And the of AEO of INOACEG: T. (Sup.). 10. Therefore, the polyg. ADFH: polyg. ILOQ=OACÉG: T. But the polygon ADF H is < O ACEG. 11.Confequently, the polygon ILOQ is < T.

But the polygon ILOQ is > T. (Arg. 9). 12.Therefore, T will be > & < the polyg. ILOQ (Arg. 9. & 11). 13.Which is impoffible.

14.Therefore, T is not <OILP.

15. From whence it follows, that the of the diameter (A E) of a (ACE G), is not to the of the diameter (IN) of another O (ACEG) to a space<the fecond (ILP). II. Suppofition.

(ILP), as the first

Let the space T be the circle ILP.

P.11. B. 5.

Ax.8. B. 1.
P.14. B. 5.

BECAUSE

ECAUSE the □ of A E: □ of IN=©ACEG: T. 16.Invertendo T:OACEG of IN: Д of A E.

But TACEG OILP: V. (II. Prep.).
Moreover, T is > OIL P. (II. Sup.).

17.Confequently, the ACEG is alfo > V.

Befides TACEG

And TO ACEG

18. Therefore, the of IN: of A E But V ACEG. (Arg. 17).

of IN: of AE (Arg. 16). OILP: V. (II. Prep.). OILP: V.

P.11. B. 5.

And it has been demonstrated (Arg.15), that the of the diameter (IN) of a (IL P), is not to the of the diameter of another O (ACEG); as the first (ILP) to a space

(ACEG).

19.Confequently, V is not < the 20. Therefore, T is not the Therefore, the space T being (Arg. 14. & 19).

21.T will be to this OIL P.

IL P.
ILP.

the fecond

neither <nor> the ILP,

22.Confequently, the OACEG: OILP☐ of AE:□ of IN. P. 7. B. 1.

CIRCLES

IRCLES are to one another as the polygons infcribed in them (P. 1. B. 12. & P. 11. B. 5).

EVER

VERY pyramid (ABCD) having a triangular base (A CD), may be divided into two equal and fimilar prifms, (I DE FLG & GLFHCE), and into two equal and fimilar pyramids, (LGIA & L F H B), which are fimilar to the whole pyramid; and the two prifms together are greater than half of the whole pyramid (A B C D).

Hypothefis.

ABCD is a pyramid whose bafe
ADC is a A.

Thefis.

1. The part IDE FLG is a prifm = && to the part GLFECH.

II. The part ALGI is a pyramid = & ∞ to the part B LF H.

III. Thofe pyramids ALGI& BLFH are ∞ to the pyramid A B C D.

IV. The prifms IDEFLG&GLFCH are together than the half of the pyr. A B C D. I. Preparation.

1. Cut all the fides of the pyramid ABCD into two equal
parts, in the points L, F, H, E, G, & I.

2. Draw the lines L F, FH, FE, GE, GI & IL, also
LG, & LH.

BECAUSE

I.

DEMONSTRATION.

ECAUSE in the ABCD the fides B D & B C are divided into two equal parts in the points F & H (Prep. 1).

3. Confequently, FH is plle. to DC.

BH HCBF: D F.

}

3. Likewife, FE is plle. to B C.

4. Therefore, FECH is a pgr.

5. It may be proved after the fame manner, that LFEG & LGCH are pgrs.

And fince F H & H L are plle. to EC & GC. (Arg. 2. & 5).

6. The planes paffing thro' LF H & ECG will be plle.

7. Therefore, LGECHF will be a prifm.

8. Likewife, LFEDIG will be also a prifm.

}

P.10. B. 1.

Pof.1. B. 1.

P.19. B. 5.

P. 2. B. 6.

D.35. B. 1.

P.15. B.11.

D.13.B. 11.

But those two prifms have the fame altitude LG, & the pgr.GIDE which is the base of the prifm LD is double of the ACEG, base of the prifm LC.

9. Therefore, the prifm L D is

P.41. B. 1.

to the prifm L C.

P.40. B.11.

Which was to be demonftrated. 1.

BECAUSE the fide B D is cut into two equal parts in F, that

FE & DE are plle. to BC & F H, each to each. (Prep. 1. &
Arg. 2. & 3).

10. The AF DE is & to ABFH.

11. The AF ED & IL G are also equal.

12. Therefore, ABFHALIG.

And fince the other fides of the pyramid A B C D are divided into two equal parts.

It may be easily proved that,

13.ABLF is to the ALAI, ABLH = AA GL, &

ALFHAAGI.

14. From whence it follows, that those parts B L HF & ALGI are

equal &

THE

pyramids.

SP.26. B. 1.
P. 7. B. 6.
D.13. B.11.
Ax.1. B. 1.

D.10. B.11.

Which was to be demonftrated. 11.

P. 2. B. 6.

HE line F H, is plle. to D C. (Arg. 2). 15.Therefore, ABFH is ABDC.

Likewife, all the triangles which form the pyramids BLHF & ALGI are to all the triangles of the whole pyramid A B C D.

16. Therefore, the pyramids BLHF & ALGI, are as to the py-
ramid A B C D.

Which was to be demonstrated. 111.
II. Preparation.

THE

Draw G H & EH.

HE line BH being

to HC (I. Prep. 1.) FHEC

P. 4.

B. I.

SP.

