## Sources of Hyperbolic GeometryThis book presents, for the first time in English, the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics. It sets out to provide recognition of Beltrami comparable to that given the pioneering works of Bolyai and Labachevsky, not only because Beltrami rescued hyperbolic geometry from oblivion by proving to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincare brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincare in their full brilliance. |

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angle angle of parallelism apply arbitrary becomes Beltrami called Cayley centre changes circle clear coincide completely condition conic section connecting considered constant construction coordinates corresponding cross-ratio curvature curve defined determined distance elliptic equal equation euclidean expression fact figure Finally finite fixed formula functions fundamental conic fundamental elements geodesic geodesic circle given gives hence hyperbolic geometry imaginary infinitely infinity interpretation intersection inversions kind Klein latter length limit line element linear transformations means measure meet memoir motions noneuclidean obtain ordinary origin orthogonal parabolic geometry parallel particular passing pencil plane Poincaré positive projective pseudosphere question radius region relation represented respect result rotation scale shows side space special measure sphere spherical substitutions suppose surface surface of constant tangential theorem theory triangle variables vertices zero