A Treatise on the Circle and the Sphere

A Treatise on the Circle and the Sphere

Author	Julian Coolidge
Publication date	1916

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 ${\displaystyle a,b,c,d}$ 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]

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, 25 (10), American Mathematical Society ({AMS}): 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

External links

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