Book XII. GR, RH, HS, SE. therefore the remainder of the cone, viz. the Mpyramid 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 of the fame altitude with the cone AL. and because as the square of AC is to 6. 1. 12. the square of EG, so is the polygon ATBYCVDQ to the polygon EOFPGRHS; and as the square of AC to the square of EG, so d. 2. 12. is d the circle A B C D to the circle EFGH; therefore the circe 6. 11. 5. ABCD ise to the circle EFGH, as the polygon ATBYCVDQ_10 L N H R 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 4. 6. 12. ATBYCVDQ_to the polygon EOFPGRHS, so is a the pyramid of which the base is the first of those 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 N. but the cone AL is greater than the pyramid contained in it; therefore f. 14. s. the solid X is greater f than the pyramid in the cone EN. but it is less, as was shown; which is absurd. therefore the circle ABCD is not not to the circle EFGH, as the cone AL to any solid which is less than Book XII. SIMILAR triplicate ratio of that which the diameters of their 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 Book XII. than the cone EFGHN. First, let it have it to a less, viz. to the folid X. make the same construction as in the preceeding Proposition, and N L; and let NES be one of the triangles containing the pyramid upon MS. because then the cone ABCDL is similar to the cone EFGHN, 2.24. Def. AC is a to EG, as the axis KL to the axis MN; and as AC to EG, so bis AK to EM; therefore as AK to EM, fo is KL to MN; and, b. 15. 5. alternately, AK to KL, as EM to MN. and the right angles AKL, EMN are equal; therefore, the sides about these equal angles being c. 6. 6. proportionals, the triangle AKL is similar to the triangle EMN. again, because AK is to KQ_, as EM to MS, and that these sides are are about equal angles AKQ, EMS, because these angles are, Book XII. Book XII. 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, lo HSEOFPGR and vertex N. but the said cone is greater than the i. 14.5. pyramid contained in it, therefore the solid X is greater i than the pyramid the base of which is the polygon HSEOFPGR, and vertex N lid which is less than the cone of which the base is the circle |