There are many simple illustrations of this case. For example, what is the shortest line between any given edge of the ceiling and the various edges of the floor of the schoolroom? If two galleries in a mine are to be connected by an air shaft, how shall it be planned so as to save labor? Make a drawing of the plan.
At this point the polyhedral angle is introduced. The word is from the Greek polys (many) and hedra (seat). Students have more difficulty in grasping the meaning of the size of a polyhedral angle than is the case with dihedral and plane angles. For this reason it is not good policy to dwell much upon this subject unless the question arises, since it is better understood when the relation of the polyhedral angle and the spherical polygon is met. Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. It should also be observed that just as we may have cross polygons in a plane, so we may have spherical polygons that are similarly tangled, and that to these will correspond polyhedral angles that are also cross, their representation by drawings being too complicated for class use.
The idea of symmetric solids may be illustrated by a pair of gloves, all their parts being mutually equal but arranged in opposite order. Our hands, feet, and ears afford other illustrations of symmetric solids.
Theorem. The sum of the face angles of any convex polyhedral angle is less than four right angles.
There are several interesting points of discussion in connection with this proposition. For example, suppose the vertex V to approach the plane that cuts the edges in A, B, C, D, ..., the edges continuing to pass through these as fixed points. The sum of the angles about V approaches what limit? On the other hand, suppose V recedes indefinitely; then the sum approaches what limit? Then what are the two limits of this sum? Suppose the polyhedral angle were concave, why would the proof not hold?
CHAPTER XX
THE LEADING PROPOSITIONS OF BOOK VII
Book VII relates to polyhedrons, cylinders, and cones. It opens with the necessary definitions relating to polyhedrons, the etymology of the terms often proving interesting and valuable when brought into the work incidentally by the teacher. "Polyhedron" is from the Greek polys (many) and hedra (seat). The Greek plural, polyhedra, is used in early English works, but "polyhedrons" is the form now more commonly seen in America. "Prism" is from the Greek prisma (something sawed, like a piece of wood sawed from a beam). "Lateral" is from the Latin latus (side). "Parallelepiped" is from the Greek parallelos (parallel) and epipedon (a plane surface), from epi (on) and pedon (ground). By analogy to "parallelogram" the word is often spelled "parallelopiped," but the best mathematical works now adopt the etymological spelling above given. "Truncate" is from the Latin truncare (to cut off).
A few of the leading propositions are now considered.