D.13. B.11.
D.10. B.11.

Ax.8. B. 1.

(Arg. 4) & VECH-VFHB (P. 29. B. 1). 17.Confequently, the AECH is to the BFH. 18.Alfo the AHGC & GEC are & to the ABLH & LHF. 19.Therefore, the pyramid L F HB is to the pyramid H G E C. But the pyramid ECHG is only a part of the prifm ECHFLG. 20. Therefore, the prifm ECHFLG is the pyramid ECHG. 21.Confequently, this prifm ECH FLG is alfo the pyramid LFHB. P. 7. B. 5. The prifm LG ECHF is to the prifm EFLGID, & the pyramid LFHB to the pyramid A IGL (Arg. 9. & 14). 22. Therefore, the prifm EFLGID is alfo > the pyramid A IGL. 23.Therefore, the two prifms ECHFLG & EFLGID together, will be the two pyramids B LFH & LAIG together. 24.From whence it follows, that the two prifms ECHFLG & EFLGID together, are > the half of the given pyr. A B C D. Which was to be demonftrated. IV.

Ax.4. B. 1.

PROPOSITION IV.

THEOREM IV.

IF there be two pyramids (A BCD & E FGH) of the fame altitude,

upon triangular bafes (A B C & E F G), and each of them be divided into two equal pyramids fimilar to the whole pyramid, (viz. the pyramid ABCD into the pyramids DL KM & ANIL, and the pyramid EFGH into the pyramids HRQS & REPT); and also into two equal prifms, (viz. the pyramid ABCD into the prifms LB & LC, and the pyramid EF G H into the prifms R F & R G); and if each of thefe pyramids (DLK M, ANIL, HRQS, & REPT) be divided in the fame manner as the first two, and fo on. The bafe (A B C), of one of the first two pyramids (A B C D), is to the bafe (E F G) of the orher pyramid (E F G H), as all the prifms contained in the first pyramid (A BCD), is to all the prifms contained in the fecond (E F G H), that are produced by the fame number of divisions.

Hypothefis.

1. The triangular pyramids ABCD & EFGH, have the fame altitude. II. Each of them are cut into two equal prisms LB & LC; alfo RF & RG, & into tuo equal pyramids fimilar to the whole pyramid. III. Each of thofe pyramids LDMK,LNIA,RTPE &RQSH, are fuppofed to be divided in the fame manner as the first two, & so on.

BECAUSE

Thefis.

The fum of all the prifms contained in the pyramid ABCD is to the sum of those contained in the pyramid EFGH, being equal in number; as the base ABC, of the pyramid ABCD is to the bafe EFG, of the pyramid E F G H.

DEMONSTRATION.

ECAUSE the pyramids A B C D & E F G H have equal altitudes, & the prifms L B, LC, RF & RG have each the half of this altitude, (Hyp. 1. & P. 3. B. 12).

1. Those prins LB, LE, RF & RG have the fame altitude.
The lines B C & F G are cut into two equal parts in the points
O & V.

Ax.7. B. 1.

P. 3. B.12.

SP.19. B. 5.
P.16. B. 5.

2. Therefore, CB: CO GF: GV. 3: Confequently, ΔΑΒΕ: ΔΙΟC=ΔΕΡG: ΔTVG. Ρ.22. Β. 6. 4. And alternando AABC:AEFG=AIOC: ATVG. P.16. B. 5. 5. Moreover, bafe IOC bafe TVG prifm LKMCOIS Cor. 3. Rem. prifm RQSG V T. ofP.35.B.11.

6. And prifm L. KOBNI: prifm LKMCOI prifin RQVFPT: prifm RQSGVT (having the fame altitude (Arg. 1.) & being equal taken two by two (Hyp. 11).

P. 7. B. 5.

7. Confequently, prifm LB + prifm LC: prifm LC

prifm R F

+prifm RG: prifin R G.

P.18. B. 5.

8. And alternando, prifin L B + prifm LC: prifm R F+ prifm RG

=prifm LC: prifm R G.

P.16. B. 5.

But prifin LC: prifm R G

9.

bafe IOC: bafe TVG (Arg. 5). And bafe IOC: bafe TVG bafe A B C : bafe EFG (Arg. 4). Therefore, the prifm L B+ pr. LC: pr. RF+ pr. RG base ABC bafe E F G.

If the remaining pyramids LK MD & LINA; alfo R QSH &
EPTR, be divided after the fame manner as the pyramids A B C D
& EFGH: it may be proved after the fame manner.

10. That the four pyramids refulting from the firft pyramids LKMD
& ANIL, will have the fame ratio to the four prifms refulting
from the laft RQSH & EPTR, that the bafes L KM & ANI
have to the bases RQS & EPT (Hyp. 111. & Arg. 9).
And it has been demonftrated, that the bafes LKM & AN I, are
each IOC; alfo RQS & EP T, each TVG.
Moreover, ▲ ABC: AEFGAIOC: ATVG (Arg.4).
11.Wherefore, the fum of all the prifms contained in the pyramid
ABC is to the fum of all the prifms contained in the pyramid
EFGH, as the bafe A B C is to the base E F G.

Which was to be demonstrated.

P.11. B. 5.

P.12. B. 5.