SPHERE, is a solid contained under one uniform round surface, such as would be formed by the revolution of a circle about a diameter thereof, as an axis. Thus the circle AE, BD (see Plate XIV. Miscel. fig. 2.) revolving about the diame ter AB, will generate a sphere, whose surface will be formed by the circum ference of the circle.
Definition,. 1. The centre and axis of a sphere are the same as the centre and diameter of the generating circle ; and as a circle has an indefinite number of diameters, so a sphere may be consider ed as having also an indefinite number of diameters, round any one of which the sphere may be conceived to be generated. 2. Circles of the sphere are those circles described on its surface, by the motions of the extremities of the chords ED, FG, IH, &c. at right angles to AB ; the diame ters of which circles are equal to those chords. 3. The poles of a circle on the sphere are those points on its surface, equally distant from the circumference of that circle : thus A and B are the poles of the circles described on the sphere by the ends of the chords ED, FG, &c. 4. A great circle of the sphere is one equally distant from both its poles; as that described by the extremities of the diameter ED, which is equally dis tant from both its poles A and B. Les ser circles of the sphere are those which are unequally distant from both their poles; as those described by the extre mities of the chords FG, HI, &c. because unequally distant from their poles A and B. See
Axioms. 1. The diameter of every great circle passes through the centre of the sphere; but the diameters of all lesser circles do not pass through the same cen tre : hence also the centre of the sphere is the common centre of all the great cir cles. 2. Every section of a sphere by a plane, is a circle. 3. A sphere is divided Into two equal parts, or hemispheres, by the plane of every great circle : and into two unequal parts, called segments, by the plane of every lesser circle. 4. The pole of every great circle is 90° distant from it on the surface of the sphere ; and no two great circles can have a common pole. 5. The poles of a great circle are the two extremities of that diameter of the sphere, which is perpendicular to the plane of that circle. 6. A plane passing through three points on the surface of the sphere, equally distant from any of the poles of a great circle, will be parallel tothe plane of that great circle. 7. The shortest distance between two points, on the surface of a sphere, is the arch of • great circle passing through these points. 8. If one great circle meets another, the angles on either side are supplements to each other; and every spherical angle is less than 180°. 9. If two circles intersect each other, the opposite angles are equal. 10. All circles on the sphere, having the same pole, are cut into similar arches, by great circles passing through that pole.