A Course in Modern Geometries pp 33-97 | 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. 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## Preview

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## Suggestions for Further Reading

- 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 - 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 - 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.zbMATHGoogle Scholar - Heath, T. L. (1956).
*The Thirteen Books of Euclid’s Elements*, 2d ed. New-York: Dover.Google Scholar - Henderson, L. D. (1983).
*The Fourth Dimension and Non-Euclidean Geometry in Modern Art*. Princeton, NJ: Princeton University Press.Google 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 abrief elementary introduction that can be used as supplementary material at the secondary-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
*Mathematics Today: Twelve Informal Essays*. Edited by L. A. Steen, 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. (Chaps. 1, 2, and 4 contain a development similar to that in this text.)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 Mathematics: People, Problems, Results*. Edited by D. M. Campbell and J. C. Higgins, Vol. 2, pp. 112–127. Belmont, CA: Wadsworth.Google Scholar - Barker, S. E (1964).
*Philosophy of Mathematics*, pp. 1–55. Englewood Cliffs, NJ: Prentice-Hall.zbMATHGoogle 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, BrownGoogle 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 - Grabiner, Judith V. (1988). The centrality of mathematics in the history of western thought.
*Mathematics Magazine*61(4): 220–230.MathSciNetzbMATHCrossRefGoogle 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.Google Scholar - Knorr, W. R. (1986).
*The Ancient Tradition of Geometric Problems*. Boston: Birkhauser.Google Scholar - 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 - 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
*Mathematics: People, Problems, Results*. Edited by D. M. Campbell and J. C. Higgins. Vol. 1, pp. 28–37. Belmont, CA: Wadsworth.Google Scholar - Torretti, Roberto (1978).
*Philosophy of Geometry from Riemann to Poincaré*. Dordrect, Holland: D. Reidel Publishing Company.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 Production Centre, Walton Hall, Milton Keynes MK7 6BH, UK.Google Scholar