# Non-Euclidean Geometry

• Judith N. Cederberg
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

Mathematics is not usually considered a source of surprises, but non-Euclidean geometry contains a number of easily obtainable theorems that seem almost “heretical” to anyone grounded in Euclidean geometry. A brief encounter with these “strange” geometries frequently results in initial confusion. Eventually, however, this encounter should not only produce a deeper understanding of Euclidean geometry but also offer convincing support for the necessity of carefully reasoned proofs for results that may have once seemed obvious. These individual experiences mirror the difficulties mathematicians encountered historically in the development of non-Euclidean geometry. An acquaintance with this history and an appreciation for the mathematical and intellectual importance of Euclidean geometry is essential for an understanding of the profound impact of this development on mathematical and philosophical thought. Thus, the study of Euclidean and non-Euclidean geometry as mathematical systems can be greatly enhanced by parallel readings in the history of geometry. Since the mathematics of the ancient Greeks was primarily geometry, such readings provide an introduction to the history of mathematics in general.

## Keywords

Ideal Point Euclidean Geometry Hyperbolic Geometry Exterior Angle Klein Model
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

1. Aleksandrov, A.D. (1969). Non-Euclidean Geometry. In: A.D. Aleksandrov, A.N. Kolmogorov, and M.A. Lavrent’ev (Eds.), Mathematics: Its Content, Methods and Meaning, Vol. 3, pp. 97–189. Cambridge, MA: M.I.T. Press. (This is an expository presentation of non-Euclidean geometry.)Google Scholar
2. Gans, D. (1973). An Introduction to Non-Euclidean Geometry. New York: Academic Press. (This is an easy-to-read and detailed presentation. )
3. Gray, J. (1979). Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Oxford: Clarendon Press.Google Scholar
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5. Lieber, L.R. (1940). Non-Euclidean Geometry: Or, Three Moons in Mathesis, 2d ed. New York: Galois Institute of Mathematics and Art. (This is an entertaining poetic presentation. )Google Scholar
6. Lockwood, J.R., and Runion, G.E. (1978). Deductive Systems: Finite and Non-Euclidean Geometries. Reston, VA: N.C.T.M. (This is a brief elementary introduction that can be used as supplementary material at the high-school level. )Google Scholar
7. Ogle, K.N. (1962). The visual space sense. Science 135: 763–771.
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9. Ryan, P.J. (1986). Euclidean and Non-Euclidean Geometry: An Analytic Approach. Cambridge: Cambridge University Press. (Uses groups and analytic techniques of linear algebra to construct and study models of these geometries. )
10. Sommerville, D. (1970). Bibliography of Non-Euclidean Geometry, 2d ed. New York: Chelsea.
11. Trudeau, R.J. (1987). The Non-Euclidean Revolution. Boston: Birkhauser. (This presentation of both Euclid’s original work and non-Euclidean geometry is interwoven with a nontechnical description of the revolution in mathematics that resulted from the development of non-Euclidean geometry. )
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## Readings on the History of Geometry

1. Barker, S.F. (1984). Non-Euclidean geometry. In: D.M. Campbell and J.C. Higgins (Eds.), Mathematics: People, Problems, Results, Vol. 2, pp. 112–127. Belmont, CA: Wadsworth.Google Scholar
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4. Bronowski, J. (1974). The music of the spheres. In: The Ascent of Man, pp. 155–187. Boston: Little, Brown.Google Scholar
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6. Gardner, M. (1966). The persistence (and futility) of efforts to trisect the angle. Scientific American 214: 116–122.
7. Gardner, M. (1981). Euclid’s parallel postulate and its modern offspring. Scientific American 254: 23–24.
8. Heath, T.L. (1921). A History of Greek Mathematics. Oxford: Clarendon Press.
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10. Hoffer, W. (1975). A magic ratio recurs throughout history. Smithsonian 6 (9): 110–124.Google Scholar
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## Suggestions for Viewing

1. A Non-Euclidean Universe (1978; 25 min). Depicts the Poincaré model of the hyperbolic plane. Produced by the Open University of Great Britain. Available in 16-mm or video format from The Media Guild, 11722 Sorrento Valley Road, Suite E, San Diego, CA 92121.Google Scholar