The word geometry is of Greek origin, γεωμετρια, to “measure the earth.” In antiquity geometric techniques were used by the Egyptians to assess taxes fairly according to area. The geometry chronicled and developed by Euclid (ca. 300 b.c.) in his great work The Elements begins with the geometry of the plane – the abstract field of a farmer. Before discussing Euclid's work and its later generalizations, let us make a short detour into spherical geometry to “measure the Earth,” idealized as a sphere. The earliest science of astronomy and the need to measure time accurately by the sun led to the development of this geometry.

Basic plane geometry is concerned with points and lines, with their incidence relations, and congruences. To study such basic notions on the sphere, we need to know what a “line” or a “line segment” means.

Definition 1.1. A great circleon a sphere is the intersection of that sphere with a plane passing through the center of the sphere (for example, the equator and the lines of constant longitude).

Early geometers understood that great circles share many formal properties with lines in the plane making them a natural choice for lines. For example, given any pair of nonantipodal points on the sphere, there is a unique great circle joining that pair of points; to construct it, take the intersection of the sphere and the plane containing the pair of points and the center of the sphere.