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, ab c, a c d, 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 mp n, np q, &c., will likewise be in the same plane. Therefore each polygonal face in the one polye dron 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, qur. 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, a n, for example, are to each other as two homologous sides AB, ab. 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, 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 a b c is parallel to the and consequently SO: So:: AB : a b. But the bases ABCDE, abc de, being similar figures, 22 ABCDE: abcde:: AB: ab (221). Multiplying the two proportions in order we shall have -3 --3 ABCDESO: abcdex So:: AB:: ab; but ABCDESO is the solidity of the pyramid SABCDE (413), and a b c d e x So is 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, ab, (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 B x H, 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 x (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 Pp::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. Fig. 220. 444. A spherical polygon is a part of the surface of a sphere terminated by several arcs of great circles. 455. A lunary surface is the part of the surface of a sphere comprehended between two semicircumferences of great circles, which terminate in a common diameter. 446. We shall call a spherical wedge the part of a sphere comprehended between the halves of two great circles, and to which the lunary surface answers as a base. 447. A spherical pyramid is the part of a sphere comprehended between the planes of a solid angle whose vertex is at the centre. The base of the pyramid is the spherical polygon intercepted by these planes. 448. A zone is the part of the surface of a sphere comprehended between two parallel planes, which are its bases. One of these planes may be a tangent to the sphere, in which case the zone has only one base. 449. A spherical segment is the portion of a sphere comprehended between two parallel planes which are its bases. One of these planes may be a tangent to the sphere, in which case the spherical segment has only one base. 450. The altitude of a zone or of a segment is the distance between the parallel planes which are the bases of the zone or segment. 451. While the semicircle DAE (fig. 220), turning about the diameter DE, describes a sphere, every circular sector as DCF, or FCH, describes a solid which is called a spherical sector. Fig. 221. THEOREM. 452. Every section of a sphere made by a plane is a circle. Demonstration. Let AMB (fig. 221) be a section, made by a plane, of the sphere of which C is the centre. From the point C draw CO perpendicular to the plane AMB, and different oblique lines CM, CM, to different points of the curve AMB which terminates the section. The oblique lines CM, CM, CB, are equal, since they are radii of the sphere; consequently they are at equal distances from the perpendicular CO (329); whence all the lines OM, OM, OB, are equal; therefore the section AMB is a circle of which the point is the centre. |