Editing Euclidean geometry (section)

====Non-Euclidean geometry====
{{Main|Non-Euclidean geometry}}
The century's most influential development in geometry occurred when, around 1830, [[János Bolyai]] and [[Nikolai Ivanovich Lobachevsky]] separately published work on [[non-Euclidean geometry]], in which the parallel postulate is not valid.<ref>Ball, p.&nbsp;485.</ref> Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.

In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the ''Elements''. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the ''Elements,'' shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third [[wikt:vertex|vertex]]. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the [[Real number#Completeness|completeness]] property of the real numbers. Starting with [[Moritz Pasch]] in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of [[Hilbert's axioms|Hilbert]],<ref>* [[Howard Eves]], 1997 (1958). ''Foundations and Fundamental Concepts of Mathematics''. Dover.</ref> [[Birkhoff's axioms|George Birkhoff]],<ref>Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)", Annals of Mathematics 33.</ref> and [[Tarski's axioms|Tarski]].<ref name="Tarski 1951">Tarski (1951).</ref>