22. The sum of the arcs of great circles drawn from any point within a spherical triangle to the extremities of any side, is less than the sum of the other two sides of the triangle. (Compare § 48.) 23. How many degrees are there in the polar distance of a circle, whose plane is 5√2 units from the centre of the sphere, the diameter of the sphere being 20 units ? (The radius of the O is a leg of a rt. A, whose hypotenuse is the radius of the sphere, and whose other leg is the distance from its centre to the plane of the O.) 24. The chord of the polar distance of a circle of a sphere is 6. the radius of the sphere is 5, what is the radius of the circle? 25. If side AB of spherical triangle ABC is a quadrant, and side BC less than a quadrant, prove ZA less than 90°. If B 26. The polar distance of a circle of a sphere is 60°. If the diameter of the circle is 6, find the diameter of the sphere, and the distance of the circle from its centre. (Represent radius of sphere by 2 x.) 27. Any point in the arc of a great circle bisecting a spherical angle is equally distant (§ 573) from the sides of the angle. (To prove arc PM: =arc PN. a pole of arc AB, and F of arc BC. ABPE and BPF are symmetrical II., and PE = PF.) B Let E be P M N A D C 28. A point on the surface of a sphere, equally distant from the sides of a spherical angle, lies in the arc of a great circle bisecting the angle. (Fig. of Ex. 27. To prove ABP = 2 CBP. and BPF are symmetrical by § 605, 2.) Spherical ABPE 29. The arcs of great circles bisecting the angles of a spherical triangle meet in a point equally distant from the sides of the triangle. (Exs. 27, 28, p. 358.) 30. A circle may be inscribed in any spherical triangle. 31. State and prove the theorem for spherical triangles analogous to Prop. IX., I., Book I. 32. State and prove the theorem for spherical triangles analogous to Prop. V., Book I. 33. State and prove the theorem for spherical triangles analogous to Prop. L., Book I. (Ex. 32.) 34. If PA, PB, and PC are three equal arcs of great circles drawn from point P to the circumference of great circle ABC, prove P a pole of ABC. (PA and PB are quadrants by Ex. 15, p. 357.) 35. The spherical polygons corresponding to a pair of vertical polyedral angles are symmetrical. (§ 456.) 36. A sphere may be inscribed in, or circumscribed about, any tetraedron. (Ex. 73, Book VII.) 37. What is the locus of points in space at a given distance from a given straight line? 38. Equal small circles of a sphere are equally distant from the centre. 39. State and prove the converse of Ex. 38. 40. The less of two small circles of a sphere is at the greater distance from the centre. 41. State and prove the converse of Ex. 40. 42. What is the locus of points on the surface of a sphere equally distant from the sides of a spherical angle? 43. If two spheres are tangent to the same plane at the same point, the straight line joining their centres passes through the point of contact. 44. The distance between the centres of two spheres whose radii are 25 and 17, respectively, is 28. Find the diameter of their circle of intersection, and its distance from the centre of each sphere. 45. If a polyedron be circumscribed about each of two equal spheres, the volumes of the polyedrons are to each other as the areas of their surfaces. (Find the volume of each polyedron by dividing it into pyramids.) 46. Either angle of a spherical triangle is greater than the difference between 180° and the sum of the other two angles. (Fig. of Prop. XX. To prove ZA > 180° · (ZB+ZC), or >(<B+<C) - 180°, according as ZB+ < C is < or > 180°. In the latter case, A'C' + A'B' > B'C'; then use § 593.) BOOK IX. MEASUREMENT OF THE CYLINDER, CONE, AND SPHERE. THE CYLINDER. DEFINITIONS. 638. The lateral area of a cylinder is the area of its lateral surface. A right section of a cylinder is a section made by a plane perpendicular to the elements of its lateral surface. 639. A prism is said to be inscribed in a cylinder when its lateral edges are elements of the cylindrical surface. In this case, the bases of the prism are inscribed in the bases of the cylinder. A prism is said to be circumscribed about a cylinder when its lateral faces are tangent to the cylinder, and its bases lie in the same planes with the bases of the cylinder. In this case, the bases of the prism are circumscribed about the bases of the cylinder. 640. It follows from § 363 that If a prism whose base is a regular polygon be inscribed in, or circumscribed about, a circular cylinder (§ 540), and the number of its faces be indefinitely increased, 1. The lateral area of the prism approaches the lateral area of the cylinder as a limit. 2. The volume of the prism approaches the volume of the cylinder as a limit. 3. The perimeter of a right section of the prism approaches the perimeter of a right section of the cylinder as a limit.* PROP. I. THEOREM. 641. The lateral area of a circular cylinder is equal to the perimeter of a right section multiplied by an element of the lateral surface. Given S the lateral area, P the perimeter of a rt. section, and E an element of the lateral surface, of a circular cylinder. Proof. Inscribe in the cylinder a prism whose base is a regular polygon, and let S' denote its lateral area, and P' the perimeter of a rt. section. Then, since the lateral edge of the prism is E, Now let the number of faces of the prism be indefinitely increased. Then, and S' approaches the limit S, P' x E approaches the limit Px E. (§ 640, 1, 3) By the Theorem of Limits, these limits are equal. (§ 188) *For rigorous proofs of these statements, see Appendix, p. 386. 642. Cor. I. is equal to the altitude. The lateral area of a cylinder of revolution circumference of its base multiplied by its 643. Cor. II. If S denotes the lateral area, T the total area, H the altitude, and R the radius of the base, of a cylinder of revolution, S=2πRH. And, T-2 RH+ 2 πR2 (§ 371) = 2πR(H+ R). PROP. II. THEOREM. (§ 368) 644. The volume of a circular cylinder is equal to the product of its base and altitude. Given V the volume, B the area of the base, and H the altitude, of a circular cylinder. Proof. Inscribe in the cylinder a prism whose base is a regular polygon, and let V' denote its volume, and B' the area of its base. Then, since the altitude of the prism is H, Now let the number of faces of the prism be indefinitely increased. Then, And, B' x H approaches the limit B x H. (§ 363, II) |