one

[by 1. 10.] there will be left fome fegments of the cone which will be less than the excefs of the cone E F G H N above the folid x. Let fuch be left, and let them be those defcribed upon EO, OF, FP, PG, GR, RH, HS, SE: Then the remaining pyramid whose base is the polygon EOFPG RHS, and vertex the point N, is greater than the folid x. Alfo in the circle A B C D defcribe the polygon ATBVCQDZ fimilar and alike fituate to the polygon EOFPGRHS; and upon it erect a pyramid, having the fame vertex as that of the cone; and let LBT be one of the triangles containing the pyramid, whose base is the polygon ATBVCQDZ, and vertex the point L, and NFO of the triangles containing the pyramid, whofe bafe is the polygon EOF PGR HS, and vertex the point N. And laftly join KT, Mo. Then because the cone A B C D L is fimilar to the cone E F G HN; [by 24. def. 11.] it will be as BD is to FH, fo is the axis K L to the axis MN. But as B D is to F H, fo is BK to F M: Wherefore as B K is to FM, fo is K L to MN. Therefore alternately as BK is to KL, fo is FM to MN: And the angles BKL, FMN are equal, as being right angles, and the fides about the equal angles BKL, FMN are proportional. Therefore [by 6. 6.] the triangle BKL is fimilar to the triangle F MN. Again, because as BK is to KT, fo is FM to м O, and they are about the equal angles B KT, F MO; for the angle BKT is the fame part of four right angles which are at the centre K, as the angle FMO is of the four right angles about the centre M: Therefore because the fides about the equal angles are proportional; the triangle BKT [by 6. 6.] will be fimilar to the triangle F MO. Again, because it has been proved that as B K is to KL, fo is F M to MN; but BK is equal to K T, and F M to MO; it will be as K T is to KL, fo is O M to MN, and the fides about the equal right angles T KL, OMN are proportional: Therefore the triangle AKT is fimilar to the triangle N MO: But because from the fimilarity of the triangles BK L, FMN, it is as A B is to B K, fo is NF to FM, and fince the triangles BKT, FMO are fimilar, as KB is to BT, fo is MF to Mo; therefore by equality of ratio [by 22. 5.] as A B is to BT, fo is NF to Fo. Again, because the triangles Ļ TK, NOм are fimilar, it is as LT is to T K, fo is NO to oм; and becaufe of the fimilar triangles K BT, OMF

as

H

Book XII, as KT is to TB, fo is MO to OF; it will be, by equality of ratio, as LT is to TB, fo is No to OF. But it has been proved that as TB is to BL, fo is oF to FN: Wherefore again, by equality of ratio, as TL is to L B, fo is on to NF: Therefore the fides of the triangles ATB, NOT are proportional; and fo [by 5. 6.] the triangles LT B, NOF are equiangular and fimilar to one another: Wherefore [by 9. def. 11.] the pyramid, whofe bafe is the triangle B TK, and vertex the point 1, is fimilar to the pyramid whose base is the triangle FMO, and vertex the point N: For they are contained under equal numbers of fimilar planes: But fimilar pyramids that have trianguJar bafes, are [by 8. 12.] in the triplicate ratio of their homologous fides: Therefore the pyramid BK TL to the pyramid FMON is in the triplicate ratio of BK to F M. În like manner drawing right lines from the points A, Zg D, Q, C, V, to K; and from E, S, H, R, G, P, to M, and erecting pyramids upon the triangles of the fame vertexes as the cones; we demonftrate that each of the pyramids of one cone is to every one of the other, in the triplicate ratio of the homologous fide BK to the homologous fide F M. that is, of BD to FH. But [by 12.5.] as one of the antecedents is to one of the confequents, fo are all the antecedents to all the confequents: Therefore as the pyramid BKTL is to the pramid FMON, fo is the whole pyramid, whofe bafe is the polygon ATBVCQDZ, and vertex the point L, to the whole pyramid, whofe bafe is the polygon EOFPGR HS, and vertex the point N: Wherefore the pyramid whose bafe is the polygon ATBVCODZ, and vertex the point L, is to the pyramid whose base is the polygon EOFPGRHS, and vertex the point N, in the triplicate ratio of BD to FH. But the cone whose bafe is the circle ABC D, and vertex the point L, is fuppofed to be to the folid x in the triplicate ratio of BD to FH: Therefore as the cone whofe bafe is the circle ABC D, and vertex the point L, is to the folid x, fo is the pyramid whose base is the polygon ATBVC Q DZ, and vertex the point L, to the pyramid whose base is the polygon EOFPG RH S, and vertex the point N. And alternately [by 16. 5.] as the cone whofe bafe is the circle ABCD, and vertex the point L, is to the pyramid.

