Book XII. PG, GR, RH, HS, SE: Therefore the remainder of the cone, viz. the pyramid of which the bafe is the polygon EOFPGRHS, and its vertex the fame with that of the cone, is greater than the folid X : In the circle ABCD defcribe the polygon ATBYCVDQ fimilar to the polygon EOFPGRHS, and upon it erect a pyramid having the fame vertex with the cone AL: And because as the fquare of AC is to the fquare of EG, fo is the polygon ATBYCVDQ to the polygon EOFPGRHS; and as the quare of AC to the fquare of EG, fo is the circle. ABCD to the circle EFGH; therefore the circle ABCD is to the circle EFGH, as the polygon ATBYCVDQ to the poly a 1. 12. b 2. 12. C II. 5. d 6. 12. € 14. 5. Z gon EOFPGRHS: But as the circle ABCD to the circle EFGH, fo is the cone AL to the folid X; and as the polygon ATBYCVDQ to the polygon EOFPGRHS, fo is the ругаmid of which the bafe is the firft of these polygons, and vertex L, to the pyramid of which the bafe is the other polygon, and its vertex N: Therefore, as the cone AL to the folid X, fo is the pyramid of which the bafe is the polygon ATBYCVDQ, and vertex L, to the pyramid the base of which is the polygon EOFPGRHS, and vertex N: But the cone AL is greater than the pyramid contained in it; therefore the folid X is greater than the pyramid in the cone EN; But it is lefs, as was shown, Which is abfurd: Therefore the circle ABCD is not to the cir- Book XII. cle EFGH, as the cone AL to any folid which is less than the cone EN. In the fame manner it may be demonftrated that the circle EFGH is not to the circle ABCD, as the cone EN to any folid less than the cone AL. Nor can the circle ABCD be to the circle EFGH, as the cone AL to any folid greater than the cone EN: For, if it be poffible, let it be fo to the folid I, which is greater than the cone EN: Therefore, by inversion, as the circle EFGH to the circle ABCD, fo is the folid I to the cone AL: But as the folid I to the cone AL, fo is the cone EN to fome folid, which must be less than the cone a 14. 5. AL, because the folid I is greater than the cone EN: Therefore, as the circle EFGH is to the circle ABCD, fo is the cone EN to a folid less than the cone AL, which was fhewn to be imposible: Therefore the circle ABCD is not to the circle EFGH, as the cone AL is to any folid greater than the cone EN: And it has been demonftrated that neither is the circle ABCD to the circle EFGH, as the cone AL to any folid less than the cone EN: Therefore the circle ABCD is to the circle EFGH, as the cone AL to the cone EN: But as the cone is to the cone, fob is the cylinder to the cylinder, because the cy- b 15. 5. linders are triple. of the cones, each to each. Therefore, as c 10. 12. the circle ABCD to the circle EFGH, fo are the cylinders upon them of the fame altitude. Wherefore cones and cylinders of the fame altitude are to one another as their bases. Q. E. D. S PROP. XII. THE OR. IMILAR cones and cylinders have to one another see N. the triplicate ratio of that which the diameters of their bafes have. Let the cones and cylinders of which the bafes are the circles ABCD, EFGH, and the diameters of the bafes AC, EG, and KL, MN the axes of the cones or cylinders, be fimilar: The cone of which the bafe is the circle ABCD, and vertex the point L, has to the cone of which the base is the circle EFGH, and vertex N, the triplicate ratio of that which AC has to EG. For, if the cone ABCDL has not to the cone EFGHN the triplicate ratio of that which AC has to EG, the cone ABCDL fhall have the triplicate of that ratio to fome folid which is lefs Book XII. or greater than the cone EFGHN. First, let it have it to a lefs, viz. to the folid X:. Make the fame conftruction as in the preceding propofition, and it may be demonftrated the very fame way as in that propofition, that the pyramid of which the base is the polygon EOFPGRHS, and vertex N, is greater than the folid X. Defcribe alfo in the circle ABCD the polygon ATBYCVDQ fimilar to the polygon EOFPGRHS, upon which erect a pyramid having the fame vertex with the cone; and let LAQ be one of the triangles containing the pyramid upon the polygon ATBYCVDQ the vertex of which is L; and let NES be one of the triangles containing the pyramid upon the a 24. def. II. b 15. 