Book XII. than the cone EFGHN. First, let it have it to a less, viz. to the solid ~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 bafe is the polygon EOFPGRHS, and vertex N is greater than the folid X. Defcribe also 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 II. L; and let NES be one of the triangles containing the pyramid upon the polygon EOFPGRHS of which the vertex is N; and join KQ, MS. because then the cone ABCDL is fimilar to the cone EFGHN, a. 24. Def. AC is a to EG, as the axis KL to the axis MN; and as AC to EG, fobis 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 these equal angles being proportionals, the triangle. AKL is fimilar to the triangle EMN, again, because AK is to KQ, as EM to MS, and that these fides b. 15. 5. C6.6. C are about equal angles AKQ, EMS, because these angles are, Book XII. each of them, the fame part of four right angles at the centers K, M; therefore the triangle AKQ_is fimilar to the triangle EMS. c. 6. 6. and because it has been fhewn 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 because 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 ES; ex aequali a, LA is to AQ, as NE to ES. again, because of the similarity 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, so MS to SE; ex aequali a, 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 equiangu e d. 22. 5. lar and fimilar to one another. and therefore the pyramid of which e. 5. 6. the base is the triangle AKQ, and vertex L, is fimilar to the pyramid the base of which is the triangle EMS, and vertex N, because their folid angles are equal f to one another, and they are contained f. B. 11. by the fame number of fimilar planes. but fimilar pyramids which have triangular bafes have to one another the triplicate & ratio of that 8. 8. 12. which their homologous fides have; therefore the pyramid AKQL has to the pyramid EMSN the triplicate ratio of that which AK has to EM. In the fame manner, if straight 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 h; h. 12. 5. therefore as the pyramid AKQL to the pyramid EMSN, fo is the whole pyramid the base 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 thefe two last 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 circle ABCD, and vertex L has to the folid X, the triplicate Book XII. ratio of that which AC has to EG; therefore as the cone of 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 bafe 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 base of which is the polygon HSEOFPGR, and vertex N. but it is also lefs; which is impoffible. therefore the cone of which the bafe is the circle ABCD, and vertex L has not to any i. 14. 5. i folid which is lefs than the cone of which the bafe is the circle EFGH and vertex N, the triplicate ratio of that which AC has to EG. In the fame manner it may be demonstrated that neither has the cone EFGHN to any folid which is less than the cone ABCDL, the triplicate ratio of that which EG has to AC. Nor can the cone ABCDL have to any folid which is greater than the cone EFGHN, the triplicate ratio of that which AC has to EG. for, if it be poffible, let it have it to a greater, viz. to the folid Z. therefore, inversely, the folid Z has to the cone ABCDL the triplicate ratio of that which EG has to AC. but as the folid Z is to the cone ABCDL, fo is the cone EFGHN to fome folid, which must be lefs than the cone Book XII. ABCDL, because the solid Z is greater than the cone EFGHN. i. 14. 5. therefore the cone EFGHN has to a solid which is less than the cone ABCDL, the triplicate ratio of that which EG has to AC, which was demonftrated to be impoffible. therefore the cone ABCDL has not to any folid greater than the cone EFGHN, the triplicate ratio of that which AC has to EG; and it was demonstrated that it could not have that ratio to any folid less than the cone EFGHN. therefore the cone ABCDL has to the cone EFGHN, the triplicate ratio of that which AC has to EG. but as the cone is to the cone, fo k k. 15. 5. the cylinder to the cylinder, for every cone is the third part of the cylinder upon the same base, and of the fame altitude. therefore also the cylinder has to the cylinder, the triplicate ratio of that which AC has to EG. Wherefore fimilar cones, &c. Q. E. D. IF [F a cylinder be cut by a plane parallel to its oppofite See N. planes, or bases; it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other. R A Let the cylinder AD be cut by the plane GH parallel to the oppofite planes AB, CD, meeting the axis EF in the point K, and let the line GH be the common fection of the plane GH and the furface of the cylinder AD. let AEFC be the parallelogram, in any position of it, by the revolution of which about the ftraight line EF the cylinder AD is described; and let GK be the common fection of the plane GH, and the plane AEFC. and because the parallel planes AB, GH are cut by the plane AEKG, AE, KG, their common fections with it, are parallela; wherefore AK is a parallelogram, T and GK equal to EA the ftraight line from the center of the circle AB. for V the fame reason, each of the ftraight lines G Book XII. drawn from the point K to the line GH may be proved to be equal to those which are drawn from the center of the circle AB to its circumference, and are therefore all equal to one anob.15 Def.1. ther. therefore the line GH is the circumference of a circle ↳ of which the center is the point K. therefore the plane GH divides the cylinder AD into the cylinders AH, GD; for they are the fame which would be described by the revolution of the parallelograms AK, GF about the ftraight lines EK, KF. and it is to be fhewn that the cylinder AH is to the cylinder HC, as the axis EK to the axis KF. L Produce the axis EF both ways; and take any number of ftraight lines EN, NL, each equal to EK; and any number FX, XM, each equal to FK; and let planes parallel to AB, CD pafs through the points L, N, X, M. therefore the common sections of these planes with the ⚫ cylinder produced are circles the centers of which are the points L, N, X, M, as was proved of the plane GH; and these planes cut off the cylinders PR, RB, DT, TQ. and because the axes LN, NE, EK are all equal, therefore c. II. 12. the cylinders PR, RB, BG are © to one another as their bafes. but their bases are equal, and therefore the cylinders PR, RB, BG are equal. and because R A N ន E B AAA HANG D Y M the axes LN, NE, EK are equal to one T |