Английская Википедия:A Treatise on the Circle and the Sphere
Шаблон: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.Шаблон:R The Chelsea Publishing Company published a corrected reprint in 1971,Шаблон:R and after the American Mathematical Society acquired Chelsea Publishing it was reprinted again in 1997.Шаблон: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 <math>a, b, c, d</math> describing the circle in the complex plane as the solutions to the equation <math>az\bar z+bz+c\bar z+d=0</math>. 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".Шаблон:R
Other topics described in the book include:
- Tangent circlesШаблон:R and pencils of circlesШаблон:R
- Steiner chains, rings of circles tangent to two given circlesШаблон:R
- Ptolemy's theorem on the sides and diagonals of quadrilaterals inscribed in circlesШаблон:R
- Triangle geometry, and circles associated with triangles, including the nine-point circle, Brocard circle, and Lemoine circleШаблон: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 triangleШаблон:R
- The work of Wilhelm Fiedler on "cyclography", constructions involving circles and spheresШаблон:R
- The Mohr–Mascheroni theorem, that in straightedge and compass constructions, it is possible to use only the compassШаблон:R
- Laguerre transformations, analogues of Möbius transformations for oriented projective geometryШаблон:R
- Dupin cyclides, shapes obtained from cylinders and tori by inversionШаблон:R
Legacy
At the time of its original publication this book was called encyclopedic,Шаблон:R and "likely to become and remain the standard for a long period".Шаблон:R It has since been called a classic,Шаблон: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.Шаблон: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".Шаблон:R
References
External links
- A Treatise on the Circle and the Sphere (1916 edition) at the Internet Archive
