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The term **non-Euclidean geometry** describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry.
The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines.
In Euclidean geometry, if we start with a point *A* and a line *l*, then we can only draw one line through *A* that is parallel to *l*.
In hyperbolic geometry, by contrast, there are infinitely many lines through *A* parallel to *l*, and in elliptic geometry, parallel lines do not exist.
(See the entries on hyperbolic geometry and elliptic geometry for more information.)

Another way to describe the differences between these geometries is as follows: consider two lines in a plane that are both perpendicular to a third line. In Euclidean and hyperbolic geometry, the two lines are then parallel. In Euclidean geometry, however, the lines remain at a constant distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular. In elliptic geometry, the lines "curve toward" each other, and eventually intersect; therefore no parallel lines exist in elliptic geometry.

While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.
The debate that eventually led to the discovery of non-Euclidean geometries began almost as soon as Euclid's work *Elements* was written.
In the *Elements*, Euclid attempted to establish a fully logical basis for the mathematics known up to his era.
In so doing, he began with a limited number of assumptions (called axioms and postulates) and sought to prove all the other results (propositions) in the work.
The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate," or simply the "parallel postulate", which in Euclid's original formulation is:

- "If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

For several hundred years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four.
Many attempted find a proof by contradiction, most notably the Italian Giovanni Gerolamo Saccheri.
In a work titled *Euclides ab Omni Naevo Vindicatus* (*Euclid Freed from All Flaws*), published in 1733, he quickly discarded elliptic geometry as a possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry.
He finally reached a point where he believed that his results demonstrated a contradiction in the system, thus showing that hyperbolic geometry is logically inconsistent.
His claim of inconsistency seems to have been based on Euclidean presuppositions, because no such contradiction was present.

A hundred years later, in 1829, the Russian Nikolai Ivanovich Lobachevsky published a treatise of hyperbolic geometry. For this reason, hyperbolic geometry is sometimes called Lobachevskian geometry. About the same time, the Hungarian Janos Bolyai also wrote a treatise on hyperbolic geometry, which was published in 1832 as an appendix to a work of his father's. The great mathematician Karl Friedrich Gauss read the appendix and revealed to Bolyai that he had worked out the same results some time earlier. Each of these men thus discovered hyperbolic geometry independently, and none of their work should be disparaged in this light. Lobachevsky's name is attached by right of earliest publication. The fundamental difference between these and earlier works, such as Saccheri's, is that they were the first to unabashedly claim that Euclidean geometry was not the only geometry, nor the only conceivable geometric structure for the universe. However, the possibility still remained that the axioms for hyperbolic geometry were logically inconsistent.

As had been mentioned, more work on Euclid's axioms needed to be done to establish elliptic geometry. Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of non-Euclidean geometries by giving a formula for a familiy of Riemannian metrics on the unit ball in Euclidean space. Sometimes he is unjustly credited with only discovering elliptic geometry, but in fact, this construction shows that his work was far-reaching, with his theorems holding for all geometries.

Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same). Even after the work of Lobachevski, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry? This question was answered by Beltrami, in 1868, who proved that a surface called the pseudosphere has the appropriate curvature to model hyperbolic geometry. His work was directly based on that of Riemann. The significance of Beltrami's work lies in showing that hyperbolic geometry was logically consistent if Euclidean geometry was.

The development of non-Euclidean geometries proved very important to physics in the 20th century. Einstein's Theory of Relativity describes space as generally flat (i.e., Euclidean), but curved (i.e., non-Euclidean) in regions near where matter is present. This kind of geometry, where the curvature changes from point to point, is called pseudo-Euclidean geometry.

- Ian Stewart; Flatterland; Perseus Publishing; ISBN 0-7382-0675-X (softcover, 2001)