THEOREM VIII. The lateral surface of a frustum of a right pyramid is equal to one half the sum of the perimeters of its bases, multiplied by the slant height. [The proof is obvious. The area of a trapezoid equals what? Find each lateral face, and add.] Db E B CHAPTER XV. THE CYLINDER. DEFINITIONS. 1. A Cylindrical Surface is a curved surface generated by the movement of a straight line which continually touches a given curve, and remains constantly parallel to its first position. Thus, the line A B, moving so as continually to touch the given curve CE, in parallel positions, generates a C cylindrical surface. 2. The moving line is the Generatrix. B 3. The curve within which the generatrix moves, is the Directrix.1 4. A line of the surface corresponding to any position of the generatrix is an Element. [Evidently, if the generatrix is of indefinite length, a cylindrical surface of indefinite extent is generated. The directrix may be any curve whatever-closed or open. are in this treatise limited to one, that in which the directrix is a circle.] 5. A Cylinder is a volume bounded. by a cylindrical surface and two parallel planes, as A H. 6. The Convex Surface of a cylinder is its cylindrical surface. 7. The Bases of a cylinder are its plane surfaces, as A B C D and E F HG. The possible cases E B G 1 The curve is conceived as directing the motion of the generatrix. H [The term cylinder, then, is assumed to denote one whose bases are circles.] 8. The Altitude of a cylinder is the perpendicular distance from one base to the plane of the other, as a. 9. The Radius of a cylinder is the radius of the base. 10. A Right Cylinder' is one whose elements are perpendicular to its bases. We may conceive a right cylinder to be generated by the revolution of a rectangle about one of its sides as an axis. 11. An Oblique Cylinder is one whose elements are oblique to its bases. 12. Similar Right Cylinders are those which are generated by similar rectangles revolving about homologous sides. 13. A prism is Inscribed in a cylinder when the bases of the prism are inscribed in those of the cylinder, the edges of the prism coinciding with elements of the cylinder. A H Cor. I.-Every section of a cylinder embracing two elements, is a parallelogram. For these elements are, by definition, parallel; and they are equal (Ch. XII, Th. VI); hence A H is a parallelogram. Cor. II. The bases of a cylinder are equal. For every position of the generatrix is parallel to its first one. Cor. III.-Any two parallel sections of a cylinder are equal. 'Right cylinder with circular base, sometimes called 'Cylinder of Revolution.' Cor. IV. All sections of a cylinder parallel to its bases, are equal circles. Cor. V. The altitude of an oblique cylin der is less than any element. For the perpendicular is the shortest distance from one base to the plane of the other. Cor. VI. The altitude of a right cylinder is equal to any element. For the elements are all perpendicular to the planes of the bases, which are parallel. THEOREM I The convex surface of a right cylinder is equal to the circumference of its base multiplied by its altitude. Let A B be a right cylinder; then will its convex surface be equal to the circumference of its base multiplied by its altitude. For, inscribe within the cylinder a prism whose bases are regular polygons. Then the lateral surface of this prism equals the perimeter of its base multiplied by its altitude, whatever be the number of sides or faces. Ch. XIII, Th. II. A But when the number of sides becomes B infinite (Ch. X, Th. VI, Sch.), the prism coincides, in every respect, with the cylinder. Hence, the convex surface of the cylinder equals the circumference of its base multiplied by its altitude. Q. E. D. Cor. I. Let S denote the lateral surface, C the circumference of the base, and a the altitude; then S=Cxa= 2 π r × а Ch. X, Th. XII. Cor. II.-Let S' denote the entire area (including the areas of the bases); then S'=S+2 + = p2 = 2πρΧα + 2 προ Ch. X, Th. II. Scholium.-The truth of this theorem is clearly evident from the consideration, that if the cylindrical surface were cut open and spread out, it would assume the form of a rectangle, whose base and altitude would be the circumference and altitude of the cylinder. THEOREM II. The volume of a cylinder is equal to the product of its base by its altitude. A Let A B be a cylinder; then B will its volume be equal to the product of its base by its altitude. For, inscribe within the cylinder a prism whose bases are regular polygons. Then the volume of the prism equals the product of its base by its altitude, whatever be the number of sides. But when the number of sides is infinite, the prism concides, in every respect, with the cylinder. Hence, the volume of the cylinder equals the product of its base by its altitude. Q. E. D. Cor.-Let V denote the volume, r the radius of the base, and a the altitude; then V= r2 × α = π r2 а. |