mac, bac; in like manner, the pyramids N-ABC, n-abc, being similar, the inclination of the planes NAC, BAC, is equal to that of the planes nac, bac; consequently, if we subtract the first inclinations respectively from the second, there will remain the inclination of the planes NAC, MAC, equal to that of the planes nac, mac. But, because the pyramids are similar, the triangle MAC is similar to ma c, and the triangle NAC is similar to nac; therefore the triangular pyramids MNAC, mnac, have two faces similar, each to each, similarly placed, and equally inclined to each other; consequently the two pyramids are similar (425); and their homologous sides give the proportion Let P and p be two other homologous vertices of the same polyedrons, and we have, in like manner, whence PN : pn :: AB : ab, MN: mn :: PN:pn :: PM: pm. Therefore the triangle PNM, formed by joining any three vertices of one polyedron, is similar to the triangle pnm, formed by joining the three homologous vertices of the other polyedron. Furthermore, let Q, q, be two homologous vertices, and the triangle PQN will be similar to pqn. We say, also, that the inclination of the planes PQN, PMN, is equal to that of the planes pqn, pтп. For, if we join QM and qm, we shall always have the triangle QNM similar to qnm, and consequently the angle QNM equal to qnm. Suppose at Na solid angle formed by the three plane angles QNM, QNP, PNM, and at n a solid angle formed by the plane angles qnm,qnp,pnm; since these plane angles are equal, each to each, it follows that the solid angles are equal. Whence the inclination of the two planes PNQ, PNM, is equal to that of the homologous planes pnq, pnm (359); therefore, if the two triangles PNG, PNM, be in the same plane, in which case we should have the angle QNM = QNP + PNM, we should have, in like manner, the angle qnm=qnp+pnm, and the two triangles qnp,pnm, would also be in the same plane. All that has now been demonstrated takes place, whatever be GEOM. 20 the angles M, N, P, Q, compared with the homologous angles m, n, p, q. Let us suppose, now, that the surface of one of the polyedrons is divided into triangles ABC, ACD, MNP, NPQ, &c., we see that the surface of the other polyedron will contain an equal number of triangles, abc, acd, mnp, npq, &c., similar to the former, and similarly placed; and if several triangles, as MPN, NPQ, &c., belong to the same face, and are in the same plane, the homologous triangles mpn, npq, &c., will likewise be in the same plane. Therefore each polygonal face in the one polyedron will correspond to a similar polygonal face in the other; and consequently the two polyedrons will be comprehended under the same number of similar and similarly disposed planes. We say, moreover, that the solid angles will be equal. For, if the solid angle N., for example, is formed by the plane angles QNP, PNM, MNR, QNR, the homologous solid angle n will be formed by the plane angles qnp, pnm,mnr, qnr. Now the former plane angles are equal to the latter, each to each, and the inclination of any two adjacent planes is equal to that of their homologous planes; therefore the two solid angles are equal, since they would coincide upon being applied. We conclude, then, that two similar polyedrons have their homologous faces similar, and their homologous solid angles equal. 432. Corollary. It follows, from the preceding demonstration, that if, with four vertices of a polyedron, we form a triangular pyramid, and also another with the four homologous vertices of a similar polyedron, these two pyramids will be similar; for they will have their homologous sides proportional (430). It will be perceived, at the same time, that the homologous diagonals (157), AN, an, for example, are to each other as two homologous sides AB, a b. THEOREM. 433. Two similar polyedrons may be divided into the same number of triangular pyramids similar, each to each, and similarly placed. Demonstration. We have seen that the surfaces of two similar polyedrons may be divided into the same number of triangles, 1 that are similar, each to each, and similarly placed. Let us consider all the triangles of one of the polyedrons, except those which form the solid angle A, as the bases of so many triangular pyramids having their vertices in A; these pyramids taken together will compose the polyedron. Let us divide likewise the other polyedron into pyramids having for their common vertex that of the angle a, homologous to A; it is evident that the pyramid, which connects four vertices of one polyedron, will be similar to the pyramid which connects the four homologous vertices of the other polyedron; therefore two similar polyedrons, &c. THEOREM. 