EFGH be defcribed in the Circle E F G H, which will be greater than one half of the Circle E FGH; and erect a Pyramid on the Square E F G H of the fame Altitude with the Cone, then that Pyramid is greater than one half of the Cone. And fo let the Circumferences EF, FG, GH, HE, be bifected in the Points O, P, R, S; and join E O, O F, FP, PG, GR, RH, HS, SE; then each of the Triangles EOF, FPG, GRH, HSE, is greater than one half of the Segment of the Circle EF GH, in which it is; and erect a Pyramid upon each of the Triangles EOF, FPG, GRH, HSE, having the fame Altitudes as the Cone: Then each of the Pyramids, thus erected, is greater than half its correfponding Segment of the Cone; wherefore, bifecting the remaining Circumferences, joining Right Lines, and erecting Pyramids upon each of the Triangles, having the fame Vertex as the Cone; and doing this continually, we fhall leave, at laft, certain Segments of the Cone, that fhall be less than the Excefs by which the Cone EF GHN exceeds the Solid X. Let thefe be the Segments that ftand on E O, O F, FP, PG, GR, RH, HS, SE; then the remaining Pyramid, whofe Bafe is the Polygon EOFPGRHS, and Vertex the Point N, is greater than the Solid X : Alfo, let the Polygon AT BY CVDQ be described in the Circle A B C D, fimilar and alike fituate to the Polygon EOFPGRHS; upon which erect a Pyramid, having the fame Altitude as the Cone; and let LBT be one of the Triangles containing the Pyramid, whofe Bafe is the Polygon ATBYCVDQ, and Vertex the Point L; as likewife NFO one of the Triangles containing the Pyramid EOFPGRHS, and Vertex the Point N; and let K T, M O, be joined: Then, because the Cone A B CDL is fimiTar to the Cone E F G HN, it fhall be, as B D is to FH, fo is the Axis KL to the Axis MN: But as BD is to FH, fo is * B K to F M; confequently, as BK is to FM, fo is K L to MN; and (by Alternation) as B K is to KL, fo is FM to MN. And fince each is perpendicular, and the Sides about the equal Angles BKL, F MN are proportional; the Triangle BKL fhall be + fimilar to the Triangle FMN. Again, because B K is to K T, as FM is

to

to MO, the Sides are proportional about the equal Angles BKT, F MO, (for the Angle BK T is the fame Part of the four Right Angles at the Centre K, as the Angle FMO is of the four Right Angles at the Centre M); therefore the Triangle BKT fhall be fimilar to the Triangle F M O. And because it 6. 6. has been proved, that BK is to KL, as FM is to MN; and BK is equal to K T; and FM to MO; it fhall be, as TK is to K L, fo is O M to MN; and the proportional Sides are about the equal Angles TKL, OMN; for they are Right Angles: Therefore the Triangle L K T thall be fimilar to the Triangle M N O. And fince by the Similarity of the Triangles BK L, F M N, it is, as L B is to B K, fo is NF to FM; and, by the Similarity of the Triangle BKT, FM O, it is, as KB is to B T, fo is MF to FO; it shall be (by Equality of Proportion), as L B is to BT, fo is NF to FO. Again, fince, by the Similarity of the Triangles LTK, NOM, it is, as L T is to TK, fo is NO to OM; and by the Similarity of the Triangles K BT, OM F, it is, as KT is to TB, fo is MO to OF; it fhall be (by Equality of Proportion), as LT is to TB, fo is NO to OF. But it has been proved, that T B is to B L, as OF is to FN; wherefore, again (by Equality of Proportion), as TL is to LB, is ON to NF; and therefore the Sides of the Triangle L T B, NO F, are proportional; and fo the Triangles LT B, NOF, are equiangular and fimilar to each other; and, confequently, the Pyramid, whofe Bale is the Triangle BK T, and Vertex the Point L, is fimilar to the Pyramid whofe Bafe is the Triangle F MO, and Vertex the Point N; for they are contained under fimilar Planes equal in Multitude: But fimilar Pyramids that have Triangular Bafes, are † to one another in the triplicate Proportion of their homologous Sides; therefore the Pyramid BKTL to the Pyramid FMON, has a triplicate Proportion of that which B K has to F M. In like manner, drawing Right Lines from the Points A, Q, D, V, C, Y, to K; as alfo others from rhe Points E, S, H, R, G, P, to M; and erecting Pyramids on the Triangles having the fame Vertices as the Cones, we demonftrate, that every Pyramid of one Cone to every one of the other

S 4

Cone,

† & of this.

$12.5,

Cone, has a triplicate Proportion of that which the Side B K has to the homologous Side M F, that is, which B D has to F H. But as one of the Antecedents is to one of the Confequents, fo are tall the Antecedents to all the Confequents. Therefore, as the Pyramid B KTL is to the Pyramid F MON, fo is the whole Pyramid, whofe Bafe is the Polygon AT BY CVDQ, and Vertex the Point L, to the whole Pyramid, whofe Bafe is the Polygon EOFPGRHS, and Vertex the Point N. Wherefore the Pyramid, whofe Bafe is the Polygon ATBYCVD Q, and Vertex the Point L, to the Pyramid whofe Bafe is the Polygon EOFPGRHS, and Vertex the Point N, has a triplicate Proportion of that which BD hath to FH. But the Cone whofe Bafe is the Circle A B C D, and Vertex the Point L, is fuppofed to have to the Solid X a triplicate Proportion of that which BD hath to FH; therefore, as the Cone, whofe Bafe is the Circle A B C D, and Vertex the Point L, is to the Solid X, fo is the Pyramid whofe Bafe is the Polygon AT BY CVDQ, and Vertex the Point L, to the Pyramid whofe Bafe is the Polygon E OF PGRHS, and Vertex the Point N. But the faid Cone is greater than the Pyramid that is in it, for it comprehends it; therefore the Solid X alfo is greater than the Pyramid, whofe Bafe is the Polygon EOFPGRHS, and Vertex the Point N; but it is alfo lefs, which is abfurd. Therefore the Cone, whofe Bafe is the Circle A B C D, and Vertex the Point L, to fome Solid less than the Cone, whofe Bafe is the Circle EFGH, and Vertex the Point N, has not a triplicate Proportion of that which PD has to FH. In like manner we demonftrate, that the Cone E F G H N, to fome Solid lefs than the Cone ABCDL, has not a triplicate Proportion of that which FH has to B D. Laftly, I fay, the Cone ABCDL, to a Solid greater than the Cone EFGHN, has not a triplicate Proportion of that which PD has to FH: For, if this be poffible, let it be fo to fome Solid Z greater than the Cone EFGHN; then (by Inverfion) the Solid Z, to the Cone ABCD L, has a triplicate Proportion of that which FH has to BD. But fince the Solid Z is greater than the Cone E F G H N, the Solid Z fhail

be

be to the Cone ABCD L, as the Cone E F GHN is to fome Solid lefs than the Cone A B C DL; and therefore the Cone E F G H N, to fome Solid less than the Cone A B C D L, bath a triplicate Propor tion of that which EA has to B D, which has been proved to be impoffible; therefore the Cone A B CDL, to fome Solid greater than the Cone EFGHN, has not a triplicate Proportion of that which B D has to FH. It has been alfo demonftrated, that the Cone A B C D L, to fome Solid lefs than the Cone EFGHN, hath not a triplicate Proportion of that which BD has to FH; wherefore the Cone ABCD L, to the Cone E F G HN, has a triplicate Proportion of that which BD has to FH. But as Cone is to Cone, fo is * Cylinder to Cylinder; # 15. 5. for a Cylinder having the fame Bafe as a Cone, and the fame Altitude, is + triple of the Cone; fince it is demonstrated that every Cone is one third Part of a Cylinder, having the tame Bale, and equal Altitude; Therefore, alío a Cylinder to a Cylinder has a triplicate Proportion of that which BD has to FH. Therefore, Amilar Cones and Cylinders are to one another in a triplicate Proportion of the Diameters of their Bafes; which was to be demonftrated.

PROPOSITION XIII.

THEOREM.

If a Cylinder be divided by a Plane parallel to the oppofite Planes; then, as one Cylinder is to the other Cylinder, fo is the Axis to the Axis.

L

ET the Cylinder AD be divided by the Plane GH, parallel to the oppofite Planes A B, CD, and meeting the Axis E Fin the Point K. I fay, as the Cylinder BG is to the Cylinder G D, fo is the Axis EK to the Axis K F.

For, let the Axis EF be both Ways produced to L and M, and put any Number of Lines EN, NL, &c. each equal to the Axis E K; and any Number of Lines F X, XM, &c. each equal to FK; and thro' the Poipts L, N, X, M, let Planes parallel to

A B,

+ 10 of this.

17

孟

A B, CD, pafs; and in those Panes from L, N, X, M, as Centres, defcribe the Circles OP, RS, TY, V Q, each equal to A B, CD; and conceive the Cylinders PR, RB, DT, TQ, to be compleated: Then, because the Axis L N, NE, E K, are equal to each other, the Cylinders P R, RB, 11 of this. B G, will be * to one another as their Bafes; and therefore the Cylinders PR, RB, BG, are equal: And fince the Axis LN, NE, E K, are equal to each other; as alfo the Cylinders PR, RB, BG; and the Number of Lines L N, NE, E K, is equal to the Number of Cylinders PR, R B, B G ; the Axis K L fhall be the fame Multiple' of the Axis EK, as the Cylinder PG is of the Cylinder GB. For the fame Reason, the Axis M K is the fame Multiple of the Axis K F, as the Cylinder G Q is of the Cylinder G D. Now, if the Axis K L be equal to the Axis K M, the Cylinder P G fhall be equal to the Cylinder GQ; if the Axis K L be greater than the Axis K M, the Cylinder P G fhall be likewife greater than the Cylinder G Q; and if lefs, lefs: Therefore, because there are four Magnitudes, viz. the Axis E K, KF, and the Cylinders BG, GD; and there are taken their Equimultiples, namely, the Axis K L, and the Cylinder P G, the Equimultiples of the Axis EK, and the Cylinder BG; and the Axis KM, and the Cylinder GQ, the Equimultiples of the Axis KF, and the Cylinder GD: And it is demonftrated, that if the Axis K L exceeds the Axis K M, the Cylinder PG will exceed the Cylinder GQ; and, if it be equal, equal; and if leis, lefs. Therefore, as the Def. 5. 5. Axis EK is to the Axis K F, fot is the Cylinder BG to the Cylinder GD. Wherefore, if a Cylinder be divided by a Plane parallel to the oppofite Planes; then, as one Cylinder is to the other Cylinder, fo is the Axis to the Axis; which was to be demonftrated.

PRO