A Treatise on the Circle and the Sphere

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 ${\displaystyle a,b,c,d}$ describing the circle in the complex plane as the solutions to the equation ${\displaystyle az{\bar {z}}+bz+c{\bar {z}}+d=0}$. 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]

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