434. Two similar pyramids are to each other as the cubes of their homologous sides. Demonstration. Two pyramids being similar, the less may be placed in the greater, so that they shall have the angle S (fig. 214) Fig 214. common. Then the bases ABCDE, abcde, will be parallel; for, since the homologous faces are similar (423), the angle Sab = SAB, as also Sbc=SBC; therefore the plane abc is parallel to the But the bases ABCDE, abcde, being similar figures, 2-2 ABCDE: abcde:: AB: ab (221). Multiplying the two proportions in order, we shall have 3-3 ABCDE X SO: abcde×So::AB: ab; but ABCDE X SO is the solidity of the pyramid SABCDE (413), and abcde× Sois the solidity of the pyramid Sabcde; therefore two similar pyramids are to each other as the cubes of their homologous sides. THEOREM. 435. Two similar polyedrons are to each other as the cubes of their homologous sides. Demonstration. Two similar polyedrons may be divided into the same number of triangular pyramids, that are similar, each to each (433). Now, the two similar pyramids APNM, apnm, Fig. 219. (fig. 219), are to each other as the cubes of their homologous sides AM, am, or as the cubes of the homologous sides AB, a b, (434). The same ratio may be shown to exist between any two other homologous pyramids; therefore the sum of all the pyramids, which compose the one polyedron, or the polyedron itself, is to the other polyedron as the cube of any one of the sides of the first is to the cube of the homologous side of the second. General Scholium. 436. We can express in algebraic language, that is, in a manner the most concise, a recapitulation of the principal propositions of this section relating to the solidity or content of polyedrons. Let B be the base of a prism, H its altitude; the solidity of the prism will be B × H, or BH. Let B be the base of a pyramid, H its altitude; the solidity of the pyramid will be BXH, or H × B, or BH. Let H be the altitude of the frustum of a pyramid, and let A, B, be the bases; then AB will be the mean proportion between them, and the solidity of the frustum will be + H × (A + B + √AB). Let B be the base of a truncated triangular prism, H, H', H", the altitudes of the three superior vertices, the solidity of the truncated prism will be + B × (H+ H + H"). Lastly, let P, p, be the solidities of two similar polyedrons, A and a, two homologous sides, or diagonals of the polyedrons, we shall have P: p :: A3: a3. SECTION THIRD. Of the Sphere. DEFINITIONS. 437. A sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within called the centre. The sphere may be conceived to be generated by the revolution of a semicircle DAE (fig. 220) about its diameter DE; Fig. 220. for the surface thus described by the curve DAE will have all its points equally distant from the centre C. 438. The radius of a sphere is a straight line drawn from the centre to a point in the surface; the diameter or axis is a line passing through the centre, and terminated each way by the surface. All radii of the same sphere are equal; the diameters also are equal, and each double of the radius. 439. It will be demonstrated, art. 452, that every section of a sphere made by a plane is a circle. This being supposed, we call a great circle the section made by a plane which passes through the centre, and a small circle the section made by a plane which does not pass through the centre. 440. A plane is a tangent to a sphere, when it has one point only in common with the surface of the sphere. 441. The pole of a circle of the sphere is a point in the surface of the sphere equally distant from every point in the circumference of the circle. It will be shown, art. 464, that every circle, great or small, has two poles. 442. A spherical triangle is a part of the surface of a sphere comprehended by three arcs of great circles. These arcs, which are called the sides of the triangle, are always supposed to be smaller each than a semicircumference. The angles, which their planes make with each other, are the angles of the triangle. 443. A spherical triangle takes the name of right-angled, isosceles and equilateral, like a plane triangle, and under the same circumstances. : |