But those two prifms have the fame altitude LG, & the pgr.GIDE which is the base of the prifm LD is double of the CEG, base P.41. B. 1. Which was to be demonftrated. 1. BECAUSE ECAUSE 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 & s to ABF H. 11. The ▲ FED & 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 = ▲AGL, & ALFHAAGI 14. From whence it follows, that thofe parts B L HF & AL GI are equal & TH pyramids. SP.26. B. 1. D.10. B.11. Which was to be demonftrated. 11. P. 2. B. 6. HE line FH, 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 BL HF & ALGI, are ramid A B C D. to the py Which was to be demonftrated. 111. II. Preparation. TH Draw G H & EH. HE line BH being to HC (I. Prep. 1.) FH EC (Arg. 4) & ECH 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 H B is to the pyramid H G E C. But the pyramid ECHG is only a part of the prifm ECHFLG. 20. Therefore, the prifm E CHFLG is the pyramid ECHG. 21.Confequently, this prifm ECHFLG is alfo the pyramid LFHB. P. 7. B. 5. The prifm LG ECHF is to the prifm EFLGID, & the pyramid LFHB to the pyramid AIGL (Arg. 9. & 14). 22. Therefore, the prifin E FLGID is alfo > the pyramid A IGL. 23.Therefore, the two prifms ECHFLG & EFLGID together, will be the two pyramids BL FH & 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 A BCD into the pyramids DL KM & ANIL, and the pyramid EFGH into the pyramids HRQS & REP T); and alfo into two equal prifms, (viz. the pyramid A B CD into the prifms LB & LC, and the pyramid EFGH into the prifms R F & RG); 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 (ABCD), is to the bafe (E F G) of the orher pyramid (E F G H), as all the prisms contained in the first pyramid (A B C D), is to all the prifms contained in the fecond (E F G H), that are produced by the fame number of divifions. Hypothefis. 1. The triangular pyramids ABCD & EFGH, have the fame altitude. II. Each of them are cut into two equal prifms LB & LC; also RF & RG, & into two 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. BECAUS Thefis. The fum of all the prifms contained in the pyramid ABCD is to the fum 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. Thofe prifms LB, LC, RF & R G have the fame altitude. Ax.7. B. 1. P. 3. B.12. 3. Confequently, 2. Therefore, CB: COGF: GV. AABC:AI0C=AEFG:AT V G. 4. And alternando AABC: AEFG=AIOC: ATVG. 6. And prifm LKOBNI: prifm LKM CÔI prism RQVFPT: {P.19. B. 5 ofP.35.B.11. P. 7. B. 5. 7. Confequently, prifm L B+ prifin LC: prifin LC prifm R F +prifm RG: prifm R G. P.18. B. 5. 8. And alternando, prifin L B+ prifm LC: prifin R F + prifin RG =prifm LC: prifm R G. P.16. B. 5. But prifin LC prifm R G bafe IOC bafe TVG (Arg. 5). And bafe IOC: bafe TVG bafe A B C : base EFG (Arg. 4). 9. Therefore, the prifm L B+ pr. L C: pr. R F + pr. R G = base ABC bafe EF G. If the remaining pyramids LK MD & LINA; alfo R QSH & 10. That the four pyramids refulting from the firft pyramids LKMD Which was to be demonftrated. P.11. B. 5. P.12. B. 5. PROPOSITION V. THEOREM V. PYRAMIDS (ABCD & E F G H) of the fame altitude, which have triangular bases (A B C & E F G): are to one another as their bases, (ABC & E F G). Hypothefis. Thefis. I. The pyramids ABCD&EFGH have for Pyram. ABCD: pyram.EFGH= bafes the ABC & EFG. bafe ABC: bafe EFG. II. They have the fame altitude. If not, DEMONSTRATION, Pyramid ABCD pyramid E F G H> bafe ABC: Preparation. 1. Take a folid X which may be > the pyramid ABCD, BECAUSE ECAUSE the two prifms refulting from the first divifion, are > the half of the pyramid A B C D; & the four following, refulting from the fecond divifion, are > than the halves of the pyramids refulting from the firft divifion, & fo on. 1. It is evident, that the fum of all the prifms contained in the pyramid A B CD, will be > the folid X, which was supposed to be < the pyramid A B C D. P. 3. B.12. Lem. B. 12. But all the prifms contained in the pyramid A B C D, are to all the prifms contained in the pyramid E F G H, as the base A B C is to the bafe E F G. And the folid X: pyramid E F G H = bafe A B C : base E F G (Prep. 1). 2. Confequently, all the prifins contained in the pyramid A B C D are to all the prifins contained in the pyramid E F G H, as the solid X is to the pyramid E F G H. P. 4. B.12. P.11. B. 5. 3. Therefore, all the prifins contained in the pyramid E F G H, are > the pyramid E F G H itself. But all the prifms contained in the pyramid A B C D, are > the folid X. (Arg. 1). 4. Which is impoffible. 5. Confequently, a folid (as X) which is the pyramid ABCD, cannot have the fame ratio to the pyramid E F G H, which the bafe ABC, has to the base E F G. And as the fame demonstration holds for any other solid greater than the pyramid A B C D. 6. It follows, that the pyramid A B CD: pyramid E F G H = bafe ABC base E F G. PYRAMID P.14. B. 5. YRAMIDS of the fame altitude, & which have equal triangles for their bafes are equal. (P. 14. & 16. B. 5.). COROLLARY II. EQUAL pyramids which bave equal triangles for their bases : bave the fame altitude. Rr |