which terminate in B" making SB" = SB'; the angle CSB" will be the third plane required. For, if a solid angle be formed of the three planes B'SA, ASC, CSB", the inclination of the planes containing the given angles ASB', ASC, will be equal to the given angle EA b. 364. Scholium. If a solid angle is quadruple, or formed by Fig. 199. four plane angles ASB, BSC, CSD, DSA (fig. 199), we cannot, by knowing these angles, determine the mutual inclination of their planes; for with the same plane angles any number of solid angles may be formed. But, if a condition be added, if, for example, the inclination of the two planes ASB, BSC, be given, then the solid angle is entirely determinate, and the inclination of any two of the planes may be found. Suppose a triple solid angle formed by the plane angles ASB, BSC, ASC; the two first angles are given as well as the inclination of their planes; we can then, by the problem just resolved, determine the third angle ASC. Afterward, if we consider the triple solid angle formed by the plane angles ASC, ASD, DSC, these three angles are known; thus the solid angle is entirely determinate. But the quadruple solid angle is formed by the union of the two triple solid angles of which we have been speaking; therefore, since these partial angles are known and determinate, the whole angle will be known and determinate. The angle of the two planes ASD, DSC, may be found immediately by means of the second partial solid angle. As to the angle of the two planes BSC, CSD, it is necessary in one of the partial solid angles to find the angle comprehended between the two planes ASC, DSC, and in the other the angle comprehended between the two planes ASC, BSC; the sum of these angles will be the angle comprehended between the two planes BSC, DSC. It will be found, in the same manner, that, in order to determine a quintuple solid angle, it is necessary to know, beside the five plane angles which compose it, two of the mutual inclinations of their planes; in a sextuple solid angle it is necessary to know three of these inclinations, and so on. SECTION SECOND. Of Polyedrons. DEFINITIONS. 365. EVERY Solid terminated by planes or plane faces is called a solid polyedron, or simply a polyedron. These planes are themselves necessarily terminated by straight lines. A solid of four faces is called a tetraedron, one of six a hexaedron, one of eight an octaedron, one of twelve a dodecaedron, one of twenty an icosaedron, &c. The tetraedron is the most simple of polyedrons; for it requires at least three planes to form a solid angle, and these three planes would leave an opening, to close which a fourth plane is neces sary. 366. The common intersection of two adjacent faces of a polyedron is called a side or edge of the polyedron. 367. A regular polyedron is one, all whose faces are equal regular polygons, and all whose solid angles are equal to each other. There are five polyedrons of this kind. 368. A prism is a solid comprehended under several parallelograms terminated by two equal and parallel polygons. To construct this solid let ABCDE (fig. 200) be any polygon, Fig. 200. if, in a plane parallel to ABC we draw the lines FG, GH, HI, &c.. equal and parallel to the sides AB, BC, CD, &c., we shall form the polygon FGHIK equal to ABCDE; if now we connect the vertices of the homologous angles by the straight lines AF, BG, CH, &c., the faces ABGF, BCHG, &c., will be parallelograms, and the solid thus formed ABCDEFGHIK will be a prism. 369. The equal and parallel polygons ABCDE, FGHIK, are called the bases of the prism. The other planes taken together, constitute the lateral or convex surface of the prism. The equal straight lines AF, BG, CH, &c., are called the sides of the prism. 370. The altitude of a prism is the distance between its bases, or the perpendicular let fall from a point in the superior base upon the plane of the inferior. 371. A right prism is one whose sides AF, BG, &c., are perpendicular to the planes of the bases; in this case, each of the Fig. 206. sides is equal to the altitude of the prism. In every other case the prism is oblique, and the altitude is less than the side. 372. A prism is triangular, quadrangular, pentagonal, hexagonal, &c., according as the base is a triangle, a quadrilateral, a pentagon, a hexagon, &c. 373. A prism whose base is a parallelogram (fig. 206), has ali its faces parallelograms, and is called a parallelopiped. A parallelopiped is rectangular, when all its faces are rectangles. 374. Among rectangular parallelopipeds is distinguished the cube or regular hexaedron comprehended under six equal squares. 375. A pyramid is a solid formed by several triangular planes proceeding from the same point and terminating in the sides of a Fig. 196. polygon ABCDE (fig. 196). Fig. 202. The polygon ABCDE is called the base of the pyramid, the point S its vertex, and the triangles ASB, BSC, &c., compose the lateral or convex surface of the pyramid. 376. The altitude of a pyramid is the perpendicular let fall from the vertex upon the plane of the base, produced if necessary. 377. A pyramid is triangular, quadrangular, &c., according as the base is a triangle, a quadrilateral, &c. 378. A pyramid is regular, when the base is a regular polygon, and the perpendicular, let fall from the vertex to the plane of the base, passes through the centre of this base. This line is called the axis of the pyramid. 