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A Treatise on the Circle and the Sphere
Julian Lowell Coolidge

AMS Chelsea Publishing
1916; 602 pp; hardcover
Volume: 236
ISBN-10: 0-8218-3488-6
ISBN-13: 978-0-8218-3488-6
List Price: US$65
Member Price: US$58.50
Order Code: CHEL/236.H
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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

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 thirty-two 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
  • Line-sphere 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
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