Book XII. Sec N. PRO P. VI. THEOR. for their bases, are to one another as their bafes. Let the pyramids which have the polygons ABCDE, FGHKL for their bafes, and their vertices in the points M, N, be of the fame altitude. as the bafe ABCDE to the base FGHKL, fo is the pyramid ABCDEM to the pyramid FGHKLN. Divide the base ADCDE into the triangles ABC, ACD, ADE; and the base FGHKL into the triangles FGH, FHK, FKL. and upon the bases ABC, ACD, ADE let there be as many pyramids of which the common vertex is the point M. and upon the remaining bases as many pyramids having their common vertex in the point N. therefore, since the triangle ABC is to the triangle FGH, as * the pyramid ABCM to the pyramid FGHN; and the triangle ACD to the triangleFGH, as the pyramid ACDM to the pyramid FGHN; and also the triangle ADE to the triangle FGH, as the pyramid a. 5. 11. 24. 5. Ć H ADEM to the pyramid FGHN; as all the first antecedents to their B. 1. Cor. common consequent, fub are all the cther ariccedents to their com mon confequent; that is, as the base ABCDE to the base FGH, fo is the pyramid ABCDEM to the pyramid FGHN. and for the fame reason, as the base FGHKL to the bafe FGH, so is the pyramid FGHKLN to the pyramid FGHN. and, by inversion, as the base FGH to the base FGHKL, so is the pyramid FGHN to the pyramid FGHKLN. then because as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN; and as the base FGH to the base FGHKL,fo is the pyramid FGHN to the pyramid FGHKLN; therefore, ex aequali', as the bale ABCDE tu the base FGHKL, so the pyramid ABCDEM to the pyramid Book XIT. FGHKLN. Therefore pyramids, &c. Q. E. D. PRO P. VII. THEO R. divided into three pyramids that have triangular bases, and are equal to one another. Let there be a prism of which the base is the triangle ABC, and let DEF be the triangle opposite to is the prism ABCDEF may be divided into three equal pyramids baving triangular bases. Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal to the ui- a. 34. to angle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal b to the pyramid of which 5.5.1, the base is the triangle EBD, and vertex the point C. but this pyramid is the same with the pyramid the base of which is the triangle EBC, and vertex the point D; for they are contained by the same planes. therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the pyramid the base of which is the triangle EBC, and vertex the point D. again, because FCBE is a parallelogram of which the diameter is CE, the triangle ECF is equal' to the triangle ECB ; therefore the pyramid D E of which the base is the triangle ECB, and vertex the point D, is equal to the pyr?mid the base of which is the triangle ECF, and vertex the point D. but the pyramid of which the base is the triangle ECB, and A vertex the point D has been proved equal to the pyramid of which the base is the triangle ABD, and vertex the point C. Therefore the prism ABCDEF is divided into three equal pyramids having triangular bafes, viz. into the pyramids ABDC, EBDC, ECFD, and because the pyramid of which the base is the triangle ABD, and vertex the point C, is the fame with the pyramid of which the base is the triangle ABC, and vertex the point D, for they are contained by the same planes ; and that the pyramid of which the base is the triangle ABD, and vertex the point C, has been demonstrated to be a third part of the prima che R Book XI base of which is the triangle ABC, and to which DEF is the op: posite triangle; therefore the pyramid of which the base is the tri- Cor. From this it is manifest, that every pyramid is the third Cor. 2. Prisms of equal altitudes are to one another as their bafes; because the pyramids upon the fame bases, and of the fame C. 6. 