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AMS Chelsea Publishing
1916; 602 pp; hardcover
List Price: US$65
Member Price: US$58.50
Order Code: CHEL/236.H
Circles and spheres are central objects in geometry. Mappings that take circles to circles or spheres to spheres have special roles in metric and conformal geometry. An example of this is Lie's sphere geometry, whose group of transformations is precisely the conformal group.
Coolidge's treatise looks at systems of circles and spheres and the geometry and groups associated to them. It was written (1916) at a time when Lie's enormous influence on the field was still widely felt. Today, there is a renewed interest in the geometry of special geometric configurations. Coolidge has examined many of the most intuitive: linear systems of circles, circles orthogonal to a given sphere, and so on. He also examines the differential and projective geometry of the space of all spheres in a given space.
Through the simple vehicles of circles and spheres, Coolidge makes contact with diverse areas of mathematics: conformal transformations and analytic functions, projective and contact geometry, and Lie's theory of continuous groups, to name a few. The interested reader will be well rewarded by a study of this remarkable book.
Graduate students and research mathematicians.
"The author has fully carried out the high aim he has set before himself: "The present work is an attempt, perhaps the first, to present a consistent and systematic account of the various theories [those of Steiner, Feuerbach, Chasles, Lemoine, Casey, ... Reye, Fiedler, Loria, Mobius, Lie, Stephanos, Castelnuovo, Cosserat, Ribaucour, Darboux, Guichard ... ].""
-- The Mathematical Gazette
"Not a list of results, but a well digested account of theories and methods ... is what he has given us for leisurely study and enjoyment."
-- Bulletin of the AMS
"The book provides a wealth of information from both a historical and mathematical perspective including many early ideas from the theory of algebraic curves and surfaces."
-- Zentralblatt MATH
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