BGFBID=2BX T, and CIH+CFE = 2CXT. But the sum of these six triangles exceeds the hemisphere by twice the triangle ABC; and the hemisphere is represented by 4T; consequently, twice the triangle ABC is equivalent to

2 A × T + 2 B × T+2C × T — 4T; therefore, once the triangle A B C is equivalent to

(A + B + C − 2) × T.

Hence the area of a spherical triangle is equal to the excess of the sum of its three angles above two right angles multiplied by the quadrantal triangle.

564. Cor. If the sum of the three angles of a spherical triangle is equal to three right angles, its area is equal to the quadrantal triangle, or to an eighth part of the surface of the sphere; if the sum is equal to four right angles, the area of the triangle is equal to two quadrantal triangles, or to a fourth part of the surface of the sphere, &c.

PROPOSITION XXI.-THEOREM.

565. The area of a spherical polygon is equal to the excess of the sum of all its angles above two right angles taken as many times as the polygon has sides, less two, multiplied by the quadrantal triangle.

E

A

C

B

Let ABCDE be any spherical polygon. From one of the vertices, A, draw the arcs A C, AD to the opposite vertices; the polygon will be divided into as many spherical triangles as it has sides less two. But the area of each of these triangles is equal to the excess of the sum of its three angles above two right angles multiplied by the quadrantal triangle (Prop. XX.); and the sum of the angles in all the triangles is evidently the same as that of all the angles in the polygon; hence the area of the polygon A B C D E is equal to the excess of the sum of all its angles above two right angles taken as many times as the polygon has sides, less two, multiplied by the quadrantal triangle.

566. Cor. If the sum of all the angles of a spherical polygon be denoted by S, the number of sides by n, the quadrantal triangle by T, and the right angle be regarded as unity, the area of the polygon will be expressed by

S — 2 (n − 2) × T = (S — 2 n + 4) × T.

BOOK X.

THE THREE ROUND BODIES.

DEFINITIONS.

567. A CYLINDER is a solid, which may be described by the revolution of a rectangle turning about one of its sides, which remains immovable; as the solid described by the rectangle ABCD revolving about its side A B.

A

B

D

The BASES of the cylinder are the circles described by the sides, AC, BD, of the revolving rectangle, which are adjacent to the immovable side, A B.

The AXIS of the cylinder is the straight line joining the centres of its two bases; as the immovable line A B.

The CONVEX SURFACE of the cylinder is described by the side CD of the rectangle, opposite to the axis A B.

568. A CONE is a solid which may be described by the revolution of a rightangled triangle turning about one of its perpendicular sides, which remains immovable; as the solid described by the right-angled triangle A B C revolving about its perpendicular side AB.

The BASE of the cone is the circle described by the revolution of the side BC, which is perpendicular to the immovable side.

B

A

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, AB. 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 AG; then its convex surface is equal to ABCDEF 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 RX (Prop. XV. Cor. 3, Bk. VI.), and the convex surface of the cylinder by 2 R XX H.

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 A B C DEF, and whose altitude is the line A G; then its solid contents are equal to the product of ABCDEF by A G.