A Course in Modern Geometries pp 25-73 | Cite as

# Non-Euclidean Geometry

## 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## Preview

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## Suggestions for Further Reading

- 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 - Gans, D. (1973).
*An Introduction to Non-Euclidean Geometry*. New York: Academic Press. (This is an easy-to-read and detailed presentation. )zbMATHGoogle Scholar - Gray, J. (1979).
*Ideas of Space: Euclidean*,*Non-Euclidean and Relativistic*. Oxford: Clarendon Press.Google Scholar - 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 - 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 - 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 - Ogle, K.N. (1962). The visual space sense.
*Science*135: 763–771.CrossRefGoogle Scholar - 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 - 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 - Sommerville, D. (1970).
*Bibliography of Non-Euclidean Geometry*, 2d ed. New York: Chelsea.zbMATHGoogle Scholar - 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 - Wolfe, H.E. (1945).
*Introduction to Non-Euclidean Geometry*. New York: Holt, Rinehart and Winston.Google Scholar - Zage, W.M. (1980). The geometry of binocular visual space.
*Mathematics Magazine*53 (5): 289–294.MathSciNetzbMATHCrossRefGoogle Scholar

## Readings on the History of Geometry

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

*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