A history of Greek mathematics - Wilbourhall.org

334 HERON OF ALEXANDRIA
The method is the same mutatis mutandis as that used in
II. 6 for the frustum of a pyramid with any triangle for base,
and it is applied in II. 9 to the case of a frustum of a pyramid
with a square base, the formula for which is
[{i(a + a')} 2 + Mi(«-a')i 2 ]*.
where a, a' are the sides of the larger and smaller bases
respectively, and h the height ; the expression is of course
easily reduced to J h(a2 + aa' + a' 1 ).
(y) Frustum of cone, sphere, and segment thereof.
A frustum of a cone is next measured in two ways, (1) by
comparison with the corresponding frustum of the circumscribing
pyramid with square base, (2) directly as the
difference between two cones (chaps. 9, 10). The volume of
the frustum of the cone is to that of the frustum of the
circumscribing pyramid as the area of the base of the cone to
that of the base of the pyramid ; i.e. the volume of the frustum
of the cone is \ it, or \\, times the above expression for
the frustum of the pyramid with a 2 , a' 2 as bases, and it
reduces to -^irh (a 2 + aa' + a' 2 ), where