Many relevant mathematicians have yielded to temptation of exploring the realm of classical and Euclidean Geometry together with his/her research activity. Jacques Hadamard, the famous French Mathematician was not an exception. Besides his profound contributions, mainly in Analysis and Number Theory, Hadamard dedicated part of his bright talent in writing a collection of two volumes, the former about plane Geometry and solid Geometry for the latter. This work made part of a collection of textbooks, resources for the teaching of mathematics at high school, edited by G. Darboux, from whom he received the invitation to write these volumes. Since their first edition, Hadamard completed and improved the exposition and the collection of the problems along his fruitful and long life.

There are several editions of Hadamard’s Leçons de Géométrie. Among them, an English translation has been recently done by M. Saul (Lessons in Geometry I. Plane Geometry, American Mathematical Society, 2008). The basic structure of all the lessons is divided in two parts: a theoretical one (including definitions, explanations, results and remarks) followed by a list of unsolved problems. The goal of the book under review is the exposition of the solution of the exercises and problems of the first volume. Together with the original work of Hadamard, the reader now has a companion (as it is explicitly said in the title) for a complete reading of Lessons in Geometry I. The solutions for the section of Miscellaneous Problems have not been included but can be found in the web site of the AMS at www.ams.org/bookpages/mbk-70.

The philosophy of this book is based upon the following points. First, the solutions (most of them done by the author) are complete and rigorous, enriched with many intuitive ideas. The rigor and the intuition are essential ingredients for a companion of Hadamard’s work. Secondly, the book is full of cross references. This simplifies the exposition of those solutions needing results included in previous problems. These cross references may be uncomfortable in a first reading of some particular solutions. A second and more laborious reading gets more benefit from them. Finally, the author incorporates remarks and software explorations at the end of some solutions to be done with the any of the current dynamic geometry software. These applications are not intended to be part of the solution itself but provide insight for both teachers and students in some particular exercises.

Hadamard’s Lessons together with this companion can be of interest to high school teachers, gifted students, students participating in Mathematical contests and any person interested in Geometry. In particular, the ambitious contents of this books is food for though about the evolution of the teaching of Mathematics since the volumes were written. The contents of school geometry have suffered, in general, successive reductions. Hadamard’s opinion about this could be summarized in a short but clear sentence contained in the Preface of the first edition of his Lessons: “Geometry reveals itself capable of exercising an undeniable influence on the activity of the mind”. This could be a good point to reflect about the evolution of this part of Mathematics in the last decades.