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*Cor. 7. of this.

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

Now let these Portions remaining be AE, EB, BF, FC, CG, GD, DH, HA. Then the Prism remaining, whofe Bafe is the Polygon AEBFCGDH, and Altitude equal to that of the Cylinders, is greater than the Triple of the Cone. But the Prim whofe Bafe is the Polygon AEBF CGDH, and Altitude the fame; as that of the Cylinder's is * triple of the Pyramid whofe Bafe is the Polygon AE BFCGDH, and Vertex the same 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 Circle ABCD; but it is leffer 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 leffer 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 ABCD. And let a Pyramid be erected on the Square ABCD having the fame Vertex as the Cone, then the Pyramid erected is greater than one half of the Cone; because, as has been already demonftrated, if a Square be defcribed about the Circle, the Square ABCD fhall be half thereof. And if folid Parallelepipedons be erected upon the Squares of the fame Altitude as the Cones, which are also called Prisms; then the Prism erected on the Square ABCD is one half of that erected on the Square described about the Circle, for they are to each other as their Bases; and fo likewise are their third Parts. Therefore the Pyramid whose Base is the Square ABCD, is one half of that Pyramid erected upon the Square defcribed about the Circle: But the Pyramid erected upon the Square described about the Circle, is greater than the Cone; for it comprehends it. Therefore the Pyramid whose Base is the Square ABCD, and Vertex the fame as that of the Cone, is greater than one half

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of the Cone. Bifect the Circumferences AB, BC, CD, DA, in the Points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA; and then each of the Triangles AEB, BFC, CGD, DHA, is greater than one half of each of the Segments they are in. Let Pyramids be erected upon each of the Triangles AEB, BFC, CGD, DHA, having the fame Vertex as the Cone: then each of these Pyramids thus erected, is greater than one half of the Segment of the Cone in which it is: And fo 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 shall at last have Segments of the Cone left, that will be less than the Excefs by which the Cone exceeds the one third Part of the Cylinder, Let these Segments be those that are on AE, EB, BF, FC, CG, GD, DH, HA, and then the remaining Pyramid whose Base is the Polygon AEBFCGDH, and Vertex the fame as that of the Cone, is greater than a third Part of the Cylinder; but the Pyramid whose Base is the Polygon AEBF CGDH, and Vertex the fame as that of the Cone, is one third Part of the Prism whose Base is the Polygon AEBFCGDH, and Altitude the fame as that of the Cylinder. Therefore the Prism, whofe 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 less 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 demonstrated.

PRO

PROPOSITION XI.

THEOREM.

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

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 AC, EG. I fay, as the Circle ABCD is to the Circle EF GH, fo is the Cone AL to the Cone EN.

For if it be not fo, it fhall be as the Circle A B C D is to the Circle EFGH, fo is the Cone AL to some Solid either lefs or greater than the Cone EN. First, 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, I; let the Square EFGH be defcribed 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 inscribed will be one half of the Pyramid circumfcribed, for they are * 6 of this. * to one another as their Bafes; and the Cone is lefs than the circumfcribed Pyramid. Therefore the Pyramid whofe Bafe is the Square EF GH, and Vertex the fame as that of the Cone, is greater than one half of the Cone. Bifect the Circumferences EF, 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, FRG, GHS, 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, GHS, of the fame Altitude as the Cone. Then each of those erected Pyramids is greater than the 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

*

the fame Altitude as that of the Cone; and doing this continually, there will at last be left Segments of the Cone that will together be lefs than the Solid I. Let those be the Segments that are on HO, OE, EP, PF, FR, RG, GS, SH. Therefore the Pyramid remaining, whofe Bafe is the Polygon HOEPFRGS, and Altitude the fame as that of the Cone, is greater than the Solid X. Let the Polygon DTAYBQCV be described 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 because the Square of AC to the Square of EG, is as the Polygon DTAY BQCV * 1 of this to the Polygon HOEPFRGS; and the Square of AC ist to the Square of EG, as the Circle ABCD †2 of this. to the Circle EFGH; it fhall be as the Circle ABCD to the Circle EFOH, fo is the Polygon DT AYBQCV to the Polygon HOEPFRGS: But as the Circle ABCD is to the Circle EF GH, fo is the Cone AL to the Solid X; and as the Polygon DTAYBQCV is to the Polygon HOEPFRGS, fo is the Pyramid whofe Bafe is the Polygon D TAYBQCV, and Vertex the Point L, to the Pyramid whose Bafe is the Polygon HOEPFRGS, and Vertex the Point N. Therefore as the Cone AL to the Solid X, fo the Pyramid whose Base is the Polygon DTAYBQCV, and Vertex the Point L, to the Pyramid whofe Bafe is the Polygon HOEPFRGS, 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 EF GH, 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 less than the Cone AL. I fay, moreover, that the Circle ABCD to the Circle EFGH, is not as the Cone AL to fome Solid greater than the Cone EN: For, if it be poffible, let it be to the Solid Z greater than the Cone; then, (by Inverfion) as the Circle EFGH is to the Circle ABCD, fo fhall the Solid Z be to the Cone AL. But fince the Solid Z is greater than the Cone EN, it shall be as the Solid S

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* 15.5.

Z is to the Cone AL, fo is the Cone EN to some Solid lefs than the Cone AL. And therefore as the Circle EFGH is to the Circle ABCD, fo is the Cone EN to fome Solid lefs than the Cone AL; which has been proved to be impoffible. Therefore the Circle ABCD to the Circle E F G H, is not as the Cone AL to fome Solid greater than the Cone EN. It has also been proved that the Circle ABCD to the Circle EF GH, is not as the Cone AL to fome Solid less than the Cone E N. Therefore as the Circle ABCD is to the Circle EFGH, fo is the Cone AL to the Cone EN: But as Cone is to Cone, so is* Cylinder to Cylinder, for each Cylinder is triple of each Cone; and therefore as the Circle ABCD is to the Circle EF GH, fo are Cylinders and Cones standing on them, of the fame Altitude. Wherefore, Cones and Cylinders of the fame Altitude, are to one another as their Bafes; which was to be demonstrated.

PROPOSITION XII.

THEOREM.

Similar Cones and Cylinders are to one another in a triplicate Proportion of the Diameters of their Bafes.

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ET there be fimilar Cones and Cylinders, whose Bafes are the Circles ABCD, EFGH, and Diameters of the Bafes BD, FH, and Axes of the Cones or Cylinders KL, MN. I fay, the Cone whose Bafe is the Circle ABCD, and Vertex the Point L, to the Cone whofe Bafe is the Circle EFGH, and Vertex the Point N, hath a triplicate Proportion of that which BD has to FH.

For if the Cone ABCDL to the Cone E F GHN, has not a triplicate Proportion of that which BD has to FH, the Cone ABCDL fhall have that triplicate Proportion to fome Solid, either lefs or greater than the Cone EFGHN. First, let it have that triplicate Proportion to the Solid X, lefs than the Cone EFGHN; and let the Square EFGH be defcribed in the Circle EFGH, which will be greater than one half of the Circle EF GH; and erect a Pyramid on

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