This book arrived during the last week of classes at Iowa State University, just as I was finishing up a two-semester senior-level geometry sequence, attended mostly by mathematics majors, many (but not all) of whom were planning to go on to teach secondary school mathematics. Because I had not taught this sequence before, and because the syllabus was fairly flexible, I had a number of decisions to make before teaching the class.

Some of these decisions were easy: I knew from the outset, for example, that I wanted to do Euclidean geometry the first semester (including topics that might be considered “advanced”, such as the nine-point circle and the theorems of Ceva and Menelaus) and then, in the second semester, talk in more detail about foundational questions and introduce the students to non-Euclidean geometry. Other decisions, though, were much more difficult.

One of the issues on which I was most conflicted was the question of how much of a formal axiomatic development of Euclidean geometry should be done. On the one hand, I am very sympathetic to the idea that students, particularly future high school students, should see a rigorous development of Euclidean geometry. Such an approach seems very valuable and also provides good practice in writing proofs. On the other hand, a formal development like this, if done in a rigorous and intellectually honest way, has some pedagogical drawbacks. It is quite time-consuming, for one thing, and, more seriously, much of that time is spent on proofs of results that the students think of as fairly obvious to begin with. This can be viewed as a learning experience, of course, but I have found that it requires some real degree of mathematical sophistication on the part of students to see why these “obvious” results even need proof in the first place.

In the end, I compromised. At the beginning of the first semester (the text for which was Isaacs’ *Geometry for College Students*) I talked a little bit about Euclid’s *Elements*, but then told the students that we would not make a fetish about a rigorous axiomatic development but would instead allow reasonable reliance on diagrams for what most people would accept as “obvious” conclusions. (For example, we did not require formal proof of the fact that in triangle ABC, the angle bisector of angle A intersected BC between B and C; for that matter, we didn’t even formalize the notion of what “between B and C” means.) Using this “naïve” approach, we went on to prove results about Euclidean geometry that went beyond what is currently done in high school, including, for example, the various triangle centers, the nine-point circle, the theorems of Menelaus and Ceva, and straightedge and compass constructions from a fairly serious viewpoint, including issues such as the classical impossible constructions.

For the second semester, I followed (but not terribly closely) the book from which I learned this material 40 years ago and which was, I was pleased to see, still in print: *Basic Concepts of Geometry* by Prenowitz and Jordan. Here again I effected a compromise: I began the course by taking a careful look at the beginning of the *Elements *and pointed out why it did not meet current mathematical standards of rigor. This entailed a serious examination of axiom systems in general, after which, to give the students a taste of modern axiomatic reasoning, I talked about incidence geometry, giving models and proving theorems.

I then pointed out to the students that additional axioms covering other aspects of geometry (betweenness, line separation, etc.) could be introduced, but that we would not take the time to do so, and instead I just wrote down a lot of very basic results (vertical angles are equal, the basic congruence criteria, etc.) from “neutral” geometry (i.e., all of the axioms of Euclidean geometry except the Fifth postulate or anything equivalent to it), all of which we accepted without detailed proof, and proceeded from there to prove less familiar results, with the goal being to show the significance that the parallel postulate played in “ordinary” Euclidean geometry. We then talked about the history of the parallel postulate and the various attempts to prove it from the other axioms, and how mathematicians eventually realized (via the use of models) that adopting a different parallel postulate (the hyperbolic one) led to a consistent mathematical theory with unusual theorems. I enjoyed both courses and think most of my students did too, but by the end of the second semester I was still conflicted about whether I had made the right choice.

And that brings me to the book under review. Jack Lee’s *Axiomatic Geometry*, devoted primarily (but not exclusively) to a rigorous axiomatic development of Euclidean geometry, is an ideal book for the kind of course I reluctantly decided not to teach. It is beautifully and carefully written, very well organized, and contains lots of examples and homework exercises. The author deals honestly with the issues raised by a formal axiomatic account of Euclidean geometry and does not take refuge in vague, imprecise axioms; sometimes the author omits a proof as being too hard for the book, but when he does so, he makes it clear what is being skipped and why, and provides a reference for a proof.

