Book XII. AEH equal d and similar to the triangle HKD: For the same reafon, the triangle AGH is equal and similar to the triangle d 4. 1. 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 € 10.1.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 base 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 base is the triangle AEG, and of which the vertex is the point H, is eEC. 11. qual f and similar to the pyramid the C. base of which is the triangle KHL, and D lar to the triangle HDK, and their 8 4. 6. fides are proportionals 8: Therefore the triangle ADB is fimilar to the triangle K L HKL, as before was proved; therefore . 25. 6. the triangle ABC is similar h to the B triangle HKL, And the pyranid of which the base is the triangle ABC, and vertex the point D, i B. 11. is therefore fiu ilar i to the pyramid of which the base is the tri&!. angle HKL, and vertex the fame point D : But the pyramid of Def. 11. which the bale is the triangle HIKI, and veriex the point D, is similar, as bas been proved, to the pyramid the base of which is the triangle AEG, and vertex the point H: Wherefore the py: samid olie base of which is the uriangle À LC, and reriex the point D, is fimilar to the pyramid of which the bafe is the triangle AEG and verex H: Therefore each of the priamids AEGH, HKLD is fozila, to the whole pyramid AECD : Ard because k 41.1. BF is <qual to TC, the parallelogram EBFG is corble k of the triangle GEC: But when there are two priso.s of the fame alti: С tude, 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 parallelogram a 40. 11. EBFG for its base, and the straight line KH opposite to it, is equal to the prisn having the triangle GFC for its base, and the triangle HKL opposite to it; for they are of the same aliitude, 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, HKL 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 the c 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 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 oppofire fide KH is equal to the prism having the triangle GFC for its bare, 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 HKL, and vertex D: Therefore the two prisms before mentioned are greater than the two pyramids of which the basis are the tri. angles 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. 0. E. D. Book XII. PRO P. IV. THEOR. Sce N. IF there be two pyramids of the same altitude, upon triangular bases, and each of them be divided into two equal pyramids fimilar to the whole pyramid, and also into two equal prisins; and if each of these pyramids be divided in the fume minner as the first two, and so on : As the base of one of the first two pyramids is to the base of the other, fo Mall all the prifüis in one of them be to all the prisons in the other, that are produced by the same number of divisions. Let there be two pyramids of the same altitude upon the triangular bates ABC, DEF, and having their veriices 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 baie DEF, so are all the prisins in the pyramid ABCG to all the prisms in the pyramid DEFH made by the same number of divifions. Make the fame construction as in the foregoing propofition : and becaule BX is equal to XC, and AL to LC, therefore XL 2. 6. 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 to CX, as EF 10 FV: And upon BC, CX are described the similar and finilarly situated rectilineal fi. ĝures ABC, LXC; and upon EF, FV, in like manner, are described the finiar figures DEF, RVF: Therefore, as the trib 22. 6.angle ABC is to the triaogle LXC, lo u is the triangle DEF to the triangle RVF, and, by permuiation, as the triangle ABC to the criangle DEF, so is the triangle LXC to the triangle RVF : And because 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 bafes ABC, DEF, which, by the hypothe lis, are equal to one another, thail be cut each into two egoal 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, RVF): Y are of the fame altitude; and therefore, as the base LXC 10 the : a cor. 32 11. the bafe RVF; that is, as the triangle ABC to the triangle Book XII. DEF, fo a is the 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 oppofite triangle STY : And because the two priíins in the pyrainid ABCG are equal to one another, and also the two prisms in the pyramid DEFH equal to one anorher, as the prism of which the base is the parallelogram. KBXL and opposite side NO, to the prism having' the triangle LXC tor je bafe, and OMN the triangle opposite to it; fo is the prifin of which the base is the parallelogramb ý $ PEVR, and oppofite fide TS, to the prism of which the base is the triangle RVF, and oppolite criangle STY Therefore, componendo, as the prisms KBXLMO, LXCOMN together are unto the prism LXCOMN; fo are the prisms PEVRTS, RVESTY to the prism RVFSTY: And, permutando, as the prifms KBXLMO, LXCOMN are to the prisms PEVRTS, RVFSTY; fo 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 (wo prisms in the pyramid ABCG to the two prisms in the pyramid DEFH : And lk-wile if the pyramids now made, for example, the two OMNG, STYH be divided in the same manner ; as the base OMN is to the bafe STY, fo fhall the two prisms in the pyramid OMNG be to the two prisms in the pyramid STYH : But the hase OMN is to the bale STY, as the base ABC to the base DEF; therefore, as the bale ABC to the base DEF, fo are ibe Book XIL. the two prisms in the pyrainid ABCG to the two prisms in the pyramid DEFH ; and so are the iwo 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 thevn of the prilins made by dividing the pyramids AKLO and DPRS, and of all made by the laine number of divisions. Q. E. D. : PRO P. V. THEOR. See N. PYRAMIDs of the fame altitude which have trian gular 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 fame altitude : As the base ABC to the base DEF, so is the py. ramid 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 f. 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 folid Q: Let thefe, 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 pyrainid ABCG in the same manner, and into as many parts, as the pyramid DEFH : Therefore, as the bale 6 4. 12. ABC to the base DEF, so b are the prisms in the pyramid ABCG to the pritims in the pyramid DEFH: But as the bale ALC to the base DEF, fo, by hypothefis, is the pyramid ABCG to the folid ; and therefore, as the pyramid ABCG to the foliu Q_, so are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH : But the pyramid ABCG is greater C 14. 5. than the prisms contained in it ; wherefore c also the folid Q is greater tban the prisms in the pyramid DEFH. But it is allo less, which is imposible. Therefore the base ABC is not to thc + This may be explained the same way as at the note in proposition a. in thc like case. |