Unless stipulated otherwise, a line can contain two distinct points without having points in-between. The boy who could not prove that a line can't intersect three sides of a triangle in their interior points, raised a deep question overlooked by generations of professionals. The updates incorporate axioms of Order , Congruence , and Continuity . In the following, I shall nonetheless relate to the set of Postulates as they appear in Elements . The approach became known as the Erlanger Programm .
Superposition is achieved by transforming one triangle onto another.
Euclid implicitly assumed that geometric figures do not change by rigid motions. Since then, study of particular kinds of transformations became an integral part of geometric research and development. On this page, for the reference sake, I shall collect short descriptions of and facts from various geometries as they become necessary for other discussions. In time, the page will serve as an index for more detailed coverage. The term Absolute Geometry had been introduced by J.Bolyai in 1832.