SPHERICS

, the Doctrine of the sphere, particularly of the several circles described on its surface; with the method of projecting the same on a plane. See Projection of the Sphere.

A circle of the sphere is that which is made by a plane cutting it. If the plane pass through the centre, it is a great circle: if not, it is a little circle.

The pole of a circle, is a point on the surface of the sphere equidistant from every point of the circumference of the circle. Hence every circle has two poles, which are diametrically opposite to each other; and all circles that are parallel to each other have the same poles.

1. If a sphere be cut in any manner by a plane, the section will be a circle. And a great circle when the section passes through the centre, otherwise it is a little circle. Hence, all great circles are equal to each other: and the line of section of two great circles of the sphere, is a diameter of the sphere: and therefore two great circles intersect each other in points diametrically opposite; and make equal angles at those points; and divide each other into two equal parts; also any great circle divides the whole sphere into two equal parts.

2. If a great circle be perpendicular to any other circle, it passes through its poles. And if a great circle | pass through the pole of any other circle, it cuts it at right angles, and into two equal parts.

3. The distance between the poles of two circles, is equal to the angle of their inclination.

4. Two great circles passing through the poles of another great circle, cut all the parallels to this latter into similar arcs. Hence, an angle made by two great circles of the sphere, is equal to the angle of inclination of the planes of these great circles. And hence also the lengths of those parallels are to one another as the sines of their distances from their common pole, or as the cosines of their distances from their parallel great circle. Consequently, as radius is to the cosine of the latitude of any point on the globe, so is the length of a degree at the equator, to the length of a degree in that latitude.

5. If a great circle pass through the poles of another; this latter also passes through the poles of the former; and the two cut each other perpendicularly.

6. If two or more great circles intersect each other in the poles of another great circle; this latter will pass through the poles of all the former.

7. All circles of the sphere that are equally distant from the centre, are equal; and the farther they are distant from the centre, the less they are.

8. The shortest distance on the surface of a sphere, between any two points on that surface, is the arc of a great circle passing through those points. And the smaller the circle is that passes through the same points, the longer is the arc of distance between them. Hence the proper measure, or distance, of two places on the surface of the globe, is an arc of a great circle intercepted between the same. See Theodosius and other writers on Spherics.