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.

Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.

Cover picture reproduced with permission of Dr. Konrad Polthier.

Readership

Graduate students and research mathematicians specializing in geometry.

Reviews

"Translations are well done and very readable ... papers ... are well chosen ... an extremely attractive and valuable book to have and to read ... fills an important niche in the mathematical literature by making these papers available to a contemporary audience ... allows the modern reader to see how the great mathematicians of another time viewed both their subject and mathematics in general, a view which can still be inspirational."

-- Bulletin of the London Mathematical Society

Table of Contents

• Introduction to Beltrami's Essay on the interpretation of noneuclidean geometry
• Translation of Beltrami's Essay on the interpretation of noneuclidean geometry
• Introduction to Beltrami's Fundamental theory of spaces of constant curvature
• Translation of Beltrami's Fundamental theory of spaces of constant curvature
• Introduction to Klein's On the so-called noneuclidean geometry
• Translation of Klein's On the so-called noneuclidean geometry
• Introduction to Poincaré's Theory of fuchsian groups, Memoir on kleinian groups, On the applications of noneuclidean geometry to the theory of quadratic forms
• Translation of Poincaré's Theory of fuchsian groups
• Translation of Poincaré's Memoir on kleinian groups
• Translation of Poincaré's On the applications of noneuclidean geometry to the theory of quadratic forms
• Index