From childhood we are taught not to judge a book by its cover; the outward appearance, indeed, cannot be an indicator of the content's value or worth. In the course of our life, we learn that most of the time this lesson is true – and we make it a rule. J.B. Shank's new book represents the exception that proves the rule. Its elegant cover decorated with a golden version of Newton's diagram of projectile motion reflects the sophistication of content and style of this insightful work.

The title of the book – Before Voltaire: The French Origins of ‘Newtonian’ Mechanics, 1680–1715 – provides the reader with enough information to frame the analysis undertaken in this dense volume: Shank explores the origins of analytical mechanics in France between 1680 and 1725, a periodization made clear by the author also in the choice of the expression ‘Before Voltaire’. In this regard, it suffices to read the first pages of the rich and exhaustive introduction to understand that the title not only indicates the span of time that the author considers in his analysis; it also represents a conceptual boundary between two different interpretations: on the one hand, there is the canonized image developed during the Enlightenment of Newton as the heroic originator of modern mathematical physics – an image that Shank aims at mitigating; on the other hand, the author proposes ‘a fresh and unsweetened reinterpretation of Newton's precise role in the historical beginnings of eighteenth-century mathematical physics’ (p. 6). This clarification, which is extensively discussed in the introduction, represents an important preliminary assumption to frame Shank's analysis. Only by downsizing – but not eliminating – the impact of the reception of Newton's work in France is it possible to satisfactorily assess the emergence of the new science of motion on the Continent. The adoption of this approach determines a broad focus on the cultural and historical context and results in a highly informative book that provides a useful account to historians interested in different facets of the scientific developments between the end of the seventeenth century and the beginning of the eighteenth.

The book is divided into three parts. Part I begins by examining the institutional origins of analytical mechanics. The intellectual and institutional dynamics of the French Académie royale des sciences, combined with the individual work of its academicians, played a crucial role in the establishment of the mathematical culture that prepared the terrain for the development of analytical mechanics in the final decades of the seventeenth century. Shank provides a detailed account on the evolving nature of mathematics – and of the role of mathematicians – both inside and outside the academic context, and observes that ‘the changing epistemological canons and the shifting hierarchies of knowledge elevated certain pursuits to the level of science while leaving others among the lower ranks of the arts and crafts’ (p. 28). He concludes this first part by introducing some key figures in the French mathematical debate of the time ‘in a way that situates analytical mechanics, and the controversies that it triggered, as a historical outcome produced, at least in part, by the new institutional politics of the post-1691 Académie Royale des Sciences’ (pp. 29–30).

Part II provides an insightful analysis of the intellectual roots of analytical mechanics, focusing on the development of the infinitesimal calculus and its reception, acceptance and use in France. The calculus was perhaps the single most important intellectual element in the development of analytical mechanics, but also the source of many controversies. This mathematical achievement, indeed, posed within the Académie ‘a range of different challenges. To academic mathematicians, it offered stunning new capacities for solving mathematical problems. But it also required the acquisition of new technical skills that needed to be mastered before these new powers could be utilized’ (p. 192). The reconstruction of the development of the calculus in the French context contributes to the reinterpretation of Pierre Varignon's new science of motion in the light of its sources – Newton, Leibniz and Malebranche. Whereas historians have traditionally attributed to Newton a major role in Varignon's accomplishments, Shank downsizes his influence and proposes a more articulated version, introducing Leibniz and Malebranche into the picture. Notwithstanding the importance of all of them, however, Shank stresses Varignon's originality and claims that he combined into an ingenious approach to motion the lessons from the great mathematicians of his time.

The institutional history of Part I and the intellectual history of Part II converge in Part III, which explores the genesis and development of the new science of analytical mechanics. The multifaceted legacy of Varignon's work constitutes the central focus of this account. His two papers of 1698 and 1700, indeed, inaugurated a new scientific project based on the infinitesimal calculus and launched a provocative epistemological challenge. According to Shank, Varignon's role was pivotal to the genesis of analytical mechanics in France; his accomplishments, however, cannot be considered separately from the institutional context and from the intellectual milieu in which he operated. The 1699 reform of the Académie, the explosion of the initial debates about analytical mechanics into an open public controversy, and the consensual acceptance of analytical mechanics in France by 1710, all contributed to the birth, development and establishment of Varignon's new science of motion.

Shank's analysis represents a rich and stimulating contribution to the history of mathematics. His research nourishes the debate on the early history of analytical mechanics, enlarges our comprehension of the dissemination and reception of the infinitesimal calculus, and provides a comprehensive and detailed portrait of French institutional and intellectual milieus in the late seventeenth and early eighteenth centuries. The richness of the contents is structured in a well-organized text that excels for clarity of style. Besides, the most interesting aspect of this book resides in its methodology. The study of mathematical and scientific achievements is strongly supported by the analysis of the institutional, intellectual, political, social and cultural contexts. A powerful combination that makes of this book a seminal work in the history of science.