Amir Alexander has made a career of finding big historical arguments in small mathematical places. His previous books have situated early modern geometry in the era's voyages of exploration – Geometrical Landscapes (2002) – and examined how Romantic stories of tragic heroes shaped the ideas and cultures of nineteenth-century mathematics – Duel at Dawn (2010). His latest book tackles his biggest theme yet in its smallest possible venue, explaining the rise of empirical science and liberal democracy through seventeenth-century controversies surrounding the mathematical concept of the infinitesimal.

In style, format and analytic approach, the book reflects the priorities of its United States publisher, the Scientific American imprint of Farrar, Straus and Giroux, which specializes in popular-science books for general audiences. But this is no breathless popularization. Alexander makes an original argument about the meaning of early modern mathematics that is well worth the attention not just of historians of mathematics but also of anyone interested in the long history of what historians of science used to call the Scientific Revolution. More than that, Alexander shows a creative and engaging way of narrating significant transformations in European history as though mathematics were not just relevant but crucially important, where rich contextual details and close readings of scientific texts are not incidental window dressing but central explanatory resources. All historians of science should take note.

Infinitesimal is Alexander's answer to a variation on the famous Needham question about why the Scientific Revolution took place in Europe rather than in China. Here, Alexander questions why the calculus and empirical science emerged in Britain and northern Europe but not in Italy, the dynamic hub of Renaissance mathematics and natural philosophy. The first part of his explanation ties the fate of Italian mathematics, science, politics and religion to the shifting fortunes of the Society of Jesus, whose ultimate triumph consigned infinitesimals and their forward-thinking users to the Italian scholarly fringe. The second part locates the Italian road not taken in the Royal Society of London, where infinitesimals formed the core of an experimental, non-dogmatic approach to mathematics that helped bring about political, religious, scientific and technological modernity as we know it.

Alexander begins with a gallivanting tour of sixteenth-century religious and political Europe, starting with the Reformation and then tracing the Jesuits' origins, expansion and eventual dominance over European scholarship and education. He then introduces the towering Jesuit geometer Christopher Clavius, whose decisive use of mathematics in the debates that culminated in the Gregorian calendar reform of 1582 helped raise mathematics' stature among the Jesuits as both a tool in itself and a model of truth and authority in Catholic theology and politics. Infinitesimals enter the story through Galileo and Cavalieri, the latter of whose ‘method of indivisibles' threatened Clavius's geometrical world view by implying that mathematics could productively harbour paradoxes and plural approaches that strayed from pure Euclidean deduction. The remainder of Part I traces the parallel doctrinal, political and mathematical battles between the intellectual descendants of Clavius and Cavalieri, culminating in indivisibles' inclusion in four of the sixty-five prohibited theses of the 1651 Jesuit Ordinatio pro studiis superioribus and the 1668 papal suppression of Cavalieri's monastic order of the Apostolic Clerics of Saint Jerome.

The book's second part, titled ‘Leviathan and the infinitesimal’, opens with the English Civil War and Restoration before introducing the fiery iconoclast Thomas Hobbes. As the title indicates, Alexander here presents a revision to the social-historical catechism of Shapin and Schaffer's Leviathan and the Air-Pump (1985). For Alexander, Hobbes embodies the pretensions of an aristocratic elite against a tolerant and liberal-minded Royal Society represented by a scrappy John Wallis, the Oxford geometer whose ‘Absurd Geometry, Rural Language, Scottish-Church-Politicks, And Barbarismes' furnished the title of one of Hobbes's less-studied philosophical polemics, and whose vicious two-decade feud with Hobbes ended only with the latter's death in 1679. It would be hard to boil this dispute down to the problem of infinitesimals, and Alexander mostly refrains from such reductionism. Instead, he offers a compelling narrative tying competing values and styles of mathematical reasoning to larger political and ideological debates in Interregnum and Restoration England, with Hobbes's shifting and contradictory geometry faring no better than his natural philosophy in Shapin and Schaffer's account.

Specialists will find plenty in Alexander's book with which to quibble. For instance, he dismisses centuries of important Islamic contributions to mathematics, including to the theory of infinitesimals, with a passing claim that the topic remained dormant between Archimedes and the Renaissance Italians. While Alexander displays a considerable sensitivity to the detailed historical context of mathematical arguments, he does allow occasional anachronisms about the concepts and social position of mathematics and mathematicians, which help him stress the differences between past and present views. Alexander omits many complicating nuances and exceptions in order to frame a more cohesive picture of the Manichean implications of infinitesimal mathematics for both its promoters and its detractors.

There is a provocative asymmetry in both parts of Alexander's story. For the Jesuit Clavius and the royalist Hobbes, a geometry purged of paradoxes was important in large part for what it could contribute to philosophy and politics. For the pair's freewheeling infinitesimal-championing opponents, indivisibles were significant primarily for what they permitted within the domain of mathematical reasoning. As Shapin and Schaffer did for Boyle, Alexander shows how such appearances of modest and pragmatic intellectual preoccupations also entailed claims about social and political order. One need not subscribe to Alexander's sweeping conclusions to appreciate what he shows with palpable drama and admirable clarity: that the histories of science, mathematics and much else can hinge on the social and political stakes of seemingly anodyne mathematical stipulations. Like Hobbes, Clavius was right.