equal each to each, these solid (s being applied to each other, coincide, (Prop. 96.) : the s AB, AC, AD, coincide with the \s GH, GK, GL, each with each. But AB coinciding with GH, and the point D with L, the |DE must fall upon LM, (Ax. 16), and BE on HM; .. the point E coincides with M, and the point F with N. Consequently the rectilineal figures which bound the one prism, coincide with, and are equal and similar to the rectilineal figures which bound the other, each to each ; the solid Zs of the one coincide with and are equal to the solid Ls of the other, each to each ; and the solids themselves are equal and similar. Cor. 1. If two triangular pyramids have the plane angles and linear sides about two of their solid angles equal, each to each, and alike situated, the pyramids are equal and similar. Cor. 2. If two triangular prisms have the linear sides about two of their solid angles both equal and parallel, each to each, the prisms are equal and similar. Cor. 3. If two pyramids have the linear sides about their vertical angles both equal and parallel, each to each, the pyramids are equal and similar. PROPOSITION XCVIII.-THEOREM. The opposite sides, as ABCD, EFGH, of a parallelopiped ABCD-EFGH, are similar and equal parallelograms, and the diagonal plane divides it into two equal and similar prisms. For, since the opposite planes, AF, DG, are parallel, and cut by a third plane BD, the common sections AB, DC, are : parallel, (Prop. 89); for a similar reason AD and BC are parallel; hence the figure ABCD is a D. In like manner, all the figures which bound the solid are Os. Hence AD =EH, and DC=HG, and the LADC is =EHG, (Prop. 85); •. the As ADC, EHG, (Prop. 5), and consequently the Òs ABCD, EFGH, are equal and similar. Again, : AE, CG, are each = and || DH; they are =and || one another; . the figure ACGE is a D, (Prop. 34, cor. 1), and divides the whole parallelopiped AG into two triangular prisms ABC-EFG, and ACD-EGH, and these prisms are equal and similar, (Prop. 97); - the plane Ls and linear sides about the solid Z8 at T and D are equal each to each. For the plane Ls EFG, ADC, are = one another, each being = the ZEHG, and the linear sides FG, DA, are = one another, each being =EH, and so of the others. Cor. 1. The opposite solid angles of a parallelopiped are equal. Cor. 2. Every rectangular parallelopiped is bounded by rectangles. Cor. 3. The ends of a prism are similar and equal figures, and their planes parallel. Cor. 4. Every parallelopiped is a quadrangular prism, of which the ends are parallelograms, and conversely. Cor. 5. If two parallelopipeds have the plane angles and linear sides about two of their solid angles equal, each to each, and alike situated, or the linear sides, about two of their solid angles, both equal and parallel, each to each, the solids are equal and similar. Cor. 6. If from the angular points of any rectilineal figure, there be drawn straight lines above its plane, all equal and parallel, and their extremities be joined, the figure so formed is a prism. If the rectilinear figure be a parallelogram, the prism is a parallelopiped. If the rectilineal figure be a rectangle, and the straight lines be perpendicular to its plane, the prism is a rectangular parallelopiped. Cor. 7. Every triangular prism is equal to another triangular prism, having its base equal to the half of one of the sides of the prism, and its altitude the perpendicular distance of the opposite edge. The prism FEG-BAC= ABF-DCG, (formed by the diagonal plane AFGD), which has for its base half the side ABFE, and for its altitude the perpendicular distance of the edge CG; for they are each half of the parallelopiped FD. N PROPOSITION XCIX. Parallelopipeds ABCD-EFGH, and ABCD-KLMN upon the same base ABCD, and between the same parallel planes AC, EM, are equal. Because the \s EF, GH, KL, MN, 5 are || AB, CD, they are || one another, and being all in the same plane, NK, ML, being produced, will .. meet both EF and HG; let them meet the former in O, P, and the lat in R, Q, and let AO, BP, CQ, DR, be joined. The re ABCD-OPQR, is a parallelopiped. For, by hyposis, the plane EQ is || AC, the plane of the parallels -HQ is || the plane of the parallels AB-EP, and plane AD-NO parallel to BC-MP. Hence AEOIR and BFP-CGQ are two triangular prisms, which e the linear sides AE, AD, AO, about the solid LA, h equal and || the linear sides BF, BC, BP, about solid [B, (Prop. 98), each to each. Consequently se prisms are equal, (Prop. 97), and each of them being en away from the whole solid ABCD-EPQH, the reinders, the parallelopipeds AG, AQ, are equal. In the he manner, the parallelopipeds AM and AQ may be ved equal. : the parallelopipeds AG, AM, are equal one another. Cor. Triangular prisms, upon the same base, and beeen the same parallel planes, are equal. For, if two planes be made to pass, the one through , EG, and the other through AC, KM, they will bisect parallelopipeds AG, AM, (Prop. 98); .. the prisms C-EHG, ADC-KNM, will be equal. PROPOSITION C.