ving the same vertex with the cone*, the pyramid inscribed Book XII, in the cone is half of the pyramid circumscribed about it, because they are to one another as their basesa : But the cone is a 6. 12. less than the circumscribed pyramid ; therefore the pyramid of which the base is the square EFGH, and its vertex the same with that of the cone, is greater than half of the cone: Divide the circumferences EF, FG, GH, HE, each into two equal parts in the points O, P, R, S, and join EO, OF, FP, PG, GR, RH, HS, SE: Therefore each of the triangles EOF, FPG, GRH, HSE is greater than half of the fegment of the circle in which it is : Upon each of these triangles erect a pyra. mid having the same vertex with the cone ; each of theie pya ramids is greater than the half of the segment of the cone in which it is : And thus dividing each of these circumferences in. to two equal parts, and from the points of division drawing Atraight lines to the extremities of the circumferences, and upon each of the triangles thus made erecting pyramids having the same vertes with the cone, and so on, there must at lenge semain fone fegments of the cone which are together leis b b Lemma. thin the folid Z: Let there be the segments upon EO, OF, FP, Vertex is put in place of altitude which is in the Greek, because the pyrapid, in what follows, is lupposed to be circumferibed about the cone, and to m ? have the same verlex. Aud the fame change is nada ia some places fol- PG, lowing Book XII. PG, GR, RH, HS, SE: Therefore the remainder of the cone, viz. the pyramid of which the base is the polygon ĘOFPGRHS, and its vertex the same with that of the cone, is greater than the folid X: In the circle ABCD describe the polygon ATBYCVDO_fimilar 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 EOFPGRHS ; and b 2. 12. as she square of AC to the. square of EG, fo is b the circle ABCD to the circle EFGH; therefore the circle ABCD is c to N, H S G M P R 1 gon EOFPGRHS: But as iħe circle ABCD to the circle EFGH, lo is the cone AL to the folid X; and as the polygon d 6. 12. ATBYCVDQ_to the polygon EOFPGRHS, fo is a che pyra mid of which the base is the first 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 folid X, 1o is the pyramid of which the base is the polygon ATBYCVOR, 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 solid X is gre ter e € 14. 5. than the pyramid in the cone EN: But it is lefs, as was thewn ; which is absurd : 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 same manner it may be demonstrated 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 tibe to the circle EFGH, as the cone AL to any folid greater 1 than the cone EN : For, if it be possible, let it be so to the folid I, which is greater than the cone EN: Therefore, by inverfion, O as the circle EFGH to the circle ABCD, fo is the solid I to is the cone AL : But as the folid I to the cone AL, so is the la cone EN to fome folid, which must be less a than the conca 14.si AL, because the folid 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 shewn 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 : Aad it bas been demonstrated that neither is the circle ABCD to the circle EFGH, as the conc AL to any solid less thao 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, fo b is the cylinder to the cylinder, becaufe the cy. b 15. si linders are triple c of the cones, each to each. Therefore, asc 10. ide the circle ABCD to the circle EFGH, so are the cylinders upon them of the fame altitude. Wherefore cones and cylinders of the fame altitude are to one another as their bases. 0. E. D. PRO P. 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, LG, and KL, MN the axes of the cones or cylinders, be timilar : The cone of which the base is the circle ABCD, and verrex the point L, has to the cone of which the base is the circle EFGH, and veries N, the criplicate 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 fall have the triplicate of that ratio to some folid which is less S 2 Book XI!. 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 propofition, and it may be demonstrated the very fame way as in that proposition, that the pyramid of which the base is the polygon EOFPGRHS, and veriex N, is greater than the folid X. Describe aiso in the circle ABCD the polygon ATBYCVDQ fimilar to the polygon EOFPGRHS, upon which ercet a pyramid having the fame vertex with the cone; and let LAQ_be one of the triangles containing the pyramid upon the polygın ATBYCVDQihe vertex of which is L ;' and let NES be one of the triangles containing the pyramid upon the N 11. b 15. 3. polygon EOFPGRHS of which the vertex is N; and join KQ, MS : Because then the cone ABCDL is fimilar to the cone a 14. def. EFGHN, AC is a to EG, as the axis KL to the axis MN; and as AC 10 EG, fo b is AK to EM; therefore as AK 10 EM, fo is KL to MN; and, alternately, AK to KL, as EM to MN: And the right angles AKL, ENN are equal; there. fore, the fides about these equal angles being proportionals , C 6. 6. the triangle AKL is similar c to the triangle EMN. Again, be. cause AK is to KQ, as EM to MS, and that these fides are about about equal angles AKQ , EMS, because these angles are, Book XII. each of them, the same part of four right angles ac che centers K, M; therefore the triangle AKQ is similar a to the tri:a 6. 6. angle EMS : And because it has been thewn that as AK 19 KL, so is EM 10 MN, and that AK is equal to KQ; and EM 10 MS, as OK to KL, fu is SM 10. MN; and therefore the sides about the right angles OKL, SMN being proportionals, the triangle LKQ_is fimilar to the triangle NMS: And because of elve fimilarity of the triangles AKL, EMN, as LA is to AK, fo is NE to EM ; and by the similarity of the triangles AKQ, EMS, as 'KA 10 AQ, so ME to ES ; ex æquaļi b, LA is b 22. 5. to AQ, as NE 10 ES. Again, because of the fimilarity of the triangles LOK, NSM, as LO to OK, 10 N5 to SM ; and from the similarity of the triangles K1Q, MES, as KQ_10 QA, fo MS to SE; ex æquali b, LQ_is to Q4, as Ng 10 SE : And it was proved that o 1 is to AL, as SE to EN; therefore, again, ex æquali , as QL IO LA, fo is SN o NE : Wherefore the triangles LQ4, NJE, having the tides about all their angles proporcionals, are equiangular c and similar to one an-c 5. 6. opher : And therefore the pyramid of which the base is the tri. angle AKQ, and vertex L, is fimilar to the pyramid the bare of which is the triangle EMS, and vertex N, because their folid angles are equal d' to one another, and they are contained a B. 11. by the same number of fimilar planes : But fimilar pyramids which have criangular bases have to one another the triplicate ratio of that which their homo'ngous fides have ; therefore e 8. 12. the pyramid AKOL has to the pyramid EMSN the triplicate ratio of that which AK has to FM. 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, Oto 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 triplicare ratio of that which the lide AK has to the Side EM ; that is, which AC has to IG : But as one antecedent to its confequent, so are all the antecedents to all the consequents f; therefore as the pyramid AKOL to the pyra• f 12. Sa mid EMSN, fo is the whole pyramid the base of which is the polygon DOATBYCV, and vertex L, to the whole pyramid of which the base is the polygon HSEOFPGR, and vertex N. Wherefore allo the first of these two last named pyramids has to the other the triplicare ratio of that which AC has to EG, S3 But, |