) , 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. by drawing the lines GI, FH; & DB. DEMONSTRATION. the same altitude. (Hyp. 11. & Prep. 2). 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. 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. { I Every PROPOSITION VII. THEOREM VII. Thesis. 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. - DEMONSTRATION. 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 Likewise, the pgr. E C is cut by its diagonal B D. (Prep. 3). P.34. B.. 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. 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). 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. 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. 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, 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. 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. 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. Eu- |