A Treatise on the Circle and the Sphere is a mathematics book on circles, spheres, and inversive geometry. It was written by Julian Coolidge, and published by the Clarendon Press in 1916.[1][2][3][4] The Chelsea Publishing Company published a corrected reprint in 1971,[5][6] and after the American Mathematical Society acquired Chelsea Publishing it was reprinted again in 1997.[7]


As is now standard in inversive geometry, the book extends the Euclidean plane to its one-point compactification, and considers Euclidean lines to be a degenerate case of circles, passing through the point at infinity. It identifies every circle with the inversion through it, and studies circle inversions as a group, the group of Möbius transformations of the extended plane. Another key tool used by the book are the "tetracyclic coordinates" of a circle, quadruples of complex numbers a , b , c , d {\displaystyle a,b,c,d} a, b, c, d describing the circle in the complex plane as the solutions to the equation a z z ¯ + b z + c z ¯ + d = 0 {\displaystyle az{\bar {z}}+bz+c{\bar {z}}+d=0} {\displaystyle az{\bar {z}}+bz+c{\bar {z}}+d=0}. It applies similar methods in three dimensions to identify spheres (and planes as degenerate spheres) with the inversions through them, and to coordinatize spheres by "pentacyclic coordinates".[7]

Other topics described in the book include:

Tangent circles[2][3] and pencils of circles[3]
Steiner chains, rings of circles tangent to two given circles[4]
Ptolemy's theorem on the sides and diagonals of quadrilaterals inscribed in circles[4]
Triangle geometry, and circles associated with triangles, including the nine-point circle, Brocard circle, and Lemoine circle[1][2][3]
The Problem of Apollonius on constructing a circle tangent to three given circles, and the Malfatti problem of constructing three mutually-tangent circles, each tangent to two sides of a given triangle[1][3]
The work of Wilhelm Fiedler on "cyclography", constructions involving circles and spheres[1][3]
The Mohr–Mascheroni theorem, that in straightedge and compass constructions, it is possible to use only the compass[1]
Laguerre transformations, analogues of Möbius transformations for oriented projective geometry[1][3]
Dupin cyclides, shapes obtained from cylinders and tori by inversion[3]


At the time of its original publication this book was called encyclopedic,[2][3] and "likely to become and remain the standard for a long period".[2] It has since been called a classic,[5][7] in part because of its unification of aspects of the subject previously studied separately in synthetic geometry, analytic geometry, projective geometry, and differential geometry.[5] At the time of its 1971 reprint, it was still considered "one of the most complete publications on the circle and the sphere", and "an excellent reference".[6]

Bieberbach, Ludwig, "Review of A Treatise on the Circle and the Sphere (1916 edition)", Jahrbuch über die Fortschritte der Mathematik, JFM 46.0921.02
H. P. H. (December 1916), "Review of A Treatise on the Circle and the Sphere (1916 edition)", The Mathematical Gazette, 8 (126): 338–339, doi:10.2307/3602790, hdl:2027/coo1.ark:/13960/t39z9q113, JSTOR 3602790
Emch, Arnold (June 1917), "Review of A Treatise on the Circle and the Sphere (1916 edition)", The American Mathematical Monthly, 24 (6): 276–279, doi:10.1080/00029890.1917.11998325, JSTOR 2973184
White, H. S. (July 1919), "Circle and sphere geometry (Review of A Treatise on the Circle and the Sphere)", Bulletin of the American Mathematical Society, American Mathematical Society ({AMS}), 25 (10): 464–468, doi:10.1090/s0002-9904-1919-03230-3
"Review of A Treatise on the Circle and the Sphere (1971 reprint)", Mathematical Reviews, MR 0389515
Peak, Philip (May 1974), "Review of A Treatise on the Circle and the Sphere (1971 reprint)", The Mathematics Teacher, 67 (5): 445, JSTOR 27959760

Steinke, G. F., "Review of A Treatise on the Circle and the Sphere (1997 reprint)", zbMATH, Zbl 0913.51004

External links

A Treatise on the Circle and the Sphere (1916 edition) at the Internet Archive

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