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PROP. XIV. THEOREM.

If a pyramid be cut by a plane parallel to its base, the section will be to the base as the fquares of their diftances from the

vertex.

B

Let EDABC be a pyramid, and mo a section parallel to the base AC; then will mo be to AC as the fquares of their diftances from the vertex.

For draw Es perpendicular to the plane of the base AC (VII. 9.); and join Ds and pr.

Then, fince mp, mn are parallel to AD, AB (VII. 12), the angle pmn will be equal to the angle DAB (VII.7.) : and pm will be to DA as Em to EA, or as mn to AB (VI. 3.)

For a like reafon each of the angles in the fection mo are equal to their correfponding angles in the base AC, and the fides about them are proportional; whence mo is fimilar to AC (VI. Def. 1.)

And because pm is parallel to DA, and pr to DS (VII. 12.), pm will be to DA as Ep to ED, or as Er to Es (VI. 3.)

1

The lines pm, DA, Er and ES being, therefore, proportional, the fquare of pm will be to the fquare of da, as the fquare of Er is to the fquare of ES (VI. 19. Cor.)

But the fquare of pm is to the fquare of DA as mo is to AC (VI. 17.); whence the fquare of Er is to the square of Es as mo is to AC (V. 11.)

Q. E. D. COR. If a pyramid be cut by a plane parallel to its base, the section will be fimilar to the base.

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Let EDABC, LKFGH be any two pyramids, of which the base AC is equal to the base FH, and the altitude Es to the altitude LP; then will EDABC be equal to

LKFGH.

For make Er equal to Lo; and draw the sections mn, vw, parallel to the bases AC, FH.

Then, by the last propofition, the fquare of Er is to the fquare of Es as mn is to AC; and the fquare of Lo to the fquare of LP as vw is to FH.

And fince the fquare of Er is equal to the fquare of Lo (Conft. and II. 2.); and the fquare of ES to the fquare

of

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, allo, 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 pyramid EDABC is equal to its correfponding section in the pyramid LKFGH, the pyramids themfelves, which are composed of those fections, must also be equal.

PRO P. XVI. THEOREM.

Q. E. D.

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

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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 whofe bafe is ABD, and vertex c, is equal to the pyramid whofe bafe is BED and vertex c (VIII. 15.)

And fince the oppofite ends of the prifm are equal to each other (VIII. Def. 3.), 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 bafe 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 prism FDABE, which is composed of them, is triple the pyramid DABC, as was to be fhewn.

COR. Every pyramid is the third part of a prism of the fame bafe and altitude; fince the base of the prifm, 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 prifms, holds equally true of pyramids ; the former being always triple the latter.

• PROP.

PROP. XVII. THEOREM.

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

D

M

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 M, 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 themselves parallel (I.29.), FC, EB will be parallel to LN; or KC, HB 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.

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