5. An OBLIQUE PRISM is one whose lateral edges are oblique to the planes of the bases. In this case, any lateral edge is greater than the altitude. за 6. Prisms are named from the number of sides of their bases; a triangular prism is one whose bases are triangles; a quadrangular prism is one whose bases are quadrilaterals ; a pentangular prism is one whose bases are pentagons, and so on. 7. A PARALLELOPIPEDON is a prism whose bases are parallelograms. A Rectangular Parallelopipedon is a right parallelopipedon, all of whose faces are rectangles; a cube is a rectangular parallelopipedon, all of whose faces are squares. 8. A PYRAMID is a polyedron bounded by a polygon called the base, and by triangles meeting at a common point, called the vertex of the pyramid. The triangles taken together make up the lateral or convex surface of the pyramid ; the lines in which the lateral faces meet, are called the lateral edges of the pyramid. 9. Pyramids are named from the number of sides of their bases; a triangular pyramid is one whose base is a triangle; a quadrangular pyramid is one whose quadrilateral, and so on. base is a 10. The ALTITUDE of a pyramid is the perpendicular distance from the vertex of the pyramid to the plane of its base. 11. A RIGHT PYRAMID is one whose base is a regular polygon, and in which the perpendicular drawn from the vertex to the plane of the base, passes through the centre of the base. This perpendicular is called the axis of the pyramid. 12. The SLANT HEIGHT of a right pyramid, is the perpendicular distance from the vertex to any side of the base. 13. A TRUNCATED PYRAMID is that portion of a pyramid included between the base and any plane which cuts the pyramid. When the cutting plane is parallel to the base, the truncated pyramid is called a FRUSTUM OF A PYRAMID, and the inter section of the cutting plane with the pyramid, is called the upper base of the frustum; the base of the pyramid is called the lower base of the frustum. 14. The ALTITUDE of a frustum of a pyramid, is the perpendicular distance between the planes of its bases. 15. The SLANT HEIGHT of a frustum of a right pyramid, is that portion of the slant height of the pyramid which lies between the planes of its upper and lower bases. 16, SIMILAR POLYEDRONS are those which are bounded by similar polygons, similarly placed. Parts which are similarly placed, whether faces, edges, or angles, are called homologous. 17. A DIAGONAL of a polyedron, is a straight line joining the vertices of two polyedral angles not in the same face. 18. The VOLUME OF A POLYEDRON is its numerical value expressed in terms of some other polyedron as a unit. The unit generally employed is a cube constructed on the linear unit as an edge. PROPOSITION I. THEOREM. The convex surface of a right prism is equal to the perimeter of either base multiplied by the altitude. Let ABCDE-K be a right prism: then is its convex surface equal to, (AB + BC + CD + DE + EA) × AF For, the convex surface is equal to the sum of all the rectangles AG, BH, CI, DK, EF, which compose it. Now, the altitude of each of the rectangles AF, BG, CH, &c., is equal to the altitude of the prism, and the area of each rectangle is equal to its base multiplied by its altitude (B. IV., P. V.): hence, the sum of these rectangles, or the convex surface of the prism, is equal to, K E (AB + BC + CD + DE + EA) × AF; H that is, to the perimeter of the base multiplied by the altitude; which was to be proved. Cor. If two right prisms have the same altitude, their convex surfaces are to each other as the perimeters of their bases. PROPOSITION II. THEOREM. In any prism, the sections made by parallel planes are equal polygons. Let the prism AH be intersected by the parallel planes. NP, SV: then are the sections NOPQR, STVXY, equal polygons. For, the sides NO, ST, are parallel, being the intersections of parallel planes with a third plane ABGF; these sides, NO, ST, are included between the parallels NS, OT: hence, NO is equal to ST' (B. I., P. XXVIII., C. 2). For like reasons, the sides OP, PQ, QR, &c., of NOPQR, are TV, VX, &c., of each; and since the N F K H Y G R E D A each to B equal to the sides equal sides are par allel, each to each, it follows that the angles NOP, OPQ, &c., of the first section, are equal to the angles STV, TVX, &c., of the second section, each to each (B. VI., P. XIII.): hence, the two sections NOPQR, STVXY, are equal polygons; which was to be proved. Cor. Every section of a prism, parallel to the bases, is equal to either base. If a pyramid be cut by a plane parallel to the base : 1o. The edges and the altitude will be divided proportionally: 2o. The section will be a polygon similar to the base. So, Let the pyramid S-ABCDE, whose altitude is be cut by the plane abcde, parallel to the base ABCDE. 1o. The edges and altitude will be divided proportionally. For, conceive a plane to be passed through the vertex & parallel to the plane of the base; then will the edges and the altitude 2o. The section abcde, will be similar to the base ABCDE. For, ab is parallel to AB, and be to BC (B. VI., P. X.): hence, the angle abc is equal to the angle ABC. In like manner, it may S E Ο B be shown that each angle of the polygon abcde is equal to the corresponding angle of the base: hence, the two polygons are mutually equiangular. Again, because ab is parallel to AB, we have, and, because be is parallel to BC, we have, bc : BC hence (B. II., P. IV.), we have, sb : SB; ab : AB : bc : BC. In like manner, it may be shown that all the sides of abcde are proportional to the corresponding sides of the polygon ABCDE: hence, the section abcde is similar to the base ABCDE (B. IV., D. 1); which was to be proved. Cor. 1. If two pyramids S-ABCDE, and S-XYZ, having a common vertex S, and their bases in the same plane, be cut by a plane abc, parallel to the plane of their bases, the sections will be to each other as the bases. |