The number of simple applications of this proposition is practically unlimited. In all such cases it is advisable to take a considerable number of numerical exercises in order to fix in mind the real nature of the proposition. Any good geometry furnishes a certain number of these exercises.
The following is an interesting property of the rectangular parallelepiped, often called the rectangular solid:
If the edges are a, b, and c, and the diagonal is d, then (a/d)2 + (b/d)2 + (c/d)2 = 1. This property is easily proved by the Pythagorean Theorem, for d2 = a2 + b2 + c2, whence (a2 + b2 + c2) / d2 = 1.
In case c = 0, this reduces to the Pythagorean Theorem. The property is the fundamental one of solid analytic geometry.
Theorem. The volume of any parallelepiped is equal to the product of its base by its altitude.
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:
