*2 Cor. of this.

For, if the Cylinder be not triple to the Cone, it fhall be greater or lefs than triple thereof. First, let it be greater than triple to the Cone, and let the Square ABCD be defcribed in the Circle A BCD; then the Square A B CD, is greater than one half of the Circle ABCD. Now let a Prifm be erected upon the Square ABCD, having the fame Altitude as the Cylinder, and this Prifm will be greater than one half of the Cylinder; becaufe, if a Square be circumfcribed about the Circle ABCD, the infcribed Square will be one half of the circumfcribed Square; and if a Prifm be erected upon the circumfcribed Square of the fame Altitude as the 7. Cylinder, fince Prifms are to * one another as their Bafes, the Prifm erected upon the Square A B CD, is one half of the Prifm erected upon the Square defcribed about the Circle A B C D. But the Cylinder is less than the Prifm erected on the Square defcribed about the Circle ABCD; therefore the Prifm erected on the Square ABCD, having the fame Height as the Cylinder, is greater than one half of the Cylinder. Let the Circumferences A B, BC, CD, DA, be bifected in the Points E, F, G, H; and join A E, E B, BF, FC, CG, GD, DH, HA: Then each of the TriThis fol- angles A EB, B FC, CG D, DH A, is † greater than lows from 2 the half of each of the Segments in which they stand. of this.

Let Prisms be erected from each of the Triangles A E B, BFC, CGD, DHA, of the fame Altitude as the Cylinder; then every one of these Prisms erected is greater than half its correfpondent Segment of the Cylinder. For, because, if Parallels be drawn thro' the Points E, F, G, H, to A B, BC, CD, DA, and Parallelograms be compleated on the faid A B, BC, CD, DA, on which are erected folid Parallelepipedons of the fame Altitude as the Cylinder; then each of thofe Prifms that are on the Triangles A E B, BFC, CGD, † 32. 11. DHA, are Halves ‡ of each of the folid Parallelepipedons; and the Segments of the Cylinder are lefs than the erected folid Parallelepipedons; and, confequently, the Prisms that are on the Triangles A E B, BFC, CGD, DHA, are greater than the Halves of the Segments of the Cylinder: And fo, bifecting the other Circumferences, joining Right Lines, and on every one of the Triangles erecting Prifms of the fame Height as

the

the Cylinder, and doing this continually, we fhall at laft have certain Portions of the Cylinder left, that are lefs than the Excess by which the Cylinder exceeds triple the Cone.

*

Now, let thefe Portions remaining be AE, EB, BF, FC, CG, GD, DH, HA; then the Prifm remaining, whose Base is the Polygon AEBFCGDH, and Altitude equal to that of the Cylinder's is greater than the Triple of the Cone. But the Prifm, whofe Bafe is the Polygon AEBFCGDH, and Altitude the fame as that of the Cylinder's, is triple of the *1 Cor. 7. Pyramid, whose Bafe is the Polygon AEBFCGDH, of this. and Vertex the fame as that of the Cone; and therefore the Pyramid, whofe Bafe is the Polygon AEBFCGDH, and Vertex the fame as that of the Cone, is greater than the Cone, whofe Bafe is the the Circle ABCD: But it is lefs alfo (for it is comprehended by it) which is abfurd; therefore the Cylinder is not greater than triple the Cone. I fay, it is neither less than triple the Cone: For, if it be poffible, let the Cylinder be less than triple the Cone; then (by Inverfion) the Cone fhall be greater than a third Part of the Cylinder: Let the Square ABCD be defcribed in the Circle ABCD; then the Square ABCD is greater than half of the Circle A BCD: And let a Pyramid be erected on the Square A B CD, having the fame Vertex as the Cone; then the Pyramid erected is greater than one half of the Cone; becaufe, as has been already demonftrated, if a Square be defcribed about the Circle, the Square ABCD shall be half thereof: And if folid Parallelepipedons, be erected upon the Squares of the fame Altitude as the Cone, which are alfo Prifms; then the Prifm erected on the Square ABCD is one half of that erected on the Square defcribed about the Circle; for they are to each other as their Bases, and fo likewife are their third Parts: Therefore the Pyramid, whofe Bafe is the Square ABCD, is one half of that Pyramid erected upon the Square described about the Circle. But the Pyramid erected upon the Square defcribed about the Circle is greater than the Cone, for it comprehends it; therefore the Pyramid, whofe Bafe is the Square ABCD, and Vertex the fame as that of the Cone,

