One of the joys of modern academic life is the multicultural camaraderie of free and robust intellectual debate. In the mathematical sciences in particular, ideally we can put aside the world¡¯s religious and political turmoil, along with our own personal and cultural differences, in order to discuss universal scientific ideas in the peace and civility of university tea rooms. This isn¡¯t just a superficial modern gloss: it is an ideal with a long and fascinating history, as Amir Alexander shows in Infinitesimal.

In the early 1640s, amid the terror and chaos of the English Civil War, a group of mathematicians and ¡°natural philosophers¡± (physicists) began to meet regularly at each other¡¯s homes in London. Eventually, their ¡°invisible college¡± would become chartered as the Royal Society of London ¨C now one of the world¡¯s oldest and most respected scientific bodies ¨C but in those early years, the purpose of these gatherings ¡°was no more than breathing freer air, and of conversing in quiet with one another, without being engaged in the passions and madness of that dismal age¡±. These poignant words are those of the Royal Society¡¯s first historian, Thomas Sprat, and Alexander effortlessly elucidates the religious, political and class ¡°passions¡± that polarised and divided England at that time. But the unique aspect of his telling of this history is his contention that such political and religious polarisation was reflected in a mathematical debate.

¡®Infinitesimals¡¯ were uncertainties that troubled both the Jesuits and Thomas Hobbes for mathematical as well as political reasons

Consequently, Infinitesimal¡¯s historical reach includes not only England but also Italy, because the end result of this debate was different in each country. Alexander begins his story in Rome, a decade before those first clandestine meetings of the Royal Society, but the backdrop is similar: a cacophony of dogmatic voices, unleashed in this case by the Reformation and Counter-Reformation, which sparked the politico-religious dramas of the Thirty Years War. It is 1632, a year that witnesses the events leading up to the trial of Galileo Galilei, charged with heresy for claiming as a proven fact Copernicus¡¯ theory that the Earth moves around the Sun. But Alexander focuses instead on a lesser-known, more subtle ¡°heresy¡±: that the continuum, or real line, is composed of tiny ¡°indivisibles¡± or ¡°atoms¡±. Five Jesuit ¡°Revisors¡± ¨C who decide what can and cannot be taught in Jesuit schools ¨C meet on 10 August 1632 to discuss this proposition, and pronounce it both ¡°improbable¡± and contrary to the teachings of Aristotle, the Catholics¡¯ favoured authority on science and philosophy. In other words, it is ¡°condemned and prohibited¡±.

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It¡¯s an astonishing way to begin a book, but this is no trite tale of anti-scientific Catholicism: Alexander then backtracks to show the intellectual and spiritual depth behind such a prohibition, through a thrilling story of the rise of the Jesuits and their desperate efforts to lead war-torn Europe back to the certainties and order of Catholic absolutism. Later, in part two, he juxtaposes this story with that of the English political philosopher and geometer Thomas Hobbes. Instead of seeking religious absolutism ¨C something he despised ¨C Hobbes¡¯ solution to the anarchy of the Interregnum was a model in which the people cede power to an absolute sovereign (a king, an aristocracy or an elected body) who will protect them from the war and chaos that would otherwise overwhelm them. As with Catholic absolutism, Hobbes¡¯ sovereign decrees what opinions are allowed, the goal being to preserve the peace. Both Hobbes and the Jesuits backed up their political ideals by appealing to Euclidean geometry, with its orderly deductive proofs yielding (apparently) absolute truths.

Rigorous geometry is then contrasted with early modern attempts to grapple with the paradoxes and uncertainties in the mathematics of ¡°infinitesimals¡± ¨C uncertainties that troubled the Jesuits and Hobbes for mathematical as well as political reasons. These paradoxes would not be resolved until the 19th century, when the ¡°limit¡± concept was made rigorous. The focus here, however, is the 17th century, when innovative mathematicians showed that infinitesimal methods were useful even if they were not yet rigorous. Extremely useful, in fact: they underlie calculus, the mathematics of change that has made possible modern theoretical and applied physics.

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Part one counterpoises the Jesuit hierarchy¡¯s insistence on teaching only time-honoured Euclidean geometry with the innovative use of infinitesimals by men such as Galileo, who used them in deducing his law of falling bodies, and Bonaventura Cavalieri, who in 1635 published a landmark book on infinitesimal methods, which he used to lay some of the foundations of calculus. The story includes a number of other contributors to this new mathematics, some of whom were Jesuits who struggled to reconcile their ideas with official doctrine. In part two, the key innovator is John Wallis, a founding member of those early Royal Society meetings. Alexander suggests that the society¡¯s scholarly ideal of rational pluralism provided an alternative vision of a stable state to Hobbes¡¯ all-powerful sovereign (or ¡°Leviathan¡±).

To help the reader engage with the debate, accessible explanations of some of the mathematics are included (about 30 pages in all) ¨C a creditable and useful addition, although I felt that occasionally 17th-century terminology was not defined clearly enough for a modern readership. There is also a rather loose use of the terms ¡°infinitesimal¡± and ¡°indivisible¡±