Book XII. PROP. IX. THEOR. THE bafes and altitudes of equal pyramids having triangular bases are reciprocally proportional: And triangular pyramids of which the bafes and altitudes are reciprocally proportional, are equal to one another. Let the pyramids of which the triangles ABC, DEF are the bafes, and which have their vertices in the points G, H, be equal to one another: The bafes and altitudes of the pyramids ABG, DEFH are reciprocally proportional, viz. the bafe ABC is to the bafe 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 folid parallelepipeds BGML, EHPO contained by b 34. 11. thefe planes and thofe oppofite to them: And because the pyramid ABCG is equal to the pyramid DEFH, and that the folid BGML is fextuple of the pyramid ABCG, and the folid EHPO fextuple of the pyramid DEFH; therefore the folid a 1. Ax. 5. BGML is equal a to the folid EHPO: But the bafes and altitudes of equal folid parallelepipeds are reciprocally proportional b; therefore as the bafe BM to the bafe EP, fo is the altitude of the folid EHPO to the altitude of the folid BGML: But as the bafe BM to the bafe EP, to is e the triangle ABC to the triangle DEF; therefore as the triangle ABC to the triangle DEF, fo is the altitude of the folid EHPO to the altitvde of the folid BGML: But the altitude of the folid EHPO is the fame with the altitude of the pyramid DEFH; and the altitude of the folid BGML is the fame with the altitude of the 615. 5. Fyramid pyramid ABCG: Therefore, as the bafe ABC to the base DEF, Book XII. fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: Wherefore the bafes and altitudes of the pyra mids ABCG, DEFH are reciprocally proportional. Again, Let the bafes and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz, the bafe ABC to the bafe 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 fame conftruction being made, becaufe 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 bafe ABC to the bafe DEF, fo 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 fame with the altitude of the folid parallelepiped EHPO; and the altitude of the pyramid ABCG is the fame with the altitude of the folid parallelepiped BGML: As, therefore, the bafe BM to the bafe EP, fo is the altitude of the folid parallelepiped EHPO to the altitude of the folid parallelepiped BGML. But folid parallelepipeds having their bafes and altitudes reciprocally proportional, are equal b to one another. Therefore the folid pa-b 34. II. rallelepiped BGML is equal to the folid parallelepiped EHPO. And the pyramid ABCG is the fixth part of the folid BGML, and the pyramid DEFH the fixth part of the folid EHPO. Therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bafes, &c. Q. E. D. EVE it. PROP. X. THEOR. VERY cone is the third part of a cylinder which has the fame bafe, and is of an equal altitude with Let a cone have the fame bafe with a cylinder, viz. the circle ABCD, and the fame 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 lefs than it. First, Let it be greater than the triple; and defcribe the fquare ABCD in the circle; this fquare is greater than the half of the circle ABCD *. As was fewn in prop. 2. of this book. Upon Book XII. Upon the square ABCD erect a prifm of the fame altitude with 'the cylinder; this prim is greater than half of the cylinder; because if a fquare be defcribed about the circle, and a prifon erected upon the fquare, of the fame altitude with the cylinder, the infcribed fquare is half of that circumfcribed; and upon thefe fquare bafes are erected folid parallelepipeds, viz. the prifms, of the fame altitude; therefore the prifm upon the fquare ABCD is the half of the prifm upon the fquare defcribed about the circle; because they are to one another as their a 32. 11. bases a: And the cylinder is lefs than the prifm upon the square defcribed about the circle ABCD: Therefore the prifm upon the fquare ABCD of the fame altitude with the cylinder, is greater than half of the cylinder. Bifect 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 fegment of the circle in which it ftands, 7. 12. as was fhewn in prop. 2. of this CD, DA, and parallelograms be F completed upon the fame AB, BC, A H D C be erected upon the parallelograms; the prifms upon the triangles AEB, BFC, CGD, DHA are the halves of the fob. Cor. lid parallelepipeds b. And the fegments of the cylinder which are upon the fegments of the circle cut off by AB, BC, CD, DA, are lefs than the folid parallelepipeds which contain them. Therefore the prifms upon the triangles AEB, BFC, CGD, DHA, are greater than half of the fegments of the cylinder in which they are; therefore if each of the circumferences be divided into two equal parts, and ftraight lines be drawn from the points of division to the extremities of the circumferences, and upon the triangles thus made, prifms be erected of the fame altitude with the cylinder, and fo on, there muft at length rec Lemma. main fome fegments of the cylinder which together are lefs c than the excess of the cylinder above the triple of the cone. Let them be thofe upon the fegments of the circle AE, EB, BF, FC, A H D 7. 