to SB, and from E draw in the faces DTE and FTE, the lines ED and EF, respectively perpendicular to TE· then will the angle DEF measure the inclination of these faces. Draw AC and DF The right-angled triangles SBA and TED, have the side SB equal to TE, and the angle ASB equal to A DA B E DTE; hence, AB is equal to DE, and AS to TD. In like manner, it may be shown that BC is equal to EF, and CS to FT. The triangles ASC and DTF, have the angle ASC equal to DTF, by hypothesis, the side AS equal to DT, and the side CS to FT, from what has just been shown; hence, the triangles are equal in all their parts, and consequently, AC is equal to DF. Now, the triangles ABC and DEF bave their sides equal, each to each, and consequently, the corresponding angles are also equal; that is, the angle ABC is equal to DEF: hence, the inclination of the planes ASB and CSB, is equal to the inclination of the planes DTE and FTE. In like manner, it may be shown that the planes of the other equal angles are equally inclined; which was to be proved. Scholium. If the planes of the equal plane angles are like placed, the triedral angles are equal in all respects, for they may be placed so as to coincide. If the planes of the equal angles are not similarly placed, the triedral angles are equal by symmetry. In this case, they may be placed so that two of the homologous faces shall coincide, the triedral angles lying on opposite sides of the plane, which is then called a plane of symmetry. In this position, for every point on one side of the plane of symmetry, there is a corresponding point on the other side. BOOK VII. POLYE DRONS. DEFINITIONS. 1. A POLYEDRON is a volume bounded by polygons. The bounding polygons are called faces of the polyedron; the lines in which the faces meet, are called edges of the polyedron; the points in which the edges meet, are called vertices of the polyedron. 2. A PRISM is a polyedron in which two of the faces are polygons equal in all their parts, and having their homologous sides parallel. The other faces are parallelograms (B. I., P. XXX.). The equal polygons are called bases of the prism; one the upper, and the other the lower base; the parallelograms taken together make up the lateral or convex surface of the prism; the lines in which the lateral faces meet, are called lateral edges of the prism. 3. The ALTITUDE of a prism is the perpendicular distance between the planes of its bases. 4. A RIGHT PRISM is one whose lateral edges are perpendicular to the planes of the bases. In this case, any lateral edge is equal to the altitude. 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. 6. Prisms are named from the number of sides of their bases; a triangular prism is one whose bases are triangles; u pentangular prism is one whose bases are pentagons, &c. 7. A PARALLELO PIPEDON is a prism whose bases are parallelograms. A Right Parallelopipedon is one whose lateral edges are perpendicular to the planes of the bases. A Rectangular Parallelopipedon is whose faces are all rectangles. . A Cube is a rectangular parallelopipedon 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 base is a quadrilateral, and so on. 10. The ALTITUDE of a pyramid is the perpendicular distance from the vertex 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 the same number of 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 perim eter 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 K F B the convex surface of the prism, is equal to, H (AB + BC + CD + DE + EA) × AF; 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. |