PROP. XII. THEOREM. 579. The intersection of two spheres is a circle, whose centre is in the straight line joining the centres of the spheres, and whose plane is perpendicular to that line. A D Given two intersecting spheres. To Prove their intersection a O, whose centre is in the line joining the centres of the spheres, and whose plane is to this line. Proof. Let O and O' be the centres of two, whose common chord is AB; draw line 00', intersecting AB at C. Then, OO' bisects AB at rt. . (§ 178) If we revolve the entire figure about 00' as an axis, the will generate spheres whose centres are O and O'. (§ 565) And AC will generate a O OC', whose centre is C, which is the intersection of the two spheres. PROP. XIII. THEOREM. (§ 402) 580. A plane perpendicular to a radius of a sphere at its extremity is tangent to the sphere. (The proof is left to the pupil; compare § 169.) 581. Cor. (Converse of Prop. XIII.) A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact. (Fig. of Prop. XIII.) (The proof is left to the pupil; compare § 170.) 582. Through four points, not in the same plane, a spherical surface can be made to pass, and but one. Given A, B, C, and D points not in the same plane. To Prove that a spherical surface can be passed through A, B, C, and D, and but one. Proof. Pass planes through A, B, C, and D, forming tetraedron ABCD, and let K be the middle point of CD. Draw lines KE and KF in faces ACD and BCD, respectively, CD; and let E and F be the centres of the circumscribed of A ACD and BCD, respectively. Then plane EKF is 1 CD. (§ 222) (§ 400) Draw line EGL ACD, and line FHL BCD; then EG and FH lie in plane EKF. (§ 439) Then EG and FH must meet at some point 0, unless they are ; this cannot be unless ACD and BCD are in the same plane, which is contrary to the hyp. (§ 418) Now O, being in EG, is equally distant from A, C, and D; and being in FH, is equally distant from B, C, and D. (§ 406, I) Then O is equally distant from A, B, C, and D; and a spherical surface described with O as a centre, and OA as a radius, will pass through A, B, C, and D. Now the centre of any spherical surface passing through A, B, C, and D must be in each of the Is EG and FH. Then as EG and FH intersect in but one point, only one spherical surface can be passed through A, B, C, and D. 583. Defs. The angle between two intersecting curves is the angle between tangents to the curves at their point of intersection. A spherical angle is the angle between two intersecting arcs of great circles. 584. A spherical angle is measured by an arc of a great circle having its vertex as a pole, included between its sides produced if necessary. B B' Given ABC and AB'C arcs of great on the surface of sphere AC, lines AD and AD' tangent to ABC and A'BC, respectively, and BB' an arc of a great O having A as a pole, included between arcs ABC and AB'C. To Prove that DAD' is measured by arc BB'. Proof. Let be the centre of the sphere, and draw diameter 40C and lines OB and OB'. Now, arcs AB and AB' are quadrants. But BOB' is measured by arc BB'. (§ 577) (?) (§§ 170, 54) (§ 426) (?) 585. Cor. I. (Fig. of Prop. XV.) Plane BOB' is 1 OA. (§ 400) Then planes ABC and BOB' are 1. (§ 441) Now a tangent to arc AB at B is 1 BOB'. (§ 439) Then it is to a tangent to arc BB' at B. (§ 398) Then, spherical Z ABB' is a rt. Z. (§ 583) That is, an arc of a great circle drawn from the pole of a great circle is perpendicular to its circumference. 586. Cor. II. The angle between two arcs of great circles is the plane angle of the diedral angle between their planes. (§ 429) SPHERICAL POLYGONS AND SPHERICAL PYRAMIDS. DEFINITIONS. 587. A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles; as ABCD. The bounding arcs are called the sides of the spherical polygon, and are usually measured in degrees. B The angles of the spherical polygon are the spherical angles (§ 583) between the adjacent sides, and their vertices are called the vertices of the spherical polygon. A diagonal of a spherical polygon is an arc of a great circle joining any two vertices which are not consecutive. A spherical triangle is a spherical polygon of three sides. A spherical triangle is called isosceles when it has two sides equal; equilateral when all its sides are equal; and right-angled when it has a right angle. 588. The planes of the sides of a spherical polygon form a polyedral angle, whose vertex is the centre of the sphere, and whose face angles are measured by the sides of the spherical polygon (§ 192). Thus, in the figure of § 587, the planes of the sides of the spherical polygon form a polyedral angle, O-ABCD, whose face AOB, BOC, etc., are measured by arcs AB, BC, etc., respectively. A spherical polygon is called convex when the polyedral angle formed by the planes of its sides is convex (§ 453). 589. A spherical pyramid is a solid bounded by a spherical polygon and the planes of its sides; as O-ABCD, figure of § 587. The centre of the sphere is called the vertex of the spherical pyramid, and the spherical polygon the base. Two spherical pyramids are equal when their bases are equal. For they can evidently be applied one to the other so as to coincide throughout. 590. If circumferences of great circles be drawn with the vertices of a spherical triangle as poles, they divide the surface of the sphere into eight spherical triangles. Thus, if circumference B'C'B" be drawn with vertex A of spherical ▲ ABC as a pole, circumference A'C"A" with B as a pole, and circumference A'B"A"B' with C as a pole, the surface of the sphere is divided into eight spherical A; A'B'C', A'B'C', A'B'C', and A"B"C" on the B B' Α' A C hemisphere represented in the figure, the others on the opposite hemisphere. Of these eight spherical A, one is called the polar triangle of ABC, and is determined as follows: Of the intersections, A' and A", of circumferences drawn with B and C as poles, that which is nearer (§ 573) to A, i.e., A', is a vertex of the polar triangle; and similarly for the other intersections. Thus, A'B'C' is the polar ▲ of ABC. |