That particular histories are written by particular authors in particular contexts for particular audiences, and that they might be so shaped by these circumstances, should surprise no one. From this banal overture, the contributors to this volume of case studies in the history of the history of mathematics engage in a thought-provoking conversation about the images, values, assumptions and purposes of historical writing about mathematics and (especially) mathematicians from the sixteenth century to the present.

A slim book without pretensions to comprehensiveness, the volume's intellectual core comes in its three middle chapters, each of which explores a different side of the contentious history of the invention of the calculus. First, Rebekah Higgitt draws on her ample expertise in the history of Newton biography to characterize a wide chronological range of portrayals of ‘Newton the mathematician’ in terms of their authors and contexts. Niccolò Guicciardini and Adrian Rice, in turn, offer detailed analysis of the positions of two authors – respectively Jean Etienne Montucla and Augustus De Morgan – on Newton's historiographically infamous priority dispute with Leibniz. Both Guicciardini and Rice use their authors and those authors' sources to provide compelling views of the dispute's historiography as viewed through the French eighteenth and English nineteenth centuries.

Preceding the volume's Newtonian core, Philip Beeley gives a rich unpacking of the content and contexts of John Wallis's Treatise of Algebra, while Benjamin Wardaugh surveys and contextualizes a variety of eighteenth-century claims about the origins and development of arithmetic. Following the Newtonians is the volume's lone essay to step entirely away from British figures and sources: Henrik Kragh Sørensen's fine encapsulation of his extensive recent work on the history of Abel-commemoration, which he presents by looking at the writings of (and sources available to) the Swedish mathematician Gösta Mittag-Leffler. The volume concludes with Jacqueline Stedall's engaging tour through the much-occluded history of sixteenth-century scholar Thomas Harriot's much-occluded writings.

As several of the authors acknowledge, the meanings and methods of both ‘history’ and ‘mathematics’ have varied considerably over the time periods considered in this volume, and even in individual chapters. The historical writing here considered ranges from assembling a ‘life and letters’ from available documents, to probing the inner psyche or moral character of one's subject, to establishing causal connections between different discoveries and developments. The volume's contributors are uniformly careful to identify the goals, traditions, competencies and biases of the historical authors they discuss, and their most stimulating analyses often come from well-informed dissections of the changing availability and relevance of different archival and published sources.

Amidst such sensitivity to historical context, the authors seem surprisingly eager not just to explain but to evaluate their sources. These evaluations range from the trivial (Wallis failed to write a modern causal history) to the puzzling or counterintuitive (Montucla's history was not nationalistic; De Morgan wrote without regard for scientific prestige or religious sectarianism). They betray a certain narrowness in the historiographical conversation of the volume and the 2010 Oxford conference from which it emerged – one strongly shaped by the rich historiography surrounding Newton and his English predecessors and by recent work on scientific biography (with an obvious mutual respect and appreciation among the volume's contributors), but less engaged with debates on mathematical presentism and reconstruction (as particularly developed in the historiography of ancient mathematics), and the politics and images of mathematical theories and labour.

One is left wondering, in particular, what mathematics has to do with the volume's case studies? Would the analyses be fundamentally different if they were histories of the history of natural history? Here, the case-study genre leaves a whole that is somewhat less than the sum of its many learned and thoughtful parts. Between studies of individual figures, works or ideas, broader considerations of the significance of the changing meanings of mathematics and history are demoted to explanatory resources rather than central topics of investigation. The shifting professional identity of mathematicians and their discipline's contested values and ideas all make appearances, to be sure, as settings for the historical writings at the book's centre. They are not, however, interrogated in their own right or built into a more comprehensive account of the definitive or distinctive characteristics of histories of mathematics.

Those who seek out this book for its treatments of any of its individual cases, especially that of Newton and the calculus, will find their fill of enriching accounts by accomplished experts. Those hoping for a concerted entry into the challenging and contentious historiography of mathematics will find far more provocations and questions than answers.