Tallelogram BK to the Parallelogram EX. Therefore three Parallelograms BM, KB, BN, are fimilar to three Parallelograms EP, EX, ER; but the three MB, BK, BN, are equal and fimilar to the three opposite ones; as also the three EP, EX, ER. Therefore the Solids BGML, EHPO, are contained under equal Numbers of fimilar and equal Planes ; and confequently, the Solid BGML is fi milar to the Solid EHPO. But fimilar folid Paral33. 11. lelepipedons are * to each other in a triplicate Pro portion of their homologous Sides. Therefore the Solid BGML to the Solid EHPO, has a Proportion triplicate of that which the homologous Side BC has to the homologous Side E F. But as the * 15.5. Solid BGML is to the Solid EHPO, fo is the Pyramid ABCG to the Pyramid DEFH; for the Pyramid is the one fixth Part of that Solid, fince the Prism, which is the half of the Solid Parallelepipedon is triple of the Pyramid. Wherefore the Pyramid ABCG to the Pyramid DEFH, shall have a triplicate Proportion to that which B C has to EF; which was to be demonstrated. Coroll. From hence it is manifeft, that similar Pyra mids having polygonous Bases, are to one another in a triplicate Proportion of their homologous Sides. For if they be divided into Pyramids having triangular Bases; because their similar polygonous Bases are divided into similar Triangles equal in Number, and homologous to the Wholes, it shall be as one Pyramid having a triangular Base in one of the Pyramids, is to a Pyramid having a triangular Base in the other Pyramid, so are all the Pyramids having triangular Bases in one Pyramid, to all the Pyramids having triangular Bases in the other Pyramid ; that is, so is one of the Pyramids having the polygonous Base, to the other, but a Pyramid having a triangular Base to a Pyramid having a triangular Base, is in a triplicate Proportion of the homologous Sides. Therefore one Pyramid having a polygonous Bafe to another Pyramid having a similar Base, is in a triplicate Proportion of their homologous Şides., PRO PROPOSITION IX. THEORE M. triangular Bases, are reciprocally proportional; portional, are equal. L ET there be equal Pyramids, having the triangu lar Bases A B C, DEF, and Vertices the Points G, H. I say, the Bafes and Altitudes of the Pyramids ABCG, DEFH, are reciprocally proportional, that is, as the Base ABC is to the Base DEF, fo is the Altitude of the Pyramid DEFH to the 'Altitude of the Pyramid ABCG. For complete the folid Parallelepipedons BGML, EHPO. Then because the Pyramid ABCG is equal to the Pyramid DEFH, and the Solid BGML is sextuple, the Pyramid ABCG, and the Solid EHPO sextuple of the Solid DEFH, the Solid B GML shall be * equal to the Solid E HP O. But the Bases * 15. 5. and Altitudes of equal folid Parallelepipedons are reciprocally proportional. Therefore,' as the Base BM is to the Base EP, fo is † the Altitude of the Solid † 34. 11. EHPO to the Altitude of the Solid B GML. But as the Base BM is to the Base EP, so is + the Triangle ABC to the Triangle DEF. Therefore, as the Triangle ABC is 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 as the Altitude of the Pyramid DEFH; and the Altitude of the Solid BGML the same as the Altitude of the Pyramid ABCG. Therefore, as the Base ABC is 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 equal Pyramids ABCG, DEFH, are reciprocally proportional; and if the Bases and Altitudes of the Pyramids ABCG, DEFH, are reciprocally proportional, that is, if the Base ABC to the Base DEF, be as the Altitude of the Pyramid DEFH DEFH to the Altitude of the Pyramid ABCG. I say, the Pyramid ABCG is equal to the Pyramid DEFH: For, the fame Construction remaining, because the Base ABC to the Basc DEF, is as the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG, and as the Base ABC is to the Base DEF, fo is the Parallelogram BM to the Parallelogram EP; the Parallelogram BM to the Parallelogram EP shall be also as the Altitude of the Pyramid DEFH is to the Altitude of the Pyramid ABCG. But as the Altitude of the Pyramid DEFH is the same as the Altitude of the folid Parallelepipedon EHPO, and the Altitude of the Pyramid ABCG the same as the Altitude of the folid Parallelepipedon BGML. Therefore the Base BM to the Base EP will be as the Altitude of the solid Parallelepipedon EHPO to the Altitude of the folid Parallelepipedon BGML. But those folid Parallelepipedons, whose Bases, and Altitudes are reciprocally proportional, are t equal to each other. Therefore the solid Parallelepipedon: B GML is equal to the solid Parallelepipedon EHPO; and the Pyramid ABCG is a fixth Part of the Solid BGML. And in like manner the Pyramid DEFH is a sixth Part of the Solid E HPO. Therefore the Pyramid AB CG is equal to the Pyramid DEFH. Wherefore the Bases and Altitudes of équal