379. The diagonal of a polyedron is a straight line which joins the vertices of two solid angles not adjacent. 380. I shall call symmetrical polyedrons two polyedrons which, having a common base, are similarly constructed, the one above the plane of this base and other below it, with this condition, that the vertices of the homologous solid angles be situated at equal distances from the plane of the base, in the same straight line perpendicular to this plane. If the straight line ST (fig. 202), for example, is perpendicular to the plane ABC, and is bisected at the point O, where it meets this plane, the two pyramids SABC, TABC, which have the common base ABC, are two symmetrical polyedrons." 381. Two triangular pyramids are similar when they have two faces similar, each to each, similarly placed, and equally inclined to each other. Thus, if we suppose the angle ABC= DEF, BAC = EDF, 1 ABS = DET, BAS = EDT, (fig. 203), if also the inclination of Fig. 203. the planes ABS, ABC, is equal to that of their homologous planes DTE, DEF, the pyramids SABC, TDEF, are similar. 382. Having formed a triangle with the vertices of three angles, taken in the same face or base of a polyedron, we can imagine the vertices of the different solid angles of the polyedrons, situated out of the plane of this base, to be the vertices of as many triangular pyramids, which have for their common base the above triangle; and these several pyramids will determine. the positions of the several solid angles of the polyedron with respect to the base. This being supposed; Two polyedrons are similar, when, the bases being similar, the vertices of the homologous solid angles are determined by triangular pyramids similar each to each.. 383. I shall call vertices of a polyedron the points situated at the vertices of the different solid angles. N. B. We shall consider only those polyedrons, which have saliant angles, or convex polyedrons. We thus denominate those, the surface of which cannot be met by a straight line in more than two points. In polyedrons of this description the plane of neither of the faces can, by being produced, cut the solid; it is impossible then, that the polyedron should be in part above the plane of one of the faces and in part below it; it is wholly on one side of this plane. THEOREM. 384. Two polyedrons cannot have the same vertices, the number also being the same, without coinciding the one with the other. Demonstration. Let us suppose one of the polyedrons already constructed, if we would construct another having the same vertices, the number also being the same, it is necessary that the planes of this last should not all pass through the same points as in the first; if they did, they would not differ the one from the other; but then it is evident that any new planes would cut the first polyedron; there would then be vertices above these planes and vertices below them, which does not consist with a convex polyedron; therefore, if two polyedrons have the same vertices, the number also being the same, they must necessarily coincide the one with the other. Fig 204. 385. Scholium. The points A, B, C, K, &c. (fig. 204), being given in position to be used as the vertices of a polyedron, it is easy to describe the polyedron. Take, in the first place, three neighbouring points D, E, H, such that the plane DEH shall pass, if there is occasion for it, through other points K, C, but leaving all the rest on the same side, all above the plane, or all below it; the plane DEH or DEHKC, thus determined, will be a face of the solid. Through one of the sides EH of this face, suppose a plane to pass, and to turn upon this line until it meets a new vertex F, or several at the same time F, 1; we shall thus have a second face FEH or FEHI. Proceed in this manner, by making planes to pass through the sides of the faces, until the solid is terminated in all directions; this solid will be the polyedron required, for there are not two which can have the same vertices. Fig. 205. THEOREM. 386. In two symmetrical polyedrons the homologous faces are equal, each to each, and the inclination of two adjacent faces in one of the solids is equal to the inclination of the homologous faces in the other. Demonstration. Let ABCDE (fig. 205) be the common base of the two polyedrons, M and N the vertices of any two solid angles of one of the polyedrons, M' and N' the homologous vertices of the other polyedron; the straight lines MM', NÑ', must be perpendicular to the plane ABC, and be bisected at the points m and n (380), where they meet this plane. This being supposed, we say that the distance MN is equal to M'N'. For, if the trapezoid m M'N' n be made to revolve about mn, until its plane comes into the position of the plane m MN n, on account of the right angles at m and n, the side m M' will fall upon its equal m M, and n N upon n N; therefore the two trapezoids will coincide, and we shall have MN= M'N'. Let P be a third vertex in the superior polyedron, and P' the homologous vertex in the other, we shall have, in like manner, MP=M'P', and NP = N'P'; consequently the triangle MNP, formed by joining any three vertices of the superior polyedron is equal to the triangle M'N'P', formed by joining the homologous vertices of the other polyedron. |