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figure EGHF is therefore a parallelogram, and it has the angle EGH a right angle; therefore it is a rectangle, and is equal to the rectangle AH, because EG is equal to AG. Therefore, when in the revolution of the rectangle AH, the straight line AG coincides with EG, the two rectangles AH and EH will coincide, and the straight line AD will coincide with the straight line EF. But AD is always in the superficies of the cylinder, for it describes that superficies ; therefore EF is also in the superficies of the cylinder.

PROPOSITION XIX. THEOREM. A cylinder and a parallelopiped having equal bases and altitudes are equal to one another.

Let ABCD be a cylinder, and EF a parallelopiped having equal bases, namely, the circle AGB and the parallelogram EH, and having also equal altitudes; the cylinder ABCD is equal to the parallelopiped EF.

If not, let them be unequal; and first, let the cylinder be less than the parallelopiped EF; and from the parallelopiped EF let there be cut off a part EQ by a plane PQ parallel to NF, equal to the cylinder ABCÍ. In the circle AGB inscribe the polygon AGKBLM that shall differ from the circle by a space less than the parallelogram PH, and a cut off from the parallelogram EH, a part OR equal to the polygon AGKBLM. The point R will fall between P and N. On the polygon ÀGKBLM let an upright prism AGBCD be constituted of the same altitude with the cylinder, which will therefore be less than the cylinder, because it is within it; and if through the point R a plane RS parallel to NF be made to pass, it will cut off the parallelopiped ES equal to the prism AGBC, because its base is equal to that of the prism, and its altitude is the same. But the prism AGBC is less than the cylinder ABCD, and the cylinder ABCD is equal to the parallelopiped EQ, by hypothesis ; therefore, ES is

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less than EQ, and it is also greater, which is impossible. The cylinder ABCD, therefore, is not less than the parallelopiped EF; and in the same manner it may be shown not to be greater than EF.

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PROPOSITION XX. THEOREM. If a cone and a cylinder have the same base and the same altitude, the cone is the third part of the cylinder.

Let the cone ABCD, and the cylinder BFKG, have the same base, namely, the circle BCD, and the same altitude, namely, 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 same altitude with the cylinder BFKG, but let the bases BCD and LIM be unequal; and first, let BCD be greater than LIM.

Then, because the circle BCD is greater than the circle LIM, a polygon may be inscribed in BCD, that I shall differ from it less than LIM does (Pl. Ge. Quad. 6), y and which, therefore, will be greater than LIM. Let this be the polygon BECFD; and upon BECFD let there be constituted the pyramid ABECFD, and the prism BCFKHG.

Because the polygon BECFD is greater than the circle LIM, the prism BCFKHG is greater than the cylinder LMNO, for they have the same altitude, but the prism has the greater base. But the pyramid ABECFD is the third part of the prism (II. 17) BCFKHG; therefore it is greater than the third part of the cylinder LMNO. Now, the cone ABECFD is, by hypothesis, the third part of the cylinder LMNO; therefore, the pyramid ABEČFD is greater than the cone ABCD, and it is also less, because it is inscribed in the cone, which is impossible ; therefore the cone ABCD is not less than the third part of the cylinder BFKG. And

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in the same manner, by circumscribing a polygon about the circle BCD, it may be shown 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.

PROPOSITION XXI. THEOREM. If a hemisphere and a cone have equal bases and altitudes, a series of cylinders may be inscribed in the hemisphere, and another series may be described about the cone, having all the same altitudes with one another, and such that their sum shall differ from the sum of the hemisphere, and the cone by a solid less than any given solid.

Let ADB be a semicircle, of which the centre is C, and let CD be at right angles to AB; let DB and DA be squares described on DC, draw DE, and let the figure thus constructed revolve about DC; then the sector BCD, which is the half of the semicircle ADB, will describe a hemisphere having C for its centre (II. Def. 10), and the triangle CDE will describe a cone, having its vertex at C, and having for its base the circle (II. Def. 12) described by DE, equal to that 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 described about the cone ECL, 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 ČX. Divide CD into such a number of equal parts, that each of them shall be less than CX; let these be CH, HG, GF, and FD. Through the points F, G, H, draw

* gm FN, GO, HP, parallel to CB, meeting the circle in the points K, L, and M; and the straight line CE in the points Q, R, and S. From the points K, L, M, draw Kf, Lg, Mb, perpendicular to GO,

HP, and CB; and from Q, R, and S, draw Qq, Rr, Ss, perpendicular to the same lines. It is evident that the figure being thus constructed, if the whole revolve about CD, the rectangles Ff, Gg, Hh, will describe cylinders (II. Def. 13) that will be circumscribed by the hemisphere BDA; and that the rectangles DN, Fq, Gr, Hs, will also describe cylinders that will circumscribe the cone ICE. Now, it may be demonstrated, as was done of the prisms inscribed in a pyramid (II. 15), that the sum of all the cylinders described within the hemisphere, is exceeded by the hemisphere by a solid less than the 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. Therefore, since the sum of the cylinders inscribed in the hemisphere, together with a solid less than 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; adding equals to equals, the sum of all these 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 series of cylinders, and the sum of the hemisphere and cone, is less than the given solid W.

Or thus : let I= the sum of the interior cylinders within the hemisphere; E= the sum of the exterior without the cone; and Y and Z two solids each less than W. Then I+Y=H, and E=C+Z; therefore, adding equals to equals, I+E+Y=H + C + Z. Hence the difference between (1 + E) and (H + C) is equal to that between Z and Y; but as these two solids are each less than W, their difference is still less than W; and hence also the difference between (I + E) and (H + C) is less than W.

PROPOSITION XXII. THEOREM. 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 with 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, and 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 CL (Pl. Ge. Quad. 8, Cor. 2) or GO; now CG is equal to GR, because CD is equal to DE, and therefore also, 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 GO about the same point G; therefore also, the cylinders that stand upon the two first of these circles having the common altitude GH, are equal to the cylinder which stands on the remaining circle, and which has the same altitude GH. The cylinders described by the revolution of the rectangles Gg and Gr are therefore equal to the cylinder described by the rectangle GP. And as the same may be shown 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 with the hemisphere.

PROPOSITION XXIII. THEOREM. Every sphere is two thirds of the circumscribing cylinder. Let the figure be constructed as in the two last proposi

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