PART SECOND. SECTION II.-Of the Round Bodies. 266. Rou ND bodies are those which are produced by the revolution of any plane figure about a straight line ; they are called bodies of revolution. Those usually discussed in the elementary treatises, are the right cone, the right cylinder and the sphere. The right cone is generated by the revolution of a right-angled triangle about one of the sides containing the right-angle, as the triangle SCA (fig. 140) about the Fig.140. side SC. The hypothen use SA by this motion generates the comical surface. Each point in the hypothen use, describes the circumference of a circle whose centre is in the line SC; the circle generated by the line CA is called the base of the cone. The line CS, upon which the generating triangle turns, is called the axis of the cone. The point S is called the summit or apex of the cone. It is evident that a plane passing through the axis will cut the conical surface in two straight lines. A plane perpendicular to the axis, will have the circumference of a circle for its section of the conical surface. 267. The cone just described is the right cone whose base is a circle. The inclined cone with a circular base (fig. 141) may be considered as generated by the motion Fig.141. of a straight line, as AS, one point of which, as S, being fixed, the other part being carried round the circumference of a circle, as ADB, situated in a plane which does not pass through S. The straight line CS joining the apex with the centre of the base, is here also called the axis. This cone is also called an oblique or scalene COlle. 268. In the right cone (fig. 140) the similar triangles Fig.149. AC / (TV / e :-- .. _ CS ACS, A/C/S, which give the proportion ATC; - ÖS AS to e e 33 A7S’ show that the radii of the circles ADB, A/D'B', are proportional to their distances from the apex of the cone; 9 Fig.140. Fig.142. but the circumferences of circles being as their radii }:{rs. We see, therefore, that it is with cones as with each other. The lateral surface of the inscribed pyra- Fig.142. mid can never be greater than the conical surface, and the lateral or convea surface of the circumscribed pyramid can never be less than the conical surface. The difference between the convex surface of the circumscribed pyramid and the conical surface, must always be less than the difference between the two pyramidal surfaces which we suppose less than any assignable magnitude. 271. The convex surface of the pyramid circumscribed about a right cone, is composed of a certain number of triangles, which, on account of the regularity of the base, and the summit being in a straight line perpendicular to the middle of the base, are isosceles and equal. The common height of all these triangles is SG the side of the cone ; one half the product of SG multiplied by the sum of their bases, will give their area. The sum of the bases of these triangles is the perimeter of the polygon; therefore, if we denote this perimeter by P, we shall have for the area of the pyramid . P × SG. This perimeter exceeds the circumference of the circle, and therefore this product exceeds the product + circ. × SG by an indefinitely small magnitude which we designate by d, and denoting the area of the cone by A, and the excess of the pyramidal surface over the conical surface by m ; we shall have the equation A + m = + circ. × (SG) -- d. The area of the inscribed pyramid, calling p the perimeter of its base, will be #p × s g : and denoting the excess of the conical surface over the lateral surface of this pyramid by m'; and also denoting by d the excess of the product circ. × (SG) over the product , p X (sg), which may be less than any assignable quantity, as by increasing the sides of the polygon, the perimeter becomes sensibly confounded with the circumference, and sg with Sū, the side of the cone : we shall have A — m' = } circ. X (SG) — d, we see, therefore, that if any magnitude, however small, be added to the area of the cone, something must be added to the product circ. X (SG) to make the equation ; and if any magnitude, however small, be subtracted from this area, a corresponding quantity must be subtracted from this product to balance the equation; from which it follows that circ. × (SG) expresses neither more nor less than the area of the conical surface. Wherefore—The con Fig.142. Fig.143. vex surface of a right cone, is measured by half the pro- SO – SO// SO// —- SO/ e whence (82) OACOVAW F O77A77 –O7A/ 2 that 1S2 O//O O' Off OA – OVAs f OWAV, O'A' Remark. By substituting the apex of the cone for the superior base, the first formula gives the area of the entire COne. 273. We have seen (270) that two pyramids may be constructed the one inscribed in a cone and the other circumscribed about it, such that the difference of their volumes shall be less than any assignable magnitude. The cone, therefore, may be considered as a pyramid of an infinite number of lateral faces; it will therefore have for the measure of its volume, one third of the product of These numerators are its height by its base. We shall however give this a less than the volume of the cone ; and we say—The vol ume of a cone has for its measure, one third of the pro- Fig.144. |