## 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. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue--not only because Beltrami rescued hyperbolic geometry from oblivion by proving it 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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77 plane angle sum arbitrary axis basic figure Beltrami called chords circle with centre coincide congruent conic section considered const constant curvature constant negative curvature construction coordinates corresponding cosh cross-ratio curve defined denote determined disc distance doubly-counted elliptic geometry equal equation euclidean geometry expression finite fixed points formula Fuchsian fuchsian groups functions fundamental conic fundamental elements fundamental points fundamental surface Gauss generalised transformation geodesic circle geodesic triangle given hence horocycles hyperbolic geometry hyperbolic plane imaginary fundamental infinitely intersection inversions Klein latter limit circle line element line pencil linear transformations Lobachevsky logarithm Matematiche mathematies memoir motions noneuclidean geometry obtain ordinary origin orthogonal parabolic geometry Poincare points at infinity projective geometry projective measure pseudosphere pseudospherical surface quadratic forms radius real points represented result rotation sinh spaces of constant sphere spherical surface of constant surface of revolution theorem variables vertices zero