Over the past decades, history of mathematics has gradually blossomed into a research field in its own right, with journals devoted to the study of the discipline and active communities of professionals around the world. Thus it was merely a matter of time until historians of mathematics would start reflecting on their own research aims, methodological approaches and practices. The edited volume I am currently reviewing, Historiography of Mathematics in the 19th and 20th Centuries, proposes to examine the numerous ways in which historians of mathematics of the last two centuries have attempted to comprehend and write about the past of their field. Drawing on the fact that, over this period, there have been noted several shifts of perspective in mathematical history writing, the general scope of this edited volume is to show how historiography of mathematics has been crucially shaped by the intellectual backgrounds, motivations and institutional settings of its practitioners.

The book comprises a brief, yet informative, introduction by the editors (Volker R. Remmert, Martina R. Schneider and Henrik Kragh Sørensen), eleven chapters by historians of mathematics of different specializations, and a helpful name index. It is worth pointing out that the essays follow a chronological order based on the historians’ time of action. As a result of this arrangement, the relevant episodes from the history of mathematics do not necessarily follow a strict chronological order.

Let me now briefly outline the contents of the volume. Maarten Bullynck's opening paper argues that around the year 1800 a shift regarding narrative pattern – that is, from an encyclopedic format to a historical narrative – took place, a change which, according to Bullynck, had a considerable impact on the way histories of mathematics were written. Through this lens, the author examines the case of Alexander von Humboldt's project on the origin and development of the decimal positional numeral system. The next paper, by Sonja Brentjes, studies methodological aspects of practices in the history of mathematical sciences in Islamicate societies in Germany and France in the nineteenth century, by emphasizing the ideological underpinnings of these efforts. The historiography of Mesopotamian mathematics is discussed in two chapters. In the first one, Jen Høyrup proposes a classification of the readers of Mesopotamian mathematics into two categories: the ‘Assyriologists’ who approached the texts from a philological perspective, and the ‘historians of mathematics’ who mainly saw it as a part of the global history of their field. In the second paper on Mesopotamian mathematics, David E. Rowe describes the efforts of Otto Neugebauer, one of the most famous historians of ancient mathematics, to break with the traditional Graeco-centric understanding of the history of science.

The title of Henrik Kragh Sørensen's contribution speaks for itself: ‘Appropriating role models for the mathematical profession: biographies in the American Mathematical Monthly around 1900’. In fact, Sørensen not only illustrates how the American Mathematical Monthly presented the biographies of famous mathematicians, but also explains how biographies as a literary genre functioned as vehicles for the mathematicians’ self-fashioning. Jeremy Gray's paper deals with histories of modern mathematics that were written in English in the middle of the twentieth century. More specifically, after examining the case of six authors whose approaches were adapted to the mathematical syllabus, Gray concludes that only Carl Boyer, among these people, made a serious effort to keep open links with contemporary history and historiography of science.

H. Floris Cohen explores the origins of the concept of ‘the mathematization of nature’ and concludes that it emerged in close relation with the equally novel concept of ‘the Scientific Revolution’. Next, Anne-Sandrine Paumier and David Aubin investigate the ways in which the writing practices of the Bourbaki group helped shape the form taken by the famous Elements of the History of Mathematics. It is in precisely this framework that they also identify the origins of the (anachronistic) idea that Euclid of Alexandria may have been a collective name of a mathematical school and not a historical person. Reinhard Siegmund-Schultze discusses the particularities of Johannes Lohne's approach to the historiography of mathematics and physics, and especially his work on Thomas Harriot and the relative notions of ‘outsider’ and ‘mainstream’. Finally, two chapters deal with ancient Greek mathematics: first, Benjamin Wardhaugh provides some information about the life and work of Sir Thomas L. Heath, the intellectual who almost single-handedly carried academic-level research on Greek mathematics in the English-speaking world at the beginning of the twentieth century; and second, Martina R. Schneider's paper – and my personal favourite – attempts to contextualize the most famous recent debate in historiography of Greek mathematics, i.e. the debate on ‘geometrical algebra’. Schneider places emphasis on the life and intellectual background of Sabetai Unguru, one of the leading protagonists of the debate, but, perhaps most importantly, attempts to grasp the broader discourse around questions of professionalism of history of mathematics.

It is evident that this volume contains a rather diverse collection of chapters, dealing with varied topics from different historical periods, and approached with different methodologies and agendas. Nevertheless, my personal impression is that the editors have managed to efficiently organize and present their overall thesis in a compelling manner, thus transforming an apparent disadvantage of the volume into a convincing argument. Having said this, one has to record that the volume is not a complete history of the historians of mathematics. However, this is perhaps some unfair criticism given the enormous amount of work that such an enterprise would require.

To conclude, the edited volume Historiography of Mathematics in the 19th and 20th Centuries includes a collection of chapters that may prove an excellent resource for scholars and students intrigued by mathematics and its history. Therefore I would strongly encourage the interested reader to have a look at it.