This is one of the few propositions in Book VII where a model is of any advantage. It is easy to make one out of pasteboard, or to cut one from wood. If a wooden one is made, it is advisable to take an oblique parallelepiped and, by properly sawing it, to transform it into a rectangular one instead of using three different solids. On account of its awkward form, this figure is sometimes called the Devil's Coffin, but it is a name that it would be well not to perpetuate. THEOREM. _The volume of any prism is equal to the product of its base by its altitude._ This is also one of the basal propositions of solid geometry, and it has many applications in practical mensuration. A first-class textbook will give a sufficient list of problems involving numerical measurement, to fix the law in mind. For outdoor work, involving measurements near the school or within the knowledge of the pupils, the following problem is a type: [Illustration] 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 _e_^3 is its volume and 2_e_^3 is the volume of the required cube. Therefore the edge of the required cube is _e_[3root]2. Now if _e_ is given, it is not possible with the straightedge and compasses to construct a line equal to _e_[3root]2, although it is easy to construct one equal to _e_[sqrt]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._