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PROP: X. THEOR.

. Every cone is a third part of a cylinder which has

the same base and an equal altituded. For let a cone have the same base aś à cylinder, viz. the circle ABC D, and an equal altitude: I say the cone is a third part of the cylinder ; that is, the cylinder is

thrice the cone. For if the cylinder be not thrice the cone; it will be either greater than thrice the cone, or

less. First let it be greater than thrice B

the conę; and describe a square ABCD
in the circle aecd: Then the square

ABCD
F

A B C D is greater than one half the с circle A B C D. And upon the square A B C D erect a prism of the fame altitude as the cylinder, which prism will be greater than one half the cylinder ; because if a square be described about the circle A B C D, the inscribed square will be one half the circumscribed square; and there are erected upon those bases solid parallelepipedons of the fame altitude, viz. the prisms themselves; and fo the prisms are to one another as their bases ; therefore the prism erected upon the squate A BCD is one half the prism erected upon the square described about the circle ABCD; and the cylinder is less than the prism erected upon the square described about the circle ABCD: Therefore the prism described upon the square A B C D of the same altitude as the cylinder, is greater than one half that cylinder. Bisect the parts A B; BC, CD, DÅ of the circumference 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 E B, BFC, CGD, DHA is greater than one half the lege ment of the circle ABCD wherein it stands, as has been already proved [fee 2. 12.] Upon each of the triangles A E B, BFC, CGD, DHA erect a prifm of the same altitude as the cylinder: Then will each of these prisms be greater than one half the respective fegment of the cylin. der; because if thro' the points E, F, G, H parallels be drawn to A B, BC, CD, DA, and parallelograms at them

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be completed, upon which folid parallelepipedons of the fame altitude as the cylinder are erected; each of these erected parallelepipeduns will be double the prisms which are in the triangles À E B, BFC, CGD, DHA; and the segments of the cylinder are less than the erected folid parallelepipedons; therefore the prisms in the triangles A E B, BFC, CGD, DHA are greater than one

half the res spective fegments of the cylinder; and to let the remaining parts of the circumference be bisected, right lines be joined, and upon each of the triangles erect prisms of the fame altitude as the cylinder, and do this always, till at last [by 1. 10.] some segments of the cylinder remain being less than the excess of the cylinder above thrice the cône. Let such segments be left, and let them be À E, EB, BF, É C, CG, GD, OH, HA. Then the remaining prism, whose base is thic polygon A E BFCGDH, and alticude the same as that of the cylinder, is greater than thrice the cone. But [by cor. 7: 12.) the prism whose base is the polygon A EBFCGDH, and altitude the same as that of the cylinder, is tbrice the pyramid whose base is the polygon A l B FCGDH;, and vertex the same as that of the cone; and therefore the pyramid whose base is the polygon A E BÉCÉDH, and vertex the same as that of the cone, is greater than the cone whose base is the circle ABC'D : but it is less too, because it is contained in it, which is imposible: Therefore the cylinder is not greater than thrice the cone. I lay moreover, that the cylinder is not less than thrice the cone. For if possible let the cylinder bę less than thrice the cone; then inversely the cone will be greater than a third part of the cylinider. Describe ihe square ABCD in the circle Á BCD; this' Square will be greater than one half the circle A B C D, and upon the square A B C D erect a pyramid, having the same vërtex as that of the cone; then the erected pyramid wil be greater than one half the cone; because, as we have already demonstrated, if a square be described about the circle, the square ABCD will be one half of it. And if solid parallelepipedons be erected upon those squares, having the same altitude with that of the cone, which are also called prisms; that erected upon the square À BCD will be one half ef that ere ted upon the square described about the Co2

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circle ; for [by 32. 11.) they are to one another as their bases; and so also are the third parts of them: Wherefore the pyramid whose base is the fquare'A B C D is one half of the 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 this is contained in that. Therefore the pyramid whose base 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, CD, 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, B FC, CGD, DHA is greater than one half of the respective segment of the circle ABCD: Upon each of the triangles A E B, BFC, CGD, DHA ereet pyramids of the fame altitude as that of the cone; then every one of these pyramids thus erected, is greater than one half the respective fegment of the cone: And so bisecting the remaining parts of the circumference, and joining right lines, and erecting a pyramid upon every one of the triangles having the same vertex as that of the cone, and doing this continually, there will at last [by 1. 10.] be left some portions of the cone that will be less than the excess of the cone abové one third part of the cylinder. Let there be such left, which let be those upon A E, E B, BF, FC, CG, GD, DH, HA. Then the remaining 'pyramid, whofe base is the polygon A E B F'CGDH, and vertex the same as that of the cone, is greater than a third part of the cylinder. But [by cor. 7. 12. ] the pyramid whose base is the polygon A E B F CGD H, 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 fame as that of the cylinder: Therefore the prism whose base is the polygon A E B FCGDH, and altitude the same as that of the cylinder, is greater than the cylinder whose bafe is the circle A BCD. But it is also less: For it is comprehended by it; which is impoffible. Therefore the cylinder is not less than thrice the cone. It has also been proved not to be greater

