ON SPHERICAL ANGLES. 710. DEF. The angle of two curves which have a common point is the angle included by the two tangents to the two curves at that point. 711. DEF. A spherical angle is the angle included between two arcs of great circles. PROPOSITION XI. THEOREM. 712. The angle of two curves which intersect on the surface of a sphere is equal to the dihedral angle between the planes passed through the centre of the sphere, and the tangents of the two curves at their point of intersection. Let the curves A B and dC intersect at A on the sur face of a sphere whose centre is 0); and let AT and AS be the tangents to the two curves respectively. We are to prove 2 TA S equal to the dihedral angle formed by the planes 0 AT and O A S. Since A T and AS do not cut the curves at A, they do not cut the surface of the sphére, and are therefore tangents to the sphere. .. A T and A S are I to the radius 0 A, drawn to the point of contact. § 186 1.LTAS measures the dihedral Z of the planes 0 AT and O AS, passed through the radius 0 A and the tangents A T and AS. § 470 But Z TA S is the 2 of the two curves A B and A C. $ 710 .:. the 2 of the two curves A B and AC= the dihedral 2 of the planes 0 AT and O A S. Q. E. D. PROPOSITION XII. THEOREM. 713: A spherical angle is equal to the measure of the dihedral angle included by the great circles whose arcs form the sides of the angle. P. PI Let BPC be any spherical angle, and BPDP and CPE P the great circles whose arcs B P and CP include the angle. We are to prove 2 BPC equal to the measure of the dihedral Z C-P P-B. Since two great © intersect in a diameter, PP is a diameter. § 685 Draw P T tangent to the O BPDP'. Then P T lies in the same plane as the O BPD P', and is I to P Pat P. In like manner draw PT' tangent to the OG PEP'. Then P T' lies in the same plane as the OC PE P', and is I to P P at P. ...ZTPT' is the measure of the dihedral Z C-PP-B. § 470 But spherical Z BPC is the same as plane < TPT'; $ 710 ... spherical Z BPC is equal to the measure of dihedral 2 C-PP-B. Q. E. D. 714. COROLLARY. A spherical angle is measured by the arc of a great circle described about its vertex as a pole and intercepted by its sides (produced if necessary). For, if BC be the arc of a great circle described about the vertex P as a pole, P B and PC are quadrants. Hence, B O and C O are perpendicular to P P'. Therefore BOC measures the dihedral angle B-P 0-C, and, hence, the spherical angle B PC. Therefore, arc BC, which measures the angle BOC, measures the spherical angle B PC. ON SPHERICAL POLYGONS AND PYRAMIDS. 715. DEF. A spherical Polygon is a portion of a surface of a sphere bounded by three or more arcs of great circles. The sides of a spherical polygon are the bounding arcs ; the angles are the angles included by consecutive sides; the vertices are the intersections of the sides. 716. DEF. The Diagonal of a spherical polygon is an arc of a great circle dividing the polygon, and terminating in two vertices not adjacent. The planes of the sides of a spherical polygon form by their intersections a polyhedral angle whose vertex is the centre of the sphere, and whose face angles are measured by the sides of the polygon. 717. Def. A spherical Pyramid is a portion of a sphere bounded by a spherical polygon and the planes of the sides of the polygon. The spherical polygon is the base of the pyramid, and the centre of the sphere is its vertex. 718. Def. A spherical Triangle is a spherical polygon of three sides. A spherical triangle, like a plane triangle, is right, or oblique ; scalene, isosceles or equilateral. 719. DEF. Two spherical triangles are equal if their successive sides and angles, taken in the same order, be equal each to each. 720. DEF. Two spherical triangles are symmetrical if their successive sides and angles, taken in reverse order, be equal each to each. 721. DEF. The Polar of a spherical triangle is a spherical triangle, the poles of whose sides are respectively the vertices of the given triangle. Since the sides of a spherical triangle are arcs, they may be expressed in degrees and minutes. PROPOSITION XIII. THEOREM. 722. Any side of a spherical triangle is less than the sum of the other two sides. Let A B C be any spherical triangle. We are to prove BC < B A + AC. Join the vertices A, B and C with the centre () of the sphere. Then, in the trihedral 20-A BC thus formed, the face & A OC, A O B and B O C are measured, respectively, by the sides A C, A B and BC. $ 202 Now, BOC< BOA +AOC, § 487 (the sum of any two & of a trihedral is greater than the third 2). ..BC < B A + AC. Q. E. D. 723. COROLLARY. Any side of a spherical polygon is less than the sum of the other sides. Ex. 1. Given a cone of revolution whose side is 24 feet, and the diameter of its base 6 feet; find its entire surface, and its volume. 2. Given the frustum of a cone whose altitude is 24 feet, the circumference of its lower base 20 feet, and that of its upper base 16 feet; find its volume. 3. The volume of the frustum of a cone of revolution is 8025 cubic inches; its altitude 14 inches; the circumference of the lower base twice that of the upper base. What are the circumferences of the bases ? 4. The frustum of a cone of revolution whose altitude is 20 feet, and the diameters of its bases 12 feet and 8 feet respectively, is divided into two equal parts by a plane parallel to its bases. What is the altitude of each part ? PROPOSITION XIV. THEOREM. 724. The sum of the sides of a spherical polygon is less than the circumference of a great circle. Let A B C D E be a spherical polygon. We are to prove A B + B C etc. less than the circumference of a great circle. Join the vertices A, B, C etc., with 0 the centre of the sphere. The sum of the face A O B, BOC etc., which form a polyhedral 2 at 0, is less than four rt. ks. § 488 .. the sum of the arcs A B, BC etc., which measure these face 6, is less than the circumference of a great circle. Q. E. D. 725. COROLLARY. If we denote the sides of a spherical triangle by a, b and c, then a +b+c< 360°. Ex. 1. The surface of a cone is 540 square inches; what is the surface of a similar cone whose volume is 8 times as great ? 2. The lateral surface of a cone is S; what is the lateral surface of a similar cone whose volume is n times as great ? |