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Book XII. GR, RH, HS, SE. therefore the remainder of the cone, viz. the

C. I. 12.

d. 2. 12.

C. II. 5.

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 of the fame altitude with the cone AL. and becaufe as the fquare of AC is to the fquare of EG, fo is the polygon ATBYCVDQ to the polygon EOFPGRHS; and as the fquare 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

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the polygon EOFPGRHS. but as the circle AECD to the circle EFGH, fo is the cone AL to the folid X; and as the polygon a. 6. 12. ATBYCVDQ to the polygon EOFPGRHS, fo is the pyramid of which the bafe is the firft of thofe 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 Lto 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 f than the pyramid in the cone EN. but it is lefs, as was fhewn; which is abfurd. therefore the circle ABCD is

f. 14. 5.

not to the circle EFGH, as the cone AL to any folid which is lefs Book XII. than the cone EN. In the fame manner it may be demonstrated that the circle EFGH is not to the circle ABCD, as the cone EN to any folid lefs 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 inverfion, 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 lefs f than the cone AL, because the folid I is greater f. 14.5. than the cone EN. therefore as the circle EFGH is to the circle ABCD, fo is the cone EN to a folid lefs than the cone AL, which was fhewn to be impoffible. 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, fo

is the cylinder to the cylinder; because the cylinders are triple g. 15. 5. h of the cones, each of each. Therefore as the circle ABCD to h. 10. 12. 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 bafes.

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 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 bafe 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 or greater

Book XII. 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 bafe is the polygon EOFPGRHS, and vertex N is greater than the folid X. Defcribe alfo in the circle ABCD the polygon ATBYCVDQ_similar 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

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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 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, b. s. S. 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

c. 6. 6.

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, so 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 ES; ex aequalid, d. 22- 5. LA is 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, so MS to SE; ex aequali ‘, 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

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lar and fimilar to one another. and therefore the pyramid of which e. 5. 6. the bafe is the triangle AKQ, and vertex L, fimilar to the pyra

mid the bafe 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 g. 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 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 firft 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 b; h. 12. 5« therefore as the pyramid AKQL to the pyramid 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 firft 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 bafe 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 bafe 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 bafe of which is the polygon HSEOFPGR, and vertex N. but it is alfo lefs; which is impoffible. therefore the cone of which the bafe is the circle ABCD, and vertex L has not to any foN

i. 14. 5.

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lid 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 demonftrated that neither has the cone EFGHN to any folid which is lefs 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, fois

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