the solid Q is greater than the prisms in the pyramid DEFH. But it is also less, which is impossible. Therefore the base ABC is not to the base DEF, as the pyramid ABCG to any solid which is less than the pyramid DEFH. In the same manner it may be demonstrated, that the base DEF is not to the base ABC, as the pyramid DEFH to any solid which is less than the pyramid ABCG. Nor can the base ABC be to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. For, if it be possible, let it be so to a greater, viz. the solid Z. And because the base ABC is to the base DEF, as the pyramid ABCG to the solid Z; by inversion, as the base DEF to the base ABC, so is the solid Z to the pyramid ABCG. But as the solid Z is to the pyramid ABCG, so is the pyramid DEFH to some solid,* which must be less (14. 5.) than the pyramid ABCG, because the solid Z is greater than the pyramid DEFH. And therefore, as the base DEF to the base ABC, so is the pyramid DEFH to a solid less than the pyramid ABCG; the contrary to which has been proved. Therefore the base ABC is not to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. And it has been proved, that neither is the base ABC to the base DEF, as the pyramid ABCG to any solid which is less than the pyramid DEFH. Therefore, as the base ABC is to the base DEF, so is the pyramid ABCG to the pyramid DEFH. Wherefore pyramids, &c. Q. E. D. PROP. VI. THEOR. PYRAMIDS of the same altitude, which have polygons for their bases, are to one another as their bases.t 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 pyramids having their common vertex in the point N: thereM N * This may be explained the same way as the like at the mark † in prop. 2. + See Note. fore, since the triangle ABC is to the triangle FGH, as (5. 12.) 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 triangle ADE to the triangle FGH, as the pyramid ADEM to the pyramid FGHN; as all the first antecedents to their common consequent, so (2. Cor. 24. 5.) are all the other antecedents to their common consequent: that is as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN: and, for the same 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 pyramid FGHKLN; therefore, ex æquali, (22. 5.) as the base ABCDE to the base FGHKL, so the pyramid ABCDEM to the pyramid FGHKLN. Therefore, pyramids, &c. Q. E. D. PROP. VII. THEOR. Every prism having a triangular base, may be 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 it: the prism ABCDEF may be divided into three equal pyramids having triangular bases. Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal (34. 1.) to the triangle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal (5. 12.) to the pyramid of which 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 F triangle ECF is equal (34. 1.) to the triangle ECB: therefore the pyramid of which the base is the triangle ECB, and vertex the point D, is D E equal to the pyramid, 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 vertex the point D, has been proved equal to the pyramid of which the base is the trian с gle ABD, and vertex the point C. Therefore A B the prism 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 same 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 prism, the base of which is the triangle ABC, and to which DEF is the opposite 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 base, viz. the triangle ABC, and DEF is the opposite triangle. Q. E. D. Cor. 1. 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 altitude, are (6. 12.) to one another as their bases. PROP. VIII. THEOR. SIMILAR pyramids having triangular bases are one to another in the triplicate ratio of that of their homologous sides. 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 side EF. Complete the parallelograms ABCM, GBCN, ABGK, and the solid parallelopiped BGML contained by these planes and those opposite to them: and, in like manner, complete the solid parallelopiped EHPO contained by the three parallelograms DEFP, HEFR, DEHX, and those opposite to them: and, because the pyramid ABCG is similar to the pyramid DEFH, the angle ABC is equal (11. def. 11.) to the angle DEF and the angle GBC to the angle HEF, and ABG to DEH: and AB is (1. def. 6.) to BC as DE to EF; that is, the sides about the equal angles are proportionals; wherefore the parallelogram 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 (24. 11.) to the three which are opposite to them, and the three EP, ER, EX, equal and similar to the three opposite to them: wherefore the solids BGML, EHPO are contained by the same number of similar planes; and their solid angles are equal (B. 11.); and therefore the solid BGML, is similar (11. def. 11.) to the solid EHPO: but similar solid parallelopipeds have the triplicate (33. 11.) ratio of that which their homologous sides have: therefore the solid BGML has to the solid EHPO the triplicate ratio of that which the side BC has to the homologous side EF: but as the solid BGML is to the solid EHPO, so is (15. 5.) the pyramid ABCG to the pyramid DEFH; because the pyramids are the sixth part of the solids; since the prism, which is the half (28. 11.) of the solid parallelopiped is triple (7. 12.) of the pyramid. Wherefore likewise the pyramid ABCG has to the pyramid DEFH the triplicate ratio of that which BC has to the homologous side EF. Q. E. D. CoR. From this it is evident, that similar pyramids which have 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 polygons; therefore as one of the triangular pyramid in the first multangular pyramids is to one of the triangular pyramids in the other, so 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 pyramid is to its similar triangular pyramid, in the triplicate ratio of their homologous sides; 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 side of the other. PROP. IX. THEOR. The bases and altitudes of equal pyramids having 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 'parallelopipeds 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 solid BGML is sectuple of the pyramid ABCG, and the solid EHPO sectuple of the pyramid DEFH; therefore the solid BGML is equal (1. Ax. 5.) to the solid EHPO : but the bases and altitudes of equal solid parallelopipeds are reciprocally proportional (34. 11.); therefore as the base BM to the base EP, so is the altitude of the solid EHPO to the altitude of the solid O R A D F BGML: but as the base BM to the base EP, so is (15. 5.) the triangle ABC to the triangle DEF; therefore as the triangle ABC to the triangle DEF, so is the altitude of the solid EHPO to the altitude of the solid BGML: but the altitude of the solid EHPO is the same with the altitude of the pyramid DEFH; and the altitude of the solid BGML is the same with the altitude of the pyramid ABCG: therefore, as the base ABC to the base DEF, so is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: wherefore the bases and altitudes of the pyramids ABCG, DEFH are reciprocally proportional. Again, let the bases and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz. the base ABC to the base DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: the pyramid ABCG is equal to the pyramid DEFH. The same construction being made, because as the base ABC to the base DEF, so is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: and as the base ABC to the base DEF, so is the parallelogram BM to the parallelogram EP : therefore the parallelogram BM is to EP, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG : but the altitude of the pyramid DEFH is the same with the altitude of the solid parallelopiped EHPO : and the altitude of the pyramid ABCG with the altitude of of the solid parallelopiped BGML: as, therefore, the base BM to the base EP, so is the altitude of the solid parallelopiped EHPO to the altitude of the solid parallelopiped BGML. But solid parallelopipeds having their bases and altitudes reciprocally proportional, are equal (34. 11.) to one another. Therefore the solid parallelopiped BGML is equal to the solid parallelopiped EHPO. And the pyramid ABCG same |