5. c 6. 6. polygon EOFPGRHS of which the vertex is N; and join KQ, MS: Because then the cone ABCDL is fimilar to the cone EFGHN, AC is to EG, as the axis KL to the axis MN; and as AC to EG, fob is AK to EM; therefore as AK to EM, fo is KL to MN; and, alternately, AK to KL, as EM to MN: And the right angles AKL, EMN are equal; therefore the fides about thefe equal angles being proportionals, the triangle AKL is fimilar to the triangle EMN. Again, becaufe AK is to KQ, as EM to MS, and that these fides are с about a 6. 6. about equal angles AKQ, EMS, becaufe thefe angles are, Book XI. each of them, the fame part of four right angles at the cen- w tres K, M; therefore the triangle AKQ is fimilar to the triangle EMS: And because it has been fhown that as AK to KL, fo is EM to MN, and that AK is equal to KQ; and EM to MS, as QK to KL, fo is SM to MN; and therefore the fides about the right angles QKL, SMN being proportionals, the triangle LKQ is fimilar to the triangle NMS: And becaufe of the fimilarity of the triangles AKL, EMN, as LA is to AK, fo is NE to EM; and by the fimilarity of the triangles AKQ, EMS, as KA to AQ, fo ME to ÉS; ex aequali, LA is b 22. 5. to AQ, as NE to ES. Again, because of the fimilarity of the triangles LQK, NSM, as LQ to QK, fo NS to SM; and from the fimilarity of the triangles KAQ, MES, as KQ to QA, fo MS to SE; ex aequalib, LQ is to QA, as NS to SE: And it was proved that QA is to AL, as SE to EN; therefore, again, ex aequali, as QL to LA, fo is, SN to NE: Wherefore the triangles LQA, NSE, having the fides about all their angles proportionals, are equiangular and fimilar to one an- e 5. 6. other: And therefore the pyramid of which the bafe is the triangle AKQ, and vertex L, is fimilar to the pyramid the bafe of which is the triangle EMS, and vertex N, becaufe their folid angles are equal to one another, and they are contained d B. 11. by the fame number of fimilar planes: But fimilar pyramids. which have triangular bafes have to one another the triplicate e ratio of that which their homologous fides have; therefore 8. 12. the pyramid AKQL has to the pyramid EMSN the triplicate ratio of that which AK has to EM. In the fame manner, if ftraight lines be drawn from the points D, V, C, Y, B, T to K, and from the points H, R, G, P, F, O to M, and pyramids be erected upon the triangles having the fame vertices with the cones, it may be demonftrated that each pyramid in the first cone has to each in the other, taking them in the fame order, the triplicate ratio of that which the fide AK has to the fide EM; that is, which AC has to EG: But as one antecedent to its confequent, fo are all the antecedents to all the confequents; therefore as the pyramid AKQL to the pyra- f 12. 5. mid EMSN, fo is the whole pyramid the bafe of which is the polygon DQATBYCV, and vertex L, to the whole pyramid of which the bafe is the polygon HSEOFPGR, and vertex N. Wherefore alfo the first of these two laft named pyramids has to the other the triplicate ratio of that which AC has to EG. But, by the hypothefis, the cone of which the base is the cir cle ABCD, and vertex L has to the folid X, the triplicate ratio of that which AC has to EG; therefore, as the cone of which Book XII. which the bafe is the circle ABCD, and vertex L, is to the folid X, fo is the pyramid the base of which is the polygon DQATBYCV, and vertex L, to the pyramid the base of which is the polygon HSEOFPGR and vertex N: But the faid cone is greater than the pyramid contained in it, therefore the folid X is greater than the pyramid, the bafe of which is the polygon HSEOFPGR, and vertex N; but it is alfo lefs; which is impoffible: Therefore the cone of which the base is the circle a 14. 5. ABCD, and vertex L, has not to any folid which is less than the the |