Book XII. PROP. VI. THEOR. See N. a 5. 12. 2. Cor. 24.5. PY YRAMIDS of the fame altitude which have polygons for their bases, are to one another as their bases. Let the pyramids which have the polygons ABCDE, FGHKL for their bales, and their vertices in the points M, N, be of the me altitude: As the bafe ABCDE to the base FGHKL, fo is the pyramid ABCDEM to the pyramid FGHKLN. Divide the bafe ABCDE into the triangles ABC, ACD, ADE; and the bafe FGHKL into the triangles FGH, FHK, FKL: And upon the bafes ABC, ACD, ADE let there be as many pyramids of which the common vertex is the point M, and upon the remaining bafes as many pyramids having their common vertex in the point N: Therefore, fince 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; and also the M a AA B D G H K triangle ADE to the triangle FGH, as the pyramid ADEM to the pyramid FGHN; as all the first antecedents to their com mon confequent; fo are all the other antecedents to their com. mon confequent; that is, as the bafe ABCDE to the bafe FGH, fo is the pyramid ABCDEM to the pyramid FGHN: And, for the fame reafon, as the base FGHKL to the bafe FGH, fo is the pyramid FGHKLN to the pyramid FGHN; And, by inverfion, as the bafe FGH to the bafe FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN: Then, because as the bafe ABCDE to the bafe FGH, fo is the pyramid ABCDEM to the pyramid FGHN; and as the bafe FGH to the base FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN; therefore, 1 therefore, ex aequali, as the bafe ABCDE to the bafe FGHKL, Book XII. fo the pyramid ABCDEM to the pyramid FGHKLN. There n fore pyramids, &c. Q. E. D. E PROP. VII. THEOR. VERY prifm having a triangular base may be divi ded into three pyramids that have triangular bases, and are equal to one another. Let there be a prifm of which the bafe is the triangle ABC, and let DEF be the triangle oppofite to it: The prifm ABCDEF may be divided into three equal pyramids having triangular bafes. b C 22. S. Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal to a 34. 1. the triangle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the b 5. 12. pyramid of which the bafe is the triangle EBD, and vertex the point C: But this pyramid is the fame with the pyramid the bafe 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 bafe is the triangle ABD, and vertex the point C, is equal to the pyramid, the bafe 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 of which the base is the triangle ECB, and vertex the point D, is equal to the pyramid, the base of which is the triangle ECF, and vertex the point D: But the pyramid of which the bafe is the triangle ECB, and vertex the point D has been proved equal to the pyramid of which the D F B bafe is the triangle ABD, and vertex the point C. Therefore the prifm ABCDEF is divided into three equal pyramids having triangular bases, 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 fame planes; and that the pyramid of which the bafe is the triangle ABD, and vertex the point C, has been demonstrated Book XII. demonftrated to be a third part of the prifm the bafe of which is the triangle ABC, and to which DEF is the oppofite triangle therefore the pyramid of which the bafe is the triangle ABC and vertex the point D, is the third part of the prifm which ha the fame bafe, viz. the triangle ABC, and DEF is the oppofit triangle. QE. D. € 6. 12. COR. 1. From this it is manifeft, that every pyramid is the third part of a prifm which has the fame base, and is of an equal altitude with it; for if the base of the prifm be any other figure than a triangle, it may be divided into prifms having triangular bafes. COR. 2. Prifms of equal altitudes are to one another as their bases; because the pyramids upon the fame bases, and of the fame altitude, are to one another as their bases. SIMIL IMILAR pyramids having triangular bases are one to another in the triplicate ratio of that of their homologous fides. Let the pyramids having the triangle ABC, DEF for their bafes, and the points G, H for their vertices, be fimilar and fimilarly fituated; the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which the fide BC has to the homologous fide EF. Complete the parallelograms ABCM, GBCN, ABGK, and the folid parallelepiped BGML contained by these planes and thofe oppofite to them: And, in like manner, complete the folid parallelepiped EHPO contained by the three parallelograms DEFP, HEFR, DEHX, and thofe oppofite to them: And be caufe a с II. cause the pyramid ABCG is fimilar to the pyramid DEFH, the Book XII. angle ABC is equal to the angle DEF, and the angle GBC to the angle HEF, and ABG to DEH: And AB is to BC, 11. def. as DE to EF; that is, the fides about the equal angles are pro- b 1. def. 6. portionals; wherefore the parallelogram BM is fimilar to EP: For the fame reason, the parallelogram BN is fimilar to ER, and BK to EX: Therefore the thrée parallelograms BM, BN, BK are fimilar to the three EP, ER, EX: But the three BM, BN, BK, are equal and fimilar to the three which are oppo, e 24. II. fite to them, and the three EP, ER, EX equal and fimilar to the three oppofite to them: Wherefore the folids BGML, EHPO are contained by the fame number of fimilar planes; and their folid angles are equal; and therefore the folid d B. II. BGML is fimilar to the folid EHPO: But fimilar folid parallelepipeds have the triplicate ratio of that which their homologous fides have: Therefore the folid BGML has to the folid EHPO the triplicate ratio of that which the fide BC has to the homologous fide EF: But as the folid BGML is to the folid EHPO, fo is f the pyramid ABCG to the pyramid DEFH; f 15. 5. because the pyramids are the fixth part of the folids, fince the prifm, which is the half of the folid parallelepiped, is triple" g 28. II, of the pyramid. Wherefore likewife the pyramid ABCG hash 7. 12. to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous fide EF. Q. E. D. 33. II. COR. From this it is evident, that fimilar pyramids which See N, have multangular bafes, are likewife to one another in the triplicate ratio of their homologous fides: For they may be divided into fimilar pyramids having triangular bafes, because the fimilar polygons, which are their bafes, may be divided into the fame number of fimilar triangles homologous to the whole polygons; therefore as one of the triangular pyramids in the firft 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, fo is the first multangular pyramid to the other: But one triangular pyramid is to its fimilar triangular pyramid, in the triplicate ratio of their homologous fides; and therefore the firft multangular pyramid has to the other, the triplicate ratio of that which one of the fides of the firft has to the homologous fide of the other. PROP. Book XII, THE PROP. IX. THE OR. HE bafes and altitudes of equal pyramids having triangular bafes are reciprocally proportional: And triangular pyramids of which the bafes and altitudes are reciprocally proportional, are equal to one another. Let the pyramids of which the triangles ABC, DEF are the bafes, 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 bafe ABC is to the bafe 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 folid parallelepipeds BGML, EHPO contained by thefe planes and thofe 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 DEFH; therefore the folid a x. Ax. 5. BGML is equal to the folid EHPO: But the bafes and altitudes of equal folid parallelepipeds are reciprocally propor b 34. II. tional; therefore as the bafe BM to the bafe EP, fo is the altitude of the folid EHPO to the altitude of the folid BGML: But as the base BM 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 alti tude of the folid BGML: But the altitude of the folid EHPO is the fame with the altitude of the pyramid DLFH; and the altitude of the folid BGML is the fame with the altitude of the e 15. 5. pyramid |