The only ones, that is, until the publication of this book. This text puts the Kleinian approach front and center and organizes a year’s worth of material in Euclidean and non-Euclidean geometry around it. The geometry studied in this group is not limited to plane geometry, but instead covers geometry in $\textit{n}$ dimensions. The author refers to this as “perhaps… the main innovation of this book.” His rationale for doing things in this generality is to help the reader gain “a complete understanding of the subject so that he/she is well prepared when he/she meets geometry again in a differentiable manifolds course, for example”.  However, recognizing the pedagogical benefits of working in, say, the case $\textit{n} =  2$, the author does occasionally focus on this special case.

It seems a bit strange to see Euclidean plane geometry relegated to the status of a “baby example”, but that’s because chapter 3 provides a sweeping generalization: here the underlying set is $\textit{n}$-dimensional Euclidean space $\mathbb{R}^{\textit{n}}$ and G is the set of isometries (relative to the usual norm induced by the dot product). These isometries are studied in some detail, with some specific ones (reflections, rotations, translations) being singled out. It is proved, for example, that any isometry can be written as an orthogonal linear transformation followed by a translation; it is also proved that any isometry of $\mathbb{R}^{\textit{n}}$ can be written as a product of at most $\textit{n}$ reflections. (This is the Cartan-Dieudonne Theorem.)

Chapter 6 is on spherical geometry: here, the underlying space is the unit sphere in $\mathbb{R}^{\textit{n}}$ and G is the group of orthogonal linear transformations. Starting with this, the author discusses such topics as great circles on the sphere, spherical trigonometry, spherical triangles (which have angle sum exceeding 180 degrees, in contrast to the Euclidean situation), and Euler’s formula $\textit{V} -\textit{E+F=2}$ for triangulations of the sphere in $\mathbb{R}^{3}$, which in turn is used to characterize three-dimensional platonic solids.

This is followed by a chapter on projective geometry, meaning the study of real projective $\textit{n}$-dimensional space. (In keeping with the Kleinian nature of the book, axioms for the projective plane, or projective space, are not discussed.) After a brief historical discussion mentioning such things as perspective drawing, projective $\textit{n}$-dimensional space is defined as the set of all subspaces of Euclidean $(\textit{n}+1)$-dimensional space, with set-theoretic inclusion corresponding to geometric incidence. Thus, for example, the real projective plane consists of projective points (1-dimensional subspaces of $\mathbb{R}^{\textit{3}}$) and lines (2-dimensional subspaces), with a point lying on a line if and only if it is a subspace. (A simple dimension-counting argument shows immediately that any two lines intersect in a point, which is, of course, the defining condition of a projective plane.) Having defined the underlying space, the projective group $P(\textit{n})$ of projective mappings is defined and studied. The chapter touches on all the key aspects of projective geometry: cross-ratio, duality, the Fundamental Theorem of Projective Geometry, and the theorems of Pappus and Desargues.

Inversive geometry is studied in chapter 8. Here the underlying space is extended Euclidean $\textit{n}$-dimensional space; i.e., $\mathbb{R}^{\textit{n}}$ with a “point at infinity” added. (In topological terms, this is the one-point compactification of $\mathbb{R}^{\textit{n}}$.) Inversions in spheres and hyperplanes are discussed, as is the larger set of Mobius transformations. The chapter ends with a discussion of inverse geometry in dimensions 1 and 2, where connections with projective geometry are made. For the case $\textit{n}$ = 2, complex numbers are put to use.

In the final (and, I thought, most demanding) chapter of the text, hyperbolic geometry is discussed. Hyperbolic geometry as defined here is the upper half-space model of Poincare, with X the upper half-space of ($\textit{n}$ + 1)- dimensional space and G the subgroup of the ($\textit{n}$ + 1)- dimensional Mobius group leaving X invariant. Hyperbolic distance is defined and isometries studied. The hyperbolic parallel postulate (through a point not on a line, there are more than one lines parallel to the given line) is established, as is the fact that the sum of the angles of a hyperbolic triangle is less than 180 degrees. (In a very unfortunate typo, the statement of the theorem that says this, Theorem 9.4.7, says “greater than”, not “less”.) After a section on hyperbolic trigonometry, the chapter ends with a detailed look at the hyperbolic plane, with some famous models of that geometry (Poincare disc, Beltrami-Klein disc model) discussed in detail.

Following this textual material, there are three appendices, two providing background material on the Euclidean space $\mathbb{R}^{\textit{n}}$ and group theory, respectively, and the other providing solutions to a selection of the problems appearing in the text. (In another serious typo, the Table of Contents lists the appendices, and things following them, as starting on page 1.) After the appendices, there is a three-page bibliography.