d 4. 1. Book XII. AEH equal d and fimilar to the triangle HKD: For the fame reason, the triangle AGH is equal and fimilar to the triangle HLD: And because the two straight lines EH, HG which meet one another are parallel to KD, DL that meet one another, and are not in the fame plane with them, they contain 10. equal e angles; therefore the angle EHG is equal to the angle 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 base KL: And the triangle EHG equal d and fimilar to the triangle KDL: For the fame reafon, the triangle AEG is alfo equal and fimilar to the triangle HKL. Therefore the pyramid of which the base is the triang'e AEG, and of which the vertex is the point H, is e C. 11. qual f and similar to the pyramid the base of which is the triangle KHL, and vertex the point D: And because HK is parallel to AB a fide of the triangle ADB, the triangle ADB is equiangular to the triangle HDK, and their 8 4. 6. fides are proportionals g: Therefore the triangle ADB is similar to the triangle HDK: And for the fame reason, the triangle DBC is similar to the triangle. DKL; and the triangle ADC to the triangle HDL; and alfo the triangle ABC to the triangle AEG: But the triangle AEG is fimilar to the triangle HKL, as before was proved; therefore h21. 6. the triangle ABC is fimilar h to the B triangle HKL, And the pyramid of Def. 11. K D L G F C which the base is the triangle ABC, and vertex the point D, i B. 11. is therefore fimilar i to the pyramid of which the base is the tri & 11. angle HKL, and vertex the fame point D: But the pyramid of which the bafe is the triangle IIKL, and vertex the point D, is fimilar, as has been proved, to the pyramid the base of which is the triangle AEG, and vertex the point H: Wherefore the pyramid the base of which is the triangle ABC, and vertex the point D, is fimilar to the pyramid of which the bafe is the triangle AEG and vertex H: Therefore each of the pyramids AEGH, HKLD is fimilar to the whole pyramid AECD: And because * 41. 1. BF is equal to FC, the parallelogram EBFG is double k of the triangle GFC: But when there are two prifms of the fame altitude, tude, of which one has a parallelogram for its base, and the other Book XII. a triangle that is half of the parallelogram, these prifms are equal a to one another; therefore the prism having the parallelograma 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 bafe, and the triangle HKL opposite to it; for they are of the same altitude, because they are between the parallel b planes ABC, 6 15. 11. HKL: And it is manifest that each of these prisms is greater than either of the pyramids of which the triangles AEG, HKL are the bafes, 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 opposite 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 the 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 prifm having the parallelogram EBFG for its base, and opposite side KH, is greater than the pyramid of which the base is the triangle AEG, and vertex the point H: And the prison of which the base is the parallelogram EBFG, and opposite side KH is equal to the prism having the triangle GFC for its base, and HKL the triangle oppofite 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 HKL, and vertex D: Therefore the two prifms before mentioned are greater than the two pyramids of which the bafes are the triangles AEG, HKL, and vertices the points H, D. Therefore the whole pyramid of which the base is the triangle ABC, and vertex the point D, is divided into two equal pyramids fimilar 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. Sce N. 2.6. PROP. IV. THEOR. IF there be two pyramids of the fame altitude, upon triangular bafes, and each of them be divided into two equal pyramids fimilar to the whole pyramid, and alfo into two equal prifins; 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 pyramids is to the base of the other, so shall all the prifms in one of them be to all the prifms in the other, that are produced by the fame number of divifions. Let there be two pyramids of the fame 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 to be divided in the like manner, and so on: As the base ABC is to the bafe DEF, so are all the prisms in the pyramid ABCG to all the prisms in the pyramid DEFH made by the same number of dis vifions. Make the fame construction as in the foregoing propofition : and because 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 fame reason, the triangle DEF is fimilar to RVF: And because BC is double of CX, and EF double of FV, therefore BC is to CX, as EF to FV: And upon BC, CX are defcribed the fimilar and fimilarly fituated rectilineal figures ABC, LXC; and upon EF, FV, in like manner, are described the fimilar figures DEF, RVF: Therefore, as the tri 6 22. 6.angle ABC is to the triangle LXC, fob is the triangle DEF to the triangle RVF, and, by permutation, as the triangle ABC to the triangle DEF, fo is the triangle LXC to the triangle RVF: And because the planes ABC, OMN, as also the planes 15.DEF, STY are parallels, the perpendiculars drawn from the points G, H to the bafes ABC, DEF, which, by the hypothefis, are equal to one another, shall be cut each into two equal 17. 11. parts by the planes OMN, STY, because the straight lines GC, HF are cut into two equal parts in the points N, Y by the fame planes: 'Therefore the prisms LXCOMN, RVFSTY are of the fame altitude; and therefore, as the base LXC to the D a cor. 32, 11. the base RVF; that is, as the triangle ABC to the triangle Book XII. are unto the prism LXCOMN; so 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 bafe DEF: Therefore, as the bafe ABC to the base DEF, so are the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH: And kewite if the pyramids now made, for example, the two OMNG, STYH be divided in the fame manner; as the base OMN is to the base STY, so shall the two prisms in the py. ramid OMNG be to the two prifms in the pyramid STYH: But the hafe OMN is to the bafe STY, as the base ABC to the bafe DEF; therefore, as the base ABC to the base DEF, fo are R3 the Book XII. the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH; and fo are the two prifms in the pyramid OMNG to the two prifms in the pyramid STYH; and so are all four to all four: And the fame thing may be theven of the prifims made by dividing the pyramids AKLO and DPRS, and of all made by the fame number of divifions. Q. E. D. See N. PROP. V. THEOR. PYRAMIDS of the fame altitude which have triangular bafes, are to one another as their bafes. Let the pyramids of which the triangles ABC, DEF are the bafes, and of which the vertices are the points G, H, be of the fame 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 folid either less than the pyramid DEFH, or greater than it ‡. First, let it be to a folid 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 prifms: Therefore these two prifms 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 folid Q: Let thefe, for example, be the pyramids DPRS, STYH: Therefore the prifms, which make the rest of the pyramid DEFH, are greater than the folid Q: Divide likewife the pyramid ABCG in the fame manner, and into as many parts, as the pyramid DEFH: Therefore, as the bate C 14.5. b 4. 12. ABC to the bafe DEF, fo bare the prifms in the pyramid ABCG to the pritoms in the pyramid DEFH: But as the bate ABC to the bafe DEF, fo, by hypothefis, is the pyramid ABCG to the folid Q; and therefore, as the pyramid ABCG to the folid Q, fo are the prifms in the pyramid ABCG to the prifms in the pyramid DEFH: But the pyramid ABCG is greater than the prifms contained in it; wherefore also the folid Qis greater than the prisms in the pyramid DEFH. But it is alfo less, which is impoffible. Therefore the base ABC is not to the This may be explained the fame way as at the note in proposition 2. in the like cafe. |