# 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

## Preview

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.)
3. Gans, D. (1973). An Introduction to Non-Euclidean Geometry. New York: Academic Press. (This is an easy-to-read and detailed presentation.)
4. Gray, J. (1979). Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Oxford: Clarendon Press.
5. Heath, T. L. (1956). The Thirteen Books of Euclid’s Elements, 2d ed. New-York: Dover.Google Scholar
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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.
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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.)
12. Sommerville, D. (1970). Bibliography of Non-Euclidean Geometry, 2d ed. New York: Chelsea.
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.)
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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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7. Gardner, M. (1981). Euclid’s parallel postulate and its modern offspring. Scientific American 254: 23–24.
8. Grabiner, Judith V. (1988). The centrality of mathematics in the history of western thought. Mathematics Magazine 61(4): 220–230.
9. Heath, T. L. (1921). A History of Greek Mathematics. Oxford: Clarendon Press.
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11. Hoffer, W. (1975). A magic ratio recurs throughout history. Smithsonian 6(9): 110–124.Google Scholar
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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