College-level courses in elementary geometry come in several flavors. One type is a “foundations” course that critically examines Euclid’s approach to Euclidean geometry, discusses the role of the Euclidean parallel postulate, examines the possibility of non-Euclidean geometries, and so forth. Texts for a course like this include Greenberg, *Euclidean and Non-Euclidean Geometries: Development and History* or *Roads to Geometry* by Wallace and West. Another approach to geometry is to emphasize the transformation point of view, as in *Continuous Symmetry* by Barker and Howe, Martin’s *Transformation Geometry* or Johnson’s *Symmetries*. Closely related to the transformation point of view is the linear-algebraic approach to geometry, as in the recent book by Tarrida, *Affine Maps, Euclidean Motions and Quadrics*, or Snapper and Troyer’s *Metric Affine Geometry*. Some books are essentially completely devoted to non-Euclidean geometries, such as Anderson’s *Hyperbolic Geometry* or Casse’s *Projective Geometry*, and then there are those (such as Stillwell’s *Four Pillars of Geometry*) that combine approaches and do a little of each.

And, of course, there is the “naïve” approach to geometry, doing it in the spirit of high school geometry but covering more advanced topics not covered in the high school curriculum. (I hasten to point out that the use of the word “naïve” is in no sense intended to be pejorative but is instead a nod to Halmos’ famous book *Naïve Set Theory*.) Isaacs’s book falls firmly in this category. It makes no attempt to discuss non-Euclidean geometries or models or to build up a completely rigorous, axiomatic approach to Euclidean geometry. Isaacs feels no need to rigorously prove, for example, that the bisector of one angle in a triangle meets the line opposite this angle at some point between the two other vertices; he is content to simply rely on a picture and our geometric intuition that this particular line can’t suddenly swerve outside the triangle before getting to the opposite side. (At times, more serious reliance is placed on intuition: the discussion on pages 64–65 of the text, for example, uses without rigorous proof certain properties of a geometric transformation defined as the composition of a scaling transformation and rotation. There are also occasional continuity arguments and arguments based on physical reasoning.)

One could generate an interesting debate as to what the “right” way to do geometry in college is. My own view is that college students, particularly math majors or students planning to teach high school geometry, should see some discussion of the foundations of the subject and be advised that non-Euclidean geometries exist, are logically consistent, and are actually occasionally useful. I confess that this view may be shaped by personal history: I took such a course in college that was taught by the best professor I have ever had, and it, more than any other course, made me understand the nature of modern mathematical reasoning. Isaacs himself seems to be a bit troubled about having students reason occasionally from diagrams, since the very first section of the text (“Introduction and Apology”) acknowledges some concerns about this. In any case, a respectable argument can be made that a very formalistic approach to geometry, including the proofs of intuitively obvious results such as the one mentioned in the preceding paragraph, bores students who simply can’t see the need for such detail.

One thing, however, seems quite clear to me: if you do teach a “naïve” course as described above, either by personal preference or because of the demands of an inflexible syllabus, I can think of no better book to use as a text than this one, originally published by Brooks Cole and now published (apparently without any change in content) by the AMS.

The strongest feature of this book is the writing style of the author. Isaacs clearly cares about language, and writes in an elegant, concise way that I have always found to be particularly appealing and a pleasure to read. I think, for example, that his graduate algebra text *Algebra: A Graduate Course* is a real gem, and his recent book on finite group theory has garnered praise as well, including right here on *MAA** Reviews*. It is remarkable that Isaacs can write for a fairly sophisticated audience (as in these two books and his book on character theory) and also write with the same beautiful style for a much less advanced audience. Any math major at the sophomore level or above should be able to read this book on his or her own, and, more importantly, would actually *want* to read this book, rather than just use it (as so many mathematics undergraduates do) as a collection of exercises.

The book is in six chapters. The first reviews the geometry that the student has seen before in high school (but may well have forgotten), the second addresses the more modern geometry of the triangle (the various triangle centers, Morley’s theorem, the nine-point circle, etc.) and the third extends the high school geometry of the circle (the Simson line, the power of a point and the radical axis, cross-ratio). Chapter 4 then returns to triangles for a discussion of the theorems of Ceva and Menelaus and some applications. A Euclidean version of Pappus’ theorem is deduced as a consequence of Menelaus’ theorem (along with a very brief discussion of projective geometry) and Desargues’ theorem appears as an exercise. The next chapter introduces vector methods (again done in a fairly naïve way, with arrows representing vectors) and the final chapter discusses geometric constructions.

In every chapter there are results that are almost sure to fascinate any student who cares about geometry; the existence of the nine-point circle is one example. The chapter on geometric constructions, as another example, not only proves the impossibility of trisecting an arbitrary angle with ruler and compass, but goes further and proves that arbitrary angles *can* be trisected if the ruler is allowed to be marked.

Each section is accompanied by a reasonable number of exercises, most of which struck me as being in the easy end of the spectrum (a contrast to some of Isaacs’ other books). A reference to a book of problems in Euclidean geometry is included in the annotated bibliography “for readers who feel we did not provide enough, or hard enough, exercises”. There are no solutions to the exercises provided, but hints are given for some of the more challenging ones. (In the bibliography Isaacs also notes that the book *Geometry Revisited* by Coxeter and Greitzer was a source of ideas for the course on which this book was based; this book therefore has substantial overlap with Coxeter and Greitzer’s book, but is, I think, better written and easier for students to follow.)

In summary, this book is an excellent choice for courses in non-axiomatic geometry. Teachers or prospective teachers of high school geometry may also want to glance at this book, since some of the results may be of interest in those classes, even if not proved.

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