Book XII. PROP. VI. THEOR. for their bases, are to one another as their bases. . See N. Let the pyramids which have the polygons ABCDE, FGHKL for their bases, and their vertices in the points M, N, be of the same altitude. as the base ABCDE to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN. Divide the base ABCDE 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 pyraniids having their common vertex in the point 1. 5. 12. 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 triangle FGH, as the pyramid ACDM to the pyramid FGHN; 24. S. and also the triangle ADE to the triangle FGH, as the pyramid ADEM to the pyramid FGHN; as all the first antecedents to their b. 2. Cor. common confequent, fo b are all the other antecedents to their com mon consequent; that is, as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN. and for the fame reason, as the base FGHKL to the base 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, so is the pyramid FGHN to the €. 22. 5. pyramid FGHKLN; therefore, ex aequali, as the base ABCDE to the the base FGHKL, so the pyramid ABCDEM to the pyramid Book XII. FGHKLN. Therefore pyramids, &c. Q. E. D. PROP. VII. THEOR. 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 it. the prism ABCDEF may be divided into three equal pyramids having triangular bafes. Join BD, EC, CD; and because ABED is a parallclogram of which BD is the diameter, the triangle ABD is equal to the tri- a. 34. 1. 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 b 5.12. the base is the triangle EBD, and vertex the point C. but this fyramid 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 fame 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 C B R bole Book XII. 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 triangle ABC, and vertex the point D, is the third part of the prism which has the same bafe, viz. the triangle ABC, and DEF is the opposite triangle. Q. É. D. CoR. I. From this it is manifest, that every pyramid is the third part of a prism which has the same base, and is of an equal altitude with it; for if the base of the prism be any other figure than a triangle, it may be divided into prisms having triangular bases. Cor. 2. Prisms of equal altitudes are to one another as their bases; because the pyramids upon the same bases, and of the same €. 6. 12. altitude, are o to one another as their bases. PROP. VIII. THEOR. another in the triplicate ratio of that of their homologous fides. Let the pyramids having the triangles ABC, DEF for their bases, and the points G, H for their vertices, be similar and similarly situated, the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which the side BC has to the homologous fide EP. Complete the parallelograms ABCM, GBCN, ABGK, and the solid parallelepiped BGML contained by these planes and those op K L X O R posite to them. and in like manner complete the solid parallelepiped EHPO contained by the three parallelograms DEFP, HEFR, DEHX, and those opposite to them. and because the pyramid 1. Def. ABCG is similar to the pyramid DEFH, the angle ABC is equal a to the the angle DEF, and the angle 'GBC to the angle HEF, and ABG Book Xii. to DEH, and AB is b.to BC, as DE to EF; that is, the sides about the equal angles are proportionals; wherefore the parallelogram b. 1. Detio. BM is similar to EP. for the same reason, the parallelogram BN is similar to ER, and BK to EX. therefore the three parallelograms BM, BN, BK are similar to the three EP, ER, EX. but the three BM, BN, BK are equal and similar to the three which are opposite c. 14. 15. to them, and the three EP, ER, EX equal and fimilar to the thrce opposite to them. wherefore the folids BGML, EHPO are contained by the same number of similar planes; and their folid angles are equal d; and therefore the folid BGML is similar a to the folid d. B.11. EHPO. but similar folid parallelepipeds have the triplicate o ratio c. 23.12. of that which their homologous fides have. therefore the folid BGML has to the solid EHPO the triplicate ratio of that which the side BC has to the homologous fide EF. but as the folid BGML is to the solid EHPO, so is f the pyramid ABCG to the pyramid f. 15. s. DEFH; because the pyramids are the sixth part of the folids, since the prism, which is the half of the folid parallelepiped, is triple b g. 28.11. of the pyramid. Wherefore likewise the pyramid ABCG has to the h. 7.12. pyramid DEFH, the triplicate ratio of that which BC has to the homologous fide EF. Q. E. D. Cor. From this it is evident, that similar pyramids which have see N. multangular bases, are likewise to one another in the triplicate ratio of their homologous sides. for, they may be divided into similar pyramids having triangular bases, because the similar polygons which are their bases may be divided into the same 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, fo are all the triangular pyramids in the first to all the triangular pyramids in the other; that is, so is the first multangular pyramid to the other, but one triangular py. , ramid 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 first has to the homologous Tide of the other. Book XII. PROP. IX. THE OR. angular 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 bafcs, and which have their vertices in the points G, H be equal to one another. the bafes 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 foli parallelepipeds BGML, EHPO contained by these planes and those opposite to them. and because the pyramid ABCG is equal to the pyramid DEFH, and that the folid BGML is fextuple of the pyramid ABCG, and the folid EHPO fextuple of the pyramid 1.1. X. 5, DEPH; therefore the folid BGML is equal to the folid EHPO. but the batis and altitudes of equal folid parallelepipeds are recipro6. 34. 11. caliy proportional b; therefore as the base EM to the base EP, so is the altitude of the folid EHPO to the altitude of the folid BGML, e. 15. 5. but as the base BM to the base EP, fo is the triangle ABC to the ti nye DEF; therefore as i he triangle ABC to the triangle DEF. 10 is iliu altitude of the found Euro to the aititude of the folid ROML, but she altitude of the fox EHPO is the same with the altitude cishe pyramid DEFH; an! the altitude of the folid |