The 19th century was an extraordinary time in the history of mathematics. This was a period that saw not only major developments in existing areas, but also the creation of entirely new branches of mathematics. Geometry, in particular, provides a good illustration of this remark. By way of example, consider the following 19th century geometric developments:

- The creation of non-Euclidean geometry by Bolyai, Lobachevsky and Gauss
- Felix Klein’s
*Erlanger Programm*, which emphasized the transformation-theoretic point of view in discussing geometries
- An increased emphasis on the foundations of the subject, including, in 1899, the publication of Hilbert’s
*Foundations of Geometry*, an axiomatic development of Euclidean Geometry

The book now under review touches on all three of these issues. This is a rigorous axiomatic development of Euclidean geometry, very much in the Hilbert mode (the authors describe their book as a “completion, updating and expansion of the core of Hilbert’s book”) but with an *Erlanger*-like viewpoint that exploits geometric transformations in the axioms themselves. And non-Euclidean geometry, though not discussed in any detail, is at least referred to, particularly when it comes to the selection of a parallel postulate as an axiom.

It is this book’s *Erlanger*-like viewpoint that constitutes the major distinction between the approach used here and the approach used by Hilbert. Hilbert’s axioms can be divided into groups: incidence, betweenness, congruence, plane separation, continuity and a parallel postulate. This text also groups together axioms, but here congruence axioms are replaced by ones related to the existence of a sufficient quantity of reflections.

Using axioms based on geometric transformations is certainly not new; it dates back at least as far as the 1950s and the work of Friedrich Bachmann. However, although Bachmann’s approach is referenced in Hartshorne’s *Geometry: Euclid and Beyond*, it is not easy to find textbooks that use reflections as part of an axiomatic system. Prior to looking at this book, the only one that I was aware of that utilized an approach like this was Ewald’s *Geometry: An Introduction*, a book that I had thought was long out of print, but which, according to the bibliography in the book under review, seems to have been reissued in 2013 by Ishi Press.

As mentioned earlier, the book under review develops the axioms for Euclidean geometry in batches, with the “subgeometries” of the title being those geometries that satisfy some axioms but not necessarily all. So, for example, geometries satisfying the basic incidence axioms are *incidence geometries*; add Playfair’s postulate (which states that given a point P not on a line \(\mathcal{L}\), there is exactly one line through P that is parallel to \(\mathcal{L}\)) and we have *affine geometry*. The first three chapters of the book discuss these geometries and their collineations, i.e., the transformations which preserve incidence.

After these chapters, Playfair’s Postulate (called Axiom PS in the text; the “S” stands for “strong”, to distinguish it from a weaker version) is put to one side; it won’t reappear until a couple of hundred pages later. The reason is that the authors want to develop *neutral geometry*, which can best be described as “geometry with no parallel axiom”. Studying neutral geometry is a great idea in a foundations of geometry course, because it clarifies the significance of the parallel postulate; it’s very interesting to see what familiar theorems of Euclidean geometry remain true in neutral geometry, and what ones require some modification. (Example: while, in Euclidean geometry, it is well known that two lines are parallel if and only if they cut off equal alternate interior angles relative to a transversal, only the “if” part of this statement is true in neutral geometry.)

Neutral geometry is, as noted above, developed in stages. Betweenness axioms are introduced, thus leading to* Incidence-Betweeness Geometry*. After this, in chapter 5, a plane separation axiom (equivalent to Pasch’s postulate) is introduced, leading to the notion of a *Pasch geometry*. Aspects of these geometries are also studied in chapters 6 and 7.

In chapter 8, neutral geometry is officially introduced. As noted earlier, in lieu of an axiom discussing congruence, the authors use the notion of reflections, defined by means of certain geometric properties that they satisfy, and introduce a postulate that, loosely speaking, guarantees that there are enough reflections in the plane.

Aspects of neutral geometry are further developed in the next two chapters, and then, in chapter 11 (about halfway through the text), axiom PS is brought back into service, resulting in Euclidean geometry, which is then studied in considerable depth over the course of the next 250 pages or so. Not only are geometric transformations (isometries, similarities, dilations) studied in detail, but it is also explained how a line in the plane can be identified with the real numbers, thereby allowing a coordinatization of the Euclidean plane.