-THEOREM. Parallelopipeds ABCD-EFGH, and ABKL-OPQR, in equal bases, and between the same parallel planes, , EQ, are equal. CASE I. When the bases have side AB common, and lie been the same parallel lines AB, , these parallelopipeds AG, , are equal. for let the planes EAL, FBK, et the plane HC of the former, the lines of common section I, KN, and the plane HF in lines EM, FN. Then since EA, AL, are || FB, BK, ir planes are parallel, and the plane EB is || the plane HK; he figure AN is a parallelopiped. But ADL-EHM, | BCK-FGN, are two triangular prisms, which have linear sides, AD, AE, AL, about the solid LA, both = 1 ll the linear sides, BC, BF, BK, about the solid _B, h to each ; . these prisms are equal, (Prop. 97, cor. 2); each of them being taken away from the whole solid, KD-EFNH, the remainders, the parallelopipeds AG, N, are equal. But AN is =AQ, (Prop. 99); 3. AG is AQ. R P G M N K G C D B CASE II. When the bases ABCD, CEFG, are equiangular, having the ZDCB = the ZGCE, place DC, CE, in one \, then GC, CB, are also in one Also let their sides be produced to meet in the points H, K. Since the Ds AC, CF, are equal, they are complements of the DAF, and the \HCK is its diagonal. Upon the base AF let a parallelopiped be erected, of the same altitude with those upon AC, CF, and let it be cut by planes || its sides, and touching the lines DCE, GCB; these planes divide the whole parallelopiped into four other parallelopipeds, upon the bases DG, AC, BE, CF. But the diagonal plane touching HK divides each of the parallelopipeds upon AF, DG, BE, into two equal prisms, (Prop. 98). is the prisms upon HGC, CEK, are together = the prisms upon HDC, CBK; and these being taken away from the = prisms upon HFK, HAK, the remainders, the parallelopipeds upon AC, CF, are equal. Consequently any two parallelopipeds upon these bases, and between the same parallel planes, are equal, (Prop. 99). Case III. When the bases are neither equiangular, nor have one side common, a parallelogram can be described on the base of the one equal to it, (Prop. 26), and equi; angular to the other, by which it is reduced to the second Therefore, universally, parallelopipeds upon equal bases, and between the same parallel planes, are equal. Cor. 1. Parallelopipeds of equal bases and equal altitudes are equal. Cor. 2. Prisms, upon equal bases, which are either both triangles, or both parallelograms, and of equal altitudes, are equal Cor. 3. Two prisms, upon equal bases, the one a triangle, and the other a parallelogram, and having the same altitude, are equal. Cor. 4. A prism upon any rectilineal base is equal to a parallelopiped having an equal base and the same altitude. PROPOSITION CI.-THEOREM. Parallelopipeds EK, AL, of equal altitudes, are to one another as their bases EFGH, ABCD. Produce EF both ways to N and R, and make the line FQ such that the altitude of the EG is to the altitude of the DAC as AB is to FQ, and complete the FW and the parallelopiped FS. Hence the OFW is = the DAC, (Prop. 64); and the parallelopipeds FS and AL case. ner it having equal bases and altitudes, are equal. Take FR any multiple of FQ, and FN any multiple of FE, and complete the OS WR, EM, and MN, and the solids QT, EP, and PN; it is manifest, that since FQ, QR, are equal, the FW, WR, are equal, (Prop. 27); and that, since FW, WR, are equal, the solids FS, QT, are also equal, (Prop. 100); :: what multiple soever the base GR is of FW, the same multiple is the solid FT of the solid FS. In the same man may be shown, that what multiple soever the base GN is of the base EG, the same multiple is the solid KN of the solid EK. Now, if the base NG be 7 the base GR, the solid NK will be the solid FT; if equal, equal; and if less, less. .. EK: FS = the base EG: the base FW; and since the base FW=AC, and the solid FS=AL; the solid EK: the solid AL= the base EG : the base AC. Cor. 1. Prisms standing on their ends, and of equal altitudes, are to one another as their bases. Cor. 2. Parallelopipeds, upon the same or equal bases, are to one another as their altitudes. For the parallelopipeds KN, KR, on the same base KF, are to one another as NG: GR, that is, as FN: FR. PROPOSITION CII.-THEOREM. Two parallelopipeds DE, KL, which have a solid angle B, of the one equal to a solid angle G of the other, are to one another in the ratio compounded of the ratios of the linear sides BA, BC, BE of the one, to the linear sides GF, GH, GL of the other, each to each, about these solid angles. For, let AB, CB, EB be produced to M, N, O, so that BM=GF, BN=GH and BO D =GL. With the lines EB, BM, and BN, let the parallelopiped EP be completed ; and with the lines OB, BM, and BN, the parallelopiped OP. Then :: the ratio DE: OP is L G |