Non-Euclidean geometry. 6th ed.

*(English)*Zbl 0909.51003
Spectrum Series. Washington, DC: The Mathematical Association of America. xviii, 336 p. (1998).

First published in (1942; Zbl 0060.32807), H. S. M. Coxeter’s book on ‘Non-Euclidean Geometry’ was and still is one of the nicest and most readable introductions to elliptic and hyperbolic geometry. In this sixth edition some small errors are corrected and a new section (15.9) is added to the last chapter.

Starting off with real projective geometry, von Staudt-Hessenberg’s method for introducing coordinates and cross-ratios in a non-metric way is used in order to introduce elliptic geometry (Chapters V to VII). In Chapter VIII Coxeter uses a more general concept called descriptive geometry as a common basis for both hyperbolic and Euclidean geometry, which is treated in Chapters IX and X. In the three succeeding chapters notions such as distance, angle and area as well as circles and triangles are introduced in both non-Euclidean geometries. Chapter XIV contains several Euclidean models for hyperbolic and elliptic geometry. The new section (15.9) intitled Inverse distance and the angle of parallelism incorporates H. S. M. Coxeter’s papers in Aequationes Math. 1, 104-121 (1968; Zbl 0159.22302), and in Ann. Math. Pura Appl., IV. Ser. 71, 73-83 (1966; Zbl 0146.16303). Unfortunately, some of the references of this section [namely, S. L. Greitzer and H. S. M. Coxeter, Geometry revisited (1967; Zbl 0166.16402), and Mirsky] are missing from the bibliography.

Starting off with real projective geometry, von Staudt-Hessenberg’s method for introducing coordinates and cross-ratios in a non-metric way is used in order to introduce elliptic geometry (Chapters V to VII). In Chapter VIII Coxeter uses a more general concept called descriptive geometry as a common basis for both hyperbolic and Euclidean geometry, which is treated in Chapters IX and X. In the three succeeding chapters notions such as distance, angle and area as well as circles and triangles are introduced in both non-Euclidean geometries. Chapter XIV contains several Euclidean models for hyperbolic and elliptic geometry. The new section (15.9) intitled Inverse distance and the angle of parallelism incorporates H. S. M. Coxeter’s papers in Aequationes Math. 1, 104-121 (1968; Zbl 0159.22302), and in Ann. Math. Pura Appl., IV. Ser. 71, 73-83 (1966; Zbl 0146.16303). Unfortunately, some of the references of this section [namely, S. L. Greitzer and H. S. M. Coxeter, Geometry revisited (1967; Zbl 0166.16402), and Mirsky] are missing from the bibliography.

Reviewer: R.Bödi (Tübingen)