Sources of Hyperbolic Geometry
About this Title
John Stillwell, Monash University, Clayton, Victoria, Australia
Publication: History of Mathematics
Publication Year: 1996; Volume 10
ISBNs: 978-0-8218-0922-8 (print); 978-1-4704-3878-4 (online)
MathSciNet review: MR1402697
MSC: Primary 01A75; Secondary 01A55, 01A60, 03B30, 51-03, 51M10
This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincaré 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 Poincaré 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 Poincaré in their full brilliance.
Graduate students and research mathematicians specializing in geometry.
Table of Contents
- Translator’s introduction (Essay on the interpretation of noneuclidean geometry)
- Essay on the interpretation of noneuclidean geometry
- Translator’s introduction (Fundamental theory of spaces of constant curvature)
- Fundamental theory of spaces of constant curvature
- Translator’s introduction (On the so-called noneuclidean geometry)
- On the so-called noneuclidean geometry
- Translator’s introduction (Theory of Fuchsian groups/Memoir on Kleinian groups/On the applications of noneuclidean geometry to the theory of quadratic
- Theory of Fuchsian groups/Memoir on Kleinian groups/On the applications of noneuclidean geometry to the theory of quadratic forms