The term ‘classical geometries’ is a catch-all phrase that could be applied to many different approaches to geometry; but the authors of this slender volume have used it to describe a well-structured introduction to a fascinating range of ‘modern’ geometrical ideas, sequentially developed within a Kleinian framework. And, although this treatment is mainly restricted to coordinatised 2-dimensional geometry, there are occasional references to 3- and n-dimensional cases. There are just four chapters, sequenced as follows:

- Euclidean Geometry
- Affine Geometry
- Projective Geometry
- Hyperbolic Geometry

First published in Mexico in 2002, the book has been translated from Spanish with a reasonable degree of clarity (by persons unspecified). It is addressed to advanced undergraduates in mathematics and also to physicists and scientists who want to learn the basics of modern geometry. The authors assume that readers will be familiar with some plane Euclidean geometry, and also with the elements of analytic geometry. There is, of course, little point in assuming more than a scant amount of pre-requisite geometrical knowledge because the vast majority of students won’t be equipped with it anyway. As for other requirements, familiarity with linear algebra (up to eigenvalues of linear transformations) and some knowledge of advanced calculus is deemed necessary.

In many ways, the approach taken by the authors is markedly different to that provided in other ‘introductory’ texts, and the book establishes its individuality in the very first chapter (which constitutes about one third of the book). This provides a succinct introduction to Euclidean transformations (isometries) but, instead of taking the backward step of deriving some of the elementary theorems from Euclid, these transformations are applied to the classification of quadric surfaces, whose isometric invariance is then established.

Other ideas, such as elliptic, hyperbolic and parabolic points are introduced, and aspects of elementary differential geometry are invoked for the examination of curvature properties of quadric surfaces (umbilic points etc). In the next section, topological properties of tori and cylinders are compared, and there is a veritable proliferation of related concepts (fundamental group, homology groups geodesics, space of orbits under group actions etc). In fact, so much is going on in this first chapter that it’s hard to see the Euclidean wood amongst a myriad of conceptual trees. However, in the last two sections of this chapter, there is a return to the more familiar Euclidean territory by means of classification of frieze patterns and the symmetry of the five Platonic solids etc. And, needless to say, ideas from elementary group theory are invoked throughout this first chapter.

Actually, the algebraic approach to geometry taken in this book utilizes a variety of coordinate systems, including Cartesian, affine, projective, and coordinate charts for P^{2}(**R**) and P^{1}(**C**) etc. For example, in chapter 2, which offers a compressed, but initially well-motivated introduction to affine geometry, transformations are defined in terms of affine coordinates, and several familiar affine invariants are established.

The third chapter (Projective Geometry) continues at the same breath-taking pace as the first. Two pages after motivational illustrations of Dürer using projective techniques to establish perspective in his drawing, the discussion centres upon the homeomorphism between P^{1}(**R**) and S^{1}, as well as topological relations such as

P^{1}(**C**) = S^{3}/S^{1}. And so, by this stage, the exciting nature of the mathematical content of the book is well and truly established. On the other hand, many of the standard ideas from projective geometry are also included in this chapter (duality, Desargues theorem, projective equivalence of the conics, poles and polars etc), and there are excursions into algebraic geometry (Bezout’s theorem) and a culminating section that defines the elliptic plane as P^{2}(**R**) together with a metric inherited from S^{2}.

The fourth and final chapter (Hyperbolic geometry) considers various models of the hyperbolic plane, such as those due to Beltrami and Poincaré. Steiner networks are introduced in the context of Möbius transformations and, following a section on hyperbolic metrics, some of the standard results in hyperbolic geometry are derived (e.g. Lambert’s result on the angles sum of a hyperbolic quadrilateral). Never boring us for a moment, the chapter closes with sections on surfaces with hyperbolic structure and tessellations of the Poincaré plane.

So, here we have a book that presents us with mathematics in its richest and most aesthetically pleasing form. Each chapter commences with elementary motivational narrative, and the book is liberally supplied with a host of interesting illustrations. In addition, the text is interspersed with occasional historical notes, although these don’t accrue to anything like a full history of non-Euclidean geometry. Exercises are also provided, but not in any great number, and teachers may need to supplement these in various places. However, solutions to these are not provided

But some caution is necessary to stem this flow of unbridled praise. For a start, the material is developed at such speed, and invokes so many ideas from other areas of mathematics, that it would be folly to recommend this book for use with those provided only with the relatively meagre mathematical background specified as pre-requisite knowledge by the authors. They suggest that the book could form the background of a one-semester course, which may be true for those who have taken courses in differential geometry and topology etc, but, otherwise, the content is so conceptually dense that a two-semester course is really needed to do justice to the range of material in this book.

Finally, although much of the text is presented with a reasonable degree of grammatical precision, there are several places where the translation obfuscates the argument (e.g. comments on the bottom of page 150 and top of page 151 in relation to the pseudosphere). Apart from such quibbles, this book comes highly recommended to those seeking structural guidelines for a stimulating course on geometry.

Peter Ruane is now retired and, having spent many years in primary and secondary mathematics teacher education, misses opportunities of teaching courses on geometry as outlined in books like this one.