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 alvitude 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 pyramid DEFH. The same construction being made, because as the base ABC to the bafe DEF, fo 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 DEFI is the fame with the altitude of the folid 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 folid parallelepiped EHPO to the altitude of the solid parallelepiped BGML. but solid parallelepipeds having their bases and altitudes reciprocally proportional, are equal b to one another. therefore the b. 34. 14. folid parallelepiped BGML is equal to the folid parallelepiped EHPO. and the pyramid ABCG is the sixth part of the solid BGML, and the pyramid DEFH the sixth part of the folid EHPO. therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bafes, &c. Q. E. D. PROP. X. THEO R. EVERY.cone is the third part of a cylinder which has the same base, and is of an oqual 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 2. 32. II. Book XII. greater than the half of the circle ABCD+. upon the square ABCD ereét a prikin of the fame altitude with the cylinder; this priim is A F त drawn to AB, PE, CD, DA, and parallelograms bc completed upon the fame AB, BC, CD, DA, and folid parallelerineds be crected upon the parallelograms; the prisins upon the triä yle's AEB, BFC, CGD, b. 2. Cor. DHA are the halves of the fshid prallelep peds b. and the leg ments of the cylinder wlich are upon the ligpients of the circle cut off by AB, BC, CD, DA, are lfs than the folid parallelepipels which contain them. therefore the prims upon the triangles AEB, BC, CGD, DITA, 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 paits, and straight lines be drawn from the points of division to the extremities of the circumferences, and upon the triangles thus made piisins be erected of the fame alti tule with the cylinder, and so on, there must at length remain some c. Lemma. sizinents of the cylinder which together are less than the excess of thcylinder above the triple of the cone. let them be those upon the fogonents of the circle AE, EB, BF, FC, CG, GD, DH, HA. there + As was shown in Prop. 2. of this Book, G 7. 12. 7. 1: fore the rest of the cylinder, that is the prilin of which the base is Book XII. 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 prifmn is triple d of the pyramid upon the fame base, of which d. 1. Cor. the vertex is the fame with the vertex of the cone ; therefore the pyramid upon the base AEBFCGDH, having the fame 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 sostible. 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 fame 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 haif of it; and if upon these squares H there be erected olid parallelepipeds of thie lame altitude with the cone, which are alto prisis, the priim upon the square ABCD shall be the half of that G which is upon the square defcribed about the circle; for they are to one another as their bases; as are also the third parts of them. therefore the pyra T mid the bałe of which is the square ABCD is half of the pyramid upon the square de!cribed about the circle. but this last pyramid is greater than the cone which it contains; therefore the pyramid upon the square APCD having the same vertex with the conc, is greater than the half of the cone. Bifeet the circumferences AB, BC, CD, DA in the poicts E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA. therefore each of the triangles AED, BFC, CGD, DHA is greater than half of the fegment of the circle in which it is. upon each of these triangles erect pyramids having the same vertex with the conc. therefore cach of thele Рramids 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 cicrumferences into two equal parts, and joining the points of division and their extremities by straight lines, and upon the triangles erecting pyramids hav, a. 32. II. Bock XII. ing their vertices the same with that of the cone, and so on, there muft at length remain fome fegments of the cone which together H G F 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 conc. 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. See N. CON ONES and cylinders of the same altitude, are to one another as their bafes. 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 bafes, be of the fame altitude, as the circle ABCD to the circle EFGH, fo 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 folid which is equal to the excess of the cope EN above the fulid X ; therefore the cone EN is equal to the solids X, Z together. in the circle EFGH describe the square EFGH, there. fore this square is greater than the half of the circle. upon the fuare EFGH erect a pyrainid of the fame altitude with the cone; this pyramid is greater than half of the cone. for if a fqnare be defcribed about the circle, and a pyramid be erected upon it, having the fame vertex with the conet, the pyramid inscribed in the cone is half Book XII. of the pyramid circumscribed about it, because they are to one another as their bases“. but the cone is less than the circumscribed py- a. 6.12. ramid; 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. divide the circumferences EF, FG, GH, HE, each into two equal parts in the points O, P, R, S, and join EO, OF, FP, PG, GR, RH, HS, SE. therefore each of the triangles EOF, FPG, GRH, HSE is greater than half of the segment of the circle in which it is. upon each of these triangles erect a pyramid having the fame vertex with the cone ; each of these pyramids is greater than the half of the segment of the cone in which it is, and thus dividing each of these circumferences into two equal parts, and from the points of division drawing Araight lines to the extremities of the circumferences, and upon each of the triangles thus made erecting pyramids having the fame vertex with the cone, and so on, there must at length remain some segments of the cone which are together less o 6. Lemma. than the solid Z. let these be the segments upon EO, OF, FP, PG, † Vertex is put in place of altitude which is in the Greek, because the pyramid, in what follows, is supposed to be circumferibed about the cone, and so must have the same vertex. and the same change is made in some places following: |