BC: be:: CD: cd, in like manner, 409. Corollary. Let S-ABCDE, S-XYZ,' be two pyramids that have a common vertex, and whose altitudes are the same, or whose bases are situated in the same plane; if these pyramids be cut by a plane parallel to their bases, the sections abcde, x y z, thus formed, will be to each other as the bases ABCDE, XYZ. For, the polygons ABCDE, abcde, being similar, their surfaces are as the squares of their homologous sides AB, a b; but AB: ab:: SA: Sa, consequently 2 --2 ABCDE: abcde:: SA: Sa. For the same reason, —12 -2 XYZ: xyz:: SX: S x. But, since a b c d e, x y z, are in the same plane, whence SA: Sa:: SX: Sx, ABCDE: abcde:: XYZ: xyz; therefore the sections abcde, xyz, are to each other as their bases ABCDE, XYZ. Fig. 215. LEMMA. 410. Let S-ABC (fig. 215), be a triangular pyramid, of which S is the vertex and ABC the base; if the sides SA, SB, SC, AB, AC, BC, be bisected at the points D, E, F, G, H, 1, and through these points the straight lines DE, EF, DF, EG, FH, EI, GI, GH, be drawn; we say that the pyramid S-ABC may be considered as composed of two prisms AGH-FDE, EGI-CFH, equivalent to each other, and two equal pyramids S-DEF, E-GBI. Demonstration. It follows from the construction, that ED is parallel to BA, and GE to AS (199); hence the figure ADEG is a parallelogram. For the same reason, the figure ADFH is also a parallelogram; consequently the straight lines AD, GE, HF, are equal and parallel; therefore the solid AGH-FDE is a prism (346). It may be shown, in like manner, that the two figures EFCI, CIGH, are parallelograms, and that thus the straight lines EF, IC, GH, are equal and parallel; therefore the solid EGI-CFH is also a prism. Now we say that these two triangular prisms are equivalent to each other. Indeed, if upon the edges GI, GE, GH, the parallelopiped GX be formed, the triangular prism EGI-CFH will be half of this parallelopiped (397); on the other hand, the prism AGH-FDE is also equal to half of the parallelopiped GX (406), since they have the same altitude, and the triangle AGH, the base of the prism, is half of the parallelogram GICH (168), the base of the parallelopiped. Therefore the two prisms EGI-CFH,AGH-FDE, are equivalent to each other. These two prisms being taken from the pyramid S-ABC there will remain only the two pyramids S-DEF, E-GBI; now we say that these two pyramids are equal to each other. Indeed, since the following sides are equal, namely, BE=SE, BG = AG = DE, EG=AD=SD, the triangle BEG is equal to the triangle ESD (43). For a similar reason, the triangle BEI is equal to the triangle ESF; moreover the mutual inclination of the two planes BEG, BEI, is the same as that of the two planes ESD, ESF, since BEG, ESD, are in the same plane, and BEI, ESF, are also in the same plane. If then, in order to apply the one pyramid to the other, we place the triangle EBG upon its equal EDS, the plane BEI must fall upon the plane ESF; and, since the triangles are equal and similarly disposed, the point I will fall upon F, and the two pyramids will coincide throughout (384). Therefore the entire pyramid S-ABC is composed of two triangular prisms AGF, GIF, equivalent to each other, and two equal pyramids S-DEF, E-GBI. 411. Corollary 1. From the vertex S let fall upon the plane ABC the perpendicular SO, and let P be the point, where this perpendicular meets the plane DEF, parallel to ABC; since SD = 1 SA, we have SP = 1 SO (408), and the triangle DEF = { triangle ABC (218); consequently the solidity of the prism AGH-FDE= | ABC × ¦ SO; 14 and the solidity of the two prisms AGH-FDE, EGI-CFH, taken together, is equal ABC SO. These two prisms are less than the pyramid S-ABC, since they are contained in it; therefore the solidity of a triangular pyramid is greater than the fourth part of the product of its base by its altitude. 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, ABC 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=SOx († 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 = DEFX SO. 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 whence ABC-DEF = DEF — DEF= } DEF ; 2S-DEF=SO × (} DEF+M), or, by taking the half of each, S-DEF SPX (DEF+M). 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 × (1⁄2 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 b c, 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 × ¦ ABC. 2. Let S-ABC be equal to SOx ( 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-abc, we shall, in like manner, have the solidity of this last equal to Sox (abc-M). But, as the bases ABC, DEF, KLM..... a b c, 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 a b c 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 Sox abc; 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 SO × ABC. We conclude then, according to the enunciation of the theorem, that the solidity of the pyramid SABC = SO × } ABC, or = ABC × 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. |