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? FOOTNOTES: [87] These may be purchased through the Leipziger Lehrmittelanstalt, Leipzig, Germany, which will send catalogues to intending buyers. [88] An excellent set of stereoscopic views of the figures of solid geometry, prepared by E. M. Langley of Bedford, England, is published by Underwood & Underwood, New York. Such a set may properly have place in a school library or in a classroom in geometry, to be used when it seems advantageous. 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. THEOREM. _The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section._ It should be noted that although some syllabi do not give the proposition that parallel sections are congruent, this is necessary for this proposition, because it shows that the right sections are all congruent and hence that any one of them may be taken. It is, of course, possible to construct a prism so oblique and so low that a right section, that is, a section cutting all the lateral edges at right angles, is impossible. In this case the lateral faces must be extended, thus forming what is called a _prismatic space_. This term may or may not be introduced, depending upon the nature of the class.