d 4. I CIO. II, Book XII. AEH equal d and similar to the triangle HKD: For the same reason, the triangle AGH is equal and similar to the triangle HLD : And because the two itraight lines EH, HG which meet one another are parallel to KD, DL that meet one ano. ther, and are not in the same plane with them, they contain equal e angles; therefore the angle EHG is equal to the an. gle KDL. Again, because EH, HG are equal to KD, DL, each to each, and the angle EHG equal to the angle KDL ; there: fore the base EG is equal to the bare KL: And the triangle EHG equal d and similar to the triangle KDL: For the fame reason, the triangle AEG is also equal and fimilar to the triangle HKL. Therefore the pyramid of which the bale is the triangle AEG, and of which the vertex is the point H, is ec. 11.qual f and similar to the pyramid the base of which is the triangle KHL, and lar to the triangle HDK, and their & 4. 6. fides are proportionals &: Therefore the triangle ADB is fimilar to the triangle HKL, as before was proved; therefore he 283 6. the triangle ABC is similar h to the B F с triangle HKL, And the Tyranid of which the bate is the triangle ABC, and vertex the point D, į B. 11. is therefore fiuilar i to the pyramid of which the base is the tri!!. angle HKL, and vertex the fame point D): But the pyramid of which the base is the triangle HKL, and veriex the point D), is similar, as has been proved, to the pyramid the bafe of which is the triangle AEG, and vertex the point H: Wherefore the pyramid ile base of which is the triangle À LC, and verlex the point D, is similar to ile pyramid of which the bafe is the triangle AEG and verex H: Therefore each of the Tamids AEGH, HKLD is fimilar to the whole pyramid AECD: Ard because K 41.1. BF is equal to FC, the parallelogram EBFG is corble k of the triangle GFC : But when there are two prisa.s of the fame alti: tude, Def. 11. tude, of which one has a parallelogram for its base, and the other Book XII. a triangle that is half of the parallelogram, these prisms are equal a to one another; therefore the prism having the parallelogram a 40. 11. EBFG for its base, and the straight line KH opposite to it, is equal to the prism having the triangle GFC for its base, and the triangle HKL opposite to it; for they are of the fame altitude, because they are between the parallel b planes ABC, b 15. 11. HKL: And it is manifest that each of these prisms is greater than either of the pyramids of which the triangles AEG, AKL are the bases, and the vertices the points H, D ; because, if EF be joined, the prism having the parallelogram EBFG for its base, and KH the straight line oppolite to it, is greater than the pyramid of which the base is the triangle EBF, and vertex the point K; but this pyramid is equal to the pyramid thec C. 11. base of which is the triangle AEG, and vertex the point H; because they are contained by equal and similar planes : Wherefore the prism having the parallelogram EBFG for its base, and opposite side KII, is greater than the pyramid of which the base is the triangle AEG, and vertex the point H: And the prism of which the base is the parallelogram EBFG, and opposite fide KH is equal to the prism having the triangle GFC for its base, and HKL the triangle opposite to it; and the pyramid of which the base is the triangle AEG, and vertex H, is equal to the pyramid of which the base is the triangle HKI, and vertex D: Therefore the two prisms before mentioned are greater than the two pyramids of which the bases are the tri. angles AEG, HKL, and vertices the points II, D. Therefore the whole pyramid of which the base is the triangle ABC, and vertex the point D, is divided into two equal pyramids similar to one another, and to the whole pyramid; and into two equal prisms; and the two prisms are together greater than half of the whole pyramid. Q. E. D. Book XII. PRO P. IV. THEOR. Sce N. there be two pyramids of the same altitude, upon triangular bases, and each of them be divided into two equal pyramids firmilar to the whole pyramid, and also into two equal prisins; and if each of these pyramids be divided in the fame manner as the first two, and so on : As the base of one of the first two pyramid is to the base of the other, fo Mall all the prisons in one of them be to all the prisons in the other, that are produced by the same number of divisions. 6. Let there be two pyramids of the faine altitude upon the triangular bates ABC, DEF, and having their vertices in the points G, H ; and let each of them be divided into two equal pyramids fimilar to the whole, and into two equal prisms; and let each of the pyramids thus made be conceived io be divided in the like manner, and so on : As the base ABC is to the bale DEF, so are all the prisins in the pyramid ABCG to all the prisms in the pyramid DEFH made by the fame number of divisions. Make the same construction as in the foregoing proposition : and becaule BX is equal to XC, and AL to LC, therefore XL is parallel a to AB, and the triangle ABC fimilar to the triangle LXC: For the same reason, the triangle DEF is fimilar to RVF: And because BC is double of CX, and EF double of FV, therefore BC is co CX, as EF 10 FV : And upon BC, CE are described the fimilar and fimilarly fituated rectilineal tġures ABC, LXC; and upon EF, FV, in like manner, are described the similar figures DEF, RVF: Therefore, as the in $ 22. 