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. Abrief 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 it should 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 Interior Angle Dynamic Geometry Software 
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.

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Suggestions for Further Reading

  1. Aleksandrov, A. D. (1969). Non-Euclidean Geometry. In Mathematics: Its Content, Methods and Meaning, A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent’ev (Eds.), Vol. 3, pp. 97–189. Cambridge, MA: M.I.T. Press. (This is an expository presentation of non-Euclidean geometry.)Google Scholar
  2. Davis, D. M. (1993). The Nature and Power of Mathematics. Princeton: Princeton University Press. (Written for the liberal arts students, Chapters 1 and 2 provide a substantial introduction to early Greek mathematics and non-Euclidean geometry.)MATHGoogle Scholar
  3. Gans, D. (1973). An Introduction to Non-Euclidean Geometry. New York: Academic Press. (This is an easy-to-read and detailed presentation.)MATHGoogle Scholar
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  5. Heath, T. L. (1956). The Thirteen Books of Euclid’s Elements, 2d ed. New-York: Dover.Google Scholar
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  12. Sommerville, D. (1970). Bibliography of Non-Euclidean Geometry, 2d ed. New York: Chelsea.MATHGoogle Scholar
  13. 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.)MATHGoogle Scholar
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Readings on the History of Geometry

  1. Barker, S. F. (1984). Non-Euclidean geometry. In Mathematics: People, Problems, Results. Edited by D. M. Campbell and J. C. Higgins, Vol. 2, pp. 112–127. Belmont, CA: Wadsworth.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 Production Centre, Walton Hall, Milton Keynes MK7 6BH, UK.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Judith N. Cederberg
    • 1
  1. 1.Department of MathematicsNorthfieldUSA

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