remain some segments of the cone which are together less Book XII. than the solid Z: Let these be the segments upon EO, OF, FP, PG, GR, RH, HS, SE: Therefore the remainder of the b Lem. l. cone, viz. the pyramid of which the base is the polygon EOFPGRHS, and its vertex the same with that of the cone, is greater than the solid X: In the circle ABCD describe the polygon ATBYCVDQ, similar to the polygon EOFPGRHS, and upon it erect a pyramid having the same vertex with the cone AL: and because as the square of AC is to the square of EG, so a is the polygon ATBYCVDQ to the polygon a 1. 12. EOFPGRHS; and as the square of AC to the square of EG, so is b the circle ABCD to the circle EFGH; therefore the b 2. 12. circle ABCD is to the circle EFGH, as the polygon c 11. 5. ATBYCVDQ to the polygon EOFPGRHS: but as the circle ABCD to the circle EFGH, so is the cone AL to the solid X; and as the polygon ATBYCVDQ to the polygon EOFPGRHS, so is d the pyramid of which the base is the first d 6. 12. of these polygons and vertex L, to the pyramid of which the base is the other polygon, and its vertex N: Therefore, as the cone AL to the solid X, so is the pyramid of which the base is the polygon ATBYCVDQ, and vertex L, to the pyramid the base of which is the polygon EOFPGRHS, and vertex Book X11. N: But the cone AL is greater than the pyramid contained in it; therefore the solid X is greater than the pyramid in e 14. 5. the cone EN. But it is less, as was shown, which is absurd : Therefore the circle ABCD is not to the circle EFGH, as the cone AL to any solid which is less than the cone EN. In the same manner, it may be demonstrated, that the circle EFGH is not to the circle ABCD, as the cone EN to any solid less than the cone AL. Nor can the circle ABCD be to the circle EFGH, as the cone AL to any solid greater than the cone EN: For, if it be possible, let it be so to the solid I, which is greater than the cone EN: Therefore, by inversion, as the circle EFGH to the circle ABCD, so is the solid I to the cone AL: But as the solid I to the cone AL, so is the cone EN to some solid which must be less a than the cone AL, because the solid 1 is greater than the cone EN: Therefore, as the circle EFGH is to the circle ABCD, so is the cone EN to a solid less than the cone AL, which was shown to be impossible; Therefore the circle ABCD is not to the circle EFGH, as the cone AL is to any solid greater than the cone EN: And it has been demonstrated, that neither is the circle ABCD to the circle EFGH, as the cone AL to any solid less than the cone EN: Therefore the circle ABCD a 14. 5. cy is to the circle EFGH, as the cone AL to the cone EN: Book XII. But as the cone is to the cone, sob is the cylinder to the linder, because the cylinders are triple the cones, each to b 15. 5. each. Therefore, as the circle ABCD to the circle EFGH, . 10. 12. so are the cylinders upon them of the same altitude. Wherefore, cones and cylinders of the same allitude are to one another as their bases.” Q. E. D. PROP. XII. THEOR. Similar cones and cylinders have to one another See N. the triplicate ratio of that which the diameters of their bases have. Let the cones and cylinders of which the bases are the circles ABCD, EFGH, and the diameters of the bases AC, EG, and KL, MN, the axes of the cones or cylinders, be similar: The cone, of which the base 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 shall have the triplicate of that ratio to some solid which is less or greater than the cone EFGHN. First, Let it have it to a less, viz. to the solid X. Make the same construction as in the preceding proposition, and it may be demonstrated, in the very same way, as in that proposition, that the pyramid of which the base is the polygon EOFPGRHS, and vertex N, is greater than the solid X. Describe also in the circle ABCD the polygon ATBYCVDQ similar to the polygon EOFPGRHS, upon which erect a pyramid baving the same 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 polygon EOFPGRHS of which the vertex is N; and join KQ, MS: Because then the cone ABCDL is similar to the cone EFGHN, AC is a to EG, as the axis KL to the axis MN;a 24.def. 11. and as AC to EG, sob is AK to EM; therefore as AK to v 15. 5. EM, so is KL to MN; and, alternately, AK to KL, as EM to MN: And the right angles AKL, EMN are equal; therefore the sides about these equal angles being proportionals, the triangle AKL is similar to the triangle EMN. Again, c 6. 6. Book XII. since AK is to KQ, as EM to MS, and since these sides are N S R each of them, the same part of four right angles at the centres a 6. 6. K, M; therefore the triangle AKQ is similar a to the triangle EMS: And because it has been shown that as AK to KL, so is EM to MN, and AK is equal to KQ; and EM to MS, as QK to KL, so is SM to MN; and therefore the sides about the right angles QKL, SMN being proportionals, the triangle LKQ is similar to the triangle NMS: And because of the similarity of the triangles AKL, EMN, as LA is to AK, so is NE to EM; and by the similarity of the triangles AKQ, b 22. 5. EMS, as K A to AQ, so ME to ES; ex æquali, LA is to AQ, as NE to ES. Again, because of the similarity of the triangles LKQ, NSM, as LQ to QK, so NS to SM; and from the similarity of the triangles KAQ, MES, as KQ to QA, so MS to SE; ex æquali , LQ is to QA, as NS to SE: And it was proved that QA is to AL, as SE to EN; therefore, again, ex æquali, as QL to LA, so is SN to NE: Where fore the triangles LQA, NSE, having the sides about all their c 5. 6. angles proportionals, are equiangular", and similar to one another: and therefore the pyramid, of which the base is the tri- Book XII. angle AKQ, and vertex L, is similar to the pyramid the base of which is the triangle EMS, and vertex N, because their solid angles are equal to one another, and they are contained d B. 11. by the same number of similar planes: But similar pyramids, which have triangular bases, have to one another the triplicate e ratio of that which their homologous sides have; therefore e 8. 12. the pyramid AKQL has to the pyramid EMSN the triplicate ratio of that which AK has to EM. In the same 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, 0, to M, and pyramids be erected upon the triangles, having the same vertices with the cones, it may be demonstrated, that each pyramid in the first cone has to each in the other, taking them in the same order, the triplicate ratio of that which the side AK has to the side EM; that is, which AC has to EG: But as one antecedent to its consequent, so are all the antecedents to all the consequents f; therefore as the pyramid AKQL to the pyra- f 12. 5. mid EMSN, so is the whole pyramid, the base of which is the polygon DQATBYCV, and vertex L, to the whole pyramid of which the base is the polygon HSEOFPGR, and vertex N. Wherefore also the first of these two last named pyramids has to the other the triplicate ratio of that which AC has to EG. But, by the hypothesis, the cone of which the base is the circle ABCD, and vertex L, has to the solid X, the triplicate ratio of that which AC has to EG; therefore, as the cone of which the base is the circle ABCD, and vertex L, is to the solid X, so 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 said cone is greater than the pyramid contained in it; therefore the solid X is greater a than the pyramid, the base of which is the poly- a 14. 5. gon #SEOFPGR, and vertex N; but it is also less, which is impossible: Therefore the cone, of which the base is the circle ABCD, and vertex L, has not to any solid which is less than the cone of which the base is the circle EFGH, and vertex N, the triplicate ratio of that which AC has to EG. In the same manner, it may be demonstrated, that neither has the cone EFGHN, to any solid 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 solid which is greater than the cone EFGHN, the triplicate ratio of that which AC bas to EG: For, if it be possible, let it have it to a greater, viz. to the solid Z: Therefore, inversely, the solid Z has to the cone ABCDL, the triplicate ratio of that which EG has to AC: |