Book VIII incide with the ftraight line EF. But AD is always in the fuperficies of the cylinder, for it defcribes that fuperficies; therefore, EF is alfo in the fuperficies of the cylinder. Therefore, &c. Q. E. D.

A

ther.

CYLINDER and a parallelepiped having equal bafes and altitudes are equal to one ano

Let ABCD be a cylinder, and EF a parallelepiped having equal bafes, viz. the circle AGB and the parallelogram EH, and having alfo equal altitudes; the cylinder ABCD is equal to the parallelepiped EF.

If not, let them be unequal; and first, let the cylinder be lefs than the parallelepiped EF; and from the parallelepiped EF let there be cut off a part EQ by a plane PQ parallel to NF, equal to the cylinder ABCD. In the circle AGB infcribe the polygon AGKBLM that fhall differ from the circle by a space lefs than the parallelogram PH, and cut off from the parallelogram EH, a part OR equal to the polygon AGKBLM. The point R,will fall between P and Ñ. On the polygon AGKBLM let an upright prifm AGBCD be conftituted of the fame altitude with the cylinder, which will

I

therefore

273

9.

herefore be less than the cylinder because it is within it b ; Book VIII. nd if through the point R a plane RS parallel to NF be b 8. nade to pafs, it will cut off the parallelepiped ES equal to he prism AGBC, because its bafe is equal to that of the rism, and its altitude is the fame. But the prism AGBC is fs than the cylinder ABCD, and the cylinder ABCD is e* ual to the parallelepiped EQ, by hypothefis; therefore, ES lefs than EQ, and it is alfo greater, which is impoffible. The cylinder ABCD, therefore, is not lefs than the paralelepiped EF; and in the fame manner it may be shewn ot to be greater than EF. Therefore they are equal. E. D.

Book VIII.

IF

F a cone and a cylinder have the fame base and the fame altitude, the cone is the third part of the 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 ABD 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.

24.8.

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 greater than LIM. Let this be the polygon BECFD; and upon BECFD let there be conftituted the pyramid ABECFD, and the prism BCFKHG.

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

2

LMNO,

LMNO, for they have the fame altitude, but the prifm has Book VIII. he greater bafe. But the pyramid ABECFD is the third art of the prifm b BCFKHG, therefore, it is greater than b 35. 7. he third part of the cylinder LMNO. Now, the cone ABECFD is, by hypothefis, the third part of the cylinder MNO, therefore, the pyramid ABECFD is greater than he cone ABCD, and it is alfo lefs, because it is infcribed in he cone, which is impoffible. Therefore, the cone ABCD s not less than the third part of the cylinder BFKG: And, In the fame manner, by circumfcribing a polygon about the ircle BCD, it may be fhewn, that the cone ABCD is not reater than the third part of the cylinder BFKG; thereore, it is equal to the third part of that cylinder. Q. E. D.

PRO P. XII. THE OR.

[F a hemifphere and a cone have equal bafes and altitudes, a series of cylinders may be inferibed n the hemifphere, and another feries may be decribed about the cone, having all the fame altitudes with one another, and fuch that their fum hall differ from the fum of the hemifphere, and the cone by a folid lefs than any given folid.

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 DE and let the figure thus conftructed revolve about DC: then, the fector BCD, which is the half of the femicircle ADB, will defcribe a hemisphere having C for its centre a, and the triangle CDE will a 14. def. defcribe a cone, having its vertex at C, and having for its 7 bafe the circle b defcribed by DE, equal to that defcribed by ↳ 18. def. 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 the r fum fhall differ from the fum of the hemifphere and the cone, by a folid less than the folid W.

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

7.

Book VIII. into fuch a number of equal parts, that each of them fhall be 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 straight 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 c 20. def. 7-cylinders that will be circumfcribed by the hemifphere BDA; and that the rectangles DN, Fq, Gr, Hs will alfo defcribe cylinders that will circumfcribe the cone ICE. Now, it may be demonftrated, as was done of the prifms inscribed in a pyramid d, that the fum of all the cylinders described within the hemisphere, is exceeded by the hemifphere by a folid lefs than the cylinder generated by the rectangle HB, that is, by

d3

133 7.

.

a folid lefs than W, for the cylinder generated by HB is lefs than W. In the fame manner, it may be demonftrated, 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 lefs than W. Therefore, fince the fum cf the cylinders infcribed in the hemil phere, together with a folid lefs than W, is equal to the he mifphere; and, fince the fum of the cylinders defcribed about the cone is equal to the cone together with a folid lefs 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 le than W. Therefore, the difference between the whole of the

cylinders