342 
HERON OF ALEXANDRIA 
and, solving for «-a, we obtain. 
x — a = 
(a + 1) (d l — $1) 
(cl +1) (dj + ci (d% — ^2) 
or 
Since <5 1; S 2 are in any case the cubes of fractions, we may 
neglect them for a first approximation, and we have 
VA =a+ , 
(a + 1) cZj + ad 2 
c 
1 \ 
X| 
A 
8 
III. 22, which shows how to cut a frustum of a cone in a given 
ratio by a section parallel to the bases, shall end our account 
of the Metrica. I shall give the general formulae on the left 
and Heron’s case on the right. Let ABED be the frustum, 
let the diameters of the bases be a, a', and the height h. 
Complete the cone, and let the height of ODE be x. 
Suppose that the frustum has to be cut b} T a plane EG in 
such a way that 
(frustum DG) : (frustum FB) — m : n. 
In the case taken by Heron 
a = 28, a' — 21, h = 12, m = 4, n = 1. 
Draw DH perpendicular to AB.