PRO P. XX. THEOREM. Every cone is equal to a pyramid of an equal base and altitude. Let DÅBC be a cone, and KEFGH a pyramid, standing upon equal bases ABC, EFGH, and having equal altitudes DP, KS; then will DABC be equal to KEFGH. For parallel to the bases, and at equal distances Dog Kr from the vertices, draw the planes nmp and vw. Then, by the last Prop. and Prop. 13; the square of Do is to the square of DP as nmp is to ABC; and the square of Kr to the square of Ks as vw to eG. And since the squares of Do, DP are equal to the squares of Kr, ks (Const. and II. 2.), nmp is to ABC as vw is to eg (V. 11.) But ABC is equal to eG, by hypothesis; wherefore nmp is, also, equal to vw (V. 10.) And, in the fame manner, it may be shewn, that any other sections, at equal distances from the vertices, are equal to each other. Since, therefore, every fection in the cone is equal to its corresponding section in the pyramid, the solids DABC, KEFGH of which they are composed, must be equal. Q. E. D. P R O P. XXI. THEOR E M. Every cone is the third part of a cylinder of the same base and altitude. Let e ao be a cone, and daec a cylinder, of the fame base and altitude; then will EAB be a third of DABC. For let KFG, KFGH be a pyramid and prism, having an equal bafe and altitude with the cone and cylinder. Then since cylinders and prisms of equal bases and altitudes are equal to each other (VIII. 18.), the cylinder DABC will be equal to the prism KFGH. And, because cones and pyramids of equal bases and altitudes are equal to each other (VIII. 20.); the cone EAB will be equal to the pyramid KFG. But the pyramid KFG is a third part of the prism KFGH (VIII. 16.), wherefore the cone E AB is, also, a third part of the cylinder DABC. Q. E. D. SCHOLIUM 1. Whatever has been demonstrated of the proportionality of pyramids, prisms, or cylinders, holds equally true of cones, these being a third of the latter. 2. It is also to be observed, that similar cones and cylinders are to each other as the cubes of their altitudes, or the diameters of their bases; the term like fides being here inapplicable. PRO P. XXII. THEOREM. If a sphere be cut by a plane the section will be a circle. B Let the sphere EBD be cut by the plane BSD; then will Bsd be a circle. For let the planes ABC, Asc.pass through the axis of the sphere ec, and be perpendicular to the plane BsD. Also draw the line .BD; and join the points A, D and res: Then since each of these planes are perpendicular to the plane BSD, their common section Ar will also be perpendicular to that plane (VII. 14.) And, because the fides AB, Ar, of the triangle abr, are equal to the sides As, Ar of the triangle Asr, and the angles Arb, Ars are right angles, the fide rb will be equal to the side rs (I. 4.) In like manner, the sides AD, Ar, of the triangle Adr, being equal to the sides As, Ar, of the triangle Asr, and the angles Ard, Ars right angles, the side rd will also be eju'l to the side rs (I. 4.) The ines re, rd and rs, are, therefore, all equal; and the fame may be shewn of any other lines, drawn from the point r to the circumference of the fection; whence BSD is a circle, as was to be shewn. Cor. The centre of every section of a sphere is always in a diameter of the sphere. PROP. XXIII. THEOREM. Every sphere is two thirds of its circumscribing cylinder. F; Let resm be a sphere, and DABC its circumscribing cylinder ; then will resM be two thirds of DABC. For let Ac be a section of the sphere through its centre and parallel to Dc, or AB, the base of the cylinder, draw the plane lh, cutting the former in n and m; and join Fe, FN, FD and Fr. Then, if the square er be conceived to revolve round the fixed axis Fr, it will generate the cylinder EC ; the quadrant fer will also generate the hemisphere EMTE; and the triangle for the cone FDC, And since fun is a right angled triangle, and FH is equal to Hm, the squares of FH, Hn, or of Hm, Hn, are equal to the square of Fn. But yn is also equal to fe or HL; whence the squares of Hm, Hn are equal to the square of Hl: or the circular fections R 3 fe&tions whose radii are Hm, Hn are equal to the circular section whose radius is HL (VIII. 5. Cor.) And as this is always the case, in every parallel position of H1, the cone fdc and cylinder Ec, which are composed of the former of these sections, are equal to the hemisphere EMre, which is composed of the latter. But the cone Foc is a third part of the cylinder ec (VIII. 21.); whence the hemisphere Emre is equal to the remaining two thirds; or the whole sphere resm to two thirds of the whole cylinder DABC, as was to be shewn. COR. I. A cone, hemisphere, and cylinder, of the same base and altitude, are to each other as the numbers 1, 2 and 3. COR. 2. All spheres are to each other as the cubes of their diameters; these being like parts of their circumscribing cylinders |