BOOK IX. MEASUREMENT OF THE THREE ROUND BODIES THE CYLINDER. 1. DEFINITION. The area of the convex, or lateral, surface of a cylinder is called its lateral area. 2. Definition. A prism is inscribed in a cylinder when its bases are inscribed in the bases of the cylinder. If a polygon ABCDEF is inscribed in the base of a cylinder, planes passed through the sides of the polygon, parallel to the elements of the cylinder, intersect the cylinder in parallelograms, ABB'A', etc. (VIII. 6), which evidently determine a prism inscribed in the cylinder. B FI E' D' 3. Definition. A prism is circumscribed about a cylinder when its bases are circumscribed about the bases of the cylinder. If a polygon ABCD is circumscribed about the base of a cylinder, planes D' d' D B' passed through the sides of the polygon, parallel to the elements of the cylinder, will evidently contain the elements, aa', bb', etc., drawn at the points of contact, and be tangent to the cylinder in these elements. The intersection of these planes with the plane of the upper base of the cylinder will therefore determine a polygon A'B'C'D', equal to ABCD, circumscribed about the upper base, and a prism will be formed which is circumscribed about the cylinder. A B 5. Definition. Similar cylinders of revolution are those which are generated by similar rectangles revolving about homologous sides. PROPOSITION I.-THEOREM. 6. A cylinder is the limit of the inscribed and circumscribed prisms, the number of whose faces is indefinitely increased. DI d' B' Let any polygon abcd be inscribed in the base of the cylinder ac' and at the vertices of this polygon let tangents be drawn to the base of the cylinder forming the circumscribed polygon ABCD. Upon these polygons as bases let prisms be formed, inscribed in, and circumscribed about, the cylinder. We shall assume, as evident, that the convex surface of the cylinder is greater than that of the inscribed prism and less than that of the circumscribed prism.* D b B Suppose the arcs ab, bc, etc., to be bisected and polygons to be formed having double the number of sides of the first; and upon these as bases suppose prisms to be constructed, inscribed and circumscribed, as before; and let this process be repeated an indefinite number of times. The difference between the convex surface of the inscribed prism and that of the corresponding circumscribed prism will continually diminish and approach to zero as its limit. There * A proof, however, can be given analogous to that of (V. 32). fore these convex surfaces themselves approach to the convex surface of the cylinder as their common limit. At the same time, it is evident that the volumes of the inscribed and circumscribed prisms approach to the volume of the cylinder as their common limit. curve. 7. Scholium. In the preceding demonstration, the base of the cylin der is not required to be a circle, but may be any closed convex We have, however, tacitly assumed that the curve is the limit of the perimeters of the inscribed and circumscribed polygons; a principle which was rigorously proved in the case of regular polygons inscribed in a circle. PROPOSITION II.-THEOREM. 8. The lateral area of a cylinder is equal to the product of the perimeter of a right section of the cylinder by an element of the surface. Let ABCDEF be the base and AA' any element of a cylinder, and let the curve abcdef be any right section of the surface. Denote the perimeter of the right section by P, the element AA' by E, and the lat eral area of the cylinder by S. b E E B C e Inscribe in the cylinder a prism ABCDEFA' of any number of faces. The right section, abcdef, of this prism will be a polygon inscribed in the right section of the cylinder formed by the same plane. Denote the lateral area of the prism by 8, and the perimeter of its right section by p; then, the lateral edge of the prism being equal to E, we have (VII. 16), 8=p X E. Let the number of lateral faces of the prism be indefinitely increased, as in the preceding proposition; then s approaches indefinitely to S as its limit, and p approaches to P; therefore, at the limit, we have (V. 31), S= PXE. 9. Corollary I. The lateral area of a right cylinder is equal to the product of the perimeter of its base by its altitude. 10. Corollary II. Let a cylinder of revolution be generated by the rectangle whose sides are R and H revolving about the side H. Then, R is the radius of the base, and H is the altitude of the cylinder. The perimeter of the base is 2R (V. 40), and hence, for the lateral area S we have the expression R H The area of each base is R' (V. 43); hence the total area T of the cylinder of revolution, is expressed by T= 2xR.H+ 2xR2 = 2′′R(H+ R). 11. Corollary III. Let S and 8 denote the lateral areas of two similar cylinders of revolution (4); T and t their total areas; R and r the radii of their bases; H and h their altitudes. The generating rectangles being similar, we have (III. 12) R H That is, the lateral areas, or the total areas, of similar cylinders of revo lution are to each other as the squares of their altitudes, or as the squares of the radii of their bases. PROPOSITION III.-PROBLEM. 12. The volume of a cylinder is equal to the product of its base by its altitude. Let the volume of the cylinder be denoted by V, its base by B, and its altitude by H. Let the volume of an inscribed prism be denoted by V', and its base by B'; its altitude will also be H, and we shall have (VII. 38) V' = B' X H. Let the number of faces of the prism be indefinitely increased, as in (8); then the limit of V' is V, and the limit of B' is B; therefore (V. 31), V=BX H. B H 13. Corollary I. Let V be the volume of a cylinder of revolution, R the radius of its base, and H its altitude; then the area of its base is R2 (V. 43); and therefore 14. Corollary II. Let V and v be the volumes of two similar cylinders of revolution; R and r the radii of their bases; H and h their altitudes; then, the generating rectangles being similar, we have that is, the volumes of similar cylinders of revolution are to each other as the cubes of their altitudes, or as the cubes of their radii. |