In 3 dimensions, there are three classes of constant curvature Geometries. All are based on the first four of Euclid's Postulates, but each uses its own version of the Parallel Postulate. The ``flat'' geometry of everyday intuition is called Euclidean Geometry (or Parabolic Geometry), and the non-Euclidean geometries are called Hyperbolic Geometry (or Lobachevsky-Bolyai-Gauss Geometry) and Elliptic Geometry (or Riemannian Geometry). It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean Geometry.

**References**

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Carslaw, H. S. *The Elements of Non-Euclidean Plane Geometry and Trigonometry.* London: Longmans, 1916.

Coxeter, H. S. M. *Non-Euclidean Geometry, 6th ed.* Washington, DC: Math. Assoc. Amer., 1988.

Dunham, W. *Journey Through Genius: The Great Theorems of Mathematics.* New York: Wiley, pp. 53-60, 1990.

Iversen, B. *An Invitation to Hyperbolic Geometry.* Cambridge, England: Cambridge University Press, 1993.

Iyanaga, S. and Kawada, Y. (Eds.). ``Non-Euclidean Geometry.'' §283 in
*Encyclopedic Dictionary of Mathematics.* Cambridge, MA: MIT Press, pp. 893-896, 1980.

Martin, G. E. *The Foundations of Geometry and the Non-Euclidean Plane.* New York: Springer-Verlag, 1975.

Pappas, T. ``A Non-Euclidean World.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, pp. 90-92, 1989.

Ramsay, A. and Richtmeyer, R. D. *Introduction to Hyperbolic Geometry.* New York: Springer-Verlag, 1995.

Sommerville, D. Y. *The Elements of Non-Euclidean Geometry.* London: Bell, 1914.

Sommerville, D. Y. *Bibliography of Non-Euclidean Geometry, 2nd ed.* New York: Chelsea, 1960.

Sved, M. *Journey into Geometries.* Washington, DC: Math. Assoc. Amer., 1991.

Trudeau, R. J. *The Non-Euclidean Revolution.* Boston, MA: Birkhäuser, 1987.

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1999-05-25