12. altitude, are o to one another as their bases. PRO P. VIII. THEO R. SIM one to another in the triplicate ratio of that of Let the pyramids having the triangles ABC, DEF for their bases, and the points G, H for their vertices, be similar and similarly firuate.l. the pyramid ADCG has to the pyramid DEFH, the triplicate ratio of that which the fide BC has to the homologous fide EF. Complete the parallelograms ABCM, CBCN, ABGK, and the folid parallelepiped BGML contained by these planes and those op К. L polite to them. and in like manner complete the folid parallelepi fed EHPO contained by the three parallelograms DEFP, HEFR, a. u. D«. DEHX, and those oprofite to them. and because the pyramid ABCG is simil:r to the pyramid DEFH, the angle ABC is equal * to the angle DEF, and the angle GBC to the angle HEF, and ABG Book xļi. to DEH. and AB is b to BC, as DE to EF; that is, the fides about the equal angles are proportionals; wherefore the parallelogram b. 1. Def.6. BM is similar to EP. for the fame reason, the parallelogram BN is fimilar to ER, and BK to EX. therefore the three parallelograms DM, BN, PX are similar to the three EP, ER, EX. but the three EM, BN, BR are equal and fimilar to the three which are op- C. 24. 17. posite to them, and the three EP, ER, EX equal and fimilar to the three opposite to them. wherefore the folids BGML, CHPO are contained by the same number of similar planes; and their folid angles are equal d; and therefore the folid BGML is fimiler a to d. B. 17. the folid EHIPO. but similar folid parallelepipeds have the triplicate e ratio of that which their homologous fides have. therefore the c. 33. IT. folid BGML has to the folid EHPO the triplicate ratio of that which the side BC has to the homologous fide EF. but as the folid BGML is to the folid EHPO, so is f the pyramid ABCG to the f. 15. 5. pyramid DEFII; because the pyramids are the fixth part of the folids, since the prism, wlich is the half & of the folid parallelepiped, s. 28. nr. is triple hof the pyramid. Wherefore likewise the pyramid ABCG h. 7. 12. nas to the pyramid DEFII, the triplicate ratio of that which BC has to the homologous fide EF. 0. E. D. Cor. From this it is evident, that liinilar pyramids which have Sec N. multangular bases, are likewise to one another in the triplicate ratio of their homologous Gdes. for, they may be divided into similar pyramids having triangular bases, because the similar polygons which are their bases may be divided into the fame number of similar triangles homologous to the whole polygons; therefore as one of the triangular pyramids in the first multangular pyramid is to one of the triangular pyramids in the other, so are all the triangular pyramids in the first to all the triangular pyramidls in the other; that is, so is the first multangular pyranaid to the other. but one triangular pyramid is to its similar triangular pyramid, in the triplicate ratio of their homologous fides; and therefore the first multangular pyramid has to the other, the triplicate ratio of that which one of the sides of the firft has to the homologous side of the other. Book XII. PRO P. IX. THEO R. triangular bases are reciprocally proportional, and triangular pyramids of which the bases and altitudes are reciprocally proportional, are equal to one another. Let the pyramids of which the triangles ABC, DEF are the bases, and which have their vertices in the points G, H be equal to one another. the bases and altitudes of the pyramids ABCG, DEFH are reciprocally proportional, viz. the base ABC is to the base DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG. Complete the parallelograms AC, AG,.GC, DF, DH, HF; and the solid parallelepipeds BGML, EHPO contained by these planes . R A B D E and those opposite to them. and because the pyramid ABCG is equal to the pyramid DEFH, and that the solid BGML is sextuple of the pyramid ABCG, and the solid EHPO fextuple of the py*. 1. Ax. 5. ramid DEFH; therefore the solid BGML is equal to the folid EHPO. but the bases and altitudes of equal folid parallelepipeds are b. 34. 11. reciprocally proportional b; therefore as the base BM to the base EP, fo is the altitude of the solid EHPO to the altitude of the solid 6. 15. 5. BGML. but as the base EM to the base EP, fo is the triangle ABC to the triangle DEF; therefore as the triangle ABC to the triangle DEF, fo is the altitude of the folid EHPO to the altitude of the solid BGML. but the altitude of the solid EHPO is the fame with the altitude of the pyramid DEFH; and the altitude of |