the vertices, be similar, and similarly situated; the pyramid ABCG has to DEFH, the triplicate ratio of that which the side BC has to the homologous side EF. A L X N H R B D E Complete the parallelograms BM, BN, BK, and the parallelepiped BGML contained by these planes and those opposite to them and, in like manner, complete the parallelepiped EHPO contained by the three parallelograms EP, ER, EX, and those opposite to them. Then, because the pyramids ABCG, DEFH are similar, the angle ABC is equal (XI. def. 8.) to DEF, GBC to HEF, and ABG to DEH: and (VI. def. 1.) AB: BC:: DE: EF; that is, the sides about the equal angles of the parallelograms BM, EP are proportionals: wherefore BM is similar to EP. For the same reason, BN is similar to ER, and BK to EX. Therefore the three parallelograms BM, BN, BK are similar to the three EP, ER, EX: and (XI. 24.) the three BM, BN, BK are equal and similar 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 (XI. B.) their solid angles are equal the solid BGML, is therefore similar (XI. def. 8.) to EHPO. But (XI. 33.) similar parallelepipeds have the triplicate ratio of that which their homologous sides have: therefore BGML has to EHPO the triplicate ratio of that which BC has to the homologous side EF. But (V. 15.) as BGML is to EHPO, so is the pyramid ABCG to the pyramid DEFH; because the pyramids are the sixth parts of the solids, since the prism, which (XI. 28.) is half of the parallelepiped, is triple (XII. 7.) of the pyramid. Therefore also the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous side EF. Similar triangular pyramids, therefore, &c. Cor. From this it is evident that similar pyramids which have bases of more sides than three; are likewise to one another in the triplicate ratio of their homologous sides. For they may be divided into similar triangular pyramids, because the similar figures which are their bases, may be divided into the same number of similar triangles, homologous to the whole bases; therefore as one of the triangular pyramids in the first pyramid is to one of the triangular pyramids in the other, so are all the triangular pyramids in the first to all in the other; that is, so is the first whole pyramid to the other. But one triangular pyramid is to a similar triangular pyramid, in the triplicate ratio of their homologous sides; and therefore the first whole pyramid has to the other, the triplicate ratio of that which a side of the first has to the homologous side of the other. PROP. IX. THEOR. THE bases and altitudes of equal triangular pyramids are reciprocally proportional: and (2.) triangular pyramids having their bases and altitudes reciprocally proportional, are equal to one another. 1. Let the pyramids of which the triangles ABC, DEF, are the bases, and G, H, the vertices, be equal: their bases and altitudes are reciprocally proportional, viz., ABC is to DEF, as the altitude of the pyramid DEFH to the altitude of ABCG. K R L N H X G Complete the parallelograms AC, AG, GC, DF, DH, HF; and the parallelepipeds BGLM, EHOP contained by these planes and those opposite to them. Then, because the pyramid ABCG is equal to DEFH, and that (I. 34. and XII. 7. cor. 1.) the solid BGLM is sextuple of ABCG, and EHOP of DEFH; therefore (V. ax. 1.) BGLM is equal to EHOP. But (XI. 34.) the bases and altitudes of equal parallelepipeds are reciprocally proportional; therefore as BM to EP, so is the altitude of EHPO to the altitude of BGML. But (V. 15.) BM is to EP, as the triangle ABC to DEF; therefore as ABC to DEF, so is the altitude of EHOP to the altitude of BGLM. But the altitude of EHOP is the same with the altitude of the pyramid DEFH; and the altitude of BGLM is the same with the altitude of the pyramid ABCG: therefore, as ABC to 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. M C B P D E 2. Again, let the bases and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz., ABC to DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: ABCG is equal to DEFH. The same construction being made, because as ABC to DEF, so is the altitude of the pyramid DEFH to the altitude of ABCG: and as ABC to DEF, so is BM to EP; therefore BM is to EP, as the altitude of the pyramid DEFH to the altitude of ABCG. But the altitude of DEFH is the same with the altitude of the parallelepiped EHOP; and the altitude of ABCG is the same with the altitude of the parallelepiped BGLM. As, therefore, the base BM to the base EP, so is the altitude of the parallelepiped EHOP to the altitude of the parallelepiped BGLM. But(XI.34.) parallelepipeds having their bases and altitudes reciprocally pro portional, are equal. Therefore the parallelepiped BGLM is equal to the parallelepiped EHOP. And the pyramid ABCG is the sixth part of BGLM, and DEHF is the sixth part of EHOP. Therefore the pyramid ABCG is equal to the pyramid DEFH: wherefore the bases, &c. PROP. X. THEOR. A CONE is a third part of a cylinder on the same base, and of equal altitude. 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 AC, as was shown in the second proposition of this book. Upon the square ABCD form 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 formed on the square, of the same altitude with the cylinder, the inscribed square is half of the circumscribed; and upon these square bases are erected parallelepipeds, viz, the prisms of the same altitude; therefore (XI. 32.) the prism upon the square ABCD is the half of the prism upon the square described about the circle. But (I. ax. 9.) the cylinder is less than the prism upon the square described about the circle AC: therefore the B B F prism upon the square ABCD of the same altitude with the cylinder, is greater than half of the cylinder. Bisect the arcs AB, BC, CD, DA, in the points E, F, G, H; and join AE, EB, &c. 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 the second proposition of this book. Describe 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 cylinder in which it is; because if, through the points E, F, G, H, parallels be drawn to AB, BC, CD, DĂ, and parallelograms be completed upon the same AB, BC, CD, DA, and parallelepipeds be formed upon the parallelograms: the prisms upon the triangles AEB, BFC, CGD, DHA are (XII. 7. cor. 2.) the halves of the parallelepipeds: and (I. ax. 9.) the segments of the cylinder which are upon the segments of the circle cut off by AB, BC, CD, DA, are less than the 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. If, therefore, each of the arcs be bisected, and straight lines be drawn from the points of division to the extremities of the arcs, and upon the triangles thus made, prisms be formed of the same altitude with the cylinder, and so on, there must at length remain some segments of the cylinder, which together (XII. lem. 1.) are less 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, &c. 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 (XII. 7. cor. 1.) this prism is triple 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; which (I. ax. 9.) is impossible. E A H F Nor can the cylinder be less than the triple of the cone. Let it be less, if possible; and consequently the cone greater than the third part of the cylinder. In the circle AC describe a square; this square is greater than the half of the circle and upon the square ABCD form 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 ABCD is the half of it; and if, upon these squares there be erected parallelepipeds of the same altitude with the cone, which are also prisms, the prism upon the square ABCD will be the half of that which is upon the square described about the circle; for (XI. 32.) they are to one another as their bases; 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 arcs AB, BC, CD, DA, in the points E, F, G, H, and Join AE, EB, &c. Then 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 form 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 was before demonstrated of the prisms and segments of the cylinder; and thus bisecting each of the arcs, and joining the points of division and their extremities by straight lines, and upon the triangles erecting pyramids having their vertices the same with that 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 Therefore the rest E H G above the third part of the cylinder. Let these be the segments PROP. XI. THEOR. F C 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 BD to the circle FH, so is the cone AL to the cone EN. If it be not so, let BD be to FH, as the cone AL to some solid either less than the cone EN, or greater than it. First, let it be to X, a solid less than EN; and let Z be the solid which is equal to the excess of EN above X; therefore the cone EN is equal to X, Z together. In the circle FH describe the square EFGH: this square is greater than the half of the circle. Upon the square FH form 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 formed upon it, having the same vertex with the cone, the pyramid inscribed in the cone is (XII. 6.) half of the pyramid circumscribed about it. But the cone is less than the circumscribed pyramid; therefore the pyramid of which the base is the square EFGH, and its vertex the same with that of the cone, is greater than half of the cone. |