angles, respectively equal to each other, and if at the same time the equal or homologous angles are disposed in the same manner in the two solid angles, these angles will be equal, and they will coincide when applied the one to the other. We have already seen that the quadrilateral SAOC may 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, those perdendiculars lie in the same direction; therefore, the point B will fall upon the point E, the line SB upon TE, and the two solid angles will wholly coincide. This coincidence, however, takes place only when we suppose that the equal plane angles are arranged in the same manner in the two solid angles; 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 impossible to make these solid angles coincide with one another. It would not, however, on this account, be less true, as our Theorem states, that the planes containing the equal angles must still be equally inclined to each other; so that the two solid angles would be equal in all their constituent parts, without, however, admitting of superposition. This sort of equality, which is not absolute, or such as admits of superposition, deserves to be distinguished by a particular name: we shall call it equality by symmetry. Thus those two solid angl, which are formed by three plane angies respectively equal to each other, but disposed in an inverse order, will be called angles equal by symmetry, or simply symmetrical angles. The same remark is applicable to solid angles, which are formed by more than three plane angles: thus a solid angle, formed by the plane angles A, B, C, D, E, and another solid angle, formed by the same angles in an inverse order A, E, D, C, B, may be such that the planes which contain the equal angles are equally inclined to each other. Those two solid angles, are likewise equal, without being capable of superposition, and are called solid angles equal by symmetry, or symmetrical solid angles. Among plane figures, equality by symmetry does not properly exist, all figures which might take this name being absolutely equal, or equal by superposition; the reason of which is, that a plane figure may be inverted, and the upper part taken indiscriminately for the under. This is not the case with solids; in which the third dimension may be taken in two different directions. F BOOK VII. Definitions. 1. THE name solid polyedron, or simple polyedron, is given to every solid terminated by planes or plane faces; which planes, it is evident, will themselves be terminated by straight lines. E POLYEDRONS. 2. The common intersection of two adjacent faces of a polyedron is called the side, or edge of the polyedron. 3. The prism is a solid bounded by several parallelograms, which are terminated at both ends by equal and parallel polygons. B C To construct this solid, let ABCDE be any polygon; then if in a plane parallel to ABCDE, the lines FG, GH, HI, &c. be drawn equal and parallel to the sides AB, BC, CD, &c. thus forming the polygon FGHIK equal to ABCDE; if in the next place, the vertices of the angles in the one plane be joined with the homologous vertices in the other, by straight lines, AF, BG, CH, &c. the faces ABGF, BCHG, &c. will be parallelograms, and ABCDE-K, the solid so formed, will be a prism. 4. The equal and parallel polygons ABCDE, FGHIK, are called the bases of the prism; the parallelograms taken together constitute the lateral or convex surface of the prism; the equal straight lines AF, BG, CH, &c. are called the sides, or edges of the prism. 5. The altitude of a prism is the distance between its two bases, or the perpendicular drawn from a point in the upper base to the plane of the lower base. 6. A prism is right, when the sides AF, BG, CH, &c. are perpendicular to the planes of the bases; and then each of them is equal to the altitude of the prism. In every other case the prism is oblique, and the altitude less than the side. 7. A prism is triangular, quadrangular, pentagoral, hexagonal, &c. when the base is a triangle, a quadrilateral, a pentagon, a hexagon, &c. 8. A prism whose base is a parallelogram, and which has all its faces parallelograms, is named a parallelopipedon. The paralelopipedon is rectangular when all its faces are rectangles. 9. Among rectangular parallelopipedons, we distinguish the cube, or regular hexaedron, bounded by six equal squares. 10. A pyramid is a solid formed by several triangular planes proceeding from the same point S, and terminating in the different sides of the same polygon ABCDE. The polygon ABCDE is called the base of the pyramid, the point S the vertex; and the triangles ASB, BSC, CSD, &c. form its convex or lateral surface. F S E 11. If from the pyramid S-ABCDE, the pyramid S-abcde be cut off by a plane parallel to the base, the remaining solid ABCDE-d, is called a truncated pyramid, or the frustum of a pyramid. A B 12. The altitude of a pyramid is the perpendicular let fall from the vertex upon the plane of the base, produced if necessary. 13. A pyramid is triangular, quadrangular, &c. according as its base is a triangle, a quadrilateral, &c. 14. A pyramid is regular, when its base is a regular polygon, and when, at the same time, the perpendicular let fall from the vertex on the plane of the base passes through the centre of the base. That perpendicular is then called the axis of the pyramid. 15. Any line, as SF, drawn from the vertex S of a regular pyramid, perpendicular to either side of the polygon which forms its base, is called the slant height of the pyramid. 16. The diagonal of a polyedron is a straight line joining the vertices of two solid angles which are not adjacent to each other 17. Two polyedrons are similar when they are contained by the same number of similar planes, similarly situated, and having like inclinations with each other. PROPOSITION I. THEOREM. The convex surface of a right prism is equal to the perimeter of its 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. F For, the convex surface is equal to the sum of all the rectangles AG, BH, CI, DK, EF, which compose it. Now, the altitudes AF, BG, CH, &c. of the rectangles, are equal to the altitude of the prism. Hence, the sum of these rectangles, or the convex surface of the prism, is equal to (AB+BC+CD+DE+EA) × AF; that is, to the perimeter of the base of the prism multi plied by its altitude. A B 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 sides of N 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 Cor. If two right prisms have the same altitude, their convex surfaces will be to each other as the perimeters of their bases F K PROPOSITION II. THEOREM. In every prism, the sections formed by parallel planes, are equal polygons. นา E Y B K H G ܂ A I C H 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 (Book VI. Prop. XIII.). Hence the two sections NOPQR, STVXY, are equal polygons. Cor. Every section in a prism, if drawn parallel to the base, is also equal to the base. PROPOSITION III. THEOREM. 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: SĂ :: So: SO, and the same for the other edges and the polygon abcde, will be similar to the base ABCDE. : B First. Since the planes ABC, abc, are parallel, their intersections AB, ab, by a third plane SAB will also be parallel (Book VI. Prop. X.); hence the triangles SAB, Sab are simllar, and we have SA: Sa :: SB: Sb; for a similar reason, we have SB Sb :: SC: Sc; and so on. Hence the edges SA, SB, SC, &c. are cut proportionally in a, b, c, &c. The altitude SO is likewise cut in the same proportion, at the point o; for BO and bo are parallel, therefore we have SO: So :: SB: Sb. E C Secondly. Since ab is parallel to AB, bc to BC, cd to CD, &c. the angle abc is equal to ABC, the angle bcd to BCD, and so on (Book VI. Prop. XIII.). Also, by reason of the similar triangles SAB, Sab, we have AB : ab :: SB: Sb; and by reason of the similar triangles SBC, Sbc, we have SB: Sb: BC: bc; hence AB: ab: BC: bc; we might likewise have BC: bc CD: cd, and so on. Hence the polygons ABCDE, abcde have their angles respectively equal and their homologous sides proportional; hence they are similar. : N |