BOOK VIII.* THE SPHERE.+ CIRCLES OF THE SPHERE AND TANGENT PLANES. DEFINITIONS. 657. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within it, called the centre. A sphere may be generated by the revolution of a semicircle ACB about its diameter AB as an axis. 658. A radius of a sphere is a A straight line drawn from the centre to any point of the surface. A diameter of a sphere is a straight C B line passing through its centre, and terminating at both ends in the surface. All the radii of a sphere are equal; and all the diameters are equal, since each is the sum of two radii. 659. A line or plane is tangent to a sphere when it has but one point in common with the surface of the sphere. 660. Two spheres are tangent to each other when their surfaces have but one point in common. 661. The common point is called the point of contact, or point of tangency. 662. Two spheres are concentric when they have the same centre. 663. Two spheres are equal when they have equal radii. *This book treats of the properties and relations of those parts of the surface of a sphere which are bounded by arcs of great circles. + In teaching this subject, as well as in teaching Spherical Trigonometry, the class-room should be furnished with a spherical black-board, on which the student should draw the diagrams of spherical surfaces. Proposition 1. Theorem. 664. Every section of a sphere made by a plane is a cir And since OA and OC are equal oblique lines from 0 to the plane ACB, .. O'A O'C. But A and C are any two pts. in the section ACB. (498) .. the curve ACB is a ✪ whose centre is O'. Q.E.D. 665. DEF. The circular section of a sphere by a plane is called a circle of the sphere. If the plane passes through the centre of the sphere, the section is a circle whose radius is equal to the radius of the sphere. This circle is called a great circle of the sphere. A section made by a plane which does not pass through the centre of the sphere is called a small circle. 666. DEF. The two points in which a diameter of the sphere, perpendicular to the plane of a circle, meets the surface of the sphere, are called the poles of the circle, and this diameter is called the axis of the circle; thus, P, P' are the poles of the circle ACB, and PP' is its axis.* *P' is said to be antipodal to P. 667. COR. 1. The line through the centre of a circle of the sphere, perpendicular to its plane, passes through the centre of the sphere. (664) Therefore, the axis of a circle passes through its centre (666), and all parallel circles have the same axis and the same poles. 668. COR. 2. All great circles of a sphere are equal. (665) 669. COR. 3. All small circles at equal distances from the centre of the sphere are equal; and of two circles unequally distant from the centre, the nearer is the larger, and conversely. 670. COR. 4. Every great circle bisects the sphere and its surface. For, if the two parts are separated, and then placed on the common base, with their convexities turned the same way, their surfaces will coincide, since all the points of either are equally distant from the centre. (657) 671. COR. 5. Any two great circles bisect each other. For, the common intersection of their planes passes through the centre of the sphere, and is a diameter of each circle. 672. COR. 6. An arc of a great circle may be drawn through any two given points on the surface of a sphere. For, the two given pts. with the centre of the sphere determine the plane of a great passing through the two given pts. (484) If the two pts. are the extremities of a diameter, the position of the circle is not determined: for the two given pts. and the centre being in the same st. line, an indefinite number of planes can be drawn through them. (482) 673. COR. 7. An arc of a circle may be drawn through any three given points on the surface of a sphere. For, the three pts. determine a plane. NOTE. (484) By the distance between two points on a sphere is meant the arc of a great circle joining them. Proposition 2. Theorem. 674. All points in the circumference of a circle of a sphere are equally distant from each of its poles. Hyp. Let P, P' be the poles of the ACB, where A, C, B are any pts. on its Oce. To prove arcs PA, PC, PB equal. Proof. Since PP' is a line through the centre of the O ACB to its plane, E P D B H P/ .. the st. lines PA, PC, PB are equal. (496) (198) Q.E.D. Similarly, the arcs P'A, P'C, P'B are equal. 675. DEF. The distance from any point in the circumference of a circle to its nearest pole is called the polar distance of the circle. 676. COR. 1. The polar distance of a great circle is a quadrant. Thus, PE, PG are quadrants, for they are the measures of the rt. s POE, POG, whose vertices are at the centre of the sphere. (237) 677. COR. 2. If a point P on the surface of a sphere is at a quadrant's distance from the two points E, G, of an are of a great circle, it is the pole of that arc. For, the s POE, POG are rt. Zs; .. the radius OP is to the plane of the arc EG (500); and .. P is the pole of the arc EG. (666) 678. SCH. In Spherical Geometry, the term quadrant usually means a quadrant of a great circle. Proposition 3. Theorem. 679. The shortest distance on the surface of a sphere, between any two points on the surface, is the arc of a great circle, not greater than a semi-circumference, joining the two points. Hyp. Let AB be an arc of a great O, not greater than a semi Oce, joining any two pts. A and B on the sphere; and let ACEB be any other line in the surface joining A and B. To prove arc AB ACEB. Proof. Take any pt. D in ACEB, pass arcs of great Os through A, D and B, D; and join O, the centre of the sphere, with A, D, B. Then, since the s AOB, AOD, DOB are the faces of the triedral whose vertex is at 0, .. ZAOD + DOB > ZAOB. (565) But the s AOD, DOB, AOB are measured by the arcs AD, DB, AB, respectively. .. arc AD arc DB > arc AB. (236) Similarly, joining any pt. in ACD with A and D by arcs of great Os, the sum of the arcs is greater than arc AD; and joining any pt. in DEB with D and B by arcs of great Os, the sum is greater than arc DB. If this process is indefinitely repeated, the line from A to B, on the arcs of the great Os, will continually increase and approach the line ACEB; that is, the sum of the arcs of the great Os will approach ACEB as the limit, and will always be greater than AB. .. arc AB < ACEB. Q.E.D. |