in it, whose base is the polygon ATBVCQD, and vertex the point L, fo is the folid x to the pyramid whose bafe is the polygon EOFPG RHS, and vertex the point N. But the faid cone is greater than the pyramid in it; for it encompaffes it: Therefore the folid x is greater than the pyramid whofe bafe is the polygon EOFPGRHS and vertex N. But it is alfo lefs; which is impoffible: Therefore the cone whose base is the circle A B C D, and vertex the point L, is not to fome folid lefs than the cone, whose base is the circle EFGH, and vertex the point N, in the triplicate ratio of B D to F H. We demonftrate alfo, after the fame manner, that the cone EFGHN is not to fame folid lefs than the cone ABCDL in the triplicate ratio of FH to BD. I fay too that the cone ABCDH to fome folid greater than the cone EFGHN, is not in the triplicate ratio of BD to F H. For if poffible, let it be so to fome folid greater, as x. Then inversely, the folid x is to the cone ABCDL in the triplicate ratio of FH to B D. But as the folid x is to the cone ABCDL, fo is the cone EFGH N, to fome folid lefs than the cone ABCDL: Therefore the cone E F G HN is to fome folid lefs than the cone A B C DL, in the triplicate ratio of FH to BD; which has been demonstrated to be impoffible: Therefore the cone A B CDL is not to fome folid greater than the cone EFG HN, in the triplicate ratio of B D to FH. But it has been also proved not to be fo to a folid lefs: Therefore the cone ABCDL is to the cone EFGHN in the triplicate ratio of B D to FH. But as one cone is to the other, fo is one cylinder to the other; for a cylinder having the same base and altitude as a cone, is thrice the cone, fince it is proved [by 10. 12.] that every cone is one third part of a cylinder of the fame base and altitude. Therefore one cylinder is to the other in the triplicate ratio of BD to F H.

Wherefore fimilar cones and cylinders are to one another in the triplicate ratio of the diameters of their bases. Which was to be demonftrated.

f The demonftration of this theorem immediately follows from that of the eighth, by fuppofing cones to be pyramids having polygonous bafes of infinite numbers of fmall equal fides, and cylinders to be prifms having fuch polygonqus bafes.

PROP,

PROP. XIII. THEOR.

If a cylinder be cut [into two cylinders] by a plane parallel to the oppofite planes; it will be as one cylinder is to the other, fo is the axis of the one to the axis of the other.

For let the cylinder AD be cut by the plane GH [int two cylinders] parallel to the oppofite planes A B, CD, meeting the axis EF in the point K; I fay as the cylinder BG is to the cylinder GD, fo is the axis EK to the axis K F.

L

N

S

For continue out the axis E F both ways to the points L, M, and take any number of right lines EN, NL, each equal to the axis EK, and any number Fx, X M equal to KF: And thro' the points L, N, X, M, draw planes parallel to the planes AB, CD, and let the circles OP, R s, TV, qz, be conceived to be defcribed in the planes drawn thro' L, N, X, M, about the centres L, N, X, M, equal to A B, CD; and conceive the cylin ders PR, RB, DT, TZ to be conftituted. Then because the axes L N, NE, E K are equal to one another; the cylinders PR, BR, BG [by 11. P 12.] will be to one another as their bafes: But the bafes are equal; and therefore the cylinders PR, RB, BG are equal to one another. And because the axes LN, NE, EK are equal to one another, and alfo the cylinders PR, RB, BG equal to one B another; and the multitude of LN, NE, EK, is equal to the multitude of PR, RB, BG; the axis KL is the fame multiple of the axis E K, as the cylinder PG is of the cylinder G B. By the fame reafon the axis MK is the fame multiple of the axis K F, as the cylinder zo is of the cylinder GD. And if the axis KL be equal to the axis K M, the cylinder PG z will be equal to the cylinder GZ. If the axis KL be greater than the

R

E

K+

C

F

T

M

ΤΗ

axis KM, the cylinder PG will be greater than the cylinder Gz; and if lefs, lefs. Therefore there are four magnitudes, viz. the axes EK, KF, and the cylinders BG, GD, and there are taken equimultiples of the axis EK, and cylinder B G, viz. the axis K L, and the cylinder PG; and equimultiples of the axis KF, and cylinder G D, viz. the axis KM and the cylinder G z. And it has been also demonftrated, that if the axis K L exceeds the axis KM, the cylinder PG will exceed the cylinder Gz; if it be equal, equal; and if less, less: Therefore [by 5. def. 5.] as the axis EK is to the axis KF, so is the cylinder B G to the cylinder GD. Which was to be demonftrated.

PROP. XIV. THEOR.

Cones and cylinders ftanding upon equal bafes, are to one another as their altitudes.

For let the cylinders FD, E B ftand upon the equal bafes AB, CD: I say as the cylinder EB is to the cylinder F D, fo is the axis GH to the axis K L.

F(K

G

CKL D

团

For produce the axis KL to the point N; and make LN equal to the axis GH; and about the axis L N let us conceive the cylinder c M to ftand. Then because the cylinders E B, C M have the fame altitude; [by 11. 12.] they are to one another as their 3 bafes: But the bafes are equal: Therefore the cylinders EE, CM will be equal. But becaufe the cylinder FM is cut. by the plane CD parallel to the oppofite planes; the cylinder CM [by 13, 12.] will be to the cylinder, FD, as the axis LN is to the axis KL. But the cylinder GM is equal to the cylinder E B, and the axis LN to the axis GH: Therefore as the cylinder E B is to the cylinder FD, fo is the axis GH to the axis KL. But as› the cylinder EB is to the cylinder. FD, fo is the cone, ABG to the cone CDK; for [by 10. 12.] they are

ACH

N

B

M