The first two chapters of the text look at Euclid’s *Elements* and incidence geometry (as a detailed example of an axiom system). These chapters are about as good an introduction to these topics as anything I have seen in print. The second chapter, in particular, is a lovely introduction to the nature of axiomatic reasoning (and the role of models) and also an excellent primer on how to do proofs. (A couple of appendices also address basic logic and proof techniques; these are also very well written and complement these chapters nicely.) Lee practically takes the reader by the hand and leads him or her through the construction of simple proofs in incidence geometry, which is an excellent example of an axiom system, since the axioms involved are simple and intuitively easy to follow and there are a lot of interesting models, a number of which are discussed.

Having explained why the *Elements *is not currently considered a rigorous example of axiomatic reasoning, the author then begins (in chapter 3) to give a formal axiomatic development of (plane) Euclidean geometry, a task that takes most of the rest of the book. The particular axioms chosen by the author, a modification of those used by the School Mathematics Study Group, are based on the real numbers; these axioms are introduced a few at a time over the course of chapters 3 through 5, with the consequences of each new set of axioms being explored before moving on to others. One axiom that is consciously not introduced in these three chapters is the Euclidean parallel postulate (that will come later), so that what is really defined at this point is neutral, not Euclidean, geometry; i.e., it is the geometry we get by remaining neutral about a parallel postulate and not assuming any particular one. (Some books call this absolute geometry, but the more recent ones use the term “neutral”, a term that, I believe, derives from the aforementioned text by Prenowitz and Jordan.)

In chapter 6, the author takes a break from proving consequences of axioms and discusses some basic models of neutral geometry, particularly the Cartesian model of Euclidean geometry and the Poincare disc model of hyperbolic geometry. Some “nonmodels” of neutral geometry are also presented, the most interesting of which is the fairly well-known “taxicab geometry”.

After the discussion of models in chapter 6, the author resumes the study of neutral geometry, looking carefully at perpendicularity and parallelism (chapter 7), polygons (chapter 8) and quadrilaterals (chapter 9). By the end of chapter 9, therefore, the reader has seen a careful and meticulous development of neutral geometry. It is a clearly written account, but does (inevitably) suffer from the pedagogical problems alluded to earlier: a fair amount of time is spent proving things that seem fairly obvious, and sometimes these proofs can be somewhat dry and technical. (As a case in point, it takes the author about a page of text to prove that if three rays emanate from a single point, the sum of the three angles formed with this point as vertex is 360 degrees.) I don’t intend this as a criticism of the text: it seems to me to be something that is inherent in any book devoted to this kind of undertaking; a problem that can be avoided only by cutting corners, which is not desirable in a book that intends to teach geometry axiomatically.

At the end of chapter 9, the reader of the book has a choice of continuing down two paths, one considerably longer than the other. The longer path consists of chapters 10 through 16 and entails a detailed study of Euclidean geometry. The Euclidean parallel postulate is introduced as an axiom in chapter 10 and its consequences are developed in some detail in chapters 11 through 16, which cover, in order, the Euclidean theory of area for triangles and parallelograms, similarity (a very nice chapter, which includes not only proofs of the theorems of Ceva and Menelaus but also a good discussion of the golden ratio), right triangles (a nice feature here is a sketch of a proof, not readily found in the elementary textbook literature, that any two models of Euclidean geometry are isomorphic), circles (including central and inscribed angles, and inscribed and circumscribed polygons), circular area (including a discussion of the definition of \(\pi\)), and constructions with straightedge and compass.

This last chapter should be singled out for special mention. It is of course not uncommon for a book on Euclidean geometry to discuss this topic, but this book’s treatment struck me as unusually comprehensive. Not only does the author go through the basic constructions (leaving many as exercises), but he also spends considerable time on the classical impossibilities (trisecting an angle, for example), even to the point of giving algebraic proofs that are often skipped in comparable books. The necessary and sufficient conditions for the construction of the regular *n*-gon are also carefully stated, but here the proof is skipped: it is, in the author’s words, “X-rated”. (Professor Lee adopts an amusing rating system for difficulty of proofs, which range from the G-rated proofs that are simple and straightforward, through the PG and R rated ones, up to the X-rated ones that are simply too difficult to include in the book.)

