 yr 4 2 4 v] 1 = b. 
"z — v x 2 4 V = x —y x ï + ÿ (Theor. 1. p. 72); 
which, contracted, are, 
2z + 2x =z a, 
Qz z 4 2v z 4 2X 1 4 2.V : = b, 
2z 3 4 6zv z 4 2x 3 4 Gnfi n c. 
2* — v z — x z — y z . 
Let x* — 2 T 4 the value of y*, in the last of these 
equations, be substituted instead of y~, in the two pre 
ceding ones, and we shall have 
22* 4 2r z 4 2x* 4 2x* — 23* 4 2v z — 6, and 
22 3 4 G2i>* 4 ?x 3 4 Gx 3 — 6xz 1 4 6x0*22 c\ 
which, abbreviated, become 
4x x 4 4o* — b, and 
22 3 4 8x 3 — 6X2* 4 Gx 4 6: X »* 22 c. 
Let $1) — x z t the value of 0*, in the former of these 
equations, be substituted, for its equal, in the latter, 
and we shall next have 2s 3 4 8x J — 6x2" 4 6x 4 6: x 
±b — x z — c; moreover, if for 2, in the last equation, 
its equal \a — x be substituted, there will come out 2 x 
ifa-hf 4 8x 3 —- Gx X \a^~:r, 1 4 3a X \b — xx 22 c;