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]