Abstract
At the end of Chapter I, we mentioned the discovery that a non-euclidean geometry could have a euclidean representation. In this chapter, we want to look at one such representation, due to H. Poincare (1854–1912), which is called “the Poincaré model of hyperbolic geometry”. Not only is this model attractively ingenious, but, as we shall explain in detail, it implies that if there is a logical inconsistency in hyperbolic geometry then there is a logical inconsistency in euclidean geometry. Thus, however non-intuitive hyperbolic geometry may appear, it cannot be refuted on logical grounds unless there is a similar refutation of the highly intuitive relations of euclidean geometry.
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© 1981 Springer-Verlag New York, Inc.
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Kelly, P., Matthews, G. (1981). A Euclidean Model of the Hyperbolic Plane. In: The Non-Euclidean, Hyperbolic Plane. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8125-9_4
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DOI: https://doi.org/10.1007/978-1-4613-8125-9_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90552-5
Online ISBN: 978-1-4613-8125-9
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