PROPOSITION XI. THE ORE M. Cones and Cylinders, of the fame Altitude, are to LET there be Cones and Cylinders of the fame For, if it be not fo, it shall be, as the Circle ABCD is to the Circle EFGH, fo is the Cone AL to fome Solid either lefs or greater than the Cone E N. 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 E N is equal to the Solids X and I. Let the Square E F G H be defcribed in the Circle E F G H, 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 thits ther as their Bafes; and the Cone is lefs than the circumfcribed Pyramid: Therefore the Pyramid, whose Bafe is the Square EFG H, 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, FRG, GSH, 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 thofe 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 * of this. +2 of this. as that of the Cone; and doing this continually, there will at last be left Segments of the Cone that will together be less than the Solid I. Let thofe be the Segments that are on H O, O E, EP, PF, FR, RG, GS, SH; therefore the Pyramid remaining, whofe Base is the Polygon HOEPFRGS, and Altitude the same 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 A C to the Square of E G, is * as the Polygon DTAYBQCV to the Polygon HOEP FRGS; and the Square of AC is + to the Square of E G, as the Circle ABCD is to the Circle EFGH; it fhall be as the Circle A B C D is to the Circle EFGH, fo is the Polygon DTAYBQCV to the Polygon HOEPFRGS. But as the Circle ABCD is to the Circle EFGH, fo is the Cone AL to the Solid X (by Hyp.): And as the Polygon DTAY BQCV 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 whofe Bafe is the Polygon HOEPFRGS, and Vertex the Point N. Therefore, as the Cone A L is to the Solid X, fo is the Pyramid whose Base is the Polygon D TAY BQC V, and Vertex the Point L, to the Pyramid whofe Bafe is the Polygon HOEPFRG S, and Vertex the Point N. But the Cone A L 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 ab furd. Therefore, the Circle A B C D to the Circle EFGH, is not as the Cone A L to fome Solid lefs than the Cone E N. In like manner it is demonftrated, that the Circle EFGH to the Circle ABCD, is not as the Cone E N to fome Solid lefs than the Cone AL: Ifay, 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 EN. 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 AL. But fince the Solid Z is greater than the t Cone Cone EN, it fhall be as the Solid Z is to the Cone AL, fo is the Cone E N to fome Solid lefs than the Cone A L; and therefore, as the Circle F F G H is to the Circle A B C D, fo is the Cone E N to fome Solid less than the Cone A L; which has been proved to be impoffible. Therefore the Solid A BCD to the Circle EF GH, is not as the Cone A L to fome Solid greater than the Cone E N. It has also been proved, that the Circle ABCD to the Circle EFGH is not as the Cone A L to fome Solid lefs than the Cone EN; therefore 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: But as Cone is to Cone, fo is Cylinder to Cylinder; for each Cylinder is triple of each Cone; and therefore, as the Circle A B C D is to the Circle EFG H, fo are Cylinders and Cones ftanding on them of the fame Altitude. Wherefore, Cones end Cylinders of the fame Altitude, are to one another as their Bafes; which was to be demonftrated. * PROPOSITION XII. THEOREM. Similar Cones and Cylinders are to one another in a triplicate Proportion of the Diameters of their Bafes. LET there be fimilar Cones and Cylinders, whole Bafes are the Circles A B C D, E F G H, and Diameters of the Bafes B D, F H, and Axes of the Cones or Cylinders K L, MN. I fay, the Cone, whofe Bafe is the Circle A B C D, 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 F H. * For, if the Cone A B CDL to the Cone EFGHN, 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 E F G H N. Firft, let it have that triplicate Proportion to the Solid X, lefs than the Cone EFGHN; and let the Square EFGH S 3 |