47. To pass a plane tangent to a sphere through a given straight line without the sphere. 48. To construct on the spherical blackboard a spherical angle of 45°; 60°; 90°; 100°; 200°. From P, the pt. where the vertex is to be placed, with a quadrant describe an arc, which will represent one side of the angle required. From P as a pole, with a quadrant describe an arc from the side before drawn, to measure the required angle. Lay off on this last are from the first arc the measure of the required angle; and through the extremity of this arc and P pass a great circle. For describing the arcs, the student can use a tape equal in length to half a great circle of the sphere, marked off into 180 equal parts. 49. To construct on the spherical blackboard a spherical triangle, having two sides 100° and 80°, and the included angle 58°. 50. To construct, as above, a spherical triangle, having a side 75°, and the adjacent angles 110° and 87°. 51. To construct, as above, a spherical triangle, having its sides 150°, 100°, 80°; also having its sides 50°, 85°, 160°. 52. To construct, as above, a spherical triangle, having two sides 120° and 88°, and the included angle 59°; then construct its polar triangle. 53. Given two points on the surface of a sphere, to describe the great circle passing through them. 54. To bisect a given arc, or a given angle, on a sphere. 55. To draw an arc of a great circle perpendicular to a spherical arc, from a given point without it. 56. To erect a perpendicular to a given arc of a great circle from a given point in the arc. 57. Given three points on a sphere, to describe a small circle to pass through them. 58. To cut a given sphere by a plane passing through a given straight line so that the section shall have a given radius. 59. At a given point in a great circle, to draw an arc of a great circle making a given angle with the first. 60. Through a given point on a sphere, to draw a great circle tangent to a given small circle. 61. Through a given point on a sphere, to draw a great circle tangent to two given small circles. 62. To inscribe a circle in a given spherical triangle, and to circumscribe a circle about the triangle. 63. To construct a right spherical triangle, having (1) a side about the right angle and the hypotenuse; and (2) an angle and the opposite side. 64. To construct a spherical triangle, having given (1) the three sides; and (2) two sides and the included angle. 65. To describe a sphere to cut orthogonally two given spheres. BOOK IX.* THE THREE ROUND BODIES. 740. The only solids bounded by curved surfaces, that are treated of in Elementary Geometry, are the cylinder, the cone, and the sphere, which are called the three round bodies. THE CYLINDER. DEFINITIONS. E 741. A cylindrical surface is a surface generated by the motion of a straight line AB, called the generatrix, which constantly touches a given curve ACDE, called the directrix, and remains paralle. to its original position. The different positions of the generatrix are called elements of the surface. 742. A cylinder is a solid bounded by a cylindrical surface and two parallel planes. The cylindrical surface is called the lateral surface, and the plane surfaces are called the bases. A ט The altitude of a cylinder is the perpendicular distance between its bases. 743. A right section of a cylinder is the section by a plane perpendicular to its elements. 744. A circular cylinder is a cylinder whose base is a circle. The axis of a circular cylinder is the straight line joining the centres of its bases. 745. A right cylinder is one whose elements are perpendicular to its bases. *This book treats of the properties and relations of the cylinder, the cone, and the sphere, and shows how to find the convex surface and volume of each of these bodies. 746. A right circular cylinder, called also a cylinder of revolution, is generated by revolving a rectangle about one of its sides. 747. Similar cylinders of revolution are those generated by similar rectangles revolving round homologous sides. Ө 748. A tangent plane to a cylinder is a plane which contains an element of the cylinder without cutting the surface. The element which the plane contains is called the element of contact. Any straight line in a tangent plane, which cuts the element of contact, is a tangent line to the cylinder. 749. A prism is inscribed in a cylinder, when its bases are inscribed in the bases of the cylinder and its lateral edges are elements of the cylinder. 750. A prism is circumscribed about a cylinder, when its bases are circumscribed about the bases of the cylinder. Proposition 1. Theorem. 751. Every section of a cylinder made by a plane passing through an element is a parallelogram. Hyp. Let the plane ABCD pass through the element AB of cylinder EH. Το prove the section ABCD a . Proof. A plane passing through the element AB cuts the Oce of the base in a second pt. D. H Through D draw DC || to AB. Then DC is in the plane BAD. ... DC is an element of the cylinder. face of the cylinder, is their intersection. E A. (68) (741) ... DC, being common to the plane and the lateral sur Also, AD is || to BC. (519) .. ABCD is a . (124) Q.E. D. 752. COR. Every section of a right cylinder made by a plane passing through an element is a rectangle. Proposition 2. Theorem. 753. The lateral area of a cylinder is equal to the perimeter of a right section of a cylinder multiplied by an element. Hyp. Let S denote the lateral area, Proof. Inscribe in the cylinder a prism ABCD-C', of any number of faces; and let s denote its lateral area and p the perimeter of its rt. section. Then, since each lateral edge is an element of the cylinder, S= = px E. A D B (749) (591) Now let the number of lateral faces of the inscribed prism be indefinitely increased. The perimeter of the right section of the prism will approach the perimeter of the right section of the cylinder as its limit. (430) ... the lateral area of the prism will approach the lateral area of the cylinder as its limit. Because, however great the number of the lateral faces, 8 = px E, and because p approaches P as its limit and s approaches S as its limit, 754. COR. 1. The lateral area of a cylinder of revolution is equal to the circumference of its base multiplied by its altitude. |