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Supplement therefore be lefs than the cylinder, because it is within it b; b16. 3. Sup. and if through the point R a plane RS parallel to NF be made to pafs, it will cut off the parallelepiped ES equal to c 2. cor. 8. the prifm AGBC, because its base is equal to that of the 3. Sup. prifm, and its altitude is the fame. But the prifm AGBC is less than the cylinder ABCD, and the cylinder ABCD is equal to the parallelepiped EQ, by hypothefis; therefore, ES is lefs than EQ, and it is alfo greater, which is impoffible. The cylinder ABCD, therefore, is not less than the parallelepiped EF; and in the fame manner, it may be shewn not to be greater than EF. Therefore they are equal. Q.E. D.

PROP.

PROP. XVIII. THEOR.

Fa cone and a cylinder have the fame base and the fame altitude, the cone is 'the third part of the

I cone

cylinder.

Let the cone ABCD, and the cylinder BFKG have the fame base, viz. the circle BCD, and the fame altitude, viz. the perpendicular from the point A upon the plane BCD, the cone ABCD is the third part of the cylinder BFKG,

If not, let the cone ABCD be the third part of another cylinder LMNO, having the fame altitude with the cylinder BFKG, but let the bafes BCD and LIM be unequal; and first, let BCD be greater than LIM.

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Book III.

Then, because the circle BCD is greater than the circle LIM, a polygon may be infcribed in BCD, that fhall differ from it lefs than LIM does, a and which, therefore, will be a 4. 1. Sup, greater than LIM. Let this be the polygon BECFD; and upon BECFD let there be conftituted the pyramid ABECFD, and the prifm BCFKHG.

Because the polygon BECFD is greater than the circle LIM, the prifm. BCFKHG, is greater than the cylinder

LMNO,

b

Supplement LMNO, for they have the fame altitude, but the prifm has the greater bafe. But the pyramid ABECFD is the third b15.3. Sup. part of the prifm BCFKHG, therefore it is greater than the third part of the cylinder LMNO. Now, the cone ARECFD is, by hypothefis, the third part of the cylinder LMNO, therefore, the pyramid ABECFD is greater than the cone ABCD, and it is also lefs, because it is infcribed in the cone, which is impoffible. Therefore, the cone ABCD is not less than the third part of the cylinder BFKG: And, in the fame manner, by circumfcribing a polygon about the circle BCD, it may be fhewn, that the cone ABCD is not greater than the third part of the cylinder BFKG; therefore, it is equal to the third part of that cylinder. QE. D.

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F a hemifphère and a cone have equal bafes and altitudes, a series of cylinders may be infcribed in the hemifphere, and another feries may be defcribed about the cone, having all the fame altitudes with one another, and fuch that their fum fhall differ from the fum of the hemifphere, and the cone, by a folid less than any given solid.

Let ADB be a femicircle, of which the centre is C, and let CD be at right angles to AB; let DB and DA be fquares defcribed on DC, draw CE, and let the figure thus conftructed revolve about DC: then, the sector BCD, which is the half of the femicircle ADB, will defcribe a hemisphere having C for its centre a, and the triangle CDE will defcribe a cone, having its vertex at C, and having for its bafe the circle b defcribed by DE, equal to that defcribed by BC, which is the base of the hemifphere. Let W be any given folid. A feries of cylinders may be infcribed in the hemifphere ADB, and another described about the cone ECL, fo that their fum fhal differ from the fum of the hemisphere and the cone, by a folid less than the folid W.

Upon the base of the hemifphere let a cylinder be conftituted equal to W, and let its altitude be CX. Divide CD

into

into fuch a number of equal parts, that each of them fhall be Book III. lefs than CX; let these be CH, HG, GF, and FD. Through the points F, G, H, draw FN, GO, HP parallel to CB, meeting the circle in the points K, L and M ; and the ftraight line CE in the points Q, R and S. From the points K, L, M draw Kf, Lg, Mh perpendicular to GO, HP and CB; and from Q, R and S, draw Qq, Rr, Ss perpendicular to the fame lines. It is evident, that the figure being thus conftructed, if the whole revolve about CD, the rectangles Ff, Gg, Hh will defcribe cylinders c that will be circumfcribed by the hemifphere c 14. def. 3. BDA; and that the rectangles DN, Fq, Gr, Hs will alfo de. Sup. fcribe cylinders that will circumfcribe the cone ICE. Now, it may be demonftrated, as was done of the prisms infcribed in

a pyramid d, that the fum of all the cylinders defcribed within d13, 3.Sap. the hemifphere, is exceeded by the hemisphere by a folid lefs than the cylinder generated by the rectangle HB, that is, by

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a folid less than W, for the cylinder generated by HB is lefs than W. In the fame manner, it may be demonstrated, that the fum of the cylinders circumfcribing the cone ICE is greater than the cone by a folid lefs than the cylinder generated by the rectangle DN, that is, by a folid less than W. Therefore, fince the fum of the cylinders infcribed in the hemifphere, together with a folid lefs than W, is equal to the hemifphere; and, fince the fum of the cylinders defcribed about the cone is equal to the cone together with a folid less than W; adding equals to equals, the fum of all these cylinders, together with a folid lefs than W, is equal to the fum of the hemifphere and the cone together with a folid lefs than W. Therefore, the difference between the whole of the

cylinders

Supplement cylinders and the fum of the hemifphere and the cone, is e qual to the difference of two folids, which are each of them lefs than W; but this difference muft alfo be lefs than W, therefore the difference between the two feries of cylinders and the fum of the hemifphere and cone is lefs than the given folid W. Q.E. D.

a 2. cor. 6. 1. Sup.

THE

PROP. XX.

HE fame things being fuppofed as in the laft propofition, the fum of all the cylinders infcribed in the hemifphere, and described about the cone, is equal to a cylinder, having the fame base and altitude with the hemifphere.

Let the figure DCB be conftructed as before, and fupposed to revolve about CD; the cylinders infcribed in the hemif phere, that is, the cylinders defcribed by the revolution of the rectangles Hh, Gg, Ff, together with thofe deferibed about the cone, that is, the cylinders defcribed by the revolution of the rectangles Hs, Gr, Fq, and DN are equal to the cylinder defcribed by the revolution of the rectangle DB.

Let L be the point in which GO meets the circle ADB, then, becaufe CGL is a right angle if CL be joined, the circles defcribed with the distances CG and GL are equal to the circle defcribed with the distance CLa or GO; now, CG is equal to GR, becaufe CD is equal to DE, and therefore alfo, the circles defcribed with the distances GR and GL are together equal to the circle defcribed with the distance GO, that is, the circles described by the revolution of GR and GL about the point G, are together equal to the circle defcribed by the revolution of GO about the fame point G; therefore alfo, the cylinders that fland upon the two first of these circles having the common altitudes GH, are equal to the cylinder which stands on the remaining circle, and which has the fame altitude GH. The cylinders defcribed by the revolution of the rectangles Gg and Gr are therefore equal to the cylinder defcribed by the rectangle GP. And as the fame may be fhewn of all the reft, therefore the cylinders defcribed by the rectangles Hh, Gg, Ff, and by the rectangles Hs, Gr, Fq, DN, are together equal to the cylinder defcribed by DB, that is, to the cylinder having the same base and altitude with the hemifphere. Q. E.D.

PROP.

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