The given line can be drawn through the point. If the parallel postulate is replaced by: It can be shown that if there is at least two lines, there are in fact infinitely 'parallel to the given line through the point'. The converse statement to the parallel postulate, as stated above, is that there are either no lines, or at least two lines Is consistent if it does not contain any contradiction. Furthermore, itĬan be shown that non-Euclidean theories can be just as consistent as Euclidean theory. Such geometries are called non-Euclidean. To build a theory of geometry where the fifth postulate is not true. The fifth postulate is independent of the other postulates.

Many attempts have been made to prove the fifth postulate using the other four postulates. The Elements of Euclid is built upon five postulates.įor over 2000 years the fifth postulate has been considered to be less intuitive than the other postulates, and not sufficiently As a statement that cannot be proven, a postulate should be self-evident. It cannot be proven using previous result. Exactly one line through the green point is parallel to the thick white line.Ī postulate (or axiom) is a statement that acts as a starting point for a theory. However, a seemingly valid statement is not good enough. The postulate is not true in 3D but in 2D it seems to be a valid statement. Parallel to the given line can be drawn through the point.

Papers on non euclidean geometry