Editing Non-Euclidean geometry (section)

==Axiomatic basis of non-Euclidean geometry==
Euclidean geometry can be axiomatically described in several ways. However, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. [[Hilbert's axioms|Hilbert's system]] consisting of 20 axioms<ref>a 21st axiom appeared in the French translation of Hilbert's ''Grundlagen der Geometrie'' according to {{harvnb|Smart|1997|loc=p. 416}}</ref> most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of [[Primitive notion|undefined terms]] obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. [[David Hilbert|Hilbert]] uses the Playfair axiom form, while [[Garrett Birkhoff|Birkhoff]], for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom 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.<ref>{{Harvard citation|Smart|1997|loc=p. 366}}</ref>

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 {{mvar|l}} not passing through P, there exist two lines through P, which do not meet {{mvar|l}}" and keeping all the other axioms, yields [[hyperbolic geometry]].<ref>while only two lines are postulated, it is easily shown that there must be an infinite number of such lines.</ref>
* 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 {{mvar|l}} not passing through P, all the lines through P meet {{mvar|l}}", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry,<ref>Book I Proposition 27 of Euclid's ''Elements''</ref> 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". To obtain a consistent set of axioms that includes this axiom about having no parallel lines, some other axioms must be tweaked. These adjustments depend upon the axiom system used. Among others, these tweaks 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.