412. Corollary 11. If we join DG, DH, we shall have a new pyramid D-AGH equal to the pyramid S-DEF; for the base DEF may be placed upon its equal AGH, and then, the angles SDE, SDF, being equal to the angles DAG, DAH, it is manifest that DS will fall upon AD (364), and the vertex S upon the vertex D. Now the pyramid D-AGH is less than the prism AGH-FDE since it is contained in it; therefore each of the pyramids S-DEF, E-GBI, is less than the prism AGH-FDE; therefore the pyramid S-ABC, which is composed of two pyramids and two prisms, is less than four of these same prisms. But the solidity of one of these prisms = ABC × SO, and its quadruple ABC × SO; hence the solidity of any triangular pyramid is less than half of the product of its base by its altitude. Fig. 215. THEOREM. 413. The solidity of a triangular pyramid is equal to a third of the product of its base by its altitude. Demonstration. Let S-ABC (fig. 215) be any triangular pyramid, BC its base, SO its altitude; we say that the solidity of the pyramid S-ABC is equal to a third part of the product of the surface ABC by the altitude SO, so that S-ABC ABC × SO, or = SO × ABC. If this proposition be denied, the solidity S-ABC must be equal to the product of SO by a surface either greater or less than ABC. 1. Let this quantity be greater, so that we shall have S-ABC = SO × (} ABC +M). If we make the same construction as in the preceding proposition, the pyramid S-ABC will be divided into two equivalent. prisms AGH-FDE, EGI-CFH, and two equal pyramids S-DEF, E-GBI. Now the solidity of the prism AGH-FDE is DEF × PO, consequently we shall have the solidity of the two prisms AGH-FDE+EGI-CFH = DEF × 2PO, or = DEF × SÓ. The two prisms being taken from the entire pyramid, the remainder will be equal to double of the pyramid S-DEF, so that we shall have 2S-DEF=SO × (} ABC+M— DEF). But, because SA is double of SD, the surface ABC is quadruple of DEF (408), and thus It appears then, that in order to obtain the solidity of the pyra. mid S-DEF, it is necessary to add to a third of the base the same surface M, which was added to a third of the base of the large pyramid, and to multiply the whole by the altitude SP of the small pyramid. If SD be bisected at the point K, and if through this point a plane KLM be supposed to pass parallel to DEF meeting the perpendicular SP in Q; according to what has just been demonstrated, S-KLM = SQ × ({ KLM +- M ). If we proceed thus to form a series of pyramids, the sides of which decrease in the ratio of 2 to 1, and the bases in the ratio 4 to 1, we shall soon arrive at a pyramid S-a bc, the base of which abc shall be less than 6M. Let So be the altitude of this last pyramid; and its solidity, deduced from that of the preceding pyramids, will be Sox (abc+M). But Mabc, and consequently abc+M> abc. It would follow then, that the solidity of the pyramid S-a bc is greater than Sox abc; which is absurd, since it was proved in the preceding proposition, corollary 11, that the solidity of a triangular pyramid is always less than half of the product of its base by its altitude; therefore it is impossible that the solidity of the pyramid S-ABC should be greater than SO x } ABC. 2. Let S-ABC be equal to SO x ( ABC-M); it may be shown, as in the first case, that the solidity of the pyramid S-DEF, the dimensions of which are less by one half, is equal to SP × (} DEF-M) ; and, by continuing the series of pyramids, the sides of which decrease in the ratio of 2 to 1, until we arrive at a term S-a bc, we shall, in like manner, have the solidity of this last equal to • Sox (abc-M). But, as the bases ABC, DEF, KLM..... a bc, form a decreasing series, each term of which is a fourth of the preceding, we shall soon arrive at a term abc equal to 12M, or which shall be comprehended between 12M and 3M; then, M being either equal to or greater than a bc, the quantity abc — M will either be equal to or less than abc; so that we shall have the solidity of the pyramid S-a b c either equal to or less than Soxabc; which is absurd, since, according to corollary 1 of the preceding proposition, the solidity of a triangular pyramid is always greater than the fourth of the product of its base by its altitude; therefore the solidity of the pyramid S-ABC cannot be less than SOX ABC. We conclude then, according to the enunciation of the theorem, that the solidity of the pyramid SABC = SO × } ABC, or ABC X SO. = 414. Corollary 1. Every triangular pyramid is a third of a triangular prism of the same base and same altitude; for ABC SO is the solidity of the prism of which ABC is the base and SO the altitude. 415. Corollary 11. Two triangular pyramids of the same altitude are to each other as their bases, and two triangular pyramids of the same base are to each other as their altitudes. Fig. 214. THEOREM. 416. Every pyramid S-ABCDE (fig. 214) has for its measure a third of the product of its base by its altitude. Demonstration. If the planes SEB, SEC, be made to pass through the diagonals EB, EC, the polygonal pyramid S-ABCDE will be divided into several triangular pyramids, which have all the same altitude SO. But, by the preceding theorem, these are measured by multiplying their bases ABE, BCE, CDE, each by a third of its altitude SO; consequently the sum of the triangular pyramids, or the polygonal pyramid S-ABCDE will have for its measure the sum of the triangles ABE, BCE, CDE, *or the polygon ABCDE, multiplied by SO; therefore every pyramid has for its measure a third of the product of its base by its altitude. 417. Corollary 1. Every pyramid is a third of a prism of the same base and same altitude. 418. Corollary II. Two pyramids of the same altitude are to each other as their bases, and two pyramids of the same base are to each other as their altitudes. 419. Scholium. The solidity of any polyedron may be estimated by decomposing it into pyramids, and this decomposition may be effected in several ways; one of the most simple is by means of planes of division passing through the vertex of the same solid angle; then we shall have as many partial pyramids, as there are faces in the polyedron excepting those which contain the solid angle from which the planes of division proceed. THEOREM. 420. Two symmetrical polyedrons are equivalent to each other or equal in solidity. Demonstration. 1. Two symmetrical triangular pyramids, as S-ABC, T-ABC (fig. 202), have each for its measure the pro- Fig. 202, duct of the base ABC by a third of its altitude SO or TO; therefore these pyramids are equivalent. 2. If we divide, in any manner, one of the symmetrical polyedrons in question into triangular pyramids, we can divide the other polyedron, in the same manner into triangular pyramids symmetrical with the former (382); but the triangular pyramids in the one case and the other being symmetrical, are equivalent, each to each; therefore the entire polyedrons are equivalent to each other or equal in solidity. 421. Scholium. This proposition seems to result immediately from a former (386), in which it was shown that, with respect to two symmetrical polyedrons, all the constituent parts of the one are equal respectively to those of the other; still it was necessary to demonstrate it in a rigorous manner. THEOREM. 422. If a pyramid is cut by a plane parallel to its base, the frustum which remains, after taking away the smaller pyramid, is equal to the sum of three pyramids, which have for their common altitude the altitude of the frustum, and whose bases are the inferior base of the frustum, its superior base, and a mean proportional between these bases. Demonstration. Let S-ABCDE (fig. 217) be a pyramid cut Fig. 217. by the plane abd parallel to the base; let T-FGH be a triangular pyramid, whose base and altitude are equal or equivalent to the base and altitude of the pyramid S-ABCDE. The two Geom. 19 Fig. 218. bases may be supposed to be situated in the same plane; and then the plane abd produced, will determine in the triangular pyramid a section fgh situated at the same altitude above the common plane of the bases; whence it follows that the section fgh is to the section ab d, as the base FGH is to the base ABD (408); and, since the bases are equivalent, the sections will be equivalent also. Consequently the pyramids S-a b c d e, T-fgh, are equivalent, since they have the same altitude and equivalent bases. The entire pyramids S-ABCDE, T-FGH, are equivalent, for the same reason; therefore the frustums ABD-da b, FGH-hƒg, are equivalent; and consequently it will be sufficient to demonstrate the proposition enunciated, with reference merely to the case of the frustum of a triangular pyramid. Let FGH-hfg (fig. 218), be the frustum of a triangular pyramid; through the points F, g, H, suppose a plane Fg H, to pass cutting off from the frustum the triangular pyramid g-FGH. This pyramid has for its base the inferior base FGH of the frustum, it has also for its altitude the altitude of the frustum, since the vertex g is in the plane of the superior base fg h. This pyramid being cut off, there will remain the quadrangular pyramid g-fh HF, the vertex of which is g and the base fh HF. Through the three points f, g, H, suppose a plane ƒg H to pass dividing the quadrangular pyramid into two triangular pyramids g-FfH, g-fh H. This last pyramid may be considered as having for its base the superior base gƒh of the frustum, and for its altitude the altitude of the frustum, since the vertex H is in the inferior base. Thus we have two of the three pyramids which compose the frustum. It remains to consider the third pyramid g-FfH. Now if we draw gK parallel to ƒ F, and suppose a new pyramid K-FƒH, the vertex of which is K, and the base FfH; these two pyramids will have the same base FfH; they will have also the same altitude, since the vertices g, K, are situated in a line g K parallel to Ff, and consequently parallel to the plane of the base; therefore these pyramids are equivalent. But the pyramid K-FƒH may be considered as having its vertex in ƒ, and thus it will have the same altitude as the frustum; as to its base FHK, we say that it is a mean proportional between the two bases FHG, fhg. Indeed the triangles FHK, fhg, have the angle Ff, and the side FK =ƒg, |