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of LP (Hyp. and II. 2.) mn will be to AC as vw is to FH (V. 9.)

But AC is equal to FH, by hypothefis; whence mn is, alfo, equal to vw (V. 10.)

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

Since, therefore, every section in the pyramid EDABC is equal to its corresponding section in the pyramid LKFGH, the pyramids themselves, which are compofed of those fections, muft alfo be equal.

Q. E. D.

PROP. XVI. THEOREM.

Every pyramid of a triangular base, is the third part of a prifm of the fame base and altitude.

E

Let DABC be a pyramid, and FDABE a prifm, ftanding upon the fame bafe ABC, and having the fame altitude; then will DABC be a third of FDABE.

For in the planes of the three fides of the prifm, draw the diagonals DB, DC and CE:

Then because DB divides the parallelogram AE into two equal parts, the pyramid whose base is ABD, and vertex c, is equal to the pyramid whose base is BED and vertex c (VIII. 15.)

And fince the oppofite ends of the prifm are equal to each other (VIII. Def. 5.), the pyramid whose base is ABC and vertex D, is equal to the pyramid whose base is DEF and vertex c (VIII. 15.)

But the pyramid whose base is ABC and vertex D, is equal to the pyramid whose base is ABD and vertex c, being both contained by the fame planes.

The three pyramids DABC, CBED and CEFD are, therefore, all equal to each other; and confequently the prifm FDABE, which is compofed of them, is triple the pyramid DABC, as was to be fhewn.

COR. Every pyramid is the third part of a prism of the same base and altitude; since the base of the prism, whatever be its figure, may be divided into triangles, and the whole folid into triangular prifms, and pyramids.

SCHOLIUM. Whatever has been demonftrated of the proportionality of prisms, holds equally true of pyramids; the former being always triple the latter.

PROP.

PRO P. XVII. THEOREM.

If a cylinder be cut by a plane parallel to its bafe, the fection will be a circle, equal to the bafe.

D

Let AF be a cylinder, and GHK a fection parallel to its bafe ABC; then will GHK be a circle, equal to ABC.

For let the planes NE, NF pafs through the axis of the cylinder LN, and meet the section GHK in м, H and K.

Then, fince the circle DEF is equal and parallel to the circle ABC (VIII. Def. 8.), the radii LF, LE will be equal and parallel to the radii NC, NB (III. 5. and VII. 12.)

And because lines which join the correfponding extremes of equal and parallel lines are themfelves parallel (I. 29.), FC, EB will be parallel to LN; or KC, Hв to

MN.

In like manner, fince the circle GHK is parallel to the circle ABC (by Hyp.), MK, MH will be parallel to NC,

NB.

And, because the oppofite fides of parallelograms are equal (I. 30.), MK will be equal to NC, and MH to NB.

But NC, NB are equal to each other, being radii of the fame circle; whence MK, 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 fhewn.

PROP. XVIII. THEOREM.

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

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Let AH be a cylinder, and DM a 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 vrw.

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

DEF.

But the bafe ACE is equal to the bafe 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 base, 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 those sections, must also be equal.

Q. E. D.

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

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PROP. XIX. THEOREM.

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 fection parallel to the base ABC; then will nmp be to ABC as the fquares of their distances from the vertex.

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

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