A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

MEASUREMENT OF SOLIDS 333
the parallelepiped (1) is on AR as base and has height h ;
the
prism (2) is equal to a parallelepiped on KQ as base and with
height h; the prism (3) is equal to a parallelepiped with NP
as base and height h; and finally the pyramid (4) is equal to
a parallelepiped of height h and one-third of RC as base.
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.
Therefore the whole solid is equal to one parallelepiped
with height k and base equal to (AR + KQ + NP + RO + ^RO)
or AO + ^RO.
Now, if AB = a,BG= b, EF = c, FG = d,
AV = ±(a + c),AT = \{b + d),RQ = |(a-c), RP = i(b-d).
Therefore volume of solid
= {f(a + c)(& + d)+-5^(c&— c) (b-d)}h.
The solid
in question is evidently the true ftcofiicrKos (' little
altar'), for the formula is used to calculate the content of
a fiii[(TKo$ in Stereom. II. 40 (68, Heib.) It is also, I think,
the