Introduction.

When we look at the stars on a clear night we have the familiar impression that they are all sparkling points of light, apparently situated on the surface of a vast sphere of which the individual observer is the centre. The eye, of course, fails to give any indication of the distances of the stars from us; however, it allows us to make some estimate of the angles subtended at the observer by any pairs of stars and, with suitable instruments, these angles can be measured with great precision. Spherical Astronomy is concerned essentially with the directions in which the stars are viewed, and it is convenient to define these directions in terms of the positions on the surface of a sphere— the celestial sphere—in which the straight lines, joining the observer to the stars, intersect this surface. It is in this sense that the usual expression “the position of a star on the celestial sphere” is to be interpreted. The radius of the sphere is entirely arbitrary. The foundation of Spherical Astronomy is the geometry of the sphere.

The spherical triangle.

Any plane passing through the centre of a sphere cuts the surface in a circle which is called a great circle. Any other plane intersecting the sphere but not passing through the centre will also cut the surface in a circle which, in this case, is called a small circle. In Fig. 1, EAB is a great circle, for its plane passes through O, the centre of the sphere. Let QOP be the diameter of the sphere perpendicular to the plane of the great circle EAB. Let R be any point in OP and suppose a plane drawn through R parallel to the plane of EAB; the surface of the sphere is then intersected in the small circle FCD. It follows from the construction that OP is also perpendicular to the plane of FCD. The extremities P and Q of the common perpendicular diameter QOP are called the poles of the great circle and of the parallel small circle.