BOOK IX.* THE THREE ROUND BODIES. 740. The only solids bounded by curved surfaces, that are treated of in Elementary Geometry, are the cylinder, the cone, and the sphere, which are called the three round bodies. THE CYLINDER. DEFINITIONS. E 741. A cylindrical surface is a surface generated by the motion of a straight line AB, called the generatrix, which constantly touches a given curve ACDE, called the directrix, and remains paralle. to its original position. The different positions of the generatrix are called elements of the surface. 742. A cylinder is a solid bounded by a cylindrical surface and two parallel planes. The cylindrical surface is called the lateral surface, and the plane surfaces are called the bases. B A The altitude of a cylinder is the perpendicular distance between its bases. 743. A right section of a cylinder is the section by a plane perpendicular to its elements. 744. A circular cylinder is a cylinder whose base is a circle. The axis of a circular cylinder is the straight line joining the centres of its bases. 745. A right cylinder is one whose elements are perpendicular to its bases. *This book treats of the properties and relations of the cylinder, the cone, and the sphere, and shows how to find the convex surface and volume of each of these bodies. 746. A right circular cylinder, called also a cylinder of revolution, is generated by revolving a rectangle about one of its sides. 747. Similar cylinders of revolution are those generated by similar rectangles revolving round homologous sides. со 748. A.tangent plane to a cylinder is a plane which contains an element of the cylinder without cutting the surface. The element which the plane contains is called the element of contact. Any straight line in a tangent plane, which cuts the element of contact, is a tangent line to the cylinder. 749. A prism is inscribed in a cylinder, when its bases are inscribed in the bases of the cylinder and its lateral edges are elements of the cylinder. 750. A prism is circumscribed about a cylinder, when its bases are circumscribed about the bases of the cylinder. Proposition 1. Theorem. 751. Every section of a cylinder made by a plane passing through an element is a parallelogram. Hyp. Let the plane ABCD pass through the element AB of cylinder EH. To prove the section ABCD a . Proof. A plane passing through the element AB cuts the Oce of the base in a second pt. D. Through D draw DC || to AB. E H Then DC is in the plane BAD. ... DC is an element of the cylinder. (68) (741) ... DC, being common to the plane and the lateral sur face of the cylinder, is their intersection. Also, AD is || to BC. (519) .. ABCD is a □. (124) Q.E.D. 752. COR. Every section of a right cylinder made by a plane passing through an element is a rectangle. Proposition 2. Theorem. 753. The lateral area of a cylinder is equal to the perimeter of a right section of a cylinder multiplied by an Now let the number of lateral faces of the inscribed prism be indefinitely increased. The perimeter of the right section of the prism will approach the perimeter of the right section of the cylinder as its limit. (430) .. the lateral area of the prism will approach the lateral area of the cylinder as its limit. Because, however great the number of the lateral faces, 8 = px E, and because p approaches P as its limit and s approaches S as its limit, 754. COR. 1. The lateral area of a cylinder of revolution is equal to the circumference of its base multiplied by its altitude. 755. COR. 2. If H denote the altitude of a cylinder of revolution, R the radius of the base, S the lateral area, and T the total area, we have S = 2πR X H, T = 2πRH+ 2πR2 = 2πR(R + H). 756. COR. 3. Let S, S' denote the lateral areas; T, T' the total areas; R, R' the radii of the bases; and H, H' the altitudes of two similar cylinders of revolution. Then, since the generating rectangles are similar, R H = (747) R' H' R' + H'' (436) (439) T and = = H' H' R R2 R(R+ H) RR+H H2 = Therefore, the lateral areas, or the total areas, of two similar cylinders of revolution are to each other as the squares of their altitudes, or as the squares of the radii of their bases. EXERCISES. 1. Required the lateral area, and also the total area, of a cylinder of revolution whose altitude is 25 inches and the diameter of whose base is 20 inches. 2. Required the convex surface of a right circular cylinder whose altitude is 16 inches, and diameter of the base 8 inches. 3. Required the altitude and radius of the base of a right circular cylinder whose lateral area is as great as a similar cylinder of which the altitude is 20 feet and diameter of the base 8 feet. Proposition 3. Theorem. 757. The volume of a cylinder is equal to the product of its base by its altitude. Hyp. Let V, B, H denote the volume of the cylinder, the area of its base, and its altitude, respectively. Proof. Inscribe in the cylinder a prism, and let V' and B' denote its volume and the area of its base. Then, since the altitude of the prism Now let the number of lateral faces of the inscribed prism be indefinitely increased. The base of the prism B' will approach the base of the cylinder B as its limit, and the volume of the prism V' will approach the volume of the cylinder V as its limit. Because, however great the number of the lateral faces, we always have 758. COR. 1. If R denotes the radius of the base of a cylinder of revolution, then В = π R2. ... V = πR'. H. (439) (757) 759. COR. 2. Let V, V' denote the volumes, R, R' the radii of the bases, and H, H' the altitudes of two similar cylinders of revolution. Then, since the generating rectangles are similar, (747) V R2H R2 HI H3 = X = R$ (758) Vi RI R/2 H' H'3 R's Therefore, the volumes of similar cylinders of revolution are to each other as the cubes of their altitudes, or as the cubes of their radii. |