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Suppl. Let ADB be a semicircle, of which the centre is C; let

CD be at right angles to AB; let DB, DA be squares described on DC; draw CE. Let the figure thus constructed révolve about DC; then the sector BCĎ, which is half of the

semicircle ADB, will describe a hemisphere having C for its a Def. 7. centreâ; and the triangle CDE will describe a cone having 3. Sup

its vertex at C, and having for its base the circleb described b. Def. 11. by DE, equal to the circle described by BC, which is the base

of the hemisphere. Let W be any given solid. A series of cylinders may be inscribed in the hemisphere ADB, and another series described about the cone ECI, so that their sum shall differ from the sum of the hemisphere and the cone by a solid less than the solid W,

Upon the base of the hemisphere let a cylinder be constituted equal to W, and let its altitude be CX.

altitude be CX. Divide CD into such a number of equal parts that each of them shall be

3. Sup.

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less than CX; let these be CH, HG, GF, FD. Through the points F, G, H, draw FN, GO, HP parallel to CB, meeting the circle in the points K, L, M, and the straight line CE in the points Q, R, S. From the points K, L, M, draw Kf, Lg, Mh perpendicular to GO, HP, CB; and from Q, R, S draw Qq, Rr, Ss perpendicular to the same lines. It is evi.

dent that if the whole figure thus constructed revolve about ş Def. 14. CD, the rectangles Ff, G, Hh will describe cylinders which

will be circumscribed by the hemisphere BDĂ; and that the rectangles DN, Fq, Gr, Hs, will also describe cylinders which

will circumscribe the cone ICE. Now it may be demon13.3.Sup. strated, as was done of the prisms inscribed in a pyramida,

that the sum of all the cylinders described within the hemi

3. Sup

sphere is exceeded by the hemisphere by a solid less than the Book III. cylinder generated by the rectangle HB, that is, by a solid less than W, for the cylinder generated by HB is less than W. In the same manner it may be demonstrated that the sum of the cylinders circumscribing the cone ICE is greater than the cone by a solid less than the cylinder generated by the rectangle DN, that is, by a solid less than W. Since the sum of the cylinders inscribed in the hemisphere, together with a solid less that w, is equal to the hemisphere, and since the sum of the cylinders described about the cone is equal to the cone together with a solid less than W; the sum of all the inscribed and circumscribed cylinders, together with a solid less than W, is equal to the sum of the hemisphere and the cone, together with a solid less than W. Therefore the difference between the whole of the cylinders and the sum of the hemisphere and the cone is equal to the difference of two solids which are each of them less than W. But this difference must also be less than W; therefore the difference between the two serieses of cylinders and the sum of the hemisphere and cone is less than the given solid W. Q. E. D.

PROP. XX.

THE" same things being supposed as in the last proposition, the sum of all the cylinders inscribed in the hemisphere and described about the cone is equal to a cylinder having the same base and altitude, as the hemisphere.

Let the figure DCB be constructed as before, and supposed to revolve about CD; the cylinders inscribed in the hemisphere, that is, the cylinders described by the revolution of the rectangles Hh, Gg, Ff, together with those described about the cone, that is, the cylinders described by the revolution of the rectangles Hs, Gr, Fq, DN are equal to the cylinder described by the revolution of the rectangle DB.

Let L be the point in which GO meets the circle ADB; then, because CGL is a right angle, if CL be joined, the circles described with the distances CG and GL are equal to the circle described with the distance CLa or GO. "Now a 1. Cor. CG is equal to GR, because the triangles CDE and CGR are

7. 1. Sup

Suppl. equiangular, and CD is equal to DE; therefore the circles

described with the distances GR and GL are together equal to the circle described 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 described by the revolution of Gò about the point G. Therefore the cylinders which stand upon the first two of these circles, having the common altitude GH, are equal to the cylinder which stands upon the remaining circle, having the same altitude GH. Therefore the cylinders described by the revolution of the rectangles Gg and Gr are equal to the cylinder described by the rectangle GP. And the same may be proved of all the rest. Therefore the cylinders described by the rectangles Hh, Gg, Ff, and by the rectangles Hs, Gr, Fq, DN, are together equal to the cylinder described by DB, that is, to the cylinder having the same base and altitude as the hemisphere. Q. E. D.

PROP. XXI.

ÉVERY sphere is two thirds of the circumscribing cylinder.

Let the figure be constructed as in the two last propositions. and if the hemisphere described by BDC be not equal to two thirds of the cylinder described by BD, let it be greater by

the solid W. Then, since the cone described by CDE is one a 18. 3. Sup. third of the cylindera described by the rectangle BD, the cone

and the hemisphere together will exceed the cylinder by W.

But that cylinder is equal to the sum of all the cylinders deb 20. 3.Sup. scribed by the rectangles Hh, Gg, Ff, Hs, Grı Fq, DN6.

Therefore the hemisphere and the cone added together exceed the sum of all these cylinders by the given solid W;

which is absurd, for the hemisphere and the cone together c19.3. Sup. differ from the sum of the cylinders by a solid less than W.

Therefore the hemisphere is equal to two thirds of the cylinder described by the rectangle BD; therefore the whole sphere is equal to two thirds of the cylinder described by twice the rectangle BD, that is, to two thirds of the circumscribing cylinder. Q. E. D.

NOTES

ON

THE ELEMENTS OF GEOMETRY.

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