Fundamental Concepts of Geometry

First published in 1955, this book has become a sort of touchstone for college geometry courses, particularly those that are aimed in part at prospective high school teachers. It carries some signs of its age, more in perspective than in treatment of its subject. The author suggests that the course is most appropriate for students with the equivalent of one or preferably two years of college mathematics. One of those years would include something like a college algebra course of the sort no longer commonly taught. No calculus background is assumed.

The subject is mostly classical geometry, broadly considered. Euclidean geometry has a place, but it doesn’t show up until more than halfway through the book. The author chooses to start — after a short discussion of preliminaries — with synthetic projective geometry, followed by analytic projective geometry. A chapter on coordinate systems is sandwiched in between. Then the author takes up affine geometry. This is followed by treatments of Euclidean and non-Euclidean geometry that are accompanied by a historical perspective on their development. The book then concludes with a brief tour of elementary topics in topology.

The author views his subject as a hierarchy. It begins with topology at the top with projective geometry immediately below, bifurcating into affine geometry on one side, and non-Euclidean geometry on the other. Euclidean geometry sits at the bottom, being just one kind of affine geometry. With the exception of topology, the organization of the book follows the same pattern. This seems odd to me because, from most students’ perspective, Euclidean geometry is the most familiar, and thus perhaps the natural place to start exploring other geometries. While the author’s approach might well appeal to stronger students, the less able are more likely to struggle if they have to start with a formal deductive system based on the unfamiliar postulates of projective geometry.

The other notable aspect of the book is its level of exposition. While it might have been written with prospective high school teachers in mind, it makes few concessions to the less sophisticated. It is not an easy book to read, and few of the exercises are straightforward. However it might have been used in the past, it is hard to imagine using this now in a class for future high school teachers. There is wonderful geometry here, though better suited to the more mathematically mature.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.