This 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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