Editing Absolute geometry (section)

{{Short description|Geometry without the parallel postulate}}
{{General geometry}}
'''Absolute geometry''' is a [[geometry]] based on an [[axiom system]] for [[Euclidean geometry]] without the [[parallel postulate]] or any of its alternatives. Traditionally, this has meant using only the first four of [[Euclid's postulates]].<ref>{{harvnb|Faber|1983|loc=pg. 131}}</ref> The term was introduced by [[János Bolyai]] in 1832.<ref>In "''Appendix exhibiting the absolute science of space: independent of the truth or falsity of Euclid's Axiom XI (by no means previously decided)''" {{harv|Faber|1983|loc=pg. 161}}</ref> It is sometimes referred to as '''neutral geometry''',<ref>Greenberg cites W. Prenowitz and M. Jordan (Greenberg, p. xvi) for having used the term ''neutral geometry'' to refer to that part of Euclidean geometry that does not depend on Euclid's parallel postulate.  He says that the word ''absolute'' in ''absolute geometry'' misleadingly implies that all other geometries depend on it.</ref> as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems (such as  [[Hilbert's axioms]] without the parallel axiom) are used instead.<ref>{{harvnb|Faber|1983|loc=pg. 131}}</ref>