COR. From the reafon given in the Cor. to the last Prop. it follows, that all prifms of equal bafes, are to each other as their altitudes.

PRO P. XII. THEOREM.

The bafes and altitudes of equal rectangular parallelepipedons are reciprocally proportional; and if the bafes and altitudes be reciprocally proportional, the parallelepipedons will be equal,

Let the rectangular parallelepipedon AR be equal to the rectangular parallelepipedon EY; then will the bafe AC be to the bafe EG, as the altitude EO is to the altitude AW.

For let AL be a rectangular parallelepipedon on the base AC, whofe altitude AP is equal to EO, the altitude of the parallelepipedon EY.

Then fince the altitudes AP, EO are equal to each other (by Conft.), the folid AL will be to the folid Ey as the base AC is to the bafe EG (VIII. 10.)

And because the folid AR is equal to the folid ET (by Hyp.), the folid AL will be to the folid AR as AC is to EG (V. 9.)

But the folid AL is to the folid AR as AP is to AW (VIII. 11.); whence, alfo, AC is to EG as AP is to AW (V. 11.), or AC to EG as EO to Aw.

Again, let AC be to EG as EO is to AW; then will Ab be equal to EY.

For, fince AL is to EY as AC to EG (VIII. 10.), and AC to EG as EO to AW (by Hyp.), AL will be to Ey as EO to AW (V. 11.)

But EO, or AP, is to AW as AL is to AR (VIII. 11.); therefore AL will be to EY as AL is to AR (V. 11.)

And fince the antecedents are equal, the confequents will also be equal; whence the folid AR is equal to the folid EY, as was to be fhewn.

COR. The fame proportion will hold of prisms in general; these being equal to rectangular parallelepipedons of equal bafes and altitudes,

PRO P. XIII. THEOREM.

Similar rectangular parallelepipedons are

to each other as the cubes of their like fides.

Let AF, KP be two fimilar rectangular parallelepipedons, whofe like fides are AB, KL; then will AF be to KP as the cube of AB is to the cube of KL.

For let AT, KW be two cubes ftanding on AX, KZ, the fquares of the fides AB, KL.

Then fince parallelepipedons on the fame base are to each other as their altitudes (VIII. 11.), AF will be to An as AH to AV, or AB; and KP to Ks as KR to KY, or

KL.

But the planes ABEH, KLOR being fimilar (VIII. Def. 2.), AH will be to AB as KR is to KL (VI. Def. 1.); whence AF is to An as KP to Ks (V. 11.); or AF to KP as An to Ks (V. 15.)

Again, fince parallelepipedons of the fame altitude are to each other as their bases (VIII. 10.), AT will be to An as AX to AC; and KW to KS as KZ to KM.

And because AX, or the fquare of AB, is to AC, as KZ, or the square of KL, is to Kм (VI. 17.); at will be to An as Kw is to Ks (V. 11.); or AT to KW as an to Ks (V. 15.)

But AF has been fhewn to be to KP as An is to Ks; therefore, alfo, AF is to KP as AT to KW (V. 11.)

Q. E. D. COR. 1. Similar rectangular parallelepipedons are to each other as the cubes of their altitudes; these being confidered as like fides of the folids.

COR. 2. Every prifm being equal to a parallelepipedon of an equal bafe and altitude (VIII. 9. Cor.), all fimilar prisms will be to each other as the cubes of their altitudes, or like fides.

PROP.

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

vertex.

S

B

Let EDABC be a pyramid, and mo a fection parallel to the base AC; then will mo be to AC as the squares 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 pnm 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 mot 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.)

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

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

PROP. XV. THEOREM.

Pyramids of equal bafes and altitudes are equal to each other.

Let EDABC, LKFGH be any two pyramids, of which the bafe 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 fections 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 square of L (Conft. and II. 2.); and the fquare of Es to the fquare

of