A Treatise on the Circle and the Sphere

Template:Short description Template:Italic title 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.Template:R The Chelsea Publishing Company published a corrected reprint in 1971,Template:R and after the American Mathematical Society acquired Chelsea Publishing it was reprinted again in 1997.Template:R

Topics

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 $a,b,c,d$ describing the circle in the complex plane as the solutions to the equation $az\overline{z}+bz+c\overline{z}+d=0$. 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".Template:R

Other topics described in the book include:

• Tangent circlesTemplate:R and pencils of circlesTemplate:R
• Steiner chains, rings of circles tangent to two given circlesTemplate:R
• Ptolemy's theorem on the sides and diagonals of quadrilaterals inscribed in circlesTemplate:R
• Triangle geometry, and circles associated with triangles, including the nine-point circle, Brocard circle, and Lemoine circleTemplate:R
• 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 triangleTemplate:R
• The work of Wilhelm Fiedler on "cyclography", constructions involving circles and spheresTemplate:R
• The Mohr–Mascheroni theorem, that in straightedge and compass constructions, it is possible to use only the compassTemplate:R
• Laguerre transformations, analogues of Möbius transformations for oriented projective geometryTemplate:R
• Dupin cyclides, shapes obtained from cylinders and tori by inversionTemplate:R

Legacy

At the time of its original publication this book was called encyclopedic,Template:R and "likely to become and remain the standard for a long period".Template:R It has since been called a classic,Template:R in part because of its unification of aspects of the subject previously studied separately in synthetic geometry, analytic geometry, projective geometry, and differential geometry.Template:R 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".Template:R

External links

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