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PROPOSITION VI. THEOREM VI. YRAMIDS (F GLIM & A B C D E) of the fame altitude, which have polygons (F GHLI, & A B C D) for their bases : are to one another as their bales. Hypothesis.

Thefis. 1. The pyramids FGHLI& ABCD, Pyram. MFGHLI: Pyram. ABCDE bave polygons for their bases.

= base FILHG : bafe ABCD 11. They have the same altitude.

Preparation.
1. Divide the bases FILHG & ABCD into triangles,

by drawing the lines GI, FH; & DB.
2. Let planes be passed thro' thofe lines & the vertices of
the pyramids, which will divide each of those pyramids
into as many pyramids as each base contains triangles.

DEMONSTRATION.
ECAUSE the triangular pyramids I L HM & A B D E have

the same altitude. (Hyp. 11. & Prep. 2).
1. The pyramid I HLM: pyr. A B D E = base HIL : base ABD.

P. 5. B.12. 2. Likewise, pyr. GIHM: pyr. ABDE= base HIG : base ABD. 3, Consequently, pyr. IHLM + pyr. GIHM : pyr. ABDE = base HIL + base HIG : bale A BD.

P.24. B. 5. 4. Moreover, pyr.

FIGM:

pyr.
ABDE = base FIG : base ABD, P. 5.

B.12. 5 Therefore, pyr. IHLM+ pyr. GIHM + pyr. FIGM: pyr.

A B D E= base HIL + base HIG + base FIG: base A B D. P.24. B. 5. But pyr. IHLM+pyr. GIHM + pyr. FIG M are = to the pyr. MFGHLI, & the base HÍL + base HIG + base Ax.1. B. 2.

FIG = base FIL HG. 6. Consequently, pyr. MFGHIL: pyr. AB DE= base FILHG base A BÚ.

P. 7. B. 5 It may be proved after the same manner, that 7. Pyr. MFGHLI: pyr. B DCE= base FILHG : base BDC. 8. Therefore, pyr. MFGHLI : pyr. ABCDE = base FILHG : base A D C B.

P.25. B. 5. Which was to be demonstrated,

BECAU 5

Ax.1. B. 2. {

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PROPOSITION VII. THEOREM VII.
VER Y triangular prism_(ADE): may be divided (by planes passing
through the ABCF & B D F) into three pyramids (A CBF, B D E F &
DCB F) that have triangular bases, and are equal to one another.
Hypothesis.

Thesis.
The given prism A D E has a

The prism A D E may be divided into triangular baje.

three equal triangular pyramids, ACB F, BDEF, DCBF.

Preparation.

1. In the pgr. D A draw any diagonal C F.
2. From the point F in the pgr. Å E, draw the diag. B F. & Poli1. B. .
3. From the point B in the pgr. C E, draw the diag. B D.
4. Let a plane be passed thro' F & B F, also thro'Bř & BD.

- DEMONSTRATION.
Because

ECAUSE A D is a pgr. cut by the diagonal CF. (Prep.1). 1. The A AC F base of the pyramid A B C F is = to the ACFD, base of the pyramid B C F D.

P.34. B. 1
But those pyramids A B CF.& BCF D, have their vertices at the
2: Therefore, the pyramid ABCF is = to the pyramid BCFD. 25, B.12.

Likewise, the pgr. E C is cut by its diagonal B D. (Prep. 3).
3. Therefore, the ACBD, base of the pyramid BCFD is = to
the ABDE, base of the pyramid D E F B.

P.34. B..
And those pyramids B CFD, &c. have their vertices at the pointF.
4. Consequently, the pyramid BCDF is = to the pyramid B DEF. SP. 5. B.12.

But the pyramid A B C F is also = to the pyramid B C DF. I Cor. 1.

(Arg. 2). 5. Therefore, the pyramids ABC F, BCDF, & BD E F are equal. Ax.1. B. 1,

point B.

Cor. 1.

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6. Consequently, the triangular prism (ADE) may þe divided into three triangular pyramids.

Which was to be demonstrated.

COROLLA R r I. From

ROM ibis it is manifeft, that every pyramid which has a triangular base, is the third part of a prism which has the same baje, & is of an equal altitude with it.

COROLLA R r II.

a ,

VERY pyramid which has a polygon for base, is the third part of a prism which has the same base, & is of an equal altitude with it ; fince it may be divided into prisms baving triangular bases.

COROLL ART III. Prisms

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

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PROPOSITION VIII. THEOREM VIII. IMILAR pyramids (A B C D & EF GH) having triangular bases

& (B D C & F G H): are to one another in the triplicate ratio of that of their homologous fides. Hypothefis.

Thesis. The e pyramids A B C D & E F G H have The pyramid ABCD is to the pyramid triangular bases DBCEGFH, whose bo EFGH, in the triplicate ratio of BD to mologous fides are BD & FG,&c. FG, that is, as DB::FG:.

Preparation.
1. Produce the planes of the ABDC, A B D & A DC;
complete the pgrs. DR, DQ & D P.

P.31. B. 1. 2. Draw P O & OQ plle. to A & A P, & produce them

P.31. B. 1. 3. Join the points O&R; & QC will be a which will

have the same altitude with the pyramid A B C D.
4. After the same manner describe the MH.
5. Infine, Join the points Q & P, also M & N, homologous

to the points B & C ; also F & H.

to O.

A H

DEMONSTRATION. BE

E CAUSE the pyramids A B C D & EF G H are Ns (Hyp.). 1. All the triangular planes which form the pyramid ABCD are o

to all the triangular planes which form the pyramid E F GH,
each to each.

9. 2. Consequently, AD: BD=EG:GF, &c.

D. 1. B. 6. 3. And the plane V AD B is = to the plane VEGF.

EGI

P. 5. B. 6. Therefore the pgr. D Q is no to the 4.

MG.
Pgr.

D. 1. B. 6. 5. Likewise, the pgr. D R&GI; DP, & G N are w; as also their opposite ones AO, EL; QR, MI.

P.24. B.u.

D.

B.II.

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Rem. I.

6. Consequently, AR & E I are

D. 9. B.u. 7. Therefore, GAR:&EI= DB:: FGS.

P.33.B.11. And fince the lines QP & BC;MN& FH, are diagonals fimilarly drawn in the equal & plle. pgrs. O A&RD; EL & IG.

(Prep. 5). 8. The parts B QAPCD & FMENH G will be s prisms : & SD. 9. B.11. each equal to the half of its e.

P.26. B.11.

(P.15. B. 5. 9. Consequently, the prisim B P QC: prism FNMH = BD® : FG: P-34. B.ii.

But the pyramid ABDC is the third part of the prism BQPC, SP. 7. B.12.

& the pyramid EFGH is the third part of the prism FMNH Cor. 1. 10. Therefore, the pyramid ABCD : pyramid.EFGH = BDS : FG. P.15. B. 5.

Which was to be demonstrated.

COROLLA R r. From

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

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