Abstract
For our proof of the fundamental formula of Bolyai — Lobachevsky geometry we shall need a lemma concerning ratios of lengths of certain arcs of horocircles. After extending the domain of definition for the critical function ∏ to the set of reals, the rest of this section is devoted to proving this lemma, our Theorem 31.17. For this extension, ∏(0) will be defined to be π/2 since ∏(x) approaches π/2 as x approaches 0. Then (0, π/2) will be made a point of symmetry for the graph of ∏ in the Cartesian plane. So the midpoint of (x,∏,(x)) and (-x,∏(-x)) will be (0, π/2). See Figure 31.1. Then for all real x, we will have ∏(x) + ∏(-x) = π.
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© 1975 Springer-Verlag New York, Inc.
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Martin, G.E. (1975). Horocircles. In: The Foundations of Geometry and the Non-Euclidean Plane. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5725-7_31
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DOI: https://doi.org/10.1007/978-1-4612-5725-7_31
Publisher Name: Springer, New York, NY
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