6.angle ABC is to the triaogle LXC, fo b is the triangle DEF !0 the triangle RVF. and, by permuiation, as the triangle : BC to the triangle DEF, fo is the triangle LXC to the triangle RVF : And becaufe the planes ABC, OMN, as also the planes $15.11. DEF, STY are parallel c, the perpendiculars drawn from the points G, H to the bales ABC, DEF, which, by the hspiebe . lis, are equal to one another, shall be cut each into two equal $!7. 11. d parts by the planes OMN, STY, because the straight lines GC, HF are cut inio two equal parts in the points N, Y by the fame planes : Theretore the prisms LXCOMN, RVFS:Y are of the fame altitude; and therefore, as the bale LXC 10 a cor. 32 11. the bafé RVF; that is, as the triangle ABC to the triangle Book Xit. DEF, so a is ihe prism having the triangle LXC for its bale, and OMN the triangle opposite to it, to the prism of which the base is the triangle RVF, and the opposite criangle STY: And because the two priims in the pyramid ABCG are equal to one another, and also the two prisms in the pyramid DEFH equal to one and her, as the prilm of which the base is the parallelogram: KBXL and opposite îide NO, to the prism having the triangle LXC for jrs base, and OMN the triangle opposite to it; fo is the prison of which the base is the parallelogram b 4. $ PEVR, and opposite fide TS, to the prism of which the base is the triangle RVF, and opposite triangle STY Therefore, componendo, as the prisms KBXLMO, LXCOMN together Are unto the prism LXCOMN; fo are the prisms PEVRTS, RVFSTY to the prism RVFSTY : And, permutando, as the prifms KBXLMO, LXCOMN are to the prisms PEVRTS, RVFSTY ; so is the prism LXCOMN to the prism RVFSTY : But as the prism LXCOMN to the prism RVFSTY, so is, as has been proved, the base ABC to the base DEF : Therefore, as the base ABC to the base DEF, so are the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH : And I kewile if the pyramids now made, for example, the two OMNG, STYH be divided in the same manner; as the base OMN is to the base STÝ, fo thall the two prisms in the pyramid OMNG be to the two prisms in the pyramid STYH : But the hase OMN is to the bałe STY, as the base ABC to the base DEF ; therefore, as the bale ABC to the base DEF, fo are R 3 ibe Book XII. the two prisms in the pyramid ABCG to the two prisis in the pyramid DEFH ; and fo are the (wo prisms in the pyramid OMNG to the two prisms in the pyramid STYH; and so are all four to all four : And the same thing may be thev:n of the prilins made by dividing the pyramids AKLO and DPRS, and of all made by the lane number of divisions. Q. E. D. PRO P. V. THEOR. See N. PYRAM AMIDS of the fame altitude which have triangular bases, are to one another as their bases. Let the pyramids of which the triangles ABC, DEF are the bases, and of which the vertices are the points G, H, be of the same altitude : As the base ABC to the base DEF, so is the pyramid ABCG to the pyramid DEFH. For, if it be not so, the base ABC must be to the base DEF, as the pyramid ABCG to a solid either less than the pyramid DEFH, or greater than it [ First, let it be to a solid less than it, viz. to the folid Q: And divide the pyramid DEFH into two equal pyramids, fimilar to the whole, and into two equal a 3. 12. prisms : Therefore these two prisms are greater a than the half of the whole pyramid. And, again, let the pyramids made by this division be in like manner divided, and so on, until the pyramids which remain undivided in the pyramid DEFH be, all of them together, less than the excess of the pyramid DEFH above the solid Q: Let these, for example, be the pyramids DPRS, STYH: Therefore the prisms, which make the rest of the pyramid DEFH, are greater than the folid Q : Divide likewise the pyramid ABCG in the same manner, and into as many parts, as the pyramid DEFH : Therefore, as the bale 64. 12. ABC to the base DEF, f) b are the prisms in the pyramid ABCG to the pritms in the pyramid DEFH: But as the bale ALC to the base DEF, so, by hypothefis, is the pyramid ABCG to the solid Q ; and therefore, as the pyramid ABCG to the foliu Q, fo are the prifas in the pyramid ABCG to the prisms in the pyramid DEFH : But the pyramid ABCG is greater C 14. s. than the prisms contained in it ; wherefore e also the folid Q is greater than the prisms in the pyramid DEFH. But it is aito less, which is imposible. Therefore the base ABC is not to the This may be explained the same way as at the note in proposition 2. in thc like case. |