A Treatise on the Circle and the Sphere

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]

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 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:

Legacy

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]

References

  1. Bieberbach, Ludwig, "Review of A Treatise on the Circle and the Sphere (1916 edition)", Jahrbuch über die Fortschritte der Mathematik, JFM 46.0921.02
  2. H. P. H. (December 1916), "Review of A Treatise on the Circle and the Sphere (1916 edition)", The Mathematical Gazette, 8 (126): 338–339, doi:10.2307/3602790, hdl:2027/coo1.ark:/13960/t39z9q113, JSTOR 3602790
  3. Emch, Arnold (June 1917), "Review of A Treatise on the Circle and the Sphere (1916 edition)", The American Mathematical Monthly, 24 (6): 276–279, doi:10.1080/00029890.1917.11998325, JSTOR 2973184
  4. White, H. S. (July 1919), "Circle and sphere geometry (Review of A Treatise on the Circle and the Sphere)", Bulletin of the American Mathematical Society, American Mathematical Society ({AMS}), 25 (10): 464–468, doi:10.1090/s0002-9904-1919-03230-3
  5. "Review of A Treatise on the Circle and the Sphere (1971 reprint)", Mathematical Reviews, MR 0389515
  6. Peak, Philip (May 1974), "Review of A Treatise on the Circle and the Sphere (1971 reprint)", The Mathematics Teacher, 67 (5): 445, JSTOR 27959760
  7. Steinke, G. F., "Review of A Treatise on the Circle and the Sphere (1997 reprint)", zbMATH, Zbl 0913.51004
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