Then fince the fection amp is parallel to the base ABC (by Hyps), and the planes. Bo, co cut them, op will be parallel to PC, and om to PB (VII. 12.) And because the triangles formed by thefe lines are equiangular, om will be to PB as Do to DP, or as op to PC (VI. 5.) But PB is equal to PC, being radii of the fame circle; wherefore om will alfo be equal to op (V. 10.) And the fame may be fhewn of any other lines drawn from the point to the circumference of the section nmp; whence nmp is a circle. Again, by fimilar triangles, Ds is to Dr as Do to DP, or as om to PB: whence the fquare of ps is to the fquare of pr as the fquare of om is to the square of PB (VI. 19.) But the fquare of am is to the fquare of PB as the circle mp is to the circle ABC (VIII. 5.); therefore the fquare of Ds is to the fquare of Dr as the circle nmp is to the circle ABC (V. 11.) Q. E. D. COR. If a cone be cut by a plane parallel to its bafe the fection will be a circle. PROP. XX. THEOREM. Every cone is equal to a pyramid of an equal bafe and altitude. Let DABC be a cone, and KEFGH a pyramid, ftanding upon equal bafes ABC, EFGH, and having equal altitudes DP, KS; then will DABC be equal to KEFGH.- For parallel to the bafes, and at equal diftances Do, Kr from the vertices, draw the planes nmp and vw. Then by the last Prop. and Prop. 13, the fquare of Do is to the fquare of DP as nmp is to ABC; and the fquare of Kr to the fquare of Ks as vw to EG. And fince the fquares of Do, DP are equal to the fquares of Kr, Ks (Conft. 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, alfo, equal to vw (V. 10.) And, in the fame manner, it may be fhewn, that any other fections, at equal diftances from the vertices, are equal to each other. Since, therefore, every section in the cone is equal to its corresponding section in the pyramid, the folids DABC, KEFGH of which they are compofed, must be equal. Q. E.D. PROP. XXI. THEOREM. Every cone is the third part of a cylinder of the fame base and altitude. Let EAB be a cone, and DABC a cylinder, of the fame bafe and altitude; then will EAB be a third of DABC. For let, KFG, KFGH be a pyramid and prism, having an equal base and altitude with the cone and cylinder. Then fince cylinders and prisms of equal bases and altitudes are equal to each other (VIII. 18.) the cylinder DABC will be equal to the prifm KFGH. And, because cones and pyramids of equal bafes 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 EAB is, alfo, a third part of the cylinder DABC. Q.E.D. SCHOLIUM 1. Whatever has been demonftrated of the proportionality of pyramids, prisms, or cylinders, holds equally true of cones, these being a third of the latter. 2. It is alfo to be obferved, that fimilar cones and cylinders are to each other as the cubes of their altitudes, or the diamaters of their bases; the term like fides being here inapplicable. PROP. XXII. THEOREM. If a sphere be cut by a plane the fection will be a circle. Let the fphere EBD be cut by the plane BSD; then will BSD be a circle. For let the planes ABC, ASC pafs through the axis of the fphere EC, and be perpendicular to the plane BSD. Alfo draw the line BD; and join the points A, D and Then fince 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 ABT are equal to the fides As, Ar of the triangle Asr, and the angles ArB, Ars are right angles, the fide ri will be equal to the fide rs (I. 4.) In like manner, the fides AD, Ar, of the triangle ADr, being equal to the fides As, Ar, of the triangle Asr, and the angles ArD, Ars right angles, the fide rD will also be equal to the fide rs (I. 4.) The lines rв, rD and rs, are, therefore, all equal; and the fame may be fhewn of any other lines, drawn from the point r to the circumference of the section; whence BSD is a circle, as was to be fhewn. COR. The centre of every fection of a sphere is always in a diameter of the sphere. PROP. XXIII. THEOREM. Every fphere is two thirds of its circumfcribing cylinder. D H Let rESM be a fphere, and DABC its circumfcribing cylinder; then will rESM be two thirds of DABC. For let AC be a fection of the fphere through its centre F; 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 fquare Er be conceived to revolve round the fixed axis Fr, it will generate the cylinder EC; the quadrant FEr will alfo generate the hemifphere EMTE; and the triangle F Dr the cone FDC. And fince FHN is a right angled triangle, and FH is equal to Hm, the fquares of FH, Hn, or of нm, нn, are equal to the fquare of Fn. But Fn is alfo equal to FE or HL; whence the fquares of нm, нn are equal to the fquare of HL: or the circular |