PROPOSITION IX. THEOREM. ing triangular Bases, are reciprocally propor- cally proportional, are equal. LET there be equal Pyramids, having the trian gular Bales A B C D E F, and Vertices the Points G, H I say, the Bases and Altitudes of the Pyramids A BCG, DEF,H, are reciprocally proportional ; that is, as the Base A B C is to the Base DEF, fo is the Altitude of the Pyramid DEFH to thie Altitude of the Pyramid A B C G. For, compleat the folid Parallelopipedon B GML, EHPO; chen, because the Pyramid A B C G is equal to the Pyramid DEFH; and the Solid BGML is sextuple the Pyramid A BCG; and the Solid EHPO sextuple the Pyramid D EFH; the Solid B GML shall be * equal to the Solid E HPO. But the Bases and Altitudes if equal folid Parallelopipedons are reciprocally proportional; therefore, as the Base B Misto the Bale EP, rois + the Altitude of the Solid EHPO to the Alitude of the Solid BGML. But as the Base B M is to the Base E P, fo is * the Triangle A BC to the Triangle DEF; therefore as the Triangle A BC is to the Triangle DEF, so is the Altitude of the SoJid E H P O to the Altitude of the Solid B G ML. But the Altitude of the Solid EHPO is the same as the Altitude of the Pyramid D.EFH; and the Altitude of the Solid BGML, the same as the Altitude of the Pyramid. ABCG; therefore, as the Base ABC is to the Base DEF, so is the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG: Wherefore the Bases and Altitudes of the equal Pyramids ABCG, DEFH, are reciprocally proportional. And if the Bases and Altitudes of the Pyramids ABCG, DEFH, are reciprocally propor. tional; that is, if the Base A B C to sbe Base DEF, + 34. 11. 15. 5. be as the Altitude of the Pyramid DEFH to the Al- PROPOSITION X. THE O'R E M. the Jame Bose, and an equal Altitude. ET a Cone have the same Base as the Cylinder ; viz, the Circle A B C D, and an Altitude equal to it. I say, the Cone is a third Part of the Cylinder ; that is, the Cylinder-is triple the Cone. For, * For, if the Cylinder be not triple to the Cone, it shall be greater or less than triple thereof. First, let it be greater than triple to the Cone, and let the Square ABCD be described in the Circle ABCD; then the Square ABCD is greater than one half of the Circle ABCD. Now let a Prism be erected upon a Square A B C D having the same Altitude as a Cylinder, and this Prism will be greater than one half of the Cylinder; because, if a Square be circumscribed about the Circle ABCD, the inscribed Square will be one half of the circumscribed Square; and if a Prism be erected upon the circumscribed Square of the fame Altitude as the 2. Cor. 7. Cylinder, fince Prisms are to one another as their of tbis. Bases, the Prism erected upon the Square A B C D, is one half of the Prism erected upon the Square described about the Circle, ABCD. But the Cylinder is less than the Prism erected on the Square described about the Circle ABCD; therefore rhe Prism erected on the Square ABCD, having the same Height as the Cym linder, is greater than one half of the Cylinder. Let the Circumferences A B, BT,CD,DA, be bifected in the Points E, F, G, H; and join AE, EB, BF, FC, CG, GD, DH, HA: Then each of the Tri+ This fol- angles A E B, BFC,CGD,DHA, is t greater than lotus from 2 the half of each of the Segments in which they stand. Let Prisions be erected from each of the Triangles AEB B F C, CGD, DHA, of the fame Altitude as the Cylinder ; then every one of these Prisms erected is greater than half its correspondent Segment of the Cy. linder. For, because, if Parallels be drawn thro' the Points E, F, G, H, to AB, BC,CD,DA, and Pa. rallelograms be compleated on the faid AB, BC, CD, DA, on which are erected folid Parallelopipedons of the same Altitude as the Cylinder; then each of those Prisms that are on the Triangles A E B, B FC, CGD, 32. 11. DHA, are Halves I of each of the solid Parallelepipe dons; and the Segments of the Cylinder are less than the erected folid Parallelopipedons; and consequently, the Prisms that are on the Triangles A E B, B FC, CGD, DHA, are greater than the Halves of the Segments of the Cylinder: And so, bisecting the other Circumferences, joining Right Lines, and on every one of the Triangles erecting Prisms of the_fame Heighe as fbe of Ibis, the Cylinder, and doing this continually, we shall at last havë certain Portions of the Cylinder lest, that are less than the Excess by which the Cylinder exceeds triple the Cone. Now, let these Portions remaining be A E, E B, BF, FC, CG, GD, DH, H A; then the Prism remaining, whose Base is the Polygon A EBFCG DH, änd Altitude equal to that of the Cylinder's is greater than the Triple of the Cone. But the Prism, whose Base is the Polygon A EBFCGDH, and Altitude the fame as that of the Cylinder's, is * triple of the Cor. 7. of tbis, Pyramid, whose Base is the Polygon A EBFCG DH, and Vertex the same as that of the Cone; and therefore the Pyramid, whose Base is the Polygon A EBFCGDH, and Vertex the same as that of the Cope, is greater than the Cone, whose Base is the Circle ABCD: But it is less also (for it is comprehended by it) which is abfurd; therefore the Cylinder is not greater than triple the Cone. I say, 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 shall be greater than a third Parc of the Cylinder: Let the Square A B C D be described in the Circle A BCD; then the Square A B C D is greater than half of the Circle A B CD: And let a Pyramid be erected on the Square A B C D, having the same Vertex as the Cone; then the Pyramld erected is greater than one half of the Cone ; because, as has been already demonstrated, if a Square be described about the Circle, the Square A B C D shall be half thereof: And if solid Parallelopipedons be erected upon the Squares of the fame Altitude as the Cone, which are allo 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 so likewise are their third Parts : Therefore the Pyramid, whose Base is the Square A B C D, is one half of that Pyramid erected upon the Square described 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 same as that of the Cone, 1 is greater than one half of the Cone. Bilect the Circumferences AB, BC, CD,D A, in the Points E, F, G, H; and join AE, EB, BF, FC, CG, GD, DH, HA ; and then each of the Triangles A E B, 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 A EB, 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 so, bifecting the remaining Circunferences, joining the Right Lines, and erecting Pyramids upon every of the Triangles having the same Altitude as the Cone, and doing this continually, we shall at last have Segments of the Cone left, that will be less than the Excess by which the Cone exceeds the one third Part of the Cyjinder : Ler these Segmenis be those that are on A E, EB, BF, FC, CG, GD, DH, HA; and then the remaining Pyramid, whose Base is the Polygon A EBFCGDH, and Vertex the same as that of the Cone, is greater than a third Part of the Cylinder: But the Pyramid, whose Base is the Polygon AEBFCGDH and Vertex the same as that of the Cone, is one third Part of the Prism whose Base is the Polygon A E B FCGDH, and Altitude the same as that of the Cylinder: Therefore the Prism, whose Base is the Polygon A EBFCGDH, and Altitude the same as that of the Cylinder, is greater than the is less allo (as being comprehended thereby); which is |