’ STEREOMETRY. 15 
to the base (pl. 3, fig. 96). The sections of two planes parallel to each 
other, but not to the base, are equal. Plate 2, fig. 64, represents a 
parallelopipedon intersected by a plane, not parallel to the base. Prisms of 
equivalent bases and equal altitudes are equal to each other (pl. 3, figs. 97- 
99). Prisms of equal bases, but of unequal altitudes, are to each other as 
their altitudes; those of equal altitudes, as their bases; those of unequal 
bases and altitudes, as the product of the two. A cube whose edge is the 
unit of length, serves as the unit of measure for determining the volume of 
a solid; it is called cubic foot, cubic inch, &c., as the edge is a foot, inch, 
&c. The volume of a cube is obtained by raising the number expressing 
the length of its edge, to the third power; that of any prism in general by 
multiplying the area of the base by the altitude, the same unit of measure 
being used in both. 
A pyramid is a solid, bounded by any rectilineal figure as base, and as 
many triangular planes meeting at the vertex as the base has sides. It is 
called three, four, or five sided, &c., as the base has three, four, five, or more 
sides (pl. 2, figs. 65, 66, 67,70). If a plane be passed through a pyramid, 
parallel to the base, the section thus formed will be similar to the base, and 
will bear to it the same proportion as the square of the perpendicular let fall 
from the vertex on the section, to the square of the altitude of the pyramid 
(pl. 3, fig. 95). A three-sided prism may be divided into three equivalent 
pyramids, of which two have the same base and altitude as the prism. 
Hence it follows that every pyramid is 4 the prism of equivalent base and 
altitude. Consequently, the solid content of a pyramid is obtained by 
taking 4 of the product of the base by the altitude. If from a pyramid we 
cut off a smaller pyramid, by a plane parallel to the base, the part that is 
left is called a truncated pyramid or a frustum. Such a solid is equivalent 
to three perfect pyramids of the same altitude with it, and having for bases 
the upper base of the frustum, the lower base, and a mean proportional 
between the two bases (jig. 101). Ifa three-sided prism be intersected by 
a plane not parallel to the base, the part remaining is equivalent to the sum 
of three pyramids of the same base as the prism, but which have for 
vertices the corners of the triangle in which the prism is intersected by the 
plane (pl. 3, fig. 100). 
3. OF THE ROUND BODIES. 
Among those solids inclosed by both plane and curved surfaces, the cy- 
linder and cone are the best known and most important, as is the sphere 
among those the whole of whose surfaces are curved; these together are 
known as the round bodies. The common or typical cylinder is bounded 
by two equal and parallel circles (forming the bases), and a curved lateral 
surface uniting their circumferences. The latter is a simple curved sur- 
face, and may be generated by the revolution of one straight line around 
the circumference of a circle, but not in its plane, and constantly parallel 
to a fixed line which then forms the axis. The cylinder is right ( pl. 2, fig. 
15 
