BOOK VIII THE CYLINDER AND CONE DEFINITIONS 480. A cylindrical surface is a surface generated by a moving straight line, which constantly intersects a given plane curve, and in all its positions is parallel to a given straight line, not in the plane of the curve. N Thus, if line AB moves so as constantly to intersect plane curve AD, and is constantly line MN, not in M the plane of the curve, it generates a cylindrical surface. B D E We call the curve the directrix, the moving line the generatrix, and any position of it, EF, an element of the surface. A cylinder is a solid bounded by a cylindrical surface, called the lateral surface, and two parallel planes called the bases. The altitude of a cylinder is the perpendicular distance between the planes of its bases. A right cylinder is a cylinder the elements of whose lateral surface are perpendicular to its bases. A circular cylinder is a cylinder whose base is a circle. о A right circular cylinder is called a cylinder of revolution because it may be generated by the revolution of a rectangle about one of its sides as an axis (§ 371). Similar cylinders of revolution are cylinders by the revolution of similar rectangles about homologous sides as axes. 481. It follows from the definitions of § 480 that the elements of the lateral surface of a cylinder are equal and parallel. (§ 365) PROP. I. THEOREM 482. A section of a cylinder made by a plane passing through an element of the lateral surface is a parallelogram. E Given ABCD a section of cylinder AF, made by a plane passing through AB, an element of the lateral surface. To Prove section ABCD a □. Proof. 1. We have AD, BC || str. lines. (§§ 348, 364) 2. If a str. line be drawn from C in plane AC|| AB, it is an element. (§§ 481, 71) 3. Then, it must be the intersection of plane AC and the cylindrical surface and coincide with CD; and CD || AB. 483. It follows from § 482 that a section of a right cylinder made by a plane perpendicular to its base is a rectangle. Proof. 1. Take E', F, G' any three points in perimeter A'B'; draw elements E'E, FF, G'G, and lines EF, FG, GE, E'F, F'G', G'E'. 2. We have EE', FF" equal and I (?); whence EE'FF is a O. 3. Then (?) (?) Similarly 4. Then A E'F'G' =▲ EFG (?); and base A'B' may be applied to AB so that E' shall fall at E, F" at F, G' at G. 5. Then, all points in perimeter A'B' will fall in perimeter AB, and bases are equal. The following are consequences of § 484. 485. The sections of a cylinder made by two parallel planes cutting all the elements are equal. For they are the bases of a cylinder. 486. The section of a cylinder made by a plane parallel to the base is equal to the base. MEASUREMENT OF CYLINDERS DEFINITIONS 487. The lateral area of a cylinder is the area of its lateral surface. A right section of a cylinder is a section made by a plane perpendicular to the elements of its lateral surface. A prism is said to be inscribed in a cylinder when its lateral edges are elements of the cylindrical surface, and its bases inscribed in the bases of the cylinder. A plane is said to be tangent to a cylinder when it contains one, and only one, element of the lateral surface. A prism is said to be circumscribed about a cylinder when its lateral faces are tangent to the cylinder, and its bases circumscribed about the bases of the cylinder. 488. If a prism whose base is a regular polygon be inscribed in, or circumscribed about, a circular cylinder (§ 480), and the number of sides of its base be indefinitely increased, 1. The lateral area of the prism approaches the lateral area of the cylinder as a limit. 2. The volume of the prism approaches the volume of the cylinder as a limit. 3. The perimeter of a right section of the prism approaches the perimeter of a right section of the cylinder as a limit.* Ex. 1. Find area of the section made by a plane drawn through an element of the lateral surface of a cylinder of revolution, whose altitude is 12, and radius of base 6, if the distance from the centre of the base to the cutting plane is 3. * Rigorous proofs of these statements are beyond the scope of the present treatise. 489. The lateral area of a circular cylinder is equal to the perimeter of a right section multiplied by an element of the lateral surface. Given S the lateral area, P the perime ter of a rt. section, E an element of the lateral surface, of a circular cylinder. Proof. 1. Inscribe in the cylinder a prism whose base is a regular polygon; let S' denote its lateral area, and P' the perimeter of a rt. section. 2. We have S'P' x E. (§ 429) (§ 488) 3. If number of sides of base be indefinitely increased, find limits of S' and P' x E. 4. We have (1) by Theorem of Limits. (§ 187) 490. By § 489, the lateral area of a cylinder of revolution is equal to the circumference of its base multiplied by its altitude. 491. If S denotes the lateral area, T the total area, H the altitude, R the radius of the base, of a cylinder of revolution, S=2π RH. (§ 334) And, T=2 RH+2π R2 (§ 337) = 2π R (H+R). π PROP. IV. THEOREM 492. The volume of a circular cylinder is equal to the product of its base and altitude. Given the volume, B the area of the base, and H the altitude, of a circular cylinder. To Prove V=Bx H. (1) Proof. 1. Inscribe in cylinder a prism whose base is a regular polygon; let V denote its volume, B' the area of its base. |