PART SECOND. SECTION II. Of the Round Bodies. 266. ROUND 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 hypothenuse SA by this motion generates the conical surface. Each point in the hypothenuse, 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 cone. 268. In the right cone (fig. 140) the similar triangles Fig.140. AC CS ACS, A'C'S, which give the proportion AS A'C' C'S show that the radii of the circles ADB, A'D'B', are A'S' Fig.140. but the circumferences of circles being as their radii (148), and their areas being as the squares of their radii area A'D'B' (A'C')2 (C'S)2 (A'S) 2 In figure 141, supposing A'D'B' to be parallel to the base, (A'C')2 (C'S)2 A'C' C'S A'S O'S' (AS)2 (A'Sj2 Fig.142. (OS)2 (1). If any cone be cut by a plane parallel to its base, the smaller cone cut off is similar to the whole. (2). Similar cones have their homologous dimensions proportional, and their bases proportional to the squares of their homologous lines; and (3). In any cone, parallel sections are to each other as the squares of their distances from the apex. 269. Remark. When we have the dimensions of any truncated cone whose bases are parallel, as ADB, A'D'B', we can find by a process analogous to that of article 233, the dimensions of the entire cone, and also of the superior cone, S- A'D'B'. 270. If we inscribe in the base of the cone (fig. 142) a regular polygon, and circumscribe also about the base a similar polygon; by drawing straight lines from the apex of the cone to the vertices of these polygons, we shall have a pyramid inscribed in the cone, and also a pyramid circumscribed about the cone. The figure, not to make confusion, exhibits but one of the lateral faces of each pyramid. The circumscribed pyramid exceeds the inscribed pyramid by a certain magnitude; but by increasing the number of sides in the polygons which are their bases, we increase the number of lateral faces in the pyramids. The inscribed pyramid is increased, and the circumscribed pyramid is diminished by this process, which may be continued till the difference between the two pyramids is less than any assignable magnitude. By this process, also, the lateral surfaces of the two pyramids are made to approach indefinitely near to each other. The lateral surface of the inscribed pyra- Fig.142. mid can never be greater than the conical surface, and the lateral or convex 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. 2 271. 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. X 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= žcirc. × (SG) +d. The area of the inscribed pyramid, calling p the perimeter of its base, will be pXsg; 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 × (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 SG, the side of the cone : we shall have A-m' circ. × (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 2 Fig.142. vex surface of a right cone, is measured by half the product of the circumference of the base multiplied by the side. 272. To find the area a truncated cone. Subtract from the area of the entire cone the area of the smaller cone cut off. Calling A' the area of the frustum Fig.143. ADB-A'D'B' (fig. 143, A the area of the entire cone, and a the area of the less, we shall have A' = A α = (circ. OA) X (SB) — (circ. O'A') x (SB'), or A' = 1 (circ. OA) × (SB) — 1 (circ. OA) × (SB') + }(circ. O'Ã1) × (SB) — (circ. Õ'A') × (SB') = (circ. OA) (SBSB')+(circ. O'A') × (SB-SB)=(circ. OA+circ.O'A')× (BB'). Hence- The area of the frustum of a right cone has for its measure the product of its side multiplied by half the sum of the circumferences of its bases. Or supposing a section A'D B"E", at equal distances from the two bases of the frustum; the similar triangles These numerators are OAO"A" O''A'' — O'A' equal by construction; the denominators are therefore equal; and OA is as much greater than O"A", as ("A" is greater than O'A'; but these radii are as their circumferences. The circumference A"D"B"E" is therefore an arithmetical mean between the other two; and is consequently equal to half their sum. Hence—The area of the frustum of a cone, has for its measure the product of its side multiplied by the circumference of the plane section made at equal distances from its bases. Remark. By substituting the apex of the cone for the superior hase, 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 its height by its base. We shall however give this a different proof. 3 . The common height of these pyramids will be SO the height of the cone (fig. 142). If we call B the base of the cone, and d the difference between the base of the cone and the base of the circumscribed pyramid, and d' the difference between the base of the cone and the base of the inscribed pyramid, V the volume of the cone, m the excess of the volume of the circumscribed pyramid over the volume of the cone, and m' the excess of the volume of the cone over the volume of the inscribed pyramid. This will give V+m } (SO) × (B + d) ; V — m' = (SO) × (B-d'). That is, if any thing be added to the volume of the one, something must be added to the product (SO) ×, to balance the equation; and if any magnitude however small be subtracted from the volume of the cone, a corresponding quantity must be subtracted from this product, to preserve the equation; from which it follows that (SO) × B, expresses neither more nor less than the volume of the cone; and we say-The volume of a cone has for its measure, one third of the product of the base multiplied by the height. 1 3 3 174. If the radius of the base of a cone be designated by R, and the height of the cone by H; we shall have for the base of the cone л. R3. Denoting its volume by V, we have V = {} ñ. R2 × H. 1 3 Remark. To find the volume of a truncated cone, subtract the volume of the less from the volume of the entire cone. Fig.142. 275. If we conceive the rectangle AC C'A' (fig. 144) Fig.144. to revolve about one of its sides CC', it will generate the body ADB-A'D'B', called a right cylinder. The straight line AA' will, by this revolution, generate the cylindrical surface. Any point, as A", in this straight line, will describe. the circumference of a circle, as A"D"B", equal and parallel to the circle ADB called the base of the cylinder and generated by AC the base of the rectangle. For A"C" perpendicular to CC', and AC also perpendicular to CC', will, by this motion, describe plane circles perpendicular to CC' (199); they are therefore parallel; A"C" being equal to AC, and they being the radii of these circles, the circles are equal. From which it follows that---Any section of a right cylinder parallel to its |