scribing prism becomes a cylinder. Hence the surface of a sphere is equal to the convex surface of its circumscribing cylinder. Cor. 2. Hence the surface of a revoloid is equal to the perimeter of its centeral conjugate section multiplied by the vertical axis, for the vertical surfaces of the circumscribing prism are equal to this product; and since in a right quadrangular revoloid its circumscribing prism has six equal sides, four of which are equal to the surface of the revoloid, it follows that the whole surface of the revoloid is to the whole surface of the prism as 2 to 3. If represent the perimeter of the central conjugate section, D the vertical axis or diameter, then D will represent the surface, and if H be the altitude of any zone by sections parallel to the conjugate axis, then will H-the curve surface of the zone or segment of the revoloid or sphere. Cor. 3. Hence also the surface of a sphere is equal to its circumference multiplied by its diameter or altitude, for the curve surface of its circumscribing cylinder is equal to this product. And since the surface of a great circle of the sphere is measured by the product of its circumference into half the radius, or by the diameter, (Prop. XVI. B. V. El. Geom.) Therefore the surface of a sphere is four times the area of its great circle: this is equal to 4R', (Prop. XIII, Sch. 3. El. S. Geom.) and because the two bases of the circumscribing cylinder are each equal to one of those circles, it follows that the whole surface of the cylinder is equal to six of those circles, and hence that the whole surface of the sphere is to the whole surface of the cylinder as 2 to 3, as before found in the elements of solid geometry. Cor. 4. Since we have shown that the surface GLMH of the polyedron is equal to the surface GRSH contained within the same parallel planes, it follows that the surface of any zone or segment LNOM either of a revoloid or sphere, is equal to the perimeter or circumference of a central conjugate section multiplied by the altitude of such zone or segment. Cor. 5. The surface of two zones taken in the same revoloid or sphere, or in equal revoloids or spheres, are to each other as the altitudes; and the surface of any zone, is to the surface of the sphere, as the altitude or diameter of the zone is to that of the sphere. Hence the surfaces of every parallel portion of equal altitude are equal. PROPOSITION IV. THEOREM. If a cylindrical ungula be cut by two planes from the same side of the cylinder, the intersection of which planes forms a diameter to the cylinder, and if the altitude of the ungula, or the extreme length of the ungula taken in the direction parallel to the axis of the cylinder, is equal to the cylinder's circumference, then the sections or ungulas so cut, will be equal to a sphere described in the cylinder, or to a sphere whose diameter is equal to that of the cylinder. Let ABCD be a cylinder, and let HK be a sphere of equal diameter described in the cylinder, and let AJB be an ungula cut from the cylinder by the two planes OLJ, MNJ, from the points A and B, whose distance AB parallel to the axis UT is equal to the circumference of the cylinder, and let the cutting planes meet in J forming a diameter to the cylinder; then will the section AJB be equal to the sphere HK. For, let planes be passed through the ungula and sphere perpendicular to the diameter formed by the intersection of the planes LOJ and NMJ; and these plane sections, formed by these planes in each solid, will be proportional to the magnitudes of the solid through such sections. Now, any section of the ungula, by a plane perpendicular to the diameter which passes through the pole J of the sphere, is a triangle; and a section through the sphere made by the same plane, is a circle; and the area of a triangle formed by any section, is equal to its base multiplied by half its altitude on such base. Thus the area of the triangle JAB=ABXHJ; the area of a parallel section Jcc, is in like manner the base ccx the altitude J3; and so for the area of any other parallel section Jbb, Jaa, &c. ; and because these triangles are equiangular, (Prop. XXII. B. IV. El. Geom.) their bases are proportional to their altitudes; thus, HJ: AB: J3: cc :: J2: bb :: J1: aa, &c. The areas of the :: = several circles formed by the same planes passing through the sphere, are equal to their circumferences multiplied by their several radii; thus the area of the great circle of the sphere HXKY is equal to the circumference HXKYX the radius HJ, the area of the circle PQ corresponding to the