PYRAM

YRAMIDS (FGLIM & ABCDE) of the fame altitude, which have polygons (F GHLI, & A B C D) for their bases: are to one another as their bafes.

1. The pyramids F G H LI& ABCD, Pyram. MFGHLI : pyram. A B C D E bave polygons for their bases. bafe FILHG bafe ABC D.

II. They have the fame altitude.

Preparation.

1. Divide the bafes FILHG & ABCD into triangles,
by drawing the lines GI, FH; & DB.

2. Let planes be paffed thro' thofe lines & the vertices of
the pyramids, which will divide each of those pyramids
into as many pyramids as each bafe contains triangles.

BECAUSE

DEMONSTRATION.

ECAUSE the triangular pyramids IL HM & ABDE have the fame altitude. (Hyp. 11. & Prep. 2).

=

2. Likewife, pyr. GIHM: pyr. ABDE bafe HIG: bafe ABD.
3, Confequently, pyr. I HLM + pyr. GIHM : pyr. ABDE
bafe HIL bafe HIG: base A B D.
4. Moreover, pyг. FIGM: pyr. ABDE bafe FIG: base ABD,
5. Therefore, pyr. IHLM+pyr. GIHM + pyr. FIGM: pyr.
ABDE bafe H I L + bafe HIG+ bafe FIG: bafe ABD.
But pyr. IHLM+pyr. GIHM + pyr. FIGM are to
the pyr. MFGHLI, & the bafe HIL+ bafe HIG+ bafe
FIG bafe FILHG.

6. Confequently, pyr. M F G HIL: pyr. ABDE=bafe FILHG base A B D.

It may be proved after the fame manner, that

{

7. Pyr. M F G HLI: pyr. B DCE= bafe FILHG: bafe BDC. 8. Therefore, pyr. M F G H LI: pyr. A BCDE= base FILHG : base A D C B.

P.24. B. 5.

Ax.1. B. 2.

P. 7. B. 5.

P.25. B. 5.

Which was to be demonftrated.

PROPOSITION VII. THEOREM VII.

EVERY triangular prifm (ADE): may be divided (by planes pafling

through the ABCF & BD F) into three pyramids (A CBF, BDEF & DCBF) that have triangular bases, and are equal to one another.

1. In the pgr. D A draw any diagonal C F.

2. From the point F in the pgr. A E, draw the diag. B F. Pof.1. B. 1.

3. From the point B in the Pgr. CE, draw the diag. B D.

4. Let a plane be paffed thro' C F & B F, alfo thro' B F & B D.

BECAUSE

- DEMONSTRATION.

ECAUSE AD is a pgr. cut by the diagonal CF. (Prep. 1). 1. The ▲ ACF base of the pyramid A B C F is to the ACF D, bafe of the pyramid B C F D.

But thofe pyramids A B C F & BC F D, have their vertices at the point B.

P.34. B. 1.

P. B.12.
5.
Cor. 1.

to

P.34. B. 1.

2: Therefore, the pyramid A B C F is to the pyramid B C F D. Likewife, the pgr. E C is cut by its diagonal B D. (Prep. 3). 3. Therefore, the ACBD, base of the pyramid BCFD is the ABDE, base of the pyramid D E F B. And those pyramids B CFD, &c. have their vertices at the pointF. 4. Confequently, the pyramid BCDF is to the pyramid BDEF. (P. 5. B.12. But the pyramid ABCF is alfo to the pyramid B C D F. Cor. 1. (Arg. 2).

Therefore, the pyramids A B C F, B C D F, & B D E F are equal. Ax.1. B. 1.

6. Confequently, the triangular prifm (ADE) may be divided into three triangular pyramids.

Which was to be demonftrated.

FROM

COROLLARY I.

ROM this it is manifeft, that every pyramid which has a triangular base, is the third part of a prifm which has the fame bafe, & is of an equal altitude with it.

COROLLARY II.

EVERY pyramid which has a polygon for base, is the third part of a prism which

has the fame bafe, & is of an equal altitude with it; fince it may be divided into prisms baving triangular bafes.

PRISM

COROLLARY III.

RISMS of equal altitudes are to one another as their bases, because pyramids upon the fame bafes, & of the fame altitude, are to one another as their bafes. (P. 6. B. 12).

SIM

THEOREM VIII.

IMILAR pyramids (A B C D & EFGH) having triangular baseş (BDC & F G H): are to one another in the triplicate ratio of that of their homologous fides.

Hypothefis.

The pyramids ABCD & E F G H have triangular bafes DBC&GFH, whose bomologous fides are BD & FG, &c.

Preparation.

Thefis.

The pyramid ABCD is to the pyramid
EFGH, in the triplicate ratio of BD to
F G, that is, as D B3 : FG?.

1. Produce the planes of the AB DC, ABD & ADC;
complete the pgrs. DR, DQ & D P.

2. Draw PO & OQ plle. to A Q & A P, & produce them

P.31. B. 1.

to O.

3. Join the points O & R ; & QC will be a

P.31. B. 1.

which will

4. After the fame manner defcribe the

MH.

have the fame altitude with the pyramid A B C D.

5. Infine, Join the points Q & P, alfo M & N, homologous
to the points B & C ; alfo F & H.

BECAUS

DEMONSTRATION.

ECAUSE the pyramids A B C D & EFGH are

(Hyp.).

1. All the triangular planes which form the pyramid A BCD are
to all the triangular planes which form the pyramid E F G H,
each to each.

2. Confequently, A D: BDEG: GF, &c.

to the plane VEGF.

DQ is

to the pgr. M G.

3. And the plane V ADB is Therefore the

4.

pgr.

5. Likewife, the pgr. oppofite ones AO,

DR & GI; DP, & G N are ; as alfo their
EL; QR, MI.

6. Confequently, AR & E I are

E.

7. Therefore, AR: EI= DB: F G3.
And fince the lines QP & BC; MN & FH, are diagonals fimi-
larly drawn in the equal & plle. pgrs. O A & RD; EL & IG.
(Prep. 5).

D. 9. B.11.
P.33. B.11.

8. The parts BQAPCD & F MEN HG will be a prifms: & SD. each equal to the half of its.

9. Confequently, the prifm BPQC : prifm FNMH

B.11.

9. P. 28. B.11. P.15. B. 5.

BD: FG.P.34. B.11.

Rem. I.

But the pyramid ABDC is the third part of the prifm BQPC, SP. 7. B.12. & the pyramid E F G H is the third part of the prifm FMNH. Cor. 1. 10. Therefore, the pyramid ABCD: pyramid EFGH➡ BD3 : FG3, P.15. B. 5. Which was to be demonftrated.

FROM

COROLLAR Y.

ROM this it is evident, that fimilar pyramids which have polygons for their bafes, are to one another in the triplicate ratio of their homologous fides, (because they may be divided into triangular pyramids; which are fimilar, taken tawo by two.

Eu-
clid.