BOOK VIII CYLINDERS AND CONES CYLINDERS 676. A cylindrical surface is a curved surface generated by a straight line which moves so as constantly to touch a given fixed curve and constantly be parallel to a given fixed straight line. B D Cylindrical surface Thus, every shadow cast by a point of light at a great distance, as by a star or the sun, approximates the cylindrical form, that is, is bounded by a cylindrical surface of light. Hence, in all radiations (as of light, heat, magnetism, etc.) from a point at a great distance, we are concerned with cylindrical surfaces and solids. 677. The generatrix of a cylindrical surface is the moving straight line; the directrix is the given curve, as CDE; an element of the cylindrical surface is the moving straight line in any one of its positions, as DF. 678. A cylinder is a solid bounded by a cylindrical surface and by two parallel planes. The bases of a cylinder are its parallel plane faces; the lateral surface is the cylindrical surface included between the parallel planes forming its bases; the altitude of a cylinder is the distance between the bases. Cylinder The elements of a cylinder are the elements of the cylindrical surface bounding it. (403) 679. Property of a cylinder inferred immediately. All the elements of a cylinder are equal, for they are parallel lines included between parallel planes (Arts. 532, 676). The cylinders most important in practical life are those determined by their stability, the ease with which they can be made from common materials, etc. 680. A right cylinder is a cylinder whose elements are perpendicular to the bases. 681. An oblique cylinder is one whose elements are oblique to the bases. 682. A circular cylinder is a cylinder whose bases are circles. 683. A cylinder of revolution is a cylinder generated by the revolution of a rectangle about one of its sides as an axis. Hence, a cylinder of revolution is a right circular cylinder. Some of the properties of this solid are derived most readily by considering it as generated by a revolving rectangle; and others, by regarding it as a particular kind of cylinder derived from the general definition. Oblique circular cylinder Cylinder of revolution 684. Similar cylinders of revolution are cylinders generated by similar rectangles revolving about homologous sides. 685. A tangent plane to a cylinder is a plane which contains one element of the cylinder, and which does not cut the cylinder on being produced. Ex. 1. A plane passing through a tangent to the base of a circular cylinder and the element drawn through the point of contact is tangent to the cylinder. (For if it is not, etc.) Ex. 2. If a plane is tangent to a circular cylinder, its intersection with the plane of the base is tangent to the base, 686. A prism inscribed in a cylinder is a prism whose. lateral edges are elements of the cylinder, and whose bases are polygons inscribed in the bases of the cylinder. Inscribed prism Circumscribed prism 687. A prism circumscribed about a cylinder is a prism whose lateral faces are tangent to the cylinder, and whose bases are polygons circumscribed about the bases of the cylinder. 688. A section of a cylinder is the figure formed by the intersection of the cylinder by a plane. A right section of a cylinder is a section formed by a plane perpendicular to the elements of the cylinder. 689. Properties of circular cylinders. By Art. 441 the area of a circle is the limit of the area of an inscribed or circumscribed polygon, and the circumference is the limit of the perimeters of these polygons; hence 1. The volume of a circular cylinder is the limit of the volume of an inscribed or circumscribed prism. 2. The lateral area of a circular cylinder is the limit of the lateral area of an inscribed or circumscribed prism. Also, 3. By methods too advanced for this book, it may be proved that the perimeter of a right section is the limit of the perimeter of a right section of an inscribed or circumscribed prism. PROPOSITION I. THEOREM 690. Every section of a cylinder made by a plane passing through an element is a parallelogram. B A Given the cylinder AQ cut by a plane passing through the element AB and forming the section ABQP. To prove Proof. ABQP a . Art. 531. It remains to prove that PQ is a straight line || AB. (for the line drawn lies in both the cutting plane and the cylindrical surface, hence, it must be their intersection). :. PQ is a straight line || AB. .. ABQP is a O. (Why ?) Q. E. D. 691. COR. Every section of a right cylinder made by a plane passing through an element is a rectangle. Ex. 1. A door swinging on its hinges generates what kind of a solid ? Ex. 2. Every section of a parallelopiped made by a plane intersecting all its lateral edges is a parallelogram, PROPOSITION II. THEOREM 692. The bases of a cylinder are equal. P B D Given the cylinder AQ with the bases APB and CQD. base APB = base CQD. To prove Proof. Let AC and BD be any two fixed elements in the surface of the cylinder AQ. Take P, any point, except A and B in the perimeter of the base, and through it draw the element PQ. Draw AB, AP, PB, CD, CQ, QD. Apply the base APB to the base CQD so that AB coin cides with CD. Then P will coincide with Q, (for ▲ APB = ▲ CQD). But P is any point in the perimeter of the base APB. every point in the perimeter of the lower base will coincide with a corresponding point of the perimeter of the upper base. ..the bases will coincide and are equal. Art. 47. Q. E. D. |