circle; for [by 32. 11.] they are to one another as their bafes; and fo alfo are the third parts of them: Wherefore the pyramid whofe bafe is the fquare A B C D is one half of the pyramid erected upon the fquare defcribed about the circle. But the pyramid erected upon the fquare described about the circle, is greater than the cone, for this is contained in that. Therefore the pyramid whose bafe is the fquare A B CD, and vertex the fame as that of the cone, is greater than one half the cone. Bifect the parts A B, BC, C D, DA of the circumference in the points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA; then each of the triangles A E B, BF C, C G D, DHA is greater than one half of the refpective fegment of the circle ABCD: Upon each of the triangles AEB, BFC, CGD, DHA erect pyramids of the fame altitude as that of the cone; then every one of thefe pyramids thus erected, is greater than one half the refpective fegment of the cone: And fo bifecting the remaining parts of the circumference, and joining right lines, and erecting a pyramid upon every one of the triangles having the fame vertex as that of the cone, and doing this continually, there will at laft [by 1. 10.] be left fome portions of the cone that will be less than the excefs of the cone above one third part of the cylinder. Let there be fuch left, which let be thofe upon A E, E B, BF, FC, CG, GD, DH, HA. Then the remaining pyramid, whofe bafe is the polygon AEB F C G D H, and vertex the fame as that of the cone, is greater than a third part of the cylinder. But [by cor. 7. 12.] the pyramid whose bafe is the polygon A E B FCGDH, and vertex the fame as that of the cone, is one third part of the prifm whose bafe is the polygon AEBFCGD H, and altitude the fame as that of the cylinder: Therefore the prifm whose base is the polygon A E B FCGDH, and altitude the fame as that of the cylinder, is greater than the cylinder whofe bafe is the circle ABCD. But it is allo lefs: For it is comprehended by it; which is impoffible. Therefore the cylinder is not lefs than thrice the cone. It has also been proved not to be greater thrice the cone: Therefore the cylinder is thrice the cone; and accordingly the cone is one third part of the cylinder. than Therefore Therefore every cone is one third part of the cylinder having the fame bafe as it, and an equal altitude. Which was to be demonstrated. d Some demonftrate this theorem thus: Every cone may be confidered as a pyramid with a polygonous base, of an infinite number of exceeding fmall equal fides; and every cylinder as a prism with fuch a polygonous bafe. And therefore fince [by cor. prop. 7.] every pyramid is a third part of a prism of the fame bafe and altitude, every cone, thus taken for a pyramid, will also be one third of a cylinder, of the fame altitude, thus taken for a prifm." PROP. XI. THEOR. Cones and cylinders having the fame altitude, are to one another as their bases. Let the cones and cylinders whofe bafes are the circles ABC D, E F G H, axes KL, MN, and diameters of the bafes are A C, E G, have the fame altitude: I fay as the circle ABCD is to the circle EFG H, fo is the cone A L, to the cone EN. For if it be not fo, it will be as the circle ABCD is to the Gircle E F G H, fo is the cone AL to some solid either lefs or greater than the cone EN. Firft let it be fo to the folid x, which is lefs. Let the folid y be cefs of the cone EN above the folid x. EN is equal to both the folids x, Y. In the circle EFGH describe the fquare EFGH: This fquare is greater than one half the circle. Upon the square EFGH erect a pyramid of the fame altitude as the cone: Then this pyramid is greater than one half the cone: for if a fquare be defcribed about the circle, and upon it be erected a pyramid of the fame altitude T equal to the exThen the cone Cc 3 4 Book XII, as the cone, the pyramid upon the infcribed fquare is one half that upon the circumfcribing fquare; for [by 6. 12.] they are to one another as their bafes; and the cone is less than the circumfcribed pyramid: Therefore the pyramid whose base is the square E FG, and vertex the fame as that of the cone, is greater than one half the cone. Bifect the parts EF, FG, GH, HE of the circumference in the points O, P, R, s; and join Ho, OE, EP, PF, FR, RG, GS, SH. Then each of the triangles HOE, EPF, FR G, GSH is greater than one half of the respective segment of the circle. Upon every one of the triangles HOE, EPF, FBG, GSH erect a pyramid of the fame altitude with that of the cope. Then each of thefe erected pyramids is greater than one half the refpective fegment of the cone; and fo bifecting the remaining parts of the circumference, joining right lines, and upon every one of the triangles erecting pyramids of the fame altitude as that of the cone, and always doing this, there will at laft [by 1. ro.] be left fome fegments of the cone that will be less than the folid v. Let fuch be left; and let them be those that stand on Ho, OE, EP, PF, FR, RG, GS, SH, and the remaining pyra mid, whose base is the polygon HOEPFRGS, and altitude the fame as that of the cone, is greater than the folid x. In the circle ABCD defcribe the polygon DTAVBQCZ, fimilar and alike fituate to the polygon HOEPFRGS, and upon it raife a pyramid of the fame altitude as the cone AL. Then because [by 20. 