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. 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.


Ideal Point Euclidean Geometry Hyperbolic Geometry Interior Angle Dynamic Geometry Software 
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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.)zbMATHGoogle Scholar
  3. Gans, D. (1973). An Introduction to Non-Euclidean Geometry. New York: Academic Press. (This is an easy-to-read and detailed presentation.)zbMATHGoogle Scholar
  4. Gray, J. (1979). Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Oxford: Clarendon Press.zbMATHGoogle Scholar
  5. Heath, T. L. (1956). The Thirteen Books of Euclid’s Elements, 2d ed. New-York: Dover.Google Scholar
  6. Henderson, L. D. (1983). The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton, NJ: Princeton University Press.Google Scholar
  7. 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
  8. Lockwood, J. R. and Runion, G. E. (1978). Deductive Systems: Finite and Non-Euclidean Geometries. Reston, VA: N.C.T.M. (This is abrief elementary introduction that can be used as supplementary material at the secondary-school level.)Google Scholar
  9. Ogle, K. N. (1962). The visual space sense. Science 135: 763–771.CrossRefGoogle Scholar
  10. Penrose, R. (1978). The geometry of the universe. In Mathematics Today: Twelve Informal Essays. Edited by L. A. Steen, pp. 83–125. New York: Springer-Verlag.CrossRefGoogle Scholar
  11. 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
  12. Sommerville, D. (1970). Bibliography of Non-Euclidean Geometry, 2d ed. New York: Chelsea.zbMATHGoogle 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.)zbMATHGoogle Scholar
  14. Wolfe, H. E. (1945). Introduction to Non-Euclidean Geometry. New York: Holt, Rinehart and Winston. (Chaps. 1, 2, and 4 contain a development similar to that in this text.)Google 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
  2. Barker, S. E (1964). Philosophy of Mathematics, pp. 1–55. Englewood Cliffs, NJ: Prentice-Hall.zbMATHGoogle 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, BrownGoogle 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. Grabiner, Judith V. (1988). The centrality of mathematics in the history of western thought. Mathematics Magazine 61(4): 220–230.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Heath, T. L. (1921). A History of Greek Mathematics. Oxford: Clarendon Press.zbMATHGoogle Scholar
  10. Heath, T. L. (1956). The Thirteen Books of Euclid’s Elements, 2d ed. New York: Dover.Google Scholar
  11. Hoffer, W. (1975). A magic ratio recurs throughout history. Smithsonian 6(9): 110–124.Google Scholar
  12. Kline, M. (1972). Mathematical Thought from Ancient to Modern Times, pp. 3–130, 861–881. New York: Oxford University Press.Google Scholar
  13. Knorr, W. R. (1986). The Ancient Tradition of Geometric Problems. Boston: Birkhauser.Google Scholar
  14. Maziarz, E., and Greenwood, T. (1984). Greek mathematical philosophy. In Mathematics: People, Problems, Results. Edited by D. M. Campbell and J. C. Higgins. Vol. 1, pp. 18–27. Belmont, CA: Wadsworth.Google Scholar
  15. Mikami, Y. (1974). The Development of Mathematics in China and Japan, 2d ed. New York: Chelsea.Google Scholar
  16. Smith, D. E. (1958). History of Mathematics, Vol. 1, pp. 1–147. New York: Dover.zbMATHGoogle Scholar
  17. Swetz, F. (1984). The evolution of mathematics in ancient China. In Mathematics: People, Problems, Results. Edited by D. M. Campbell and J. C. Higgins. Vol. 1, pp. 28–37. Belmont, CA: Wadsworth.Google Scholar
  18. Torretti, Roberto (1978). Philosophy of Geometry from Riemann to Poincaré. Dordrect, Holland: D. Reidel Publishing Company.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 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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