section Jec, is equal to the circumference PQ× radius JP, and the areas of the circles RS, &c., their several circumferences multiplied by their several radii. And because the circumfer ence of circles are to each other as their radii, (Prop. XV. B. V. El. Geom.) as radius JK: circle HK: : radius J3 : circle PQ radius J2: circumference RS, &c.; and, since this is the ratio of the lines AB, cc, bb, &c., as shown above, the several circumferences are in the same ratio of those lines as bases of their several triangles; but the line AB by hypothesis, is equal to the circumference HK; hence the several circumferences PQ, RS, 1J, &c., are respectively equal to the several bases cc, bb, aa, &c.; hence, also, the areas of the several circles being sections of the sphere, are respectively equal to their seve ral corresponding triangles, being sections of the ungula made by the same planes; and as this is true whatever may be the number of the parallel sections, or in whatever position they are taken, it follows that the solidity of the ungula is equal to that of the sphere. Cor. 1. Since the section AJB may be regarded as composed of the two unglas AJH, BJH, regarding JH as their common base, and because ungulas of the same base are proportional to their altitudes, it follows that if we cut the ungula HgJ, whose altitude Hg= HB, and the ungula HJf, whose altitude HfHA, then the section fJg including those ungulas together with an equal opposite section, hJi are equal to the section AJB, consequently equal to the sphere HK. Or if we take the section CTD, whose base CD=AB, this section will also be equal to the sphere HK. Cor. 2. Since, it may be shown, that all sections of a spheroid or ellipsoid, by planes parallel to its axis of revolu tion, are similar ellipses; and since ellipses are to their inscribed circles, as the diameter of the circle to the major axis of the ellipse, (Proposition IX of the Ellipse, B. I,) the solidity of an ellipsoid HWKV is to the solidity of the sphere HYKX, as the axis of revolution VW of the ellipsoid is to the axis of revolution XY of the sphere, and hence if an ungula, whose base is equal to half the base of the cylinder be taken, and whose altitude is to that of the ungula CTD as the axis of revolution VW of the ellipsoid to the axis of revolution XY of the sphere, that ungula will be equal to the ellipsoid HWKV made by the revolution of the ellipse on its axis ŴW. Cor. 3. As the ellipsoid HWKV is to the sphere HYKX as the axis VW of the ellipsoid to the axis XY of the sphere, and as the cylinder fhig circumscribing the ellipsoid is to the cylinder, pqrs circumscribing the sphere, as the same axis VW to the same axis XY, or as the length of those cylinders resrespectively; the ellipsoid HWKV bears the same ratio to its circumscribing cylinder fhig, as the sphere HYKX to its circumscribing cylinder pqrs. If S=the sphere, and E=the ellipsoid, and if P=the cylinder circumscribing the ellipsoid, and Q=the cylinder circumscribing the sphere, then, and, XY:S:: VW: E XY: Q:: VW: P, hence, (Prop. XIX. B. I. El. Geom.) S: Q:: E P. Scholium. The segment AJB being equal to the sphere HK, the annexed figure, may represent the manner in which they are convertible into each other, as there exists no mathematical reason why the segment AJB may not be changed, as partly represented in the figure. Cor. 4. It is also evident that an ungula BCDH, whose base BCD is the segment of a circle, similar to a segment AHB of a vertical section through this circular spindle AHBG; if the altitude DH of the ungula is equal to the circumference of the conjugate section HG of the spindle, the solidity of the ungula will be equal to that of the spindle, so also will its curve surface. And, if a cylinder segment ABCDH be so cut that the length AB shall be equal to the circumference of a circle of which HG is the radius, and AG and BG lie in the planes ACE, BFD, then may the G A H H segment be converted into a ring whose outside diameter is equal 2HG; and every section of the ring will be equal to a section through the segment. Cor, 5. Hence, also, if a cylindric segment ABC be cut by two planes meeting in the surface of the cylinder at C, and A D B terminating at A and B on the opposite And if instead of the cutting planes meeting in the surface of the cylinder, they meet at a distance EC from the cylinder, C and at such an angle that the distance FG shall be equal to the circumference of a circle whose radius is CD, and the section AFGB will form a cylindric ring, whose inner diameter is equal to twice CE. |