The CONVEX SURFACE of a cone is described by the hypothenuse, A C, of the revolving triangle.

The VERTEX of the cone is the point A, where the hypothenuse meets the immovable side.

The AXIS of the cone is the straight line joining the vertex to the centre of the base; as the line A B.

The ALTITUDE of a cone is a line drawn from the vertex perpendicular to the base; and is the same as the axis, A B. The SLANT HEIGHT, or SIDE, of a cone, is a straight line drawn from the vertex to the circumference of the base; as the line A C.

569. The FRUSTUM of a cone is the part of a cone included between the base and a plane parallel to the base; as the solid CD-F.

F A! E

D

B

The AXIS, or ALTITUDE, of the frustum, is the perpendicular line AB included between the two bases; and the SLANT HEIGHT, or SIDE, is that portion of the slant height of the cone which lies between the bases; as FC.

570. SIMILAR CYLINDERS, or CONES, are those whose axes are to each other as the radii, or diameters, of their bases.

571. The sphere, cylinder, and cone are termed the THREE ROUND BODIES of elementary Geometry.

PROPOSITION I. THEOREM.

572. The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.

Let ABCDEF-G be a cylinder, whose circumference is the circle ABCDEF, and whose altitude is the line. A G; then its convex surface is equal to ABC D E F multiplied by AG.

In the base of the cylinder inscribe any regular polygon, A B C D E F, and on this polygon construct a right prism of the same altitude with the cylinder. The prism will be inscribed in the convex surface of the cylinder. The convex surface of this prism is equal to the perimeter of its base multiplied by its altitude, A G (Prop. I. Bk. VIII.).

Conceive now the arcs subtending the sides of the polygon to be continually bisected, until a polygon is formed having an indefinite number of sides; its perimeter will then be equal to the circumference of the circle ABCDEF (Prop. XII. Cor., Bk. VI.); and thus the convex surface of the prism will coincide with the convex surface of the cylinder. But the convex surface of the prism is always equal to the perimeter of its base multiplied by its altitude; hence, the convex surface of the cylinder is equal to the circumference of its base multiplied by its altitude.

573. Cor. 1. If two cylinders have the same altitude, their convex surfaces are to each other as the circumferences of their bases.

574. Cor. 2. If H represent the altitude of a cylinder, and R the radius of its base, then we shall have the circumference of the base represented by 2 R × я (Prop. XV. Cor. 3, Bk. VI.), and the convex surface of the cylinder by 2 R XTX H.

PROPOSITION II.-THEOREM.

575. The solid contents of a cylinder are equal to the product of its base by its altitude.

Let ABCDEFG be a cylinder whose base is the circle ABCDEF, and whose altitude is the line A G; then its solid contents are equal to the product of ABCDEF by A G.

In the base of the cylinder inscribe any regular polygon, A B C D E F, and on this polygon construct a right prism of the same altitude with the cylinder. The prism will be inscribed in the convex surface of the cylinder. The solid. contents of this prism are equal to the product of its base by its altitude (Prop. XIII. Bk. VIII.).

Conceive now the number of the sides of the polygon to be indefinitely increased, until its perimeter coincides with the circumference of the circle ABCDEF (Prop. XII. Cor., Bk. VI.), and the solid contents of the prism will equal those of the cylinder. But the solid contents of the prism will still be equal to the product of its base by its altitude; hence the solid contents of the cylinder are equal to the product of its base by its altitude.

576. Cor. 1. Cylinders of the same altitude are to each other as their bases; and cylinders of equal bases are to each other as their altitudes.

577. Cor. 2. Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases. For the bases are as the squares of their radii (Prop. XIII. Bk. VI.), and the cylinders being similar, the radii of their bases are to each other as their altitudes (Art. 570); therefore the bases are as the squares of the altitudes; hence, the products of the bases by the altitudes, or the cylinders themselves, are as the cubes of the altitudes.

578. Cor. 3. If the altitude of a cylinder be represented by H, and the area of its base by R2 × (Prop. XV. Cor. 2, Bk. VI.), the solid contents of the cylinder will be represented by R2 × × H.

579. The convex surface of a cone is equal to the circumference of the base multiplied by half the slant height.

Let A B C DEF-S be a cone whose base is the circle ABCDEF, and whose slant height is the line SA; then its convex surface is equal to ABCDEF multiplied by SA.

In the base of the cone inscribe any regular polygon, ABCDEF, and on this polygon construct a regular pyramid having the same vertex, S, with the cone. scribed in the cone.

F

B

S

D

H

с

Then a right pyramid will be in

From S draw SH perpendicular to B C, a side of the polygon. The convex surface of the pyramid is equal to the perimeter of its base, multiplied by half its slant height, SH (Prop. XV. Bk. VIII.). Conceive now the arcs subtending the sides of the polygon to be continually bisected, until a polygon is formed having an indefinite number of sides; its perimeter will equal the circumference of the circle ABCDEF; its slant height, SH, will equal that of the cone, and its convex surface coincide with the convex surface of the cone. But the convex surface of every right pyramid is equal to the perimeter of its base, multiplied by half the slant height; hence the convex surface of the cone is equal to the circumference of its base multiplied by half its slant height.

580. Cor. If SA represent the slant height of a cone, and R the radius of the base, then, since the circumference of the base is represented by 2 R × π (Prop. XV. Cor. 3, Bk. VI.), the convex surface of the cone will be represented by 2 R× × SA, equal to X RX SA.

PROPOSITION IV.-THEOREM.

581. The convex surface of a frustum of a cone is equal to half the sum of the circumference of the two bases multiplied by its slant height.

Let ABCDEF-M be the frustum of a cone, and AG its slant height; then the convex surface is equal to half the sum of the circumferences of the two bases ABCDEF, GHIKLM, multiplied by A G.

A

M L

HO

D

B

C

For, inscribe in the bases of the frustum two regular polygons of the same number of sides, having their sides parallel, each to each. Draw the straight lines AG, BH, CI, &c., joining the vertices of the corresponding angles, and these lines will be the edges of the frustum of a pyramid inscribed in the frustum of the cone. The convex surface of the frustum of the pyramid is equal to half the sum of the perimeters of the two bases multiplied by its slant height, ON (Prop. XVII. Bk. VIII.).

Conceive now the number of sides of the inscribed polygons to be indefinitely increased; the perimeters of the polygons will then coincide with the circumferences of the circles ABCDEF, GHIKLM; and the slant height, ON, of the frustum of the pyramid, will equal the slant height, AG, of the frustum of the cone; and the surfaces of the two frustums will coincide.

But the convex surface of every frustum of a right pyramid is equal to half the sum of the perimeters of its two bases, multiplied by its slant height; hence, the convex surface of the frustum of the cone is equal to half the sum of the circumference of its two bases multiplied by half its slant height.

582. Cor. Through R, the middle point of the side KD,