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 describing the circle in the complex plane as the solutions to the equation . 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:
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]