This book was first published in 1970 under the title A Course of Geometry for Colleges and Universities. At the time it appeared the author was part of a group of eminent geometers who worked on the College Geometry Project. This was a program funded by the National Science Foundation with the goal of producing short films for classroom use on specific topics in geometry. Twelve films were produced. Although this happened more than forty years ago, the graphics and animation are remarkably good. The films are still available (now on DVD) from the MAA. Did they boost overall appreciation for classical geometry in the mathematical community? Did they inspire any future mathematicians? I’d like to hope they did.
At about the same time as the film project, the author wrote this book based on a course for advanced undergraduates and first-year graduate students, and also for a special institute for college teachers of geometry. The subject is “elementary geometry”, defined by the author to mean geometry up to, but not including the rigorous study of algebraic curves. It includes projective geometry in two and three dimensions as well as some geometry in n dimensions. As the title indicates, it is a very comprehensive work, as complete a volume on the subject that I’m aware of.
The emphasis here is on the use of algebraic methods to study geometry, and linear algebra in particular is used extensively. The author does use synthetic (or, as he says, Euclidean) geometric proofs on occasion, when he finds them more appealing. Basic elements of linear algebra — the idea of a vector space, solutions of linear homogeneous equations, a little matrix theory — are prerequisites.
After preliminaries and a chapter on the use of vector methods in Euclidean and affine geometry, the author explores the geometry of circles extensively in three long chapters. The focus here is on aspects of the theory of circles important in other areas of geometry. This also includes a detailed study of coaxal systems, the simplest linear system of curves (besides systems of lines). The concept of inversion is also introduced here.
The next two chapters study first mappings of the plane. The first looks at mappings of the Euclidean plane, especially isometries, using the algebra of complex numbers. Some group theory is introduced. Then the following chapter focuses on mappings of the inversive plane — Möbius transformations. Here the author also introduces the Poincaré model of hyperbolic geometry.
Most of the rest of the book is devoted to projective geometry, first the projective plane and projective space, next some results in n dimensions, and then the projective geometry of conics and quadrics. A final chapter offers a prelude to algebraic geometry. There are more than 500 exercises spread throughout the book at a variety of difficulty levels from relatively straightforward to challenging.
There is a great deal of material in this book, so much that an unguided student could easily get lost. The author has chosen comprehensiveness over selectivity, and the book — as a textbook — suffers for that. In addition, while the power of the algebraic approach to geometry is clear, it often seems inelegant. Much of the inherent beauty of the subject just doesn’t come through.
Bill Satzer (firstname.lastname@example.org) 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.