Non-Euclidean Geometry - Axiomatic Basis of Non-Euclidean Geometry

Axiomatic Basis of Non-Euclidean Geometry

Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid's original system of five postulates (axioms) is not one of these as his proofs relied on several unstated assumptions which should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. In all approaches, however, there is an axiom which is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom which says that "there exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom which is equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry. As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.

To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways. Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement "In a plane, given a point P and a line ℓ not passing through P, there exist two lines through P which do not meet ℓ" and keeping all the other axioms, yields hyperbolic geometry. The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a line ℓ not passing through P, all the lines through P meet ℓ", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, but this statement says that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the "obtuse angle case". In order to obtain a consistent set of axioms which includes this axiom about having no parallel lines, some of the other axioms must be tweaked. The adjustments to be made depend upon the axiom system being used. Among others these tweaks will have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. Riemann's elliptic geometry emerges as the most natural geometry satisfying this axiom.