6. and 1. 12.] as the fquare of AC is to the fquare of EG, so is the polygon DTA уBQCZ to the polygon HOEPF RGS, and as the fquare of Ac is to the fquare of G H, fo is [by 2. 12.] the circle ABCD to the circle EFGB; [by 11. 5.] as the circle ABCD is to the circle EFGH, fo is the polygon DTAуBQCZ to the polygon HOEP FRG S. But by conftr.] as the circle ABCD is to the circle EFGH, fo is the cone AL to the folid x; and [by 6. 12.] as the polygon DTA VBQCZ is to the polygon HOEPFRGS, fo is the pyramid whole bafe is the polygon DT AV BQCK and vertex the point L, to the pyramid whose base is the polygon HOEPFRGS, and vertex the point N: Therefore as the cone A L is to the folid x, fo is the pyramid whose base is the polygon DTA V BQCZ, and vertex the point 1, to the pyramid whofe bafe is the polygon HOEPFRGS, В HOEPFRGS, and vertex the point N: Wherefore alternately as the cone AL is to the pyramid which is in it, fo is the folid x to the pyramid which is in the cone EN. But the cone AL is greater than the pyramid which is in it: Therefore the folid x is greater than the pyramid which is in the cone EN: But it is lefs too; which is abfurd. Therefore it is not as the circle ABCD to the circle EFGH, fo is the cone AL to fome folid lefs than the cone EN. In like manner we demonstrate that it is not as the circle E F G H is to the circle ABCD, fo is the cone EN to fome folid less than the cone AL: I fay also that it is not as the circle ABCD is to the circle EFGH, fo is the cone AL to fome folid greater than the cone EN. For if poffible, let it be so to some solid x, which is greater. Then inversely, as the circle EFGH is to the circle A B C D, fo will the folid x be to the cone AL. But as the folid x is to the cone AL, so is the cone EN to fome folid lefs than the cone AL: And therefore as the circle E F G H is to the circle A B C D, fo is the cone EN to fome folid less than the cone AL; which has been demonstrated to be impoffible; therefore it is not as the circle A B C D is to the circle EFGH, fq is the cone AL to fome folid greater than the cone E N. It has also been proved that it is not fo to any folid which is lefs. Therefore as the circle ABCD is to the circle EFGH, fo is the cone AL to the cone E N. But as cone is to cone, fo is cylinder to cylinder; for [by 10. 12.] the one is thrice the other; and therefore as the circle ABCD is to the circle EFGH, fo are the cones and cylinders of the fame altitude ftanding upon them. Wherefore cones and cylinders of the fame altitude, are to one another as their bafes. Which was to be demonftrated. e This may be demonstrated shorter, by the method of indivisibles, and from the notes upon the fixth and tenth theọrems. PROP. XII. THEOR. Similar cones and cylinders are to one another in the triplicate ratio of the diameters of their bases. Let there be fimilar cones and cylinders, whofe bafes are the circles ABCD, EFGH, diameters of the bases BD, FH, and axes of the cones or cylinders KL, MN; I fay the cone whofe bafe is the circle A B CD, and vertex the point L, will be to the cone whofe bafe is the circle E F G H, and vertex the point L, in the triplicate ratio of BD to FH. For if the cone ABCDL be not to the cone E F G H N in the triplicate ratio of BD to FH, the cone ABCDL will have a triplicate ratio to fome folid either lefs than the cone ratio to the folid x S which is lefs; and in the circle EFGH describe the squareEFGH. Therefore the fquare EFGH is greater than one half the circle EFGH. Again, upon the fquare E F G H erect a pyramid of the fame altitude as that of the cone and fo this R erected pyramid is greater than one half the cone. Divide the parts EF, FG, GH, HE of the circumference into equal parts in the points O, P, R, S, and join E o, o F, FP, PG, GR, RH, HS, SE: Then each of the triangles E OF, FPG, GRH, OSE is greater than one half of the refpective fegment of the circle E F G H. Again, upon each of the triangles EOF, FPG, GRH, HSE erect a pyramid, having the fame vertex as the cone, and each of the erected pyramids is greater than one half of the refpective fegment of the cone. Therefore bifecting the remaining fegments of the circumference, and joining right lines, and erecting pyramids upon every one of the triangles having the fame vertex as the cone, and always doing this, at last |