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 sextuple of the pyramid ABCG, and the solid EHPO sextuple 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 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 propor tional. 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 is the same with the altitude 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 is the sixth part of the solid BGML, and the pyramid DEFH is the sixth part of the solid EPHO. 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. E A H 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 parallelopipeds, viz. the prisms of the same altitude; therefore the prism upon the square ABCD is half of the prism upon the square described about the circle because they are to one another as their bases (32. 11.); 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: then each of the triangles AEB, BFC, CGD, DHA is greater than the half of the segment of the circle in which it stands, as was shown in prop. 2. of this book. Erect prisms upon each of these triangles, of the same altitude with the cylinder; each of these prisms is greater than half of the segment of the cylin- B der in which it is; because if, through the points E, F, G, H, parallels be drawn to AB, BC, CD, DA, and parallelograms be completed upon the same AB, BC, CD, DA, and solid parallelopipeds be erected upon the parallelograms; the prisms upon the triangles AEB, BFC, CGD, DHA are the halves of the solid parallelopipeds (2. Cor. 7. 12.) And the segments of the cylinder which are upon the segments of the circle cut off by AB, BC, CD, DA, are less than the solid parallelopipeds 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 altitude with the cylinder, and so on, there must at length remain some segments of the cylinder which together are less (Lem.) 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, G C D LA D FC, CG, GD, DH, HA. Therefore the rest of the cylinder, that is, the prism of which the base is the polygon AEBFCGDH, and of which the altitude is the same with that of the cylinder, is greater than the triple of the cone: but this prism is triple (1. Cor. 7. 12.) of the pyramid upon the same base, of which the vertex is the same with the vertex of the cone; therefore the pyramid upon the base AEBFCGDH, having the same vertex with the cone, is greater than the cone, of which the base is the circle ABCD but it is also less, for the pyramid is contained within the cone; which is impossible. 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 describe a square; this square is greater than the half of the circle: and upon the square ABCD erect a pyramid having the same vertex with the cone: this pyramid is greater than the half of the cone; because, as was before demonstrated, if a square be described about the circle, the square H ABCD is the half of it; and if, upon these squares, there be erected solid parallelopipeds of the same altitudes with the cone, which are also prisms, the prism upon the square ABCD shall be the half of that which is upon the square described about the circle; for they are to one another as their bases (32. 11.); as are also the third parts of them; therefore the pyramid, the base of which is the square ABCD, is half of the pyramid upon the square described about the circle: but this last pyramid is greater than the cone which it contains; therefore the pyramid upon the square ABCD, having the same vertex with the cone, is greater than the half of the cone. Bisect the circumferences AB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA: therefore each of the triangles AEB, BFC, CGD, DHA is greater than half of the segment of the circle in which it is upon each of these triangles erect pyramids having the same vertex with the cone. Therefore each of these pyramids is greater than the half of the segment of the cone in which it is, as before was demonstrated of the prisms and segments of the cylinder: and thus dividing each of the circumferences into two equal parts, and joining the points of division and their extremities by straight lines, and upon the triangles erecting pyramid's having their vertices the same with that : E G B C F A H D G of the cone, and so on, there must at length remain some segments of the cone, which together shall be less than the excess of the cone above the third part of the cylinder. Let these be the segments upon AE, EB, BF, FC, CG, GD, DH, HA. Therefore the rest of the cone, that is, the pyramid, of which the base is the polygon AEBFCGDH, and of which the vertex is the same with that of the cone, is greater than the third part of the cylinder. But this pyramid is the third part of the E prism upon the same base AEBFCGDH, and of the same altitude with the cylinder. Therefore this prism is greater than the cylinder of which the base is the circle ABCD. But it is also less ; for it is contained within the cylinder; which is impossible. Therefore the cylinder is not less than the triple of the cone. And it has been demonstrated that neither is it greater than the triple. Therefore the cylinder is triple of the cone, or the cone is the third part of the cylinder. Wherefore every cone, &c. Q. E. D. PROP. XI. THEOR. B F CONES and cylinders of the same altitude, are to one another as their bases.* Let the cones and cylinders, of which the bases are the circles ABCD, EFGH, and the axes KL, MN, and AC, EG the diameters of their bases, be of the same altitude. As the circle ABCD to the circle EFGH, so is the cone AL to the cone EN. If it be not so, let the circle ABCD be to the circle EFGH, as the cone AL to some solid either less than the cone EN, or greater than it. First, let it be to a solid less than EN, viz. to the solid X; and let Z be the solid which is equal to the excess of the cone EN above the solid X; therefore the cone EN is equal to the solids X, Z together. In the circle EFGH describe the square EFGH, therefore this square is greater than the half of the circle: upon the square EFGH erect a pyramid of the same altitude with the cone; this pyramid is greater than half of the cone. For, if a square be described about the circle, and a pyramid be erected upon it, having the same ver * See Note. |