570.

Cor. By Ex. 497 can you show that the entire surface of Fig. (1) is to entire surface of Fig. (2) as

Can you prove that the volume of any cylinder equals the product of its base by its altitude?

Sug. 1. How do you find the volume of any prism? If the lateral faces of the prism be indefinitely increased, to what does the prism approach as its limit?

H

Sug. 2. Let G C be any cylinder and inscribe a prism within the cylinder. Now let the number of sides increase indefinitely.

Sug. 3. For convenience call volume of the cylinder

v, altitude h, base b, and volume of prism v', altitude h', base b'.

v = ? For any number of sides?

v = ? b = ?

Draw and state conclusion, and call it Prop. V.

572.

Cor. I. Call r the radius of a cylinder of revolution, and show that v = r2 h.

573.

Cor. II. Can you show that the volumes of any two similar cylinders of revolution are to each other as the cubes of their radii, or as the cubes of their altitudes ?

Cor. III. Can you show that similar cylinders of revolution are to each other as the cubes of any like dimensions?

THE CONE.

Definitions.

575.

A conical surface is the surface generated by a moving straight line called the generatrix, passing through a fixed point and constantly touching a fixed curve.

576.

The fixed point is called the vertex, and the fixed curve is called the directrix. Let the figure represent a conical surface.

The generatrix in any position, as A A' C C', is called an element of the surface.

578.

When the generatrix is of indefinite length, the surface is composed of two portions, one above the vertex, the other below the vertex. These parts are called the upper and lower nappes, respectively.

579.

A cone is a solid bounded by a conical surface and a plane. The plane is called the base of the cone, the conical surface is called the lateral, or convex surface.

580.

The altitude of a cone is the perpendicular distance from the vertex to the base, or the base produced. The axis of a cone is the line joining its vertex to the center of its base.

581.

A circular cone is one whose base is a circle.

582.

If the axis is perpendicular to the base, the cone is called a right cone.

cone.

583.

If the axis is oblique to the base, it is called an oblique

584.

A right circular cone is called a cone of revolution, since it may be generated by revolving a right triangle about one of its legs as an axis.

What does the hypotenuse generate? the leg not used for the axis? What is the hypotenuse in any position?

The element of a right cone is the slant height of the cone.
All the elements of a cone of revolution are equal. Why?
Has an oblique cone a slant height?

585.

Similar cones of revolution are cones generated by similar right triangles revolving about homologous legs.

586.

A tangent line to a cone is a line, not an element, which touches the cone, but does not cut it, no matter how far it is produced.

587.

A tangent plane contains an element of the cone, but does not, when produced, cut the surface.

The element of contact is the element contained by the tangent plane.

A pyramid is inscribed in a cone when its lateral edges are elements of the cone and its base is inscribed in the base of the cone.