AMS Chelsea Publishing 1916; 602 pp; hardcover Volume: 236 ISBN10: 0821834886 ISBN13: 9780821834886 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. Readership Graduate students and research mathematicians. Reviews "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 Table of Contents The Circle in Elementary Plane Geometry  Fundamental definitions and notation
 Inversion
 Mutually tangent circles
 Circles related to a triangle
 The Brocard figures
 Concurrent circles and concyclic points
 Coaxal circles
The Circle in Cartesian Plane Geometry  The circle studied by means of trilinear coordinates
 Fundamental relations, special tetracyclic coordinates
 The identity of Darboux and Frobenius
 Analytic systems of circles
Famous Problems in Construction  Lemoine's geometrographic criteria
 Problem of Apollonius, number of real solutions
 Construction of Apollonius
 Construction of Gergonne
 Steiner's problem
 Circle meeting four others at equal or supplementary angles
 Malfatti's problem, Hart's proof of Steiner's construction
 Analytic solution, extension to thirtytwo cases
 Examples of Fiedler's general cyclographic methods
 Mascheroni's geometry of the compass
The Tetracyclic Plane  Fundamental theorems and definitions
 Cyclics
The Sphere in Elementary Geometry  Miscellaneous elementary theorems
 Coaxal systems
The Sphere in Cartesian Geometry  Coordinate systems
 Identity of Darboux and Frobenius
 Analytic systems of spheres
Pentaspherical Space  Fundamental definitions and theorems
 Cyclides
Circle Transformations  General theory
 Analytic treatment
 Continuous groups of transformations
Sphere Transformations  General theory
 Continuous groups
The Oriented Circle  Elementary geometrical theory
 Analytic treatment
 Laguerre transformations
 Continuous groups
 Hypercyclics
 The oriented circle treated directly
The Oriented Sphere  Elementary geometrical theorems
 Analytic treatment
 The hypercyclide
 The oriented sphere treated directly
 Linesphere transformation
 Complexes of oriented spheres
Circles Orthogonal to One Sphere  Relations of two circles
 Circles orthogonal to one sphere
 Systems of circle crosses
Circles in Space, Algebraic Systems  Coordinates and identities
 Linear systems of circles
 Other simple systems
The Oriented Circle in Space  Fundamental relations
 Linear systems
 The Laguerre method of representing imaginary points
Differential Geometry of Circle Systems  Differential geometry of the \(S_6^{\,5}\) of all circles
 Parametric method for circle congruences
 The Kummer method
 Complexes of circles
 Subject index
 Index of proper names
