Amir Alexander's latest entry in the history of mathematics aimed at a general audience (a surprisingly robust genre) focuses on the role of infinitesimals – those paradoxical objects smaller than any magnitude yet still greater than zero – in early modern religious and political thought. This is a story about not the calculus of Newton and Leibniz, but rather the myriad infinitistic problem-solving techniques – most prominently Bonaventura Cavalieri's method of indivisibles and John Wallis's arithmetic of infinities – that laid the groundwork for Newton and Leibniz's grand syntheses. Alexander's argument is not modest: he asserts in his introduction that ‘the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms on a scale unknown in human history’ (p. 14). This audacious claim turns out to refer chiefly to the late seventeenth-century rivalry between Thomas Hobbes and John Wallis over indivisibles and the squaring of the circle, a topic previously covered in depth by Douglass M. Jesseph. Unlike other scholars, Alexander focuses less on the mathematical and scientific results of infinitesimals (the full power of calculus would not be realized until the eighteenth century) and more on what he calls the ‘war over the infinitely small’ (p. 174) in Italy and England. Like any war, the superficial conflict arose from far deeper and more profound cultural struggles, and Alexander illuminates the political, social, theological and philosophical dimensions of the early modern mathematics of infinitesimals.

Chapter 1 provides a crash course on the Counter-Reformation and the rise of the Jesuits, while Chapter 2 homes in on the greatest Jesuit mathematical master – Christopher Clavius – and his worship of geometry as a virtually divine source of infallible truth. Clavius strove to consolidate mathematics, and indeed all knowledge, in the form of rigorous demonstration from first principles. Introducing Cavalieri's indivisibles and their adoption by Galileo and Torricelli in their new science of mechanics, Chapter 3 showcases Alexander's considerable skill in explicating historical mathematics to modern readers. Chapters 4 and 5 offer reviews of the Jesuits’ denunciation of indivisibles as a threat to mathematical order and their crusade to stamp out the scourge of the infinitely small in Italy. Thus far more than a mathematical curiosity, Alexander portrays the infinitely small as a battleground upon which ‘the path of Italian modernity was fought over and decided’ (p. 180).

In Chapter 6 Alexander shifts his focus to England during the Civil War and to the deep mark it had left on influential thinkers such as Thomas Hobbes and his generation. Here Alexander offers his most intriguing and original argument by contending that Hobbes's and Wallis's attitudes towards mathematical method mirrored their philosophies on governance and the proper English state. As Clavius did with theology, in the Leviathan Hobbes formed his political theory in the image of Euclid's Elements, idolizing geometry as an unshakable bastion of rigour and certain truth to be emulated by all forms of knowledge. Hobbes could not tolerate the infinitely small for reasons similar to the Jesuits’. In contrast, Wallis, as the first mathematician admitted to the Royal Society, advocated a pluralistic, freethinking society wherein uncompromising certainty had no place. In his Arithmetica Infinitorum (1656) Wallis took as its patron saint not Euclid but Francis Bacon, and arrived at his famous formula for π through a kind of Baconian induction (not to be confused with ‘induction’ proofs that mathematicians employ today) that embraced infinitesimals while making no claim to Euclidean rigour. Alexander argues that Wallis's mathematics – deliberately non-rigorous, embracing a plurality of approaches – suited the English cultural moment: fresh out of the Civil War and Interregnum, with no appetite for Hobbes's unbending dogmatism. Hence Wallis invented a uniquely English mathematics that would soon factor heavily into Newtonian calculus and mathematized natural philosophy. The rest, as they say, was history.

Overall, Alexander provides an excellent and readable overview of this pivotal chapter in the history of mathematics while advancing bold and well-defended arguments that would keep historians of science interested. However, the direct causal connection between infinitesimals and political/religious order asserted in the introduction is nowhere to be found: at most Alexander shows, first, how the Jesuits nipped Italian infinitesimal techniques in the bud and, second, how the English political and religious climate proved much more conducive to non-traditional mathematics. While his conclusion somewhat underwhelms, Alexander nevertheless brilliantly links mathematics, theology and political philosophy while revealing the pull that mathematics exerted on early modern philosophers and theologians. The book could have benefited from a more nuanced discussion of mathematical analysis (ways of finding solutions to problems) and synthesis (the rigorous demonstration of solutions) as understood in the sixteenth and seventeenth centuries. While the exclusion of such detailed examinations for the sake of accessibility is understandable, without this groundwork Alexander occasionally has difficulty explaining how Cavalieri or Wallis conceptualized their methods (strictly analysis) and why they did not see the lack of Euclidean rigour as a problem. Quibbles aside, Alexander's work exemplifies an increasingly accepted truism within the history of mathematics: that mathematics is never just mathematical. Like anything else, it shapes and is shaped by historically contingent social, cultural and intellectual factors. This book should appeal to professional historians of science and mathematics as well as to more general audiences.