27, line 5 from bottom, for fig. 11, read fig. 12. ,, 33, Ex. 7, for a=b, read b = c. ,, 52, Ex. 14, for (), read 2 SPHERICAL TRIGONOMETRY. CHAPTER I. THE SPHERE. 1. A Sphere is a solid body such that all points on its surface are equally distant from a certain point within it, called the Centre. Any straight line drawn from the centre of a sphere to the surface is called a Radius, and any straight line drawn through the centre and terminated both ways by the surface is called a Diameter. A sphere may be generated by the revolution of a semicircle round its diameter. 2. Great and Small Circles.-Let a sphere be gene D Ο E B rated by the revolution of the semicircle ACB (fig. 1) B round its diameter AB. Let C be any point on the semicircle, and let CD be a perpendicular from C on AB. It is obvious that as ACB revolves round AB, the point C describes a circle round D as centre; and also that O, the middle point of AB, being equally distant from all points on ACB, is the centre of the sphere. When the plane of the circle CC' described by C passes through O it divides the sphere into two equal parts, and the curve of section is called a Great Circle. When its plane does not pass through the centre it is called a Small Circle. Example. The Meridians, Equator, and Ecliptic, are great circles. The Parallels of Latitude are small circles. The angular distance AC is called the Spherical Radius of the circle CC'. It is obvious that the spherical radius of a great circle is a quadrant. 3. Only one Great Circle can be drawn through two given points on the surface of a sphere; for its plane must also pass through the centre; and three points not in the same right line are sufficient to determine a plane completely. If the two given points be diametrically opposite, the right line joining them passes through the centre of the sphere, and an infinite number of great circles can be drawn through them, as, for example, the meridians on the surface of the Earth. The shortest distance that can be traced on the surface of a sphere between two points on it is the arc of the great circle passing through them. For, when the radius of a sphere is indefinitely increased, its surface, at any part, may be considered as a plane (see Chap. II., Art. 18), and a great circle passing through two points on it becomes a right line joining two points in a plane; but the right line joining two points is the shortest distance between them; therefore, etc.* It is for this reason, and because only one great circle can be drawn through two points that distances on the surface of a sphere are measured along arcs of great circles. Advantage is taken of this property of the great circle in Navigation. 4. Axes and Poles.-The line AB (fig. 1) is called the Axis of the circle CC' (described by any point C of the semicircle ACB during its revolution round AB), and the extremities A and B of its axis are called its Poles. Any point, and the great circle of which it is the pole, are termed pole and polar, with respect to one another. It is obvious, from the manner in which CC' was generated, that AB is perpendicular to the plane of CC', for CD remained perpendicular to AB during its entire revolution. Hence the Axis of a circle may be defined as the diameter *The following method of looking at this question is also instructive :— If a string be stretched between two points on the surface of a sphere (or on any surface) it will evidently be the shortest distance that can be traced on the surface between the points, since by pulling the ends of the string its length, between the points, will be shortened as much as the surface will permit. Now any part of the string being acted on by two terminal tensions, and by the reaction of the surface, which is everywhere normal to it, must lie in a plane containing the normal to the surface. Hence, the plane of the string contains the normals to the surface at all points of its length; i.e. the string lies in the form of a great circle. of the sphere perpendicular to the plane of the circle, or the line joining the centre of the sphere with the pole of the circle. Cor. The pole of a circle is equidistant from all points on the circumference of the circle. For AC2 = AD2 + CD2 = constant. In the case of a great circle, D becomes the centre of the sphere, and hence the poles of a great circle are equidistant from its circumference. 5. Primary and Secondary Circles.—Any circle is, called a Primary in relation to those great circles which cut it at right angles. These latter are called Secondaries; e.g. parallels of latitude are primaries, and the meridians are secondaries to them. In fig. 1 regarding the circle CC' as a primary, the circle ACB during its revolution round AB is in all positions a secondary to CC'. Hence it follows that :— (1). The plane of any secondary contains the axis of the primary. (2). All the secondaries pass through the poles of the primary. (3). The planes of all the secondaries have a common line of intersection, viz., the axis of the primary. (4). If there can be drawn common secondaries to two circles, the planes of those circles are parallel. For, by (3), the two circles have the same axis. The distance of any point on the surface of a sphere from a circle traced thereon is measured by the arc of the secondary intercepted between the point and the circle. |