Proposition 19. Theorem. 736. The area of a spherical triangle is equal to its spherical excess. Hyp. Let A, B, C denote the A numerical measures of the s of the spherical ▲ ABC, the rt. being the unit of s, and the tri-rectangular A the unit of areas. To prove area ABC=A+B+C-2. Proof. Continue any one side as AB so as to complete the great O B B ABA'B'. Continue the other two sides AC and BC till they meet this O in A' and B'. Then area ABC÷area A'BC=lune ABA'C=2A, and area ABC+area AB’C=lune AB'CB=2B. (735) Also, since the As ABC and A'B'C are together equivalent to a lune whose angle is C, ... area ABC + area A'B'C = 2C. (729) But the sum of the areas of the As ABC, A'BC, AB'C, A'B'C is equal to the area of the hemisphere, or 4. .*. adding these three equations, we have 2 area ABC + 4 = 2A + 2B + 2C. ... area ABC=A+B+C - 2. Q.E.D. 737. COR. 1. By a method similar to that in (736), in connection with (730) and (735), it may be proved that The volume of a triangular spherical pyramid is equal to the spherical excess of its base, (the volume of the trirectangular pyramid being the unit of volume). 738. COR. 2. If the three vertices of a triangle are on a great circle, its three sides must coincide with that circle, and each angle must equal 180°. The area of the triangle is then a hemisphere, and its spherical excess 4 rt. /s, or 360°. Therefore the area of the surface of the whole sphere is 720°. Hence: The area of a spherical triangle is to that of the surface of the sphere as its spherical excess, in degrees, is to 720°. Proposition 20. Theorem. 739. The area of a spherical polygon is equal to its spherical excess. Proof. From any vertex, as A, draw diagonals, dividing the polygon into (n-2) As. Then, since the area of each ▲ the sum of its s minus 2 rt.s(736); and since the sum of the s of the (n − 2) ▲ s = the sum of the s of the polygon, or S, ... K = S − 2(n − 2). Q. E.D. NOTE. In the last three propositions, only the ratios of the areas are expressed. If the absolute area is required, the area of the surface of the sphere must be known. EXERCISES. THEOREMS. 1. The intersection of the surfaces of two spheres is a circle whose plane is at right angles to the line joining the centres of the spheres, and whose centre is in that line. 2. If lines be drawn from any point of the surface of a sphere to the ends of a diameter, they will form with each other a right angle. 3. If any number of lines in space pass through a point, the feet of the perpendiculars upon these lines from another point lie upon the surface of a sphere. 4. If two straight lines are tangent to a sphere at the same point, the plane of these lines is tangent to the sphere. 5. On spheres of different radii, mutually equiangular triangles are similar. See (721), note. 6. The sum of the two arcs of great circles drawn from the extremities of one side of a spherical triangle to a point within it, is less than the sum of the other two sides. 7. If from any point on the surface of a sphere two arcs of great circles are drawn perpendicular to a circumference, the shorter of the two arcs is the shortest arc that can be drawn from the given point to the circumference. 8. Any lune is to a tri-rectangular triangle as its angle is to half a right angle. 9. Spherical polygons are to each other as their spherical excesses. 10. Two oblique arcs drawn from the same point to points of the circumference equally distant from the foot of the perpendicular are equal. 11. Of two oblique arcs, the one which meets the circumference at the greater distance from the foot of the perpendicular is the longer. 12. The arc of a great circle, tangent to a small circle, is perpendicular to the radius of the sphere at the point of contact. 13. If three spheres intersect one another, their planes of intersection intersect in a right line perpendicular to the plane containing the centres of the spheres. 14. Prove that this line is the locus of points from which tangent lines to the three spheres are equal. 15. Through a fixed point, within or without a sphere, three lines mutually at right angles intersect the sphere: prove that the sum of the squares of the three chords is constant, depending only on the radius of the sphere and the distance of the point from the centre. 16. Prove also that the sum of the squares of the six segments is constant. 17. Prove that the area of a spherical triangle, each of whose angles is of a right angle, is equal to the surface of a great circle. 18. If through a point O any secant OPP' is drawn to cut a sphere in P, P', prove that OP. OP' is constant. 19. Find the radii of the spheres inscribed and circumscribed to a regular tetraedron. NUMERICAL EXERCISES. 20. If the sides of a spherical triangle are respectively 65°, 112°, and 85°, how many degrees are there in each angle of its polar triangle? 21. If the angles of a spherical triangle are respectively 90°, 115°, and 70°, how many degrees are there in each side of its polar triangle? 22. Given the spherical triangle whose sides are respectively 80°, 90°, and 140°, to find the angles of its polar triangle. 23. What part of the surface of a sphere is a lune whose angle is 45° 54°? 80°? 24. What part of the volume of a sphere is an ungula whose angle is 72° ? 36°? 25. If the angle of a lune is 50°, find its area on a sphere whose surface is 72 square inches. Ans. 10 square inches. 26. Find the area of a spherical triangle whose angles are respectively 75°, 100°, and 115°, on a sphere whose surface is 72 square inches. Ans. 11 square inches. 27. Find the area of a spherical triangle each of whose angles is 70°, on a sphere whose surface is 144 square inches. Ans. 6 square inches. 28. Find the area of a spherical triangle whose angles are 60°, 90°, and 120°, on a sphere whose surface is 64 square inches. Ans. 8 square inches. 29. Find the area of a spherical polygon of six sides each of whose angles is 150°, on a sphere whose surface is 100 square inches. Ans. 25 square inches. 30. Find the area of a bi-rectangular triangle whose vertical angle is 108°, on a sphere whose surface is 100 square inches. Ans. 15 square inches. 31. Find the area of a spherical pentagon whose angles are respectively 138°, 112°, 131°, 168°, and 153°, on a sphere whose surface is 40 square feet. Ans. 9 square feet. 32. Find the area of a spherical triangle whose angles are 61°, 109°, and 127°, on a sphere whose surface is 10 square inches. Ans. 1.625 square inches. 33. Find the area of a spherical quadrangle whose angles are 170°, 139°, 126°, and 141°, on a sphere whose surface is 400 square inches. 34. Find the area of a spherical pentagon whose angles are 122°, 128°, 131°, 160°, and 161°, on a sphere whose surface is 150 square feet. 35. Find the angles of an equilateral spherical triangle whose area is equal to that of a great circle. Ans. 120°. 36. Find the angles of an equilateral spherical triangle whose area is equal to that of an equilateral spherical hexagon, each of whose angles is 150°. Ans. 80°. |