proaches, if the number of its lateral faces is indefinitely increased.

The volume of a cylinder is the limit which the volume of the inscribed prism approaches if the number of its faces is increased indefinitely.

PROPOSITION XXV. THEOREM

654. The lateral area of a cylinder is equal to the product of the perimeter of a right section of the cylinder by an element.

Given L the lateral area, P the perimeter of a right section, and E an element of the cylinder AK; L' the lateral area, p' the perimeter of a section of a prism with a regular polygon as base, inscribed in cylinder AK.

Proof. The edge of the inscribed prism coincides with an element of the cylinder.

If the number of faces of the inscribed prism be increased, L' will approach L as a limit, and P' will approach P as a limit.

655. DEF. A cylinder of revolution is a right circular cylinder because it may be generated by a rectangle revolving about one of its sides as an axis.

656. DEF. Similar cylinders of revolution are cylinders generated by similar rectangles revolving about homologous sides

as axes.

657. COR. 1. The lateral area of a cylinder of revolution is the product of the circumference of its base by its altitude. 658. COR. 2. If I denote the lateral area, T the total area, H the altitude, and R the radius of a cylinder of revolution, L=2 TRH, and

T=2 TR (H+R).

Ex. 1593. Find the total area of a cylinder of revolution, if H = 6, and R = 2.

659. The volume of a circular cylinder is equal to the product of its base by its altitude.

K

Given

A

the volume, B the base, and I the altitude, of cylinder AK; ' the volume, B′ the regular polygon forming the base of a prism inscribed in AK.

HINT. Use the Theorem of Limits as in preceding proposition.*

* Assume that B is the limit which B' approaches, if the number of sides is indefinitely increased.

R,

660. COR. For a cylinder of revolution with radius of base V=TR2 X H.

661. The lateral areas, or the total areas, of similar cylinders of revolution are to each other as the squares of their radii, or as the squares of their altitudes; and their volumes are to each other as the cubes of their radii, or as the cubes of their altitudes.

Given L, L', the lateral areas; T, T', the total areas; V, v', the volumes; R, R', the radii; and H, H', the altitudes of two similar cylinders of revolution.

L : L' = T : T' = R2 : R'2 = H2 : H12,
V: V' = R3: R'3 = H3 : H'3.

To prove

and

Ex. 1594. Find the ratio of the volumes of two similar cylinders of revolution, if their altitudes are to each other as 3 to 4.

Ex. 1595. Find the radius of the base of a cylinder of revolution if 22

the lateral surface equals 440, and the altitude equals 7 (

assume π=

7

Ex. 1596. Find the radius of the base of a cylinder if its lateral surface is equivalent to the sum of the lateral surfaces of three cylinders of revolution, whose radii are 3, 4, and 5, and all four cylinders have equal altitudes.

Ex. 1597. Two cylinders of revolution have equal altitudes, and their radii are respectively 3 and 4. Find a third cylinder of revolution of the same altitude and equivalent to the sum of the two given cylinders.

Ex. 1598. Find volume of a circular cylinder if the radius of the base is 3, an element, E, equals 4, and the inclination of E to the base is 45°.

CONES

662. DEF. A conical surface is a surface generated by a moving straight line which continually intersects a given fixed curve and constantly passes through a fixed point not in the same plane with the curve.

663. DEF. The generatrix of the surface is the moving straight line; the directrix is the given curve; and the vertex is the fixed point.

664. DEF. An element of a conical surface is the generatrix in any position.

665. DEF. The upper and lower nappes are the portions of the conical surface formed above and below the vertex when the length of the generatrix is unlimited.

666. DEF. A cone is a solid bounded by a conical surface and a plane cutting all its elements.

667. DEF. The lateral surface of the cone is the conical surface; the base of the cone is the plane surface; the vertex of the cone is the vertex of the conical surface; and the elements of the cone are the elements of the conical surface.

668. DEF. A circular cone is a cone whose base is a circle; the axis of a circular cone is the straight line joining the vertex and the center of the base.

669. DEF. A right circular cone is one whose axis is perpendicular to the base; an oblique circular cone is one whose axis is not perpendicular to the base.

NOTE. In this book only circular cones are treated of.

670. DEF. The altitude of a cone is the perpendicular from the vertex to the plane of the base.

671. DEF. A cone of revolution is a right circular cone, for this latter may be generated by a right triangle revolving about one of its arms as an axis.