Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/314

290 two convex solid figures are equal if they are contained by equal plane figures similarly arranged; see Catalan's Théorèmes et Problèmes de Géométrie Elémentaire. This result was first demonstrated by Cauchy, who turned his attention to the point at the request of Legendre and Malus; see the Journal de I' École Polytechnique, Cahier 16.

XI. Def. 26. The word tetrahedron is now often used to denote a solid bounded by any four triangular faces, that is, a pyramid on a triangular base; and when the tetrahedron is to be such as Euclid defines, it is called a regular tetrahedron.

Two other definitions may conveniently be added.

A straight line is said to be parallel to a plane when they do not meet if produced.

The angle made by two straight lines which do not meet is the angle contained by two straight lines parallel to them, drawn through any point.

XI. 21. In XI. 21 the first case only is given in the original. In the second case a certain condition must be introduced, or the proposition will not be true; the polygon BCDEF must have no re-entrant angle. See note on I. 32.

The propositions in Euclid on Solid Geometry which are now not read, contain some very important results respecting the volumes of soHds. We will state these results, as they are often of use; the demonstrations of them are now usually given as examples of the Integral Calculus.

We have already explained in the notes to the second Book how the area of a figure is measured by the number of square inches or square feet which it contains. In a similar manner the volume of a solid is measured by the number of cubic inches or cubic feet which it contains; a cubic inch is a cube in which each of the faces is a square inch, and a cubic foot is similarly defined.

The volume of a prism is found by multiplying the number of square inches in its base by the number of inches in its altitude; the volume is thus expressed in cubic inches. Or we may multiply the number of square feet in the base by the number of feet in the altitude; the volume is thus expressed in cubic feet. By the base of a prism is meant either of the two equal, similar, and parallel figures of XI. Definition 13; and the altitude of the prism is the perpendicular distance between these two planes.