† 34. 11. 15. 5. PROPOSITION IX. The Bafes and Altitudes of equal Pyramids, hav- LET there be equal Pyramids, having the trian gular Bales AB C, D E F, and Vertices the Points G, H. I fay, the Bafes and Altitudes of the Pyramids A BCG, DEFH, are reciprocally proportional; that is, as the Bafe A B C is to the Bafe DEF, fo is the Altitude of the Pyramid DEFH to the Altitude of the Pyramid A B C G. * For, compleat the folid Parallelopipedon B G M L, EHPO; then, because the Pyramid A B C G is equal to the Pyramid DEFH; and the Solid BG ML is fextuple the Pyramid A BCG; and the Solid EHPO fextuple the Pyramid DEFH; the Solid BG ML fhall be equal to the Solid E HPO. But the Bafes and Altitudes if equal folid Parallelopipedons are reciprocally proportional; therefore, as the Bafe B M is to the Bafe E P, fo is + the Altitude of the Solid EHPO to the Altitude of the Solid BGML. But as the Base BM is to the Base E P, fo is the Triangle ABC to the Triangle D EF; therefore as the Triangle ABC is to the Triangle DEF, fo is the Altitude of the Solid EHPO to the Altitude of the Solid B G M L. But the Altitude of the Solid EH PO is the fame as the Altitude of the Pyramid DEFH; and the Altitude of the Solid BG ML, the fame as the Altitude of the Pyramid ABCG; therefore, as the Base ABC is to the Bafe DEF, fo is the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG: Wherefore the Bafes and Altitudes of the equal Pyramids ABCG, DEF H, are reciprocally proportional. And if the Bafes and Altitudes of the Pyramids A BCG, DEFH, are reciprocally propor tional; that is, if the Bafe A B C to the Bafe DEF, ba be as the Altitude of the Pyramid DEFH to the Altitude of the Pyramid A B C G; I fay, the Pyramid A BCG is equal to the Pyramid DEFH: For, the fame Conftruction remaining, because the Base ABC to the Base DEF, is as the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG; and as the Base ABC is to the Base DEF, so is the Parallelogram BM to the Parallelogram EP; therefore the Parallelogram B M to the Parallelogram EP, shall be alfo as the Altitude of the Pyramid D E F H is to the Altitude of the Pyramid A B C G. But as the Altitude of the Pyramid DEFH is the fame as the Altitude of the folid Parallelopipedon EPHO, and the Altitude of the Pyramid ABCG, the fame as the Altitude of the folid Parallelopipedon BGML; therefore the Base BM to the Bafe E P, will be as the Altitude of the folid Parallelopipedon EH PO to the Altitude of the folid Parallelopipedon BG ML. But thofe folid Parallelopipedons, whofe Bales and Altitudes are reciprocally proportional, are † equal to each other; therefore the folid† 34. 18. Parallelopipedon BG ML, is equal to the folid Parallelopipedon EHPO: Now the Pyramid A B ̊C G is a fixth Part of the Solid BGML; and, in like manner, the Pyramid DEFH is a fixth Part of the Solid EHPO; therefore the Pyramid ABCG is equal to the Pyramid DEFH. Wherefore, the Bafes and Altitudes of equal Pyramids, having triangular Bafes, are reciprocally proportional; and thofe Pyramids having triangular Bafes, whofe Bafes and Altitudes are reciprocally proportional, are equal; which was to be demonftrated. PROPOSITION X. THEOREM. Every Cone is a third Part of a Cylinder, having the Jame Bafe, and an equal Altitude. L ET a Cone have the fame Bafe as the Cylinder; viz. the Circle A B C D, and an Altitude equal to it. I fay, the Cone is a third Part of the Cylinder; that is, the Cylinder is triple the Cone. For, 2. of this. * For, if the Cylinder be not triple to the Cone, it fhall be greater or lefs than triple thereof. Firft, let it be greater than triple to the Cone, and let the Square ABCD be defcribed in the Circle AB CD; then the Square A BCD is greater than one half of the Circle ABCD. Now let a Prifm be erected upon a Square ABCD having the fame Altitude as a 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 Prism be erected upon the circumfcribed Square of the fame Altitude as the Cor. 7. Cylinder, fince Prisms are to one another as their Bafes, the Prifm erected upon the Square A BC D, is one half of the Prism erected upon the Square described about the Circle, ABCD. But the Cylinder is lefs than the Prism erected on the Square described about the Circle A B C D; therefore the Prifm erected on the Square ABCD, having the fame Height as the Cy linder, is greater than one half of the Cylinder. Let the Circumferences A B, B T, 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 Triangles A EB, BFC,CG D, DHA, is + greater than the half of each of the Segments in which they ftand. Let Prifins be erected from each of the Triangles AEB BF C, CGD, DHA, of the fame Altitude as the Cylinder; then every one of thefe Prifms 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, C D, D A, and Parallelograms be compleated on the said AB, BC, CD, DA, on which are erected folid Parallelopipedons of the fame Altitude as the Cylinder; then each of thofe Prisms that are on the Triangles A E B, B F C, C G D, 32. 11. DHA, are Halves ‡ of each of the folid Parallelepipedons; and the Segments of the Cylinder are lefs than the erected folid Parallelopipedons; and confequently, 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 fo, bifecting the other Circumferences, joining Right Lines, and on every one of the Triangles erecting Prifms of the fame Height as + This follotus from 2 of this. the the Cylinder, and doing this continually, we fhall at laft have certain Portions of the Cylinder left, that are lefs than the Excefs by which the Cylinder exceeds triple the Cone. this. Now, let thefe Portions remaining be A E, E B, BF, FC, CG, G D, D H, HA; then the Prifm remaining, whose Base is the Polygon A E B FCGD H, and Altitude equal to that of the Cylinder's is greater than the Triple of the Cone. But the Prifm, whose Bafe is the Polygon A EBFCGDH, and Altitude the fame as that of the Cylinder's, is triple of the Cor. 7. f Pyramid, whofe Bafe is the Polygon A EBFCG DH, and Vertex the fame as that of the Cone; and therefore the Pyramid, whofe Bafe is the Polygon AEBFCGD H, and Vertex the fame as that of the Cone, is greater than the Cone, whofe Bafe is 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 A B C D is greater than half of the Circle A B C D And let a Pyramid be erected on the Square A B C D, 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 A B C D fhall be half thereof: And if folid Parallelopipedons 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 Bafes, and fo likewife are their third Parts: Therefore the Pyramid, whofe Bafe is the Square A B CD, is one half of that Pyramid erected upon the Square defcribed 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, S : is is greater than one half of the Cone. Bifect the Circumferences AB, BC, CD, DA, in the Points E, F, G, H; and join AE, E B, BF, F C, C G, 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, C G D, DHA, 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 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 fhall 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 thefe Segments be those that are on A E, E B, B F, F C, C G, G D, DH, HA; and then the remaining Pyramid, whofe Bafe 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 AEBFCGDH and Vertex the fame as that of the Cone, is one third Part of the Prifm whofe Bafe is the Polygon AEBFCGDH, and Altitude the fame as that of the Cylinder: Therefore the Prifm, whose Bafe is the Polygon A EBF C G D H, and Altitude the fame as that of the Cylinder, is greater than the Cylinder, whofe Bafe is the Circle A B C D; 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 |