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Book XII. PG, GR, RH, HS, SE: Therefore the remainder of the cone, m 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, fois

the polygon ATBYCVDQ to the polygon EOFPGRHS; and b 2. 12. as the square of AC to the square of EG, so is the circle ,

ABCD to the circle EFGH ; therefore the circle ABCD ise to cil. 5.

the circle EFGH, as the polygon ATBYCVDQ to the polye

a 1. 12.

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gon EOFPGRHS : But as the circle ABCD to the circle EFGH, fo 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 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, fo 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 the solid X is greater than the pyramid in the cone EN; But it is less, as was shown,

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#hich is absurd : Therefore the circle ABCD is not to the cir. Book XII. cle EFGH, as the conc 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 poflible, let it be fo to the solid I, which is greater than the cone EN : Therefore, by inversion, as the circle EFGH to the circle ABCD, fo 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 • than the cone a 14. 5. AL, because the solid I is greater than the cone EN: Therefore, as the circle EFGH is to the circle ABCD, fo is the cone EN to a solid less than the cone AL, which was shewn to be impofáble : 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 is to the circle EFGH, as the cone AL to the cone EN : But as the core is to the cone, so b is the cylinder to the cylinder, because the cy- b 15. s. linders are triple.. of the cones, each to each. Therefore, as c 1o. 12. the circle ABCD to the circle EFGH, so are the cylinders upon them of the fame altitude.

Wherefore cones and cylinders of the same altitude are to one another as their bases. Q: E. D.

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IMILAR cones and cylinders have to one another Sce N.

the triplicate ratio of that which the diameters of their bases have.

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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 fimilar : The cone of which the base is the circle ABCD, and verrex the point L, bas 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

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Book XII. or greater than the cone EFGHN. First, let it have it to a less, m viz. to the solid X :. Make the same construction as in the pre

ceding propofition, and it may be demonstrated the very fame way as in that propofition, that the pyramid of which the base is the polygon EOFPGRHS, and vertex N, is greater than the folid X. Describe also in the circle ABCD the polygon ATBYCVDQ 6milar to the polygon EOFPGRHS, upon which ere&t a pyramid having 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

N

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II.

polygon EOFPGRHS of which the vertex is N ; and join KQ,

MS: Because then the cone ABCDL is similar to the cone a 24. def. EFGHN, AC is * to EG, as the axis KL to the axis MN;

and as AC to EG, so bis AK to EM; therefore as AK to b 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; there

fore the sides about these equal angles being proportionals, © 6. 6. the triangle AKL is similar to the triangle EMN. Again, ber cause Ai is to KQ, as EM to MS, and that these lides are about equal angles AKQ, EMS, because thefe angles are, Book XII. each of them, the fame part of four right angles at the cen- m tres K, M; therefore the triangle AKO is similar to the tri. * 6. 6. angle EMS: And because it has heen shown 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 sides 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 AKO, EMS, as KA to AQ, fo ME to ES; ex aequalib, LA is b 22. 5. to AQ, as NE to ES. Again, because of the fimilarity of the triangles LQK, NSM, as LQ to QK, so NS to SM; and from the Gmilarity of the triangles KAQ, MES, as KQ_to QA, fo MS to SÉ ; 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 Gides about all their angles proportionals, are equiangular and similar to one an. c 5.6. other : And therefore the pyramid of which the base is the tri. angle AKQ, and vertex L, is similar to the pyramid the base of which is the triangle EMS, and vertex N, because their folid angles are equald to one another, and they are contained d B. 11. by the saine number of limilar planes : But similar pyramids which have triangular bases have to one another the triplicate e ratio of that which their hoinologous fides have ; therefore e 8. 12. the pyramid AKQI, has to the pyramid EMSN the triplicate Tatio 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, O 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 tide AK has to the fide EM; that is, which AC has to EG; But as one antecedent to its consequent, so are all the antecedents to all the consequents i ; therefore as the pyramid AKOL to the pyra- [ 12. s. 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 HSEUFPGR, and vertex N. Wherefore also the first of these two last named pyramids has to the olher the triplicate salio 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 folid X, the triplicate ratio of that which AC has to EG; therefore, as the cone of S 3:

about which

Book XII. 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 polygoa
DQATBYCV, and vertex L, to the pyramid the base of which
is the polygon HSEOFPGR and vertex N: But the said cone

greater than the pyramid contained in it, therefore the solid a 14. S. X is greater than the pyramid, the base of which is the poly

gon HSEOFPGR, and vertex N; but it is also less; which is
impossible : Therefore the cone of which the base is the circle

N

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ABCD, and vertex L, has not to any folid which is less than the
cone of which the base is the circle EFGH and vertes N, the tri-
plicate 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 ĄC. Nor can the conė 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 pol-
Gble, let it have it to a greater, viz. to the solid 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

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