The other path that a reader can take from chapter 9 is to proceed directly to chapters 17 through 19. Chapter 17 talks about statements, in neutral geometry, that are equivalent to Euclid’s parallel postulate, thus clarifying the role of that postulate in Euclidean geometry. It is proved, for example, that any neutral geometry is either Euclidean or hyperbolic. The next two chapters (18 and 19) are on hyperbolic geometry and explore the consequences of assuming the hyperbolic parallel postulate, rather than the Euclidean one, as an axiom. It occurred to me as I read these chapters that chapters 1 through 9, followed by 17 through 19, provided a pretty good approximation to my own second semester of geometry as described earlier, though to save time in my course I did not develop neutral geometry as rigorously as does this text. I’m not sure that in a single semester it would be possible to cover all these 12 chapters; I suppose that one could use this text for a one-semester foundations course by cutting a few corners on the rigorous development (covering chapters 1 and 2, and 17 through 19, carefully, but fudging a bit on the development in chapters 3 through 9), but given the fact that an axiomatic development is this book’s *raison d’etre*, I suspect that the author might not approve of this approach.

In addition to all of the preceding, there is a final chapter (chapter 20) that surveys without proof some additional topics in geometry, including, for example, area in hyperbolic geometry, differential geometry, and geometric transformations. No proofs are given here, but the author does provide references for further reading on these topics.

As the above description of the book should make clear, a lot of material is covered. Certainly there is enough for a two-semester course, and that, perhaps, raises a potential problem for those instructors who teach at a university that only offers one semester of geometry. I think that either of the two paths through the book that I described above (chapters 1 through 16, or chapters 1 through 9 and then 17 through 19) probably contain more material than can be comfortably done in a single semester, and it does seem something of a pity to start an axiomatic development of geometry and not see it through to some reasonable conclusion. Perhaps, however, by judicious picking and choosing an instructor could chart a reasonable one-semester path through the material; the author does give some suggestions as to topics that can be omitted. I would certainly hope so, because this book has a great deal to recommend it; it is, as I noted earlier, very nicely written with the needs of the student front and center at all times. At the very least, it makes for an excellent reference for people wanting to see a careful development of Euclidean geometry, and definitely belongs on the shelf of any good college or university library.

Are there things that I wish had been done differently? Sure; it is a rare reviewer who will answer this question “no”. I thought, for example, that a disproportionate number of the homework exercises were of the “prove theorem 14.7” variety. I also would have liked to have seen a more extended discussion of the Poincaré half-plane model of hyperbolic geometry, particularly given that model’s importance in other areas of mathematics. And, although hyperbolic area theory is discussed briefly in chapter 20, I would have liked to have seen a more expanded account of this topic as well, and would have also liked to see more of a discussion of elliptic geometry. (The author does say that he intends to supplement the text with additional material on the book’s website; perhaps these are some things that may appear there.)

Competition for this text include books like Gerard Venema’s *Foundations of Geometry* and Marvin Greenberg’s *Euclidean and non-Euclidean Geometries: Development and History.* The book under review, however, is readily distinguishable from both of these. For one thing, both Greenberg and Venema place more of an emphasis on non-Euclidean geometry than does Lee. The book by Greenberg is also somewhat more sophisticated and demanding than is this book and may, in fact, be a bit too demanding for the average college student (though it does contain what I believe to be an unparalleled discussion of the history of non- Euclidean geometry, and really gives the reader a sense of the excitement of this huge revolution in mathematical thought.)

In summary, this is an interesting and valuable book. Apart from its obvious value as a reference to anybody who is interested in geometry, it offers food for thought to any prospective teacher of such a course, and is making me rethink the approach I used in my own two-semester sequence.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University. The junior-level foundations of geometry course that he took in 1970 is without question the most vivid and exciting mathematics course that he ever took.