This proposition is one of the most important in Book VII, because it is the basis of the mensuration of the cylinder as well as the prism. Practical applications are easily suggested in connection with beams, corridors, and prismatic columns, such as are often seen in school buildings. Most geometries supply sufficient material in this line, however. THEOREM. _An oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism._ This is a fundamental theorem leading up to the mensuration of the prism. Attention should be called to the analogous proposition in plane geometry relating to the area of the parallelogram and rectangle, and to the fact that if we cut through the solid figure by a plane parallel to one of the lateral edges, the resulting figure will be that of the proposition mentioned. As in the preceding proposition, so in this case, there may be a question raised that will make it helpful to introduce the idea of prismatic space. THEOREM. _The opposite lateral faces of a parallelepiped are congruent and parallel._ It is desirable to refer to the corresponding case in plane geometry, and to note again that the figure is obtained by passing a plane through the parallelepiped parallel to a lateral edge. The same may be said for the proposition about the diagonal plane of a parallelepiped. These two propositions are fundamental in the mensuration of the prism. THEOREM. _Two rectangular parallelepipeds are to each other as the products of their three dimensions._ This leads at once to the corollary that the volume of a rectangular parallelepiped equals the product of its three dimensions, the fundamental law in the mensuration of all solids. It is preceded by the proposition asserting that rectangular parallelepipeds having congruent bases are proportional to their altitudes. This includes the incommensurable case, but this case may be omitted. 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 _d_^2 = _a_^2 + _b_^2 + _c_^2, whence (_a_^2 + _b_^2 + _c_^2) / _d_^2 = 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._