This introductory text from 1970 offers two broad paths by which to discover plane projective geometry. A profusion of exercises (over 800) are present, with answers to the odd-numbered ones in the back. Combine this with thorough presentation of preliminary material for chapters well-seasoned with worked proofs and examples and you have a self-contained book that is ideal for self-study. Alternatively, either of the two major portions of the book could serve as the text for an undergraduate course.

Initially, the reader encounters a historical review of the development of elementary ideas of three-dimensional perspective geometry and the development of basic plane perspective geometry through such steps as the image and object planes suggested by rabattement. Then, the first major portion of the book occurs over three chapters. This is a development of analytic projective geometry by extending planar Euclidean geometry through such ideas as *ideal* points and lines. In order to provide the underpinnings to conics, cross ratios, perspectivities, and more this portion includes the basics of matrix theory and linear algebra. This analytical approach concludes with linear transformations and an introduction to group theory that supports investigation of the projective group and its subgroups. This portion includes such fundamentals of projective geometry as the Theorems of Desargues and Pappus.

The next three chapters are the second major portion of the book. This portion presents the same ideas as the first, but through an axiomatic approach. Augmenting this portion is an introduction to field theory which allows, among other topics, the discussion of coordinatization. The final two chapters introduce the concept of a metric into the projective plane in order to obtain elliptic and hyperbolic geometries. (This was nteresting to me. Based on earlier chapters, I would have expected an introduction to measure theory in order to motivate the definition of point-to-point distance as a theoretically sound starting point.)

Adding to this complete, two-part introduction to projective geometry is one appendix on determinant fundamentals and another supporting investigation of an alternative geometry for which the Theorem of Desargues does not hold. This makes for an intriguing and illuminating conclusion to a classic text.

Tom Schulte engineers software and survives in metro Detroit as an adjunct professor in the Oakland Community College mathematics department.