Editing Absolute geometry (section)

==Relation to other geometries==
The theorems of absolute geometry hold in [[hyperbolic geometry]], which is a [[non-Euclidean geometry]], as well as in [[Euclidean geometry]].<ref>Indeed, absolute geometry is in fact the intersection of hyperbolic geometry and Euclidean geometry when these are regarded as sets of propositions.</ref> Absolute geometry is inconsistent with [[elliptic geometry]] or [[spherical geometry]]: the notion of ordering or betweenness of points on lines, used to axiomatize absolute geometry, is inconsistent with these other geometries.<ref>{{citation|first=G.|last=Ewald|title=Geometry: An Introduction|year=1971|publisher=Wadsworth|page=53|url=https://archive.org/details/geometryintroduc0000ewal/page/52}}</ref>

Absolute geometry is an extension of [[ordered geometry]], and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with [[affine geometry]], which does not assume Euclid's third and fourth axioms.
(3: "To describe a [[circle]] with any centre and distance [[radius]].",
4: "That all [[right angle]]s are equal to one another." ) 
Ordered geometry is a common foundation of both absolute and affine geometry.<ref>{{harvnb|Coxeter|1969|loc=pp. 175–6}}</ref>

The [[geometry of special relativity]] has been developed starting with nine axioms and eleven propositions of absolute geometry.<ref>[[Edwin B. Wilson]] & [[Gilbert N. Lewis]] (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the [[American Academy of Arts and Sciences]] 48:387–507</ref><ref>[https://web.archive.org/web/20090926011519/http://ca.geocities.com/cocklebio/synsptm.html], a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by [[Wayback Machine]]</ref> The authors [[Edwin B. Wilson]] and [[Gilbert N. Lewis]] then proceed beyond absolute geometry when they introduce [[hyperbolic rotation]] as the transformation relating two [[frames of reference]].