*De grands défis mathématiques d’Euclide à Condorcet,* edited by Evelyne Barbin, 2010, 176 pp., Vuibert-Adapt, Paris, ISBN 2311000195 / ISBN 9782311000191

The inter-IREM Commission on Epistemology and History of Mathematics[1] has partnered with the publishers Vuibert and SNES-Adapt, well-known for their support of teacher training, to produce *De grands défis mathématiques d’Euclide à Condorcet* (*Major Mathematical Challenges from Euclid to Condorcet*). This short book (180 pages) is not an abridged history of great historical or mathematical problems. Rather, it offers us nine experiences from the everyday teaching practice of authors whose common goal is the introduction of a historical perspective in college and university mathematics education. Even if these experiences could really be considered as challenges (in an historical sense for the first mathematicians and in an educational sense for today's teachers), they are not limited to the period (already very large) from Greek Antiquity to the aftermath of the French Revolution. If we have to give a range for the mathematical practices presented in this book, it would extend from the Mesopotamian scribes to mathematicians and engineers of the 19th and 20th centuries!

The nine chapters of the book correspond to nine experiences organized into four distinct sections representing the major areas of mathematics taught today: analysis, algebra, geometry, and probability. In the spirit of the work of IREM, the chapters have not been designed as models to follow, but rather are presented as evidence of teaching practice by second level or university teachers. They are written so that the first third (at least) of each chapter is a historical introduction putting the mathematical challenge both in its scientific context and in its historical context. The remaining two-thirds give a description, as precise as possible, of the manner in which the history of mathematics is introduced in the classroom, focusing on reading original texts. Thus, numerous excerpts (some previously unpublished in French translation) are provided to readers. Each chapter ends with a list of references that distinguishes original texts from historical studies, enabling the reader to complete his knowledge whatever his profile (math teacher, teacher trainer, or person curious about the history of mathematical ideas and practices).

The book provides various answers to difficult questions constantly debated in the IREM group, and also in the HPM group,[2] about the relationship between history of mathematics and mathematics education, and in particular the pedagogical profit of using original texts. Reading of original texts in class is attractive and may appear beneficial in the long process of learning a new concept. Although it is encouraged in official instructions for the maths curriculum of several European countries, the book under review reminds us that it is also extremely difficult to implement. Anyone who considers it asks: What kind of texts? How could I do it? For what results and when? Numerous answers are suggested in this book. In particular, E. Barbin outlines a synthesis of several types of readings and possible analysis based on objectives that could be useful to teachers willing to consider original texts as a tool of their pedagogy (*De grands défis mathématiques…*, pp. 65-66). Chapter after chapter, the history of mathematics is explicitly introduced at various levels of secondary and postgraduate education in order to be integrated into, and not merely added to, traditional teaching.

For those with some reading knowledge of French, reading this book would be extremely fruitful. Although our purpose is not to give an exhaustive list of problems presented in the book or even an inventory of texts that illustrate them, we cannot resist revealing some of them. Measurement of angles useful for navigation and surveying are illustrated by several excerpts from the *Encyclopedia* edited by Diderot and D'Alembert and *The Elements of Geometry* written by Clairaut. Gambling provides an opportunity to read selections from Leibniz and Condorcet, allowing the reader an epistemological reflection on probability and statistics. There are also problems of ballistics and celestial mechanics that led Euler to approximate the solution of a differential equation. As a final example, curves are studied in an original way: with focus on the challenges of *drawing* them from Pacioli and Dürer grappling with typographical problems in the early days of printing up to the digital definition of a curve with the famous example of Bezier curves useful to design cars.

From an epistemological point of view, this book provides a real opportunity to help teachers and students capture the richness of mathematics and its practice, together with their inherent difficulties. Thus, the reader realizes that the representation of mathematical objects is not an obstacle only for pupils or students but also for mathematicians themselves. The problem of irrationality arose in Antiquity and still was widely debated in the Renaissance. Excerpts from Euclid, Nicomachus of Gerasa, Theon of Smyrna, and Jacques Pelletier du Mans are here offered as evidence. Another example is worth noting: the representation of a vector in the plane in the relatively unknown work of the 19^{th} century French mathematician Mourey. Let us finally mention reflections on rigor, reasoning and proof in mathematics. These are common to all nine chapters and two authors consider them explicitly. The first one proposes a reading of the *Elements* of Euclid, always profitable for teaching geometry and the hypothetical-deductive approach. The second one invites us to compare several resolutions to a common problem: how to construct a *square** *inscribed in a given *triangle*. This last study is illustrated not only by the original texts (Chuquet, Marolois, and al-Khwarizmi) but also by pupils’ work.

In conclusion, written for the non-specialist, *De grands défis mathématiques d’Euclide à Condorcet* provides the fundamentals of the historical approach in the classroom, focusing on the subtle invention of notions and concepts from several historical problems. It is interesting for anyone curious about the history of mathematics. Moreover, it allows the reader, whatever his background, to plunge into the unique atmosphere of a classroom imbued with the collective desire of sharing knowledge. Finally, we note that the Commission inter-IREM soon will invite us to read nine more pedagogical experiences in another volume with the same editor, *Des mathématiques éclairées par l’histoire. Des arpenteurs aux ingénieurs* (*Mathematics highlighted by history. **From Surveyors to Engineers,* Vuibert-Adapt, Paris, 2011). Unfortunately for the English reader, these two books are available only in French. Perhaps this could be rectified.

[1] This commission works on the introduction of a historical perspective in mathematics education. The members are mathematicians, historians of mathematics, researchers in mathematics education, and teachers of mathematics involved in academic groups of the IREM (Institute for Research on Mathematics Education).

[2] HPM is the International Study Group on the Relations between History and Pedagogy of Mathematics affiliated with the International Commission on Mathematical Instruction* *(ICMI).

________________________