than thrice the cone: Therefore the cylinder is thrice the cone; and accordingly the cone is one third part of the cylinder.

Therefore

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Therefore every cone is one third part

of the cylinder having the same base as it, and an equal altitude. Which was to be demonstrated.

d Some demonstrate this theorem thus: Every cone may be considered as a pyramid with a polygonous base, of an infinite number of exceeding small equal fides; and every cylinder as a prism with such a polygonous base. And therefore since [by cor. prop. 7.) every pyramid is a third part of a prism of the same base 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 prism.'

PRO P. XI. THEOR. Cones and cylinders baving the same altitude, are to

one another as their bafese. Let the cones and cylinders whose bases are the circles ABC D, E F G H, axes KL, MN, and diameters of the bases are A C, E G, have the same altitude; I say as the circle ABCD is to the circle EFGH, so is the cone AL, to the cone En.

For if it be not so, it will be as the circle ABCD is to the circle E F G H, so is the cone al to some solid either less or greater than the cone En. First let it be so to the solid x, which is less. Let the folid y be equal to the excess of the cone en above the folid x. Then the cone EN is equal to both the

L folids x, y. In the cir

N cle E F G H describe the

D square EFGH: This

T

H

Z square is greater than one half the

circle. Upon the square EFGH

СЕ
K

M
erect a pyramid of the
fame altitude as the
cone: Then this

pyra

B mid is greater than one half the cone: for if ą Square be described about the circle, and up

: on it be erected a pyra: mid of the same altitude

P

x

Cc

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as the cone, the pyramid upon the inscribed square is one half that upon the circumscribing square; for (by 6. 12.] they are to one another as their bales; and the cone is less than the circumscribed pyramid: Therefore the pyramid whose kafe is the square E FGH, and vertex the same as that of the cone, is greater than one half the cone. Biseer the parts E F, FG, GH, HE of the circumference in the points 0, P, R, S; and join Hò, O'E, E P, FF, FR, RG; GS, $ H. Then each of the triangles HoE, E PF, FRG, Gsh is greater than one half of the respective segment of the circle. Upon every one of the triangles HOE, E PF, FRG, GSH erect a pyramid of the fame alitude with that of the core. Then each of these erected pyramids is greater than one half the respective segment 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 same altitude as that of the cone, and always doing this, there will at laft (by 1. 10.] be left some segments of the cone that will be less than ihe solid , Let fuch be left; and let them be those that stand on Ho, O E, E P, PP, FR, RG, GS, SH, and the remaining pyra: mid, whose base is the polygon HO E P FR G , and altitude the same as that of the cone, is greater than the folid X.

In the circle ABCD describe the polygon DTAVBQC%, fimilar and alike fituate to the polygon HOEPFRGS, and upon it raise a pyramid of the same altituda as the cone Al. Then because [by 20.6. and 1. 12.) as the square of ac is to the square of EG, fo is the polygon DTA Y BQcz to the polygon HOEPF RGS, and as the square of Ac is to the aquare of G H, fo is (by 2. 12.) the circle A B C D to the circle E F G H; [by JI. 5.) as the circle a'r cd is to the circle EfGH, fo is the polygon DTA Y BQcz to the polygon HOEPFRGS. But [by constr.] as the circle ABCD is to the circle EFGH, so is the cone AL to the solid x; and (by 6. 12.) as the polygon D Ţ AV BO'cz is to the pol, gon HOEPFRGS, so is the pyramid whole base is the polygon DTA V BOCY and vertex the point i, to the pyramid whose base is the polygon HO EPFRGs, and vertex the point n: Therefore as the cone Al is to the solid x, so is the pyramid whose base is the polygon DTA V BOCz, and vertex the point 1, to the pyramid whose base is the polygon

HOEPF & G $,

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