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Any theory of the nature of geometric space differing from the traditional view held since Euclid's time. These geometries arose in the 19th century when several mathematicians working independently explored the possibility of rejecting Euclid's parallel postulate. Different assumptions about how many lines through a point not on a given line could be parallel to that line resulted in hyperbolic geometry and elliptic geometry. Mathematicians were forced to abandon the idea of a single correct geometry; it became their task not to discover mathematical systems but to create them by selecting consistent axioms and studying the theorems that could be derived from them. The development of these alternative geometries had a profound impact on the notion of space and paved the way for the theory of relativity. *Seealso* Nikolay Lobachevsky, Bernhard Riemann.

Learn more about non-Euclidean geometry with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

Study of points, lines, angles, surfaces, and solids based on Euclid's axioms. Its importance lies less in its results than in the systematic method Euclid used to develop and present them. This axiomatic method has been the model for many systems of rational thought, even outside mathematics, for over 2,000 years. From 10 axioms and postulates, Euclid deduced 465 theorems, or propositions, concerning aspects of plane and solid geometric figures. This work was long held to constitute an accurate description of the physical world and to provide a sufficient basis for understanding it. During the 19th century, rejection of some of Euclid's postulates resulted in two non-Euclidean geometries that proved just as valid and consistent.

Learn more about Euclidean geometry with a free trial on Britannica.com.

Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

In mathematics, plane geometry may mean:

- geometry of a plane,
- geometry of the Euclidean plane,

or sometimes a plane is any flat surface that extends without end in all directions.

- geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;
- geometry of the hyperbolic plane or two-dimensional spherical geometry.

See also: plane curve.

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Last updated on Monday July 28, 2008 at 12:10:57 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday July 28, 2008 at 12:10:57 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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