ABC, and DEF 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 its 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 to one another as their bases. • 6. 12. PROPOSITION 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 shall have 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 to the angle DEF, and the 11 Def.11. angle GBC to the angle HEF, and ABG to DEH: and AB is * similar to the three EP, ER, EX: but the three BM, BN, BK * 1 Def. 6. are equal and similar to the three which are opposite to 24. 11. • B. 11. 33. 11. 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; and therefore the solid BGML • 11 Def.11. is similar to the solid EHPO: but similar solid parallelopipeds have the triplicate 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 * the pyramid ABCG to the pyramid DEFH; because the pyramids are the sixth part of the solids, since the prism, which is the half of the solid parallelopiped, is triple* 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. 15. 5. * 28. 11. • 7. 12. † 12.5. 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 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 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. PROPOSITION 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 BOOK XII. PROP. IX. equal to one another: the bases and altitudes of the pyramids & 7.12. • 34. 11. the 15. 5. Complete the parallelograms AC, AG, GC, DF, DH, HF; and the solid parallelopipeds BGML, EHPO, contained by these planes and those which are 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 † 28. 11. the solid EHPO sextuple of the pyramid DEFH; therefore the solid BGML is equal to the solid EHPO: but the bases 1 Ax. 5. and altitudes of equal solid parallelopipeds are reciprocally proportional *; 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 EP, so is 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. K X H R B F D E Again, let the bases and altitudes of the pyramids ABCG, DEPH, 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 shall be 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 is the same with the altitude * 34. 11. † 2 Ax. 5. * of the solid parallelopiped BGML: therefore as 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 to one another; therefore the solid parallelopiped BGML is equal to the solid parallelopiped 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. † 2. 12. * 32. 11. PROPOSITION 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 shall be the third part of the cylinder; that is, the cylinder shall be triple of the cone. B F Ꮋ 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 inscribe the square ABCD in the circle: this square is greater † than the half of the circle ABCD. Upon the square ABCD, erect a prism of the same altitude with the cylinder; this prism shall be greater than half of the cylinder: for let a square be described about the circle, and let a prism be erected upon the square, of the same altitude with the cylinder: then the inscribed square is half of that circumscribed; and upon these square bases are erected solid parallelopipeds, viz. the prisms of the same altitude; therefore the prism upon the square ABCD is the half of the prism upon the square described about the circle; because they are to one another as their bases and the cylinder is less than the prism upon the square described about the circle ABCD, therefore the prism upon the square ABCD of the same altitude with the cylinder, is greater than half of the cylinder. Bisect the circumferences AB, BC, CD, DA, in the points E, F, G, H; and join AE, EB, BF, FC, CG, GD, DH, HA: EX B G then, each of the triangles AEB, BFC, CGD, DHA is greater Nor can the cylinder be less than the triple of the cone. Let it be less, if possible; therefore, inversely, the cone is greater than the third part of the cylinder. In the circle ABCD, inscribe a square: this square is greater than the half of the circle: and upon the square ABCD erect a pyramid, • 2 Cor. 7. 12. • Lemma. 1 Cor. 7. 12. |