B. XII. 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. 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 c 6. 12. same altitude, are c 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 parallelepiped BGML contained by these planes and B E F those opposite 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, be. cause the pyramid ABCG is similar to the pyramid DEFH, the B. XII. angle ABC is equal a to the angle DEF, and the angle GBC to the angle HEF, and ABG to DEH: and AB is b to BC, as DE to EF ; a. 11. def. that is, the sides about the equal angles are proportionals ; where. 11. fore the parallelogram BM is similar to EP : for the same reason, b 1. def.6. 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 e to the three which are opposite to them, and the three c 24. 11. 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 d; d B. 11. and therefore the solid BGML, is similar a to the solid EHPO : but similar solid parallelepipeds have the triplicate e ratio of that e 33. 11. 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 f the pyramid ABCG to the pyramid f 15. 5. DEFH; because the pyramids are the sixth part of the solids, since the prism, which is the half 8 of the solid parallelepiped, g 28. 11. is triple h of the pyramid. Wherefore likewise the pyramid h 7. 12. 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 See Note. 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 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 pyramids in the other; that is, so is the first multangular pyramid to the other : but une 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. B. XII. 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 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 solid BGML is sextuple of the pyramid ABCG, and the solid EHPO sextuple al. Ax.5. of the pyramid DEFH ; therefore the solid BGML is equal a to the solid EHPO : but the bases and altitudes of equal solid pab 34. 11. rallelepipeds are reciprocally proportional b; therefore as the base BM to the base EP, so is the altitude of the solid EHPO to the altitude of the solid BGML: but as the base BM to the base c 15.5. 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 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 B. XII. the base DEF, so is the altitude of the pyramid DEFH to them 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 altiude 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 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, so 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 b to b 34. 11. one another. Therefore the 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 is the sixth part of the solid EHPO. Therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bases, &c. Q. E.D. PROP. X. THEOR. EVERY 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 same 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 must 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 greater than the half of the circle ABCD". * As was shown in prop. 2. of this book. B. XII. Upon the square ABCD erect a prism of the same 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 circumscribed ; and upon these square bases are erected solid parallelepipeds, viz. the prisms of the same altitude; therefore the prism upon the square ABCD is the half of the prism upon the square descri bed about the circle: because they are to one another as their a 32. 11. bases a : and the cylinder is less than the prism upon the square described about the circle ABCD: therefore the prism upon A H B of the cylinder in which it is: be D cause if, through the points E, F, G, H, parallels be drawn to AB, BC, I G CD, DA, and parallelograms be completed upon the same AB, BC, с CD, DA, and solid parallelepipeds be erected upon the parallelograms; the prisms upon the triangles AEB, BFC, CGD, DHA are the halves of the solid 62 Cor. parallelepipeds b. And the segments of the cylinder which are 7. 12. 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 segments 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 same c Lem. altitude with the cylinder, and so on, there must at length re main some segments of the cylinder which together are less c. than the excess of the cylinder above the triple of the cone. Let them be those upon the segments of the circle AE, EB, BF, ma. |