Coxeter's Non-Euclidean Geometry begins with a wonderful historical overview of the development of non-Euclidean geometry in the first chapter. Only a few proofs are given or sketched in this chapter. They flow with the prose and play an integral part in the understanding of the beginnings of hyperbolic, spherical, elliptic and differential geometry, among others. The mathematician, as well as the non-mathematician, is able to gain insights into these various types of geometry by the end of this chapter.
For the remainder of the book, the historical aspects are never far from the mathematics. Many chapters begin with an introduction which puts the contents of the chapter into historical perspective. Coxeter uses comparisons, especially to Euclidean geometry, to aid in the understanding of other geometries. An example is found in chapter 8 where descriptive geometry (described as "high school geometry with congruence and parallelism left out") is compared to projective geometry.
Coxeter takes us through the developments of real projective geometry, elliptic geometry in one, two and three dimensions, descriptive geometry and hyperbolic geometry in one and two dimensions. Though he introduces the elliptic metric "by means of absolute polarity" and could have introduced the hyperbolic metric in a similar fashion, he chose to "reverse the process" in order to "follow the historical development more closely". This not only gives the reader a different perspective, but once again reminds us of the historical perspective.
Chapters eleven through thirteen give us a comparative look at the geometries, by looking at circles, triangles and areas in elliptic and hyperbolic geometry, among other things. Chapter fourteen gives an analogy between ellipses and hyperbolas in Euclidean space, and elliptic an hyperbolic geometry in terms of "polarity".
Though this would not be an easy book for the undergraduate, one can get a view of the field by reading the first chapter and the last two chapters. Non-Euclidean Geometry gives both an overview and an in-depth treatment of geometry.
Robert Stolz (email@example.com) is associate professor of mathematics at the University of the Virgin Islands. He works in the area of probability and functional analysis and is currently looking at applications to financial mathematics.