Pyramids, having triangular Bafes, are procally proportional; and those Pyramids, having triangular Bajes, whose Bases and Altitudes are reciprocally proportional, are equal; which was to be demonftrated, + 34. II. THEOREM. ibe Jame Base, and an equal Altitude. to it. I say the Cone is a third Part of the Cylinder; For * For if the Cylinder be not triple to the Cone, it shall be greater or less than triple thereof. First, let it be greater than triple to the Cone, and let the Square ABCD be described in the Circle ABCD, then the Square ABCD is greater than one half of the Circle ABCD. Now let a Prism be erected upon the Square ABCD, having the same Altitude as the Cylinder, and this Prism will be greater than one half of the Cylinder; because, if a Square be circumscribed about the Circle ABCD, the infcrib'd Square will be one half of the circumscribed Square ; and if a Prism be erected upon the circumscribd Square of the fame Altitude as the Cylinder, since Prisms are to one * 2 Cor. 7. another as their Bases, the Prism erected upon the of this. Square ABCD is one half of the Prism erected upon the Square described about the Circle ABCD. But the Cylinder is lesser than the Prism erected on the Square described about the Circle ABCD. Therefore the Prism erected on the Square ABCD, having the same Height as the Cylinder, is greater than one half of the Cylinder. Let the Circumferences AB, BC, CD, D'A, be bifected in the Points E, F, G, H, and join AE, E B, BF, FC, CG, GD, DH, HA. Then each of the Triangles AEB, BFC, CGD, DHA, is † greater than the half of each of the Seg- † This fol lows from 2p ments in which they stand. Let Prisms be erected of this. from each of the Triangles A E B, BF C, CGD, DHA, of the same Altitude as the Cylinder, then every one of these Prisms erected is greater than its correspondent Segment of the Cylinder. For because, if Parallels be drawn through the Points E, F, G, H, to AB, BC, CD, DA, and Parallelograms be compleated on the said AB, BC, CD, DA, on which are erected solid Parallelepipedons of the same Altitude as the Cylinder ; then each of those Prisms that are on the Triangles A E B, BFC, CGD, DHA, are Halves + of each of the solid Parallelepipedons; and the Segments of the Cylinder are less than the erected folid Parallelepipedons; and consequently the Prisms that are on the Triangles AEB, BFC, CGD, DHA, are greater than the Halves of the Segments of the Cylinder; and so bisecting the other Circumferences, joining Right Lines, and on every one of the Triangles erecting Prisms of the fame Height as the Cylinderi Cylinder; and doing this continually, we shall at last have certain Portions of the Cylinder left, that are less than the Excess by which the Cylinder exceeds triple the Cone. Now let these Portions remaining be AE, E B, BF, FC, CG, GD, DH, HA. Then the Prism remaining, whose Base is the Polygon AFBFCGDH, and Altitude equal to that of the Cylinders, is greater than the Triple of the Cone. But the Prism whose Base is the Polygon AEBF CGDH, and * 1 Cor. 7. Altitude the fame; as that of the Cylinder's is * triple of ebis. of the Pyramid whose Bafe is the Polygon AEBFCGDH, and Vertex the same as that of the Cone. And therefore the Pyramid whose Base is the Polygon AEBFCGDH, and Vertex the same as that of the Cone, is greater than the Cone whose Base is the Circle ABCD; but it is lesser also; (for it is comprehended by it) which is absurd. Therefore the Cylinder is not greater than triple the Cone. I say it is neither lesser than triple the Cone; For if it be possible, let the Cylinder be less than triple the Cone: Then (by Inversion) the Cone shall be greater than a third Part of the Cylinder. Let the Square ABCD be described in the Circle ABCD; then the Square ABCD is greater than half of the Circle ABCD. And let a Pyramid be erected on the Square ABCD having the same Vertex as the Cone, then the Pyramid erected is greater than one half of the Cone; because, as has been already demonstrated, if a Square be described about the Circle, the Square ABCD shall be half thereof. And if folid Parallelepipedons be erected upon the Squares of the same Altitude as the Cones, which are also called Prisms; then the Prism erected on the Square ABCD is one half of that erected on the Square described about the Circle, for they are to each other as their Bases; and so likewise are their third Parts. Therefore the Pyramid whose Base is the Square ABCD, is one half of that Pyramid erected upon the Square described about the Circle: But the Pyramid erected upon the Square described about the Circle, is greater than the Cone; for it comprehends it. Therefore the Pyramid whose Base is the Square ABCD, and Vertex the same as that of the Cone, is greater than one half of ) |