Then, because AF, FL are each equal to AB, and NT, TX are each equal to MN (by Conft.), the parallelograms FE, FK will be each equal to BE, and the parallelograms NV, TZ to NQ (II. 5.)

And, fince the folids AG, FH have equal bases and altitudes with the folid AC, they will be each equal to AC (VIII. 9.); and, for the fame reason, the folids NW, TY, will be each equal to NR.

Whatever multiple, therefore, the base BK is of the bafe BE, the fame multiple will the folid вH be of the folid AC; and, for the fame reason, whatever multiple the base Mz is of the base NQ, the fame multiple will the folid My be of the folid NR.

If, therefore, the base BK be equal to the base мz, the folid BH will be equal to the folid MY; and if greater, greater; and if lefs, lefs; whence the bafe BE is to the bafe NO, as the folid AC is to the folid NR (V. Def. 5.) Q. E. D.

COR. From this demonftration, and the Cor. to the laft Prop. it appears that all prisms of equal altitudes, are to each other as their bafes; every prifm being equal to a rectangular parallelepipedon of an equal base and altitude.

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PROP. XI. THEOREM.

Rectangular parallelepipedons of equal bafes are to each other as their altitudes.

K

M

W

S

H

E

Let AK, EP be two rectangular parallelepipedons ftanding on the equal bases AC, EG; then will AK be to EP as the altitude AM is to the altitude Es.

For let Aw be a rectangular parallelepipedon on the base AC, whose altitude AV is equal to the altitude Es of the parallelepipedon EP:

Then, fince the bafe AC is equal to the bafe EG (by Hyp.), and the altitude AV is equal to the altitude ES (by Conft.), the folid Aw will be equal to the folid EP (VIII.9.)

And if AL, AY be confidered as bafes, the folid ak will be to the folid Aw as the bafe AL is to the base AY (VIII. 10.)

But the bafe AL is to the base AY as the fide AM is to the fide AV (VI. 1.); whence by equality the folid ak will be to the folid Aw as the altitude AM is to the altitude AV (V. 11.)

Since, therefore, the folid Aw is equal to the folid EP, and the altitude AV to the altitude Es, the folid AK will alfo be to the folid EP as AM is to ES (V. 9.)

Q. E. D.

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.

PROP. 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 base AC be to the base EG, as the altitude EO is to the altitude AW.

For let AL be a rectangular parallelepipedon on the base AC, whose 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 Ex as the base AC is to the base EG (VIII. 10.)

And because the folid AR is equal to the folid EY (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 Ar 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 ar 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.

PROP. 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 Kİ.

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 bafes (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 fquare 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, also, AF is to KP as AT to KW (V. 11.)

1

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 prifms will be to each other as the cubes of their altitudes, or like fides.

PROP.