ving the fame vertex with the cone*, the pyramid infcribed Book XII, in the cone is half of the pyramid circumfcribed about it, because they are to one another as their bafesa: But the cone is a 6. 12. lefs than the circumfcribed pyramid; therefore the pyramid of which the bafe is the fquare EFGH, and its vertex the fame 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 pyramid having the fame vertex with the cone; each of thele py ramids is greater than the half of the fegment of the cone in which it is: And thus dividing each of these circumferences into two equal parts, and from the points of divifion drawing traight lines to the extremities of the circumferences, and upou each of the triangles thus made erecting pyramids having the fame vertex with the cone, and fo on, there muft at length remain fome fegments of the cone which are together lefs bb Lemma. thin the folid Z: Let thefe be the fegments upon EO, OF, FP,

Vertex is put in place of altitude which is in the Greek, because the pyra mid, in what follows, is fuppofed to be circumfcribed about the cone, and fo me have the fame vertex. And the fame change is made in fome places following.

Book XII, PG, GR, RH, HS, SE: Therefore the remainder of the cone, viz. the 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 having the fame vertex with the cone AL: And because as the fquare of AC is to the fquare of EG, fo a is the polygon ATBYCVDQ to the polygon EOFPGRHS; and as the fquare of AC to the fquare of EG, fo is b the circle ABCD to the circle EFGH; therefore the circle ABCD is to the circle EFGH, as the polygon ATBYCVDQ to the poly

a 1. 12.

b 2.

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6. 12.

€ 14. 5.

gon EOFFGRHS: But as the circle ABCD to the circle EFGH, fo is the cone AL to the folid X; and as the polygon ATBYCVDQ to the polygon EOFPGRHS, fo is d the pyra mid of which the bafe is the firft of thefe polygons, and ver tex 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 L, to the pyramid the bafe 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 e than the pyramid in the cone EN: But it is lefs, as was fhewn;

which is abfurd: Therefore the circle ABCD is not to the cir. Book XII. cle EFGH, as the cone AL to any folid which is lefs than the cone EN. In the fame manner it may be demonftrated 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 less than the cone a 14. §i AL, because the folid I is greater than the cone EN: Therefore, as the circle EFGH is to the circle ABCD, fo is the cone EN to a folid less 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 lefs 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-b is the cylinder to the cylinder, becaufe the cy⋅ b 15. s. linders are triple of the cones, each to each. Therefore, asc 10. 1* the circle ABCD to 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 bases 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 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 fhall have the triplicate of that ratio to fome folid which is lefs

Book XII or greater than the cone EFGHN. First, let it have it to a lefs, viz. to the folid X: Make the fame construction 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 fimilar 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 L; and let NES be one of the triangles containing the pyramid upon the

11.

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polygon EOFF GRHS 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 to EG, fo b is AK to EM; therefore as AK to EM, fo is KL to MN; and, alternately, AK to KL, as EM to MN: And the right angles AKL, EMN are equal; there fore, the fides about thefe equal angles being proportionals, the triangle AKL is fimilar to the triangle EMN. Again, be caufe AK is to KQ, as EM to MS, and that thefe fides are

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about

about equal angles AKQ, EMS, because thefe 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 a to the tria 6. 6. angle EMS: 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 OK to KL, fo 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 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 AKQ, EMS, as KA to AQ, fo ME to ES; ex æquali b, LA is b 22. 5. to AQ, as NE to ES. Again, becaufe of the fimilarity of the triangles LOK, NSM, as LQ to QK, fo NS to SM; and from the fimilarity of the triangles KAQ, MES, as KQ to QA, fo MS to SE; ex æquali b, LQ is to Q4, as No to SE: And it was proved that QA is to AL, as SE to EN; therefore, again, ex æquali, as QL to LA, fo is SN to NE : Wherefore the triangles LOA, NSE, having the tides about all their angles proportionals, are equiangular and fimilar to one an-e 5. 6. other: And therefore the pyramid of which the base is the triangle AKQ, and vertex L, is fimilar to the pyramid the bafe of which is the triangle EMS, and vertex N, because their folid angles are equal d to one another, and they are contained d B. 11. by the fame number of fimilar planes: But fimilar pyramids which have triangular bafes have to one another the triplicate

e ratio of that which their homologous fides 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 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 first 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 f; therefore as the pyramid AKOL to the pyraf 12. mid 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 allo the firft of thefe two laft named pyramids has to the other the triplicate ratio of that which AC has to EG,