In the autumn of 1656, the Savilian professor of geometry at the University of Oxford, John Wallis, published an attack on the mathematics of Thomas Hobbes entitled Due correction for Mr Hobbes or schoole discipline for not saying his lessons right. Wallis took aim at several targets presented by Hobbes in his Leviathan, which had appeared five years earlier. These included the use made by Hobbes of the principles of Euclidean geometry, the criticism that Hobbes advanced of the English universities (which he supposed to be stifled by the bonds of Aristotelian or school philosophy), and the attack mounted by Hobbes on the political and intellectual power of the clergy, which was accompanied by suggestions that the author of Leviathan held heterodox ideas about the nature of God. In particular, Wallis focused on the implications of the claim that Hobbes had made elsewhere that he was able to provide a rigorous method of constructing a square equal in area to a given circle. At the heart of the discussion lay the mathematical and philosophical term infinite. What did it mean to suggest that God might not be infinite; how should mathematicians describe the infinite divisions of a line or area; what was implied by talk about an infinite number of terms in a mathematical series? The stance taken by Wallis defined his positions: in defense of a divine being that could not be limited or purely material and in favor of a mathematics based on the principles of arithmetic, as much as those of geometry, and inductive in its reasoning.

The quarrel between Wallis and Hobbes (which has been superbly treated in print by Douglas Jesseph) shapes the second part of Amir Alexander’s quirky and readable book about the early seventeenth-century history of the use of infinitesimals. The first part concentrates on an Italian story, led by the geometrician Bonaventura Cavalieri, and focusing on the decision in 1632 of the Jesuit order to forbid the teaching of the infinite division of the continuum of material particles into ever smaller units. As in the debate between Wallis and Hobbes, the point of this discussion was not confined to geometry but reached out into questions of the makeup of the physical world, which had profound consequences for teaching. Those consequences extended to issues of the understanding of doctrine as well as nature, not only in terms of the language used about God, but also through consideration of the real presence in the Eucharist.

Alexander wants to tell a broad story. His readers are treated to lengthy and digressive potted accounts of the Reformation, the foundation of the Jesuits, and the causes of the English Civil Wars. They are also told that the history of infinitesimals is related directly to these themes. The defense of the faith supposedly encouraged Jesuit mathematicians to reject infinitesimals, in a way that was also tied up with their competition with other religious orders (including that to which Cavalieri belonged). The resulting banishment of advanced mathematics from Italian education (Alexander asserts) helped to explain the translation of economic power from Italy to England. The triumph of a democratic science in the Royal Society, exemplified in the arguments with which Wallis made infinitesimals acceptable, helped to lay the foundations of a new mathematical age, embodied in the successes of Newton.

This makes for an exciting but often rather crude tale. It can be faulted on points of detail (from the misdating of the publication of the work of Copernicus to the claim that the Church of England was the only Protestant church to have bishops), but its main shortcomings are more profound. In part, they consist of particular sins of omission: whether of Islamic or medieval considerations of the problem of infinite divisibility or the role of French and early seventeenth-century English mathematicians in shaping a debate that did not derive entirely from Galileo’s physics. There are also misunderstandings, be they of Zeno’s paradoxes or of the relationship of Wallis’s arguments of the 1650s to the Royal Society (founded in 1662). It is unfortunate that Alexander coins the term “Jesuat” to describe Cavalieri’s order, the Jesuati, and (without a decisive secondary literature on which to rely) attributes their suppression in 1668 to mathematical rather than moral failings. But the real problem with this book is its failure to explain the contemporary relationship between mathematics and natural philosophy, in the context of widespread institutional and intellectual (rather than political or economic) change. It is thus unclear to the reader why in fact infinitesimals mattered so much in the period between 1630 and 1670 and how they came to be so important in the development of mathematics over the following century.