is greater than one half of the Cone, Bifect the Circumferences A B, BC, CD, DA, in the Points E, F, G, H; and join A E, EB, BF, FC, CG, G Ð, DH, HA; and then each of the Triangles A E B, B FC, CGD, DHA, is greater than one half of each of the Segments they are in. Let Pyramids be erected upon each of the Triangles A E B, B FC, CG D, D HA, having the fame Vertex as the Cone; then each of thefe Pyramids, thus erected, is greater than one half of the Segment of the Cone in which it is; and so, bifecting the remaining Circumferences, joining the Right Lines, and erecting Pyramids upon every of the Triangles having the fame Altitude as the Cone, and doing this continually, we fhall at laft have Segments of the Cone left, that will be less than the Excess by which the Cone exceeds the one third Part of the Cylinder: Let thefe Segments be those that are on A Ë, EB, BF, FC, CG, GD, DH, HA; and then the remaining Pyramid, whofe Base is the Polygon AEBFCGD H, and Vertex the fame as that of the Cone, is greater than a third Part of the Cylinder: But the Pyramid, whofe Bafe is the Polygon AEBFCGD H, and Vertex the fame as that of the Cone, is one third Part of the Prifm whose Base is the Polygon A E B F C G D H, and Altitude the fame as that of the Cylinder: Therefore the Prifm, whose Bafe is the Polygon AEBFCGDH, and Altitude the fame as that of the Cylinder, is greater than the Cylinder, whofe Bafe is the Circle ABCD; but it is lefs alfo (as being comprehended thereby); which is abfurd; therefore the Cylinder is not lefs than triple of the Cone: But it has been proved alfo not to be greater than triple of the Cone; therefore the Cylinder is neceffarily triple of the Cone. Wherefore, every Cone is a third Part of a Cylinder, having the fame Bafe, and an equal Altitude; which was to be demonftrated.

PRO.

PROPOSITION XI.

THE ORE M.

Cones and Cylinders, of the fame Altitude, are to
one another as their Bases.

LET there be Cones and Cylinders of the fame Al-
titude, whofe Bafes are the Circles ABCD,
EFGH, Axes KL, MN, and Diameters of the
Bases A C, EG. I fay, as the Circle A B C D is to the
Circle E F G H, fo is the Cone A L to the Cone EN.

For, if it be not fo, it fhall be, as the Circle A B C D is to the Circle EFG H, fo is the Cone A L to fome Solid either lefs or greater than the Cone EN. Firft, let it be to the Solid X lefs than the Cone; and let the Solid I be equal to the Excefs of the Cone EN above the Solid X: Then the Cone EN is equal to the Solids X and I. Let the Square E F G H be described in the Circle EFGH, which Square is greater than one half of the Circle, and erect a Pyramid upon the Square EFGH, of the fame Altitude as the Cone; therefore the Pyramid erected is greater than one half of the Cone. For if we defcribe a Square about the Circle, and a Pyramid be erected thereon, of the fame Altitude as the Cone; the Pyramid infcribed will be one half of the Pyramid circumfcribed; for they are* to one ano- * 6 of this ther as their Bafes; and the Cone is lefs than the circumfcribed Pyramid: Therefore the Pyramid, whofe Bafe is the Square EFGH, and Vertex the fame as that of the Cone, is greater than one half of the Cone. Bifect the Circumferences E F, FG, GH, HE, in the Points P, R, S, O; and join HO, OE, EP, PF, FR, RG, GS, SH; then each of the Triangles HOE, EPF, FR G, GS H, is greater than one half of the Segment of the Circle wherein it is. Let a Pyramid be raised upon every one of the Triangles HOE, EPF, FRG, GSH, of the fame Altitude as the Cone; then each of those erected Pyramids is greater than one half of its correfpondent Segment of the Cone: And fo bifecting the remaining Circumferences, joining the Right Lines, and erecting Pyramids upon each of the Triangles, of the fame Altitude

I

+2 of this.

as that of the Cone; and doing this continually, there will at laft be left Segments of the Cone that will together be less than the Solid I. Let those be the Segments that are on HO, OE, EP, PF, FR, RG, GS, SH; therefore the Pyramid remaining, whose Base is the Polygon HOEPFRGS, and Altitude the fame as that of the Cone, is greater than the Solid X. Let the Polygon DTAYBQCV be defcribed in the Circle ABCD, fimilar and alike fituate to the Polygon HOEPFRGS; and let a Pyramid be erected thereon of the fame Altitude as the Cone AL: Then, * 1 of this. because the Square of A C to the Square of EG, is * as the Polygon DTAYBQCV to the Polygon HOEPFRGS; and the Square of AC is + to the Square of EG, as the Circle ABCD is to the Circle EFGH; it fhall be, as the Circle ABCD is to the Circle EFGH, fo is the Polygon DTAYBQCV to the Polygon HOEPFRGS. But as the Circle ABCD is to the Circle E F GH, fo is the Cone AL to the Solid X (by Hyp.): And as the Polygon DTAYBQCV is to the Polygon HOEPFRGS, 16 of this. fo is the Pyramid, whofe Bafe is the_Polygon DTAYBQC V, and Vertex the Point L, to the Pyramid whose Base is the Polygon HOEP FRGS, and Vertex the Point N. Therefore, as the Cone AL is to the Solid X, fo is the Pyramid, whofe Base is the Polygon DTAYBQCV, and Vertex the Point L, to the Pyramid whofe Bafe is the Polygon HOEPFRG S, and Vertex the Point N. But the Cone AL is greater than the Pyramid that is in it; therefore the Solid X is greater than the Pyramid that is in the Cone EN; but it was put lefs, which is abfurd. Therefore the Circle ABCD to the Circle EFGH, is not as the Cone AL to fome Solid lefs than the Cone EN. In like manner it is demonftrated, that the Circle EFGH to the Circle ABCD, is not as the Cone EN to fome Solid lefs than the Cone AL: I say, moreover, that the Circle A B C D to the Circle EFGH, is not as the Cone AL to fome Solid greater than the Cone E N. For, if it be poffible, let it be to the Solid Z greater than the Cone; then (by Inverfion), as the Circle E F G H is to the Circle ABCD, fo fhall the Solid Z be to the Cone A L. But fince the Solid Z is greater than the

Cone