As the summary above indicates, this book covers a lot of material — much more material, in fact, than could ever be covered in a single semester (or, I suspect, even two semesters). Thus, an instructor teaching a course in the foundations of geometry using this book will have to make hard choices about what to cover and what to omit. One proposed syllabus suggested by the authors is to cover the first five chapters in detail. While this would give the students a good rigorous introduction to the axiomatic method in geometry (albeit with less emphasis on models than I would like, a point I’ll return to later), it would not, I think, give them much of a payoff; they would never even see neutral geometry, much less a complete set of axioms for Euclidean geometry. Alternatively, the authors point out, some of the material in chapters 4 and 5 could just be summarized, and then the class can move directly to chapter 8 on neutral geometry. This approach, too, suffers from the defect of not giving the student a full set of axioms for the Euclidean plane, and it seems a pity to start an undertaking like this without seeing it through to its conclusion.

The authors do a commendable job of writing out proofs in detail and attempting to make the text accessible to undergraduates. However, there is an inherent pedagogical problem in undertakings like this one: if you do things correctly, the details can occasionally get fairly technical and tedious. Things that most students view as “obvious” and would, for that reason, just be inclined to take for granted, must, unless you choose to cut corners, be proved in detail, and the resulting proofs are sometimes fairly difficult and intricate. The authors recognize this fact, and have chosen to not cut corners; this is an honest development of the subject. This is admirable, and increases the value of this book as a reference, but a potential downside to this approach is that the attendant pedagogical issues referred to in the previous paragraph are, of necessity, present.

It’s not only the mathematical proof details that can cause some consternation to students. The authors are quite fond of acronyms and use them often: every definition and theorem, and all but one chapter title, is given an acronym. Reading through a list of words like COBE, MMI, ORD, QX, and DLN sometimes made me feel as though I were reading a list of airport abbreviations. This also led to statements that might be heavy-going for a student, such as this one, taken verbatim from the text: “It should be noted that in the presence of property PE (which was proved as Theorem NEUT.48(B)), Axiom PW is equivalent to Axiom PS.”

Similar potential problems occur with notation. The authors use the fairly standard notation \(\overset{\leftrightarrow}{AB}\) to denote the unique line containing the distinct points \(A\) and \(B\), but, in addition to lines, they also use, and have separate (but similar) notation for, open segments, closed segments, half-open segments, open rays, and closed rays; in each case the double arrow on top of the \(AB\) is replaced by a different symbol. The notation can occasionally make a reader somewhat starry-eyed, a point the authors themselves are apparently aware of; on page 230, for example, they refer to one equation, with notation on top of notation, as a “rather odd-looking equality”.

I’m not sure I can think of any way to avoid the issues discussed above and still present a rigorous, honest account of the subject, but, here again, prospective readers should nonetheless at least note that these issues are present. As a result, “drop in” reading of the book may not be very successful.

One other concern that I have, that I think the authors *could *have done something about, involves the use of models. When discussing axiomatic geometry, I like to introduce models early and use them often. Unfortunately, with the exception of one very simple finite model of incidence geometry that is discussed very briefly in chapter 1, this text delays the introduction of models until the very last chapter. Particularly since the authors introduce axioms in stages, I think it would have been valuable to introduce models early on and then see, with each new batch of axioms, which ones have to be discarded.

In view of the sheer mass of material covered in this text, and the technical detail in which it is presented, I would, if I were to teach a foundations of geometry course by doing a rigorous development of Euclidean geometry (something I have always resisted doing, precisely because of the pedagogical difficulties referred to earlier), probably use a book like Lee’s *Axiomatic Geometry* instead of this book. Lee is not as persnickety as are the authors of this text, and occasionally skips the proofs of certain results altogether, but his book does offer a good look at the axiomatic method, complete with an earlier and better look at the use of models, and allows the student to see, in a semester, a fairly complete development of the subject. It also contains more information on non-Euclidean geometry than is found in this book.

Though I probably wouldn’t select the book under review as a text, I’m certainly glad I own it. It makes a very useful reference source, and, as mentioned previously, there aren’t very many current textbooks that discuss geometry from this particular point of view. I commend this book to the attention of instructors with an interest in the foundations of geometry, and to university librarians.

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