L’algèbre au temps de Babylone: Quand les mathématiques s’écrivaient sur de l’argile [Algebra in the Time of Babylon: When Mathematics Were Written on Clay.]
Jens Høyrup. Preface by Karine Chemla. Paris. Vuibert/ADAPT-SNES. 2010. xiv + 162 pp.
A new book on the history of mathematics has appeared in the very useful collection “Inflexion” directed by Jean Rosmorduc, a French historian of physics, in Vuibert/ADAPT-SNES Edition. After Arabic algebra , it is Babylonian mathematics that is honored. This is an opportunity for Jens Høyrup (Roskilde University, Denmark) to give us the French translation of a revised and expanded version of a book he published in 1998 for Danish high-school students and teachers . According to the author, this book is intended “[for] those interested in the history of mathematics even if they have no mathematical knowledge beyond what they learned in high school” . The sources for the author are clay tablets dating from the second half of the Old Babylonian period (between 1800 and 1600 BC). It is important to note the mysteries that are contained in these tablets. We have no information on their authors. Their origin is also very difficult to determine. Most of them, according to the author, were exhumed during clandestine excavations and/or were purchased by various museums from antique shops in Baghdad and elsewhere .
The work is divided into a preface written by Karine Chemla, a foreword by the author himself, an introduction, eight chapters, two appendices, several bibliographic elements, and a general index. The book is obviously written with a special pedagogical emphasis. To achieve its goal, Jens Høyrup overlooked nothing. In particular, several reading levels are clearly detailed for readers more or less initiated in mathematics and in Babylonian texts. The introduction immerses the reader very gradually in the historical, socio-cultural, and scientific context of Babylonian mathematics. Special inserts dedicated to the history of Mesopotamia are presented, including one on cuneiform writing and another on the explanation of the sexagesimal system. The first four chapters constitute the most extensive part of the book (approximately 100 pages). The author shares with us secrets from clay tablets that he studied over many years. These tablets are currently stored in, for example, the Louvre Museum in Paris (AO), the British Museum in London (BM), the Staatliche Museum in Berlin (VAT), or in collections of several universities, such as Yale (YBC) in the United States or Strasbourg (Str.) in France.
The main goal of this book is to provide a deep discussion of the very typical Old Babylonian algorithms for solving linear or quadratic equations. Thus, we are invited to a new reading of a wide range of issues following a geometric interpretation of these algorithms. This is obviously not a surprise for those who know the research of the author . For him, the algebraic interpretation, well known since the seminal work of François Thureau-Dangin  and Otto Neugebauer , gives a good reading of the numbers but “it does not pierce [the] mathematical thinking [of Babylonian scribes]” . Additionally, it does not explain what they did and how they completed their calculations during the Old Babylonian period. The purpose of the book is plainly to provide key notions in order to help the reader think about mathematics as it was practiced in this early period. The reconstruction of the scribe’s step-by-step process is amply illustrated by geometrical figures, through which our understanding is greatly facilitated. Chapter 4 deals entirely with the use of ‘Babylonian algebra’ for solving geometric problems, such as division of plane figures. It is not until the next chapter that Høyrup finally outlines his “reasoned opinion”  on ‘Babylonian algebra’ and its specificities.
In more than one way, this book echoes the work of today’s mathematics teachers. This is particularly true in Chapter 6, where the author describes aspects of the context in which the “refined problems although irrelevant [to] practice”  that are omnipresent in the book were presented and developed. These are pseudo-concrete problems; that is to say, problems artificially constructed from a convenient practice (e.g., measurement and sharing of fields, brick production, loans with interest). In the formation of Babylonian scribes, the main goal of the use of such problems would have been to justify the requirements of their education and in particular their training in sexagesimal calculations. The penultimate chapter deals with the inevitable questions of origin and transmission of knowledge and other scientific practices. Like an etiological tale that might bear on the origin of algebra, the main part of the book ends with a moral: “mathematics can be thought of [in] several ways” .
The two appendices following the final chapter are distinctly written for the more involved reader wanting to fully satisfy his or her curiosity. French translations of additional problems are provided, as well as transliterations of the main problems of the book with a glossary of all terms used.
Jens Høyrup, a recognized specialist in Babylonian mathematics, has written a book in which he has made his vast knowledge accessible to everyone. The primary audience is undoubtedly the secondary mathematics teacher who will be delighted to find many examples of the way in which algorithms can be introduced and practiced in high school. Many other topics such as the sticky magnitude-number pair, proof in mathematical activities, or the place of diagrams in reasoning, are extensively covered. Thus, this book can also be easily used in teacher training on epistemological themes. Indeed, its reading and use naturally suggest exercises based on historical texts, or at least inspired by them. Finally, I am sure that the French translations  of many problems with their detailed mathematical analysis and the historiographical point of view of the author contribute better knowledge of Old Babylonian mathematics to the community of historians of mathematics. To conclude, if the wager of Jens Høyrup was to remove the veil for many of us from some of the most characteristic features of Old Babylonian mathematics, it is here, definitively and without exaggeration, a won bet.
 A. Djebbar, L’algèbre arabe. Genèse d’un art [The Arabic Algebra. Genesis of an Art]. Paris. Vuibert/ADAPT-SNES. 2005. 224p. (French)
 J. Høyrup. Matematik på lertavler. København. Matematiklærerforeningen. 2010. (Danish) http://www.akira.ruc.dk/~jensh/Publications/1998%7Bc%7D_AlgebraPaaLertavler.pdf
 J. Høyrup, L’algèbre au temps de Babylone. Quand les mathématiques s’écrivaient sur de l’argile. Paris. Vuibert/ADAPT-SNES. 2010. p. xiii.
 J. Høyrup, L’algèbre au temps de Babylone…, op. cit., p. 21.
 In particular, the problems presented in the book under report are all extracted from J. Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin. New York. Springer. 2002. 459p.
 F. Thureau-Dangin, Textes mathématiques babyloniens, Société Orientale, Leyde, 1938, 320p. (French)
 O. Neugebauer, Mathematische Keilschrift-Texte, Springer-Verlag, Berlin/New York, 1973, 516p. (German)
 J. Høyrup, L’algèbre au temps de Babylone…, op. cit., p. 10.
 J. Høyrup, L’algèbre au temps de Babylone…, op. cit., p. 103.
 J. Høyrup, L’algèbre au temps de Babylone…, op. cit., p. 107.
 J. Høyrup, L’algèbre au temps de Babylone…, op. cit., p. 122.
 As mentioned above (note 5), for those who do not read fluently French, all the problems are available in English in J. Høyrup, Lengths, Widths, Surfaces…, op. cit.