Non-Euclidean Geometry

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


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.


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.


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Suggestions for Further Reading

  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. )zbMATHGoogle Scholar
  3. Gray, J. (1979). Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Oxford: Clarendon Press.Google Scholar
  4. Heath, T.L. (1956). The Thirteen Books of Euclid’s Elements, 2d ed. New York: Dover. Henderson, L.D. (1983). The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar
  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.CrossRefGoogle Scholar
  8. Penrose, R. (1978). The geometry of the universe. In: L.A. Steen (Ed.), Mathematics Today: Twelve Informal Essays, pp. 83–125. New York: Springer-Verlag.CrossRefGoogle Scholar
  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. )CrossRefGoogle Scholar
  10. Sommerville, D. (1970). Bibliography of Non-Euclidean Geometry, 2d ed. New York: Chelsea.zbMATHGoogle Scholar
  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. )zbMATHGoogle Scholar
  12. Wolfe, H.E. (1945). Introduction to Non-Euclidean Geometry. New York: Holt, Rinehart and Winston.Google Scholar
  13. Zage, W.M. (1980). The geometry of binocular visual space. Mathematics Magazine 53 (5): 289–294.MathSciNetzbMATHCrossRefGoogle Scholar

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
  2. Barker, S.F. (1964). Philosophy of Mathematics, ipp. 1–55. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  3. Bold, B. (1969). Famous Problems of Geometry and How to Solve Them. New York: Dover.Google Scholar
  4. Bronowski, J. (1974). The music of the spheres. In: The Ascent of Man, pp. 155–187. Boston: Little, Brown.Google Scholar
  5. Eves, H. (1976). An Introduction to the History of Mathematics, 4th ed. New York: Holt, Rinehart and Winston.zbMATHGoogle Scholar
  6. Gardner, M. (1966). The persistence (and futility) of efforts to trisect the angle. Scientific American 214: 116–122.CrossRefGoogle Scholar
  7. Gardner, M. (1981). Euclid’s parallel postulate and its modern offspring. Scientific American 254: 23–24.CrossRefGoogle Scholar
  8. Heath, T.L. (1921). A History of Greek Mathematics. Oxford: Clarendon Press.zbMATHGoogle Scholar
  9. Heath, T.L. (1956). The Thirteen Books of Euclid’s Elements, 2d ed. New York: Dover.Google Scholar
  10. Hoffer, W. (1975). A magic ratio recurs throughout history. Smithsonian 6 (9): 110–124.Google Scholar
  11. Kline, M. (1972). Mathematical Thought from Ancient to Modern Times, pp. 3–130, 861–881. New York: Oxford University Press.zbMATHGoogle Scholar
  12. Knorr, W.R. (1986). The Ancient Tradition of Geometric Problems. Boston: Birkhauser. Maziarz, E., and Greenwood, T. ( 1984 ). Greek mathematical philosophy. In: D.M.Google Scholar
  13. Campbell and J.C. Higgins (Eds.), Mathematics: People, Problems, Results,Vol. 1, pp. 18–27. Belmont, CA: Wadsworth.Google Scholar
  14. Mikami, Y. (1974). The Development of Mathematics in China and Japan, 2d ed. New York: Chelsea.Google Scholar
  15. Smith, D.E. (1958). History of Mathematics, Vol. 1, pp. 1–147. New York: Dover.zbMATHGoogle Scholar
  16. Swetz, F. (1984). The evolution of mathematics in ancient China. In: D.M. Campbell and J.C. Higgins (Eds.), Mathematics: People, Problems, Results, Vol. 1, pp. 28–37. Belmont, CA: Wadsworth.Google Scholar

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

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Judith N. Cederberg
    • 1
  1. 1.Department of MathematicsSt. Olaf CollegeNorthfieldUSA

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