But NC, NB are equal to each other, being radii of the fame circle; whence мK, MH are alfo equal to each other.

And the fame may be fhewn of any other lines, drawn from the point M, to the circumference of the fection GHK; confequently GHK is a circle, and equal to ABC, as was to be shewn.

PROP. XVIII. THEOREM.

Every cylinder is equal to a prifm of an equal base and altitude.

Let AH be a cylinder, and DM á prism, standing upon equal bases ACB, DEF, and having equal altitudes; then will AH be equal to DM.

For parallel to the bafes, and at equal diftances from them, draw the planes onm, and urw.

Then, by the laft Prop. and Prop. 8, the fection onm is equal to the base ACB, and the section vrw to the base DEF.

But the base ACB is equal to the base DEF, by hypothefis; whence the fection onm is alfo equal to the fection vrw.

And,

And, in the fame manner, it may be fhewn, that any other fections, at equal distances from the bafe, are equal to each other.

Since, therefore, every section of the cylinder is equal to its correfpondent fection in the prifm, the folids themfelves, which are compofed of thofe fections, must also be equal.

Q. E. D.

SCHOLIUM. Whatever has been demonftrated of the proportionality of prisms, holds equally true of cylinders; the former being equal to the latter.

If a cone be cut by a plane parallel to its bafe, the fection will be to the bafe as the fquares of their distances from the vertex.

D

Let DABC be a cone, and nmp a section parallel to the base ABC; then will nmp be to ABC as the fquares of their diftances from the vertex,

For draw the perpendicular Dr; and let the planes CDP, BDP pass through the axis of the cone, and meet the section in o, p, and m.

Then fince the section nmp is parallel to the bafe ABC (by Hyp.), and the planes Bo, co cut them, op will be parallel to PC, and om to PB (VII. 12.)

And because the triangles formed by thefe lines are equiangular, om will be to PB as Do to DP, or as op tọ PC (VI. 5.)

But PB is equal to PC, being radii of the fame circle; wherefore om will also be equal to op (V. 10.)

And the fame may be fhewn of any other lines drawn - from the point o to the circumference of the fection nmp; whence nmp is a circle.

Again, by fimilar triangles, Ds is to Dr as Do to DP, or as om to PB; whence the fquare of Ds is to the fquare of Dr as the fquare of om is to the fquare of PB (VI. 19.) But the fquare of om is to the fquare of PB as the circle nmp is to the circle ABC (VIII. 5.); therefore the fquare of Ds is to the square of Dr as the circle nmp is to the circle ABC (V. II.)

Q. E. D. COR. If a cone be cut by a plane parallel to its base the fection will be a circle.

PROP. XX. THEOREM.

Every cone is equal to a pyramid of an equal bafe and altitude.

A A

B

Let DABC be a còne, and KEFGH a pyramid, standing upon equal bafes ABC, EFGH, and having equal altitudes DP, KS; then will DABC be equal to KEFGH.

For parallel to the bases, and at equal distances Do, Kr from the vertices, draw the planes nmp and vw.

Then, by the laft Prop. and Prop. 13, the fquare of Do is to the fquare of DP as nmp is to ABC; and the fquare of Kr to the fquare of Ks as vw to EG.

And fince the fquares of Do, DP are equal to the squares of Kr, Ks (Confi. and II. 2.), nmp is to ABC as vw is to EG (V. 11.)

But ABC is equal to EG, by hypothefis; wherefore nmp is, alfo, equal to vw (V. 10.)

And, in the fame manner, it may be fhewn, that any other fections, at equal diftances from the vertices, are equal to each other.

Since, therefore, every fection in the cone is equal to its correfponding fection in the pyramid, the folids DABC, KEFGH of which they are compofed, muft be equal. Q. E. D.

PROP. XXI. THEOREM.

Every cone is the third part of a a cylinder of the fame bafe and altitude.

Let EAB be a cone, and DABC a cylinder, of the fame base and altitude; then will EAB be a third of DÁBĊ.

For let KFG, KFGH be a pyramid and prifm, having an equal bafe and altitude with the cone and cylinder.

Then fince cylinders and prifms of equal bafes and altitudes are equal to each other (VIII. 18.), the cylinder DABC will be equal to the prism KFGH.

And, because cones and pyramids of equal bases and altitudes are equal to each other (VIII. 20.); the cone EAB will be equal to the pyramid KFG.

But the pyramid KFG is a third part of the prifm KFGH (VIII. 16.), wherefore the cone EAB is, also, a third part of the cylinder DABC.

Q. E. D.

SCHOLIUM I. Whatever has been demonftrated of the proportionality of pyramids, prifms, or cylinders, holds equally true of cones, these being a third of the latter.

2. It is alfo to be observed, that fimilar cones and cylinders are to each other as the cubes of their altitudes,