A history of Greek mathematics - Wilbourhall.org

332 HERON OF ALEXANDRIA
view of their surprising character, referred to Archimedes by
certain writers who give the traditional account of their
origin. But whether they belong to Archimedes or another,
it is necessary to give a sketch of these methods as well.'
The Book begins with generalities about figures all the
sections of which parallel to the base are equal to the base
and similarly situated, while the centres of the sections are on
a straight line
through the centre of the base, which may be
either obliquely inclined or perpendicular to the base ;
whether
the said straight line (' the axis ') is or is not perpendicular to
the base, the volume is equal to the product of the area of the
base and the perpendicular height of the top of the figure
'
from the base. The term height ' is thenceforward restricted
to the length of the perpendicular from the top of the figure
on the base.
(a) Cone, cylinder, parallelepiped (prism), pyramid, and
II.
frustum.
1-7 deal with a cone, a cylinder, a 'parallelepiped' (the
base of which is not restricted to the parallelogram but is in
the illustration given a regular hexagon, so that the figure is
more properly a prism with polygonal bases), a triangular
prism, a pyramid with base of any form, a frustum of a
triangular pyramid ;
the figures are in general oblique,
(f3) Wedge-shaped solid (ficDfjLiorKos or crcprjvio-Kos).
II. 8 is a case which is perhaps worth giving. It is that of
a rectilineal solid, the base of which is a rectangle ABCD and
has opposite to it another rectangle EFGH, the sides of which
are respectively parallel but not necessarily proportional to
those of ABCD. Take AK equal to EF, and BL equal to FG.
Bisect BK, CL in V, W, and draw KRPU, VQOM parallel to
AD, and LQRN, WOPT parallel to AB. Join FK, GR, LG,
GU, BK
Then the solid is divided into (1) the parallelepiped with
AR, EG as oppqsite faces, (2) the prism with KL as base and
FG as the opposite edge, (3) the prism with NU as base and
GH as opposite edge, and (4) the pyramid with RLGU as base
and G as vertex. Let h be the
'
height ' of the figure. Now