Let's start by laying out some of the reviewer's biases. First, I like mathematics books to have a personality, and particularly like books that have a sense of humor. Second, much as I appreciate the charms and historical importance of synthetic geometry, I do not think it must play a central role in how we teach undergraduates. Third, I am sympathetic to the idea that mathematics students should build on what they know; in particular, I like the idea of making serious use of linear algebra (and even matrix groups) in a geometry text.
From where I stand, then, there is a lot to like about this new book. The authors have put together an introduction to geometry that is modern, interesting, and open. It is modern in that it uses linear algebra, coordinates, and metrics from the very beginning. It is interesting both in terms of the content and in its presentation. And it is open in the sense that it creates many options for further study and development of the material.
The authors have tried to be very informal in style, to the extent of addressing the reader as "you" and referring to themselves as "I", "despite the fact that there are more than two of me." So the book is full of sentences such as "My aims are..." or "I usually choose the angle to be between 0 and π." There is no sacrifice of mathematical precision, except for instances where something is beyond the level of the text; in such cases, the author (!) points out that this is the case and sketches the essence of the proof. (For example, this happens when he proves that the distance between two points in Euclidean space is the length of the shortest curve connecting them.) The result of all this is that the book is a pleasure to read.
At the same time, students will by no means find this book easy. In particular, quite a lot of linear algebra is present from the first page on. The dot product appears on page 5, orthogonal matrices on page 10, eigenvalues (real and/or complex) on page 12. I think this is all to the good. After all, if we spend the time teaching linear algebra students these ideas, we owe them the courtesy of making use of them when doing so makes things easier.
I heartily approve the inclusion of some basic topology and transformation groups. The material included is interesting, ambitious, and a good foundation for further study.
I haven't, of course, tested the book in class. I'm certain that students would find it hard in parts, and that I would have to create a lot of motivation to keep them going. I suspect the results would repay the effort involved.
Finally, anyone that has the confidence to do what the author has done in problem A5 (on page 181) has my instant respect. How often has a mathematics book made you laugh?
Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. He is the editor of FOCUS, FOCUS Online, and MAA Reviews. Somehow, he manages to stay sane. Humor helps.