plane angles, respectively equal to each other, and if, at the same time, the equal or homologous angles are disposed in the same order, the two triedral angles will coincide when applied the one to the other, and consequently, are equal (a. 14). may For, we have already seen that the quadrilateral SAOC be placed upon its equal TDPF; thus, placing SA upon TD, SC falls upon TF, and the point O upon the point P. But because the triangles AOB, DPE, are equal, OB, perpendicular to the plane ASC, is equal to PE, perpendicular to the plane TDF; besides, these perpendiculars lie in the same direction; therefore, the point B will fall upon the point E, the line SB upon TE, and the two angles will wholly coincide. Scholium 3. The equality of the triedral angles does not exist, unless the equal faces are arranged in the same manner. For, if they were arranged in an inverse order, or, what is the same, if the perpendiculars OB, PE, instead of lying in the same direction with regard to the planes ASC, DTF, lay in opposite directions, then it would be impossi ble to make these triedral angles coincide the one with the other. The theorem would not, however, on this account, be less true, viz.: that the faces containing the equal angles must be equally inclined to each other; so that the two triedral angles would be equal in all their constituent parts, without, however, admitting of superposi tion. This sort of equality, which is not absolute, or such as admits of superposition, ought to be distinguished by a particular name: we shall call it, equality by symmetry. Thus, those two triedral angles, which are formed by faces respectively equal to each other, but disposed in an inverse order, will be called triedral angles equal by symme try, or simply symmetrical angles. BOOK VII. POLYEDRONS. DEFINITIONS. 1. POLYEDRON is a name given to any solid bounded by polygons. The bounding polygons are called faces of the polyedron; and the straight line in which any two adjacent faces meet each other, is called an edge of the polyedron. 2. A PRISM is a polyedron in which two of the faces are equal polygons with their planes and homologous sides parallel, and all the other faces parallelograms. 3. The equal and parallel polygons are called bases of the prism-the one the lower, the other, the upper baseand the parallelograms taken together, make up the lateral or convex surface of the prism. 4. The ALTITUDE of a prism is the distance between its two bases, and is measured by a line drawn from a point in one base, perpendicular to the plane of the otì r. 5. A right prism is one whose edges, formed by the intersection of the lateral faces, are perpendicular to the planes of the bases. Each edge is then equal to the altitude of the prism. In every other case, the prism is oblique, and each edge is then greater than the altitude. 6. 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: a hexangular prism is one whose bases are hexagons, &c. *. 7. A PARALLELOPIPEDON is a prism whose bases are parallelograms. 8. A RECTANGULAR PARALLELOPIPEDON is one whose faces are all rectangles. When the faces are squares, it is called a cube, or regular hexaedron. The 9. A PYRAMID is a solid bounded by a polygon, and by triangles meeting at a common point, called the vertex. polygon is called the base of the pyramid, and the triangles, taken together, The pyrathe convex, or lateral surface. mid, like the prism, takes different names, according to the form of its base: thus, it may be triangular, quadrangular, pentangular, &c. 10. The ALTITUDE of a pyramid is the perpendicular let fall from the vertex on the plane of the base. 11. A RIGHT PYRAMID is one whose base is a regular polygon, and in which the perpendicular let fall from the vertex upon the base passes through the centre of the base. This perpendicular is then called the axis of the pyramid. 12. The SLANT HEIGHT of a right pyramid, is the perpendicular let fall from the vertex to either side of the polygon which forms the base. 13. If a pyramid is cut by a plane parallel to its base, forming a second base, the part lying between the bases, is called a truncated pyramid, or frustum of a pyramid. 14. The altitude of a frustum is the perpendicular distance between its bases: and the slant height, is that portion of the slant height of the pyramid intercepted between the bases of the frustum. 15. The diagonal of a polyedron is a line joining the vertices of any two of its angles, not in the same face. 16. Similar polyedrons are those whose polyedral angles are equal, each to each, and which are bounded by the same number of similar faces. 17. Parts which are like placed, in similar polyedrons, whether faces, edges, or angles, are called homologous. 18. A regular polyedron is one whose faces are equal and regular polygons, and whose polyedral angles are equal. PROPOSITION I. THEOREM. The convex surface of a right prism is equal to the perimeter of either base multiplied by its altitude. Let ABCDE-K be a right prism: then will its convex surface be 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, K the altitudes AF, BG, CH, &c., of the rectangles, are 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. 5). Hence, the sum of these rectangles, or the convex surface of the prism, is equal to E (AB+BC+CD+DE+EA)×AF; F A D B that is, to the perimeter of the base of the prism multiplied by the altitude. 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 every prism, the sections formed by parallel planes, are equal polygons. Let the prism AH be intersected by the parallel planes NP, SV; then are the polygons NOPQR, STVXY, equal. For, the sides ST, NO, are parallel, being the intersections of two parallel planes with a third plane ABGF; these same sides, ST, NO, are included between the parallels NS, OT, which are edges of the prism: hence, NO is equal to ST. For like reasons, the sides OP, PQ, QR, &c., of the section NOPQR, are equal to the sides TV, VX, XY, &c., of the section STVXY, each to each; and since the equal sides are at the same time parallel, it follows that the angles NOP, OPQ, &c., of the first section, are equal to the angles STV, TVX, &c., of the second, each to each (B. VI., P. 13). Hence, the two sections NOPQR, STVXY, are equal polygons. S PROPOSITION III. THEOREM. N F Y E K R T B G X I H 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 its base: 1st. The edges and the altitude will be divided proportionally: 2d. The section will be a polygon similar to the base. Let the pyramid S-ABCDE, of which SO is the altitude be cut by the plane abcde; then will SA :: So: SO, Sa and the same for the other edges; and the polygon abcde, will be similar to the base ABCDE. |