BOOK VIII THE SPHERE 671. DEF. A sphere is a solid bounded by a surface, all the points of which are equally distant from a point within called the center. 672. DEF. The radius of a sphere is a straight line drawn from the center to any point in the surface. 673. DEF. The diameter of a sphere is a straight line passing through the center and terminated at either end by the surface. 674. From the definitions it follows that (1) All the radii of a sphere are equal, and all diameters are equal. (2) A semicircle rotating about its diameter generates a sphere. (3) Two spheres are equal if their radii are equal, and conversely. (4) A point is without a sphere if its distance from the center is greater than the radius. Ex. 1168. The radii of two spheres are respectively 10 in. and 4 in., their lines of centers (i.e. the line joining their centers) is 7 in. Is every point of the smaller sphere lying within the larger one ? PROPOSITION I. THEOREM 675. Every section of a sphere made by a plane is a circle. Hyp. CBD is the intersection of plane MN, and a sphere whose center is 0. Since any two points B and C in CBD are equidistant from A, all points must be equidistant from A. Or CBD is a circle. Q.E.D. 676. COR. 1. A circle nearer to the center of a sphere is greater than one more remote. = For since AC OC2 - AO, AC is the smaller, the greater AO. 677. DEF. A great circle of a sphere is a section made by a plane passing through the center. 678. DEF. A small circle of a sphere is a section made by a plane not passing through the center. 679. DEF. The axis of a circle of a sphere is the diameter perpendicular to the plane of the circle; its ends are the poles of the circle. 680. COR. 2. The axis of a circle passes through the center of the circle, and conversely. 681. COR. 3. All great circles of a sphere are equal. 682. COR. 4. Any two great circles of a sphere bisect each other. For since the plane of each contains the center of the sphere, their intersection is a diameter and bisects both circles. 683. COR. 5. Every great circle bisects the sphere. 684. COR. 6. One and only one circle may be drawn through any three points in the surface of a sphere. (A plane is determined by three points.) 685. COR. 7. A great circle may be drawn through two points B and C in the surface of a sphere. (Three points, B, C, and the center O, determine a plane.) Generally there is only one great circle which passes through two given points, but if the given points are ends of a diameter, any number of great circles can be passed through these points. 686. DEF. The distance between two points on the surface of a sphere is the length of the minor arc of a great circle between them. Ex. 1169. What is the radius of a small circle, if the distance of its plane from the center of the sphere is 9 in., and the radius of the sphere is 15 in. PROPOSITION II. THEOREM 687. All points in the circumference of a circle of a sphere are equidistant from a pole of the circle. Hyp. P and P' are the poles of circle ABC of a sphere. To prove and arc PA arc PB arc PC, arc P'A arc P'B: =arc P'C. HINT. - Prove by means of (473) the equality of straight lines PA, PB, and PC. 688. DEF. The polar distance of a circle of a sphere is the distance of a point in the circumference from the nearer pole. 689. SCHOLIUM. A quadrant in Spherical Geometry is the fourth part of the circumference of a great circle. 690. COR. The polar distance of a great circle is a quadrant. Ex. 1170. The polar distance of a circle of a sphere is 60°, and the radius of the sphere is 13 in. Find (a) the distance of its plane from the center. PROPOSITION III. THEOREM 691. On the surface of a sphere, a point at a quadrant's distance from two other points, not the extremities of a diameter, is the pole of a great circle passing through these two points. Hyp. P, A, and B are three points on the surface of a sphere, and PA and PB are quadrants. To prove P is the pole of a great circle AB. HINT. - Prove that a line OP is perpendicular to plane ABO by means of (461). 692. SCHOLIUM. Theorem III enables us to construct a great circle through two points on the surface of a material sphere by means of the compasses. From the given points A and B as centers draw arcs with a radius equal to the chord of a quadrant, intersecting in P. From P, with the same radius, draw a circle which is the required one. 693. DEF. A plane is tangent to a sphere when it has one and only one point common with the surface of a sphere. 694. DEF. A line is tangent to a sphere when it has one and only one point common with the surface of a sphere. |