12. FC, CG, GD, DH, HA. Therefore the rest of the cylin-Book XII. der, that is, the prifm of which the bafe is the polygon AEBFCGDH, and of which the altitude is the fame with that of the cylinder, is greater than the triple of the cone But this prifm is triple d of the pyramid upon the fame bafe, of which a 1. Cor.. the vertex is the fame with the vertex of the cone; therefore the pyramid upon the bafe AEBFCGDH, having the fame vertex with the cone, is greater than the cone, of which the bafe is the circle ABCD: But it is alfo lefs, for the pyramid is contained within the cone; which is impoffible. Nor can the cylinder be less than the triple of the cone. Let it be less, if poffible: Therefore, inverfely, the cone is greater than the third part of the cylinder. In the circle ABCD defcribe a fquare; this fquare is greater than the half of the circle: And upon the fquare 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 demonftrated, if a fquare be defcribed about the circle, the fquare ABCD is the half of it; and if, upon thefe fquares there be erected folid parallelepipeds of the fame altitude with the cone, which are alfo prifms, the prifm upon the fquare ABCD fhall be the half of that which is upon the fquare defcribed about the circle; for they are to one another as their bafes a; as are alfo the third parts of them: Therefore the pyramid, the bafe of which is the fquare ABCD, is half of the pyramid upon the fquare defcribed about the circle: But this laft pyramid is greater than the cone which it contains; therefore the pyramid upon the fquare ABCD having the fame vertex with the cone, is greater than the half of the cone. Bifect 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 fegment of the circle in which it is: Upon each of thefe triangles erect pyramids having the fame vertex with the cone. Therefore each of thefe pyramids is greater than the half of the segment of the cone in which it is, as before was demonftrated of the prifms and fegments of the cylinder; and thus dividing each of the circumferences into two equal parts, and joining the E B F G a 32. 18. 1 A H D G Book XII. points of divifion and their extremities by ftraight lines, and upon the triangies erecting pyramids having their vertices the fame with that of the cone, and fo on, there muft at length remain fome fegments of the cone, which together fhall be lefs than the excefs of the cone above the third part of the cylinder. Let thefe be the fegnients upon AE, EB, BF, FC, CG, GD, DH, HA. Therefore the rest of the cone, that is, the pyramid, of which the bafe is the polygon ALBFCGDH, and of which the vertex is the fame with that of the cone, is greater that the third part of the cylinder. But this pyramid is the third part of the prifm upon the fame bafe AEBFCGDH, and of the fame altitude with the cylin der. Therefore this prifm is greater than the cylinder of which the bafe is the circle ABCD. But it is alfo lefs, within the cylinder; which is impossible. linder is not less than the triple of the cone. demonftrated that neither is it greater than the triple. Therefore the cylinder is tiple of the cone, or, the cone is the third part of the cylinder. Wherefore every cone, &c. Q. E. D. Scc N. CONES E B F for it is contained Therefore the cyAnd it has been PROP. XI. THEOR. ONES and cylinders of the fame altitude, are to one another as their bafes. Let he cones and cylinders, of which the bafes 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 fo, let the circle ABCD be to the circle EFGH, as the cone AL to fome folid either lefs than the cone EN, or greater than it. Firft, let it be to a folid lefs than EN, viz. to the folid X; and let Z be the folid which is equal to the excefs of the cone EN above the folid X; therefore the cone EN is equal to the folids X. Z together. In the circle EFGH defcribe the fquare EFGH, therefore this fquare is greater than the half of the circle: Upon the fquare EFGH erect a pyra mid of the fame altitude with the cone; this pyramid is greater than half of the cone. For, if a fquare be defcribed alout the circle, and a pyramid be elected upon it, ha I ving |