See W. Ullmann, Origins of the Great Schism (1948, repr. 1972); B. Tierney, Foundations of the Conciliar Theory (1955, repr. 1969); E. F. Jacob, Essays in the Conciliar Epoch (3d ed. 1963); M. Gail, The Three Popes (1969); J. H. Smith, The Great Schism (1970).
See E. Walker, The Great Trek (5th ed. 1965).
Between any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the Riemannian circle.
Between two points which are directly opposite each other, called antipodal points, there are infinitely many great circles, but all great circle arcs between antipodal points have the same length, i.e. half the circumference of the circle, or , where r is the radius of the sphere.
Because the Earth is approximately spherical (see Earth radius), the equations for great-circle distance are important for finding the shortest distance between points on the surface of the Earth, and so have important applications in navigation.
Let be the geographical latitude and longitude of two points (a base "standpoint" and the destination "forepoint"), respectively, the longitude difference and the (spherical) angular difference/distance, or central angle, which can be constituted from the spherical law of cosines:
This arccosine formula above can have large rounding errors for the common case where the distance is small, however, so it is not normally used. Instead, an equation known historically as the haversine formula was preferred, which is much more accurate for small distances: