As we continually double the number of sides of the inscribed polygon, its semi-perimeter approaches the semicircle as limit, and its surface of revolution approaches the sphere as limit, while CQ, its apothem, approaches r, the radius of the sphere, as limit. Representing the sum of the surfaces of the frustums by ΣF, and BD by 2r, we have

That is, the variable sum is to the variable CQ in the constant ratio 4; therefore, by 798, Principle of Limits, their limits have the same ratio,

835. The last proof gives also the following rule:

To find the area of a zone.

RULE. Multiply the altitude of the segment by twice π times the radius of the sphere.

FORMULA. 2 = 2αrπ.

CHAPTER IV.

SPACE ANGLES.

836. A Plane Angle is the divergence of two straight lines which meet in a point.

837. A Space Angle is the spread of two or more planes which meet in a point.

IV

838. Symmetrical Space Angles are those which cut out symmetrical spherical polygons on a sphere, when their vertices are placed at its center.

839. A Steregon is the whole amount of space angle round about a point in space.

840. A Steradian is the angle subtended at the center by that part of every sphere equal to the square of its radius.

841. The space angle made by only two planes corresponds to the lune intercepted on any sphere whose center is in the common section of the two planes.

842. A Spherical Pyramid is a portion of a globe bounded by a spherical polygon and the planes of the sides of the polygon. The center of the sphere is the apex of the pyramid; the spherical polygon is its base.

843. Just as plane angles at the center of a circle are proportional to their intercepted arcs, and also sectors, so space

angles at the center of a sphere are proportional to their intercepted spherical polygons, and also spherical pyramids.

EXAMPLE. Find the ratio of the space angles of two right cones of altitude a, and a2, but having the same slant height, h.

These space angles are as the corresponding calots (or zones of one base) on the sphere of radius h. Therefore, by 835, the required ratio

For the equilateral and right-angled cones this becomes

844. To construct a space angle of two faces equivalent to any polyhedral angle, only involves constructing a lune equivalent to a spherical polygon, as in 731.

845. To find the area of a lune.

RULE. Multiply its angle in radians by twice its squared radius.

FORMULA. L = 2r2u.

PROOF. By 703, lunes are as their angles,

a lune is to a hemisphere as its angle is to a straight angle,

846. COROLLARY I. A lune measures twice as many steradians as its angle contains radians.

847. COROLLARY II. If two-faced space angles are equal, their lune angles are equal; so a dihedral angle may be measured by the plane angle between two perpendiculars, one in each face, from any point of its edge.

848. Suppose the vertex of a space angle is put at the center of a sphere, then the planes which form the space angle will cut the sphere in arcs of great circles, forming a spherical polygon, whose angles may be taken to measure the dihedral angles of the space angle, and whose sides measure its face angles.

B B

A

F

Hence from any property of spherical polygons we may infer an analogous property of space angles.

For example, the following properties of trihedral angles have been proved in our treatment of spherical triangles :—

I. Trihedral angles are either congruent or symmetrical which have the following parts equal: