In three 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). Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry.

# Non-Euclidean Geometry

## See also

Absolute Geometry, Elliptic Geometry, Euclid's Postulates, Euclidean Geometry, Hyperbolic Geometry, Parallel Postulate, Spherical Geometry## Explore with Wolfram|Alpha

## References

Bolyai, J. "Scientiam spatii absolute veritam exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) indepentem: adjecta ad casum falsitatis, quadratura circuli geometrica." Reprinted as "The Science of Absolute Space" in Bonola, R.*Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai.*New York: Dover, 1955.Bonola, R.

*Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai.*New York: Dover, 1955.Borsuk, K.

*Foundations of Geometry: Euclidean and Bolyai-Lobachevskian Geometry. Projective Geometry.*Amsterdam, Netherlands: North-Holland, 1960.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.Greenberg, M. J.

*Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed.*San Francisco, CA: W. H. Freeman, 1994.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.Lobachevski, N. Reprinted as "Theory of Parallels" in Bonola, R.

*Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai.*New York: Dover, 1955.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.Weisstein, E. W. "Books about Non-Euclidean Geometry." http://www.ericweisstein.com/encyclopedias/books/Non-EuclideanGeometry.html."Welcome to the Non-Euclidean Geometry Homepage." http://members.tripod.com/~noneuclidean/.Woods, F. S. "Non-Euclidean Geometry." Ch. 3 in

*Monographs on Topics of Modern Mathematics Relevant to the Elementary Field*(Ed. J. W. A. Young). New York: Dover, pp. 93-147, 1955.

## Referenced on Wolfram|Alpha

Non-Euclidean Geometry## Cite this as:

Weisstein, Eric W. "Non-Euclidean Geometry."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Non-EuclideanGeometry.html