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 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 c to one another as their bases. c. 6. 12. IMILAR pyramids having triangular bases, are one to 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 fintilarly 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 Tolid parallelepiped BGML contained by these planes and those opK L Х. 0 R B posite to them. and in like manner complete the solid parallelepi ped EHPO contained by the three parallelograms DEFP, HEFR, 2. 11. Def. DEHX, and those opposite to them. and because the pyramid ABCG is similar to the pyramid DEFH, the angle ABC is equal a f. to the angle DEF, and the angle GBC to the angle HEF, and ABG Book XII. to DEH. and AB is o to BC, as DE to EF; that is, the sides about b. I. Def. 6. the equal angles are proportionals; wherefore the parallelogram BM is fimilar 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 c to the three which are op C. 24. II, posite to them, and the three EĎ, ER, EX equal and fimilar to the three opposite to them. wherefore the solids BGML, EHPO are contained by the fame number of similar planes; and their solid angles are equal d; and therefore the solid BGML is similar a to d. B. 11. the solid EHPO. but similar solid parallelepipeds have the triplicate € ratio of that which their homologous fides have. therefore the e. 33. II. solid 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 folid EHPO, so is f the pyramid ABCG to the 15. 5. Pyramid DEFH; because the pyramids are the sixth part of the folids, since the prism, which is the half of the folid parallelepiped, g. 28. 11. is triple h of the pyramid. Wherefore likewise the pyramid ABCG h. 7. 12. has to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous side EF. 0. 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 fides. 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 fimilar 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, fò 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 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 fide of the other. R. 2 Book XII. PROP. IX. THEOR. HE 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 bafes, 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 0 R А D E of the pyramid ABCG, and the folid EHPO sextuple of the pya. I. As. 5. ramid DEFH ; therefore the solid BGML is equal a to the solid EHPO. but the bases and altitudes of equal folid parallelepipeds are b. 34. II. reciprocally proportional 6; therefore as the base BM to the base EP, so is the altitude of the folid EHPO to the altitude of the folid C. 15. 5. BGML. but as the base BM to the base EP, so is c the triangle ABC to the triangle DEF; therefore as the triangle ABC to the triangle DEF, so 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 e.py the solid BGML is the same with the altitude of the pyramid Book XII. 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 ramid DEFH. The fame 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 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 parallelepiped EHPO; and the altitude of the pyramid ABCG is the same with the altitude of the solid parallelepiped BGML. as, therefore, the base BM to the base EP, fo is the altitude of the solid parallelepiped EHPO to the altitude of the solid parallelepiped BGML. but solid parallelepipeds having their bases and altitudes reciprocally proportional, are equal to one another. therefore the b. 34. 11. solid parallelepiped BGML is equal to the solid parallelepiped EHPO. and the pyramid ABCG is the sixth part of the solid BGML, and the pyramid DEFH the fixth part of the folid EHPO. therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bases, &c. Q. E. D. E VERY cone is the third part of a cylinder which has the same base, and is of an equal altitude with it. Let a cone have the fame base with a cylinder, viz. the circle ABCD, and the same altitude. the cone is the third part of the cylinder; that is, the cylinder is triple of the cone. If the cylinder be not triple of the cone, it muft either be greater than the triple, or less than it. First, Let it be greater than the triple; and describe the square ABCD in the circle; this square is Book XII. greater than the half of the circle ABCD +. upon the square ABCD bop erect a prism of the fame altitude with the cylinder; this prism is greater than half of the cylinder; because if a square be described about the circle, and a prism erected upon the square, of the same altitude with the cylinder, the inscribed square is half of that cir-. cumscribed; and upon these square bases are erected solid parallelepipeds, viz. the prisms, of the fame altitude; therefore the prism upon the square ABCD is the half of the prism upon the fquare de scribed about the circle; because they are to one another as their R. 32. II. bases a. and the cylinder is less than the prism upon the square de scribed about the circle ABCD. therefore the prism upon the square А. E H C parallelograms; the prisms upon the triangles AEB, BFC, CGD, D. 2. Cor. DHA are the halves of the folid parallelepipeds 6. and the seg ments of the cylinder which are upon the segments of the circle cut off by AB, BC, CD, DA, are less than the solid parallelepipeds which contain them. therefore the prisms upon the triangles AEB, BFC, CGD, DHA, are greater than half of the fegments of the cylinder in which they are. therefore if each of the circumferences be divided into two equal parts, and straight lines be drawn from the points of division to the extremities of the circumferences, and upon the triangles thus made prisms be erected of the fame alti tude with the cylinder, and so on, there must at length remain some c. Lemma. segments of the cylinder which together are lefs than the excess of the cylinder above the triple of the conę. let them be those upon the † As was hewn in Prop. 2. of this Book. 7. 12. |