This book, a Dover reprint of a text first published in 1974, is a text for a first, fairly low-level, college course in geometry. The emphasis is on axiomatic development of the basic facts of Euclidean plane geometry, with some attention paid to solid geometry and coordinate geometry at the end. Non-Euclidean geometry is not covered.

The preface speaks of the fact that many college geometry courses are “swifter version[s] of a high school geometry course” and therefore miss the “opportunity to exploit the fuller measure of maturity and sophistication possessed by college students.” This sounds promising, but unfortunately the goal of exploiting this opportunity does not appear to be fully realized in this book. The level of material that is covered here rarely rises, either in substance or in tone, much above the high school level. Topics that one might refer to as “advanced Euclidean geometry” and that are not generally seen in high school (such as the theorems of Menelaus and Ceva, the nine-point circle, Morley’s theorem, the Euler line, Miguel’s theorem, etc.) are not mentioned, and many proofs in the book appear in two or three column format, as in high school.

The first and most basic theorem of Euclidean geometry — that if two distinct lines intersect, they do so in exactly one point — is stated early on but not proved until page 116, because the author defers any discussion of the concept of indirect proofs until that point. The chapter on analytic geometry, likewise, does not contain any results that would not be familiar to a good high school student (e.g., two oblique lines are perpendicular if and only if the product of their slopes is –1.) Finally, although the book has many exercises, they are all on the easy end of the spectrum for a college textbook.

True, this book does develop Euclidean geometry more rigorously than might be typically done in a regular high school course, but it does so in a very elementary way, at a level that would not be out of line in, say, an honors high school course. The idea of presenting a fairly precise development of Euclidean geometry to high school students is not new. The 1958 books by the School Mathematics Study Group, for example, invoke a pedagogically-friendly set of axioms (a deliberately non-independent set) and do geometry fairly carefully from them, but these books are intended for a secondary school audience. (These books can be found online at http://ceure.buffalostate.edu/~dltl/NewMath/.)

The book under review, in fact, strikes me as being somewhat on the same level as the SMSG texts. It begins with a chapter on basic logic (truth tables, converse and contrapositive of a statement, etc.) and then proceeds in the second chapter to set out various axioms (called postulates here). Over the course of the next seven chapters, the book addresses basic incidence properties, angles, triangles, parallel lines, similarity, polygons and area, and circles. Although the postulates, as stated, allow for the possibility of multiple planes, most or all of what takes place here is plane geometry. Chapter 9, however, is on “space geometry”, and the last chapter, chapter 10, is on coordinate geometry. The topics covered in these ten chapters are basically those that are covered in the two SMSG volumes. Very little, if anything, beyond the high school level is proved in these chapters.

The axioms that are used here are also similar to the ones used in the SMSG volumes. A few words of elaboration are perhaps appropriate. In 1899, David Hilbert gave a fairly lengthy set of axioms for Euclidean geometry based on the concepts of incidence, order, congruence, continuity, and parallelism. More than thirty years later, Birkhoff gave a much shorter set of axioms; he was able to cut down the number by exploiting properties of the real numbers and postulating both a “ruler axiom” (establishing a bijection between the points on a line and the set of real numbers), and an axiom for the measure of an angle. (Interestingly, Birkhoff did not postulate a Euclidean parallel postulate, as did Hilbert, but he was able to prove that fact as a theorem from his remaining axioms.) The SMSG axiom system is a hybrid of the Hilbert and Birkhoff approaches, and tries to avoid a pedagogical problem that looms large in any rigorous axiomatic development of Euclidean geometry: namely, that some “obvious” results may require long and tedious proofs, the need for which may not be apparent to beginning students. The SMSG axioms avoid much of these problems by deliberate redundancy, asserting things as axioms that could actually, albeit with some effort, be proved as theorems. The axiom system used in this book, which also combines features of Hilbert’s and Birkhoff’s axioms in a way very similar to the SMSG axioms, also has this redundancy, but I think that’s entirely appropriate for a text at this level, provided — as is the case here — that the author is honest about things.

A recent text that also rigorously develops Euclidean geometry is Lee’s *Axiomatic Geometry*. Lee’s book, I think, is more suited to a college-level audience than is the text under review. Lee’s book motivates the need for a rigorous development of geometry by looking critically at Euclid’s *Elements *and explaining in some detail why a more formal approach is necessary. It also makes use of the concept of models to illustrate the various axioms as they are introduced, and thereby exposes the student to non-Euclidean geometry as well as Euclidean. The failure of the book under review to do these things seems like a missed opportunity.

I do not mean to suggest, however, that this is, in any sense, a bad book. What it does, it does quite competently; the basic facts of Euclidean geometry are set out in a clear and readable manner. My only concern is whether it does enough to warrant consideration as a text for a geometry course at the college level. Perhaps the appropriate audience might be geometry students at a two-year college, or students at a university that offers a course in geometry that is not intended for majors; the kind of course, for example, for which *College Geometry: A Problem-Solving Approach with Applications *by Musser, Trimpe and Maurer might be suitable. In addition, this book might be a useful text for education majors who aspire to teach geometry at the secondary level.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.