If this represents the cross section of a railway embankment that is l feet long, h feet high, b feet wide at the bottom, and b´ feet wide at the top, find the number of cubic feet in the embankment. Find the volume if l = 300, h = 8, b = 60, and b´ = 28.
The mensuration of the volume of the prism, including the rectangular parallelepiped and cube, was known to the ancients. Euclid was not concerned with practical measurement, so that none of this part of geometry appears in his "Elements." We find, however, in the papyrus of Ahmes, directions for the measuring of bins, and the Egyptian builders, long before his time, must have known the mensuration of the rectangular parallelepiped. Among the Hindus, long before the Christian era, rules were known for the construction of altars, and among the Greeks the problem of constructing a cube with twice the volume of a given cube (the "duplication of the cube") was attacked by many mathematicians. The solution of this problem is impossible by elementary geometry.
If e equals the edge of the given cube, then e3 is its volume and 2e3 is the volume of the required cube. Therefore the edge of the required cube is e∛2. Now if e is given, it is not possible with the straightedge and compasses to construct a line equal to e∛2, although it is easy to construct one equal to e√2.
The study of the pyramid begins at this point. In practical measurement we usually meet the regular pyramid. It is, however, a simple matter to consider the oblique pyramid as well, and in measuring volumes we sometimes find these forms.
Theorem. The lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base.
This leads to the corollary concerning the lateral area of the frustum of a regular pyramid. It should be noticed that the regular pyramid may be considered as a frustum with the upper base zero, and the proposition as a special case under the corollary. It is also possible, if we choose, to let the upper base of the frustum pass through the vertex and cut the lateral edges above that point, although this is too complicated for most pupils. If this case is considered, it is well to bring in the general idea of pyramidal space, the infinite space bounded on several sides by the lateral faces, of the pyramid. This pyramidal space is double, extending on two sides of the vertex.
Theorem. If a pyramid is cut by a plane parallel to the base:
1. The edges and altitude are divided proportionally.
2. The section is a polygon similar to the base.
To get the analogous proposition of plane geometry, pass a plane through the vertex so as to cut the base. We shall then have the sides and altitude of the triangle divided proportionally, and of course the section will merely be a line-segment, and therefore it is similar to the base line.
The cutting plane may pass through the vertex, or it may cut the pyramidal space above the vertex. In either case the proof is essentially the same.