This text arose from a one-semester course for future teachers at the Humboldt-Universität in Berlin. This edition is a translation commissioned by the American Mathematical Society and is a volume in the *Student Mathematical Library* series. The motivating idea is to present elementary geometry from an advanced perspective in a mathematically systematic form. In Germany future teachers would take such a course in their second year following courses in calculus and linear algebra.

*Elementary Geometry* has four main sections. The first treats elementary geometrical figures — lines, triangles, circles, conic sections, Platonic solids — of ordinary Euclidean geometry. Two- and three-dimensional symmetries are discussed in the second section. Topics here include affine mappings, projections, central dilations and translations, isometries and similarity transformations. The third section focuses on hyperbolic geometry. The authors present axioms for elementary geometry, discuss the axiom of parallels and then introduce hyperbolic geometry as an alternative to Euclid. The Poincaré model in the upper half plane is developed first, and then the Cayley transform is used to pass to the disc model. The final section is a brief treatment of spherical geometry.

The treatment of geometry in this book is more sophisticated — sometimes a lot more sophisticated — than that customarily seen by future teachers in the United States. For example, the authors use group theory rather freely. The section on symmetries refers to exact sequences of groups and group extensions, and the section on hyperbolic geometry includes a classification of all elementary Fuchsian groups. Beyond these specific topics, the general level of the authors’ presentation is likely to be intimidating to many students training to be high school teachers.

A better matched audience for this book would be mathematics undergraduates who have completed at least a semester of abstract algebra. The sections on symmetries and hyperbolic geometry offer several opportunities for special projects or independent study. The treatment of symmetries includes a nice discussion of what the authors call frieze and ornament groups.

The text has a large set of exercises, very few of which are routine. There are many helpful drawings and the authors have made good use of a four color format. The book has its own website here, with both supplementary material and errata.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.