A quarter century ago I noticed that some of the graduate physics students in my university were carrying around copies of Scientific American. Armed with that clue, I dug out every article on the newly discovered fundamental particles. Within the space of a week of fairly easy reading I was able to acquire a good sense of what this subject was all about. These articles explained the basic stuff our professors assumed we must know (but most of us surely didn't).
Brannan, Esplen, and Gray's Geometry accomplish for math what those Scientific American articles did for physics: speaking at a level accessible to anyone with a good high school education, they bring the interested reader up to speed in affine, projective, hyperbolic, inversive, and spherical geometry. They provide the simple explanations, diagrams, and computational details you are assumed to know-but probably don't-when you take advanced courses in topology, differential geometry, algebraic geometry, Lie groups, and more. I wish I had had a book like this when I learned those subjects.
Individual chapters of about 50 pages focus on distinct geometries. Each one is written to be studied in the course of five evenings: a week or two of work apiece. Although they build sequentially, just about any of them can be read after mastering the basic ideas of projective geometry (chapter 3) and inversive geometry (chapter 5). This makes the latter part of the book relatively accessible even to the less-committed reader and an effective handbook for someone looking for just an overview and basic formulas.
The approach is surprisingly sophisticated. The authors do not shy away from introducing and using a little bit of group theory, even at the outset. (Scientific American, even in its heyday, never dared do that.) They present all geometries from a relatively modern point of view, as the study of the invariants of a transitive group of transformations on a set. Many explanations and proofs are based on exploiting properties of these transformations. This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry").
One nice aspect is the authors' evident awareness of and appreciation for the history of mathematics. Marginal notes begin at Plato and wind up with Felix Klein's Erlangen program some 2300 years later. Although the text does not necessarily follow the historical development of geometry, its references to that development provide a nice context for the ideas. This is an approach that would improve the exposition of many math texts at all levels.
The authors are British and evidently write for students with slightly different backgrounds than American undergraduates. Obvious prerequisites are a mastery of algebra and a good high school course in Euclidean geometry. Synopses of the limited amounts of group theory and linear algebra needed appear in two brief appendices. However, readers had better be intuitively comfortable with matrix operations, including diagonalization and finding eigenspaces, because matrices and complex numbers are used throughout the book for performing computations and developing proofs. A knowledge of calculus is not needed. Indeed, calculus is not used in the first two-thirds of the book, appearing only briefly to derive a distance formula for hyperbolic geometry (a differential equation for the exponential map is derived and solved). During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). Definitions of these functions are not routinely provided, but algebraic identities appear in marginal notes where they are needed.
Now for the quibbles. The book has lots of diagrams, but not enough of them. The problems are usually trivial, tending to ask for basic calculations to reinforce points in the text. The text itself does not go very deeply into any one geometry, being generally content with a few illustrative theorems. An opportunity exists here to create a set of gradually more challenging problems that would engage smarter or more sophisticated readers, as well as show the casual reader where the theories are headed.
This book is the work of three authors and it shows, to ill effect, in Chapter 6 ("non-euclidean geometry"). Until then, the text is remarkably clean and free of typographical and notational errors. This chapter contains some glaring errors. For example, a function s(z) is defined at the beginning of a proof on page 296, but the proof confusingly proceeds to refer to "s(0,c)", "s(a,b)", and so on.
The written-by-committee syndrome appears in subtler ways. There are few direct cross-references among the chapters on inversive, hyperbolic, and spherical geometry, despite the ample opportunities presented by the material. Techniques used in one chapter that would apply without change to similar situations in another are abandoned and replaced with entirely different techniques. Within the aberrant Chapter 6, some complex derivations could be replaced by much simpler proofs based on material earlier in the chapter.
The last chapter attempts to unify the preceding ones by exhibiting various geometries as sub-geometries of others. It would have been better to make the connections evident as the material was being developed. It is disappointing, too, that nothing in this book really hints at the truly interesting developments in geometry: differentiable manifolds, Lie groups, Cartan connections, complex variable theory, quaternion actions, and much more. Indeed, any possible hint seems willfully suppressed: the matrix groups in evidence, such as SL(2, R), SU(1,1, C), PSL(3, R), O(3), and so on, are always given unconventional names, for instance. Even where a connection is screaming out, it is not made: the function abstractly named "g" on pages 296-97 is the exponential map of differential geometry, for instance.
Despite these limitations, Brannan et al. is a good and enjoyable book for anyone from high school through first-year graduate level in mathematics.