A history of Greek mathematics - Wilbourhall.org

8
342 HERON OF ALEXANDRIA
and, solving for x — a, we obtain
"
(a+l)(d l
-8l )-\-a(d — 8 2 2y
(a+l)^-^)
or #A =a +
(a + 1) (d x
-
X ) + a(d 2
-8
2 )
Since 8 V 8 2
neglect them for a first approximation, and we have
are in any case the cubes of fractions, we may
(a + l)d 1
+ ad 2
C
i \
D
Xl \
i \
/
/, - -
h
1 * "^^***** "'"
,
a
H K z a
B
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 GDE be x.
Suppose that the frustum has to be cut by a plane FG in
such a way that
(frustum DG) : (frustum
In the case taken by Heron
FB) — m :
n.
a = 28, a'= 21, h = 12, m =^4, n = 1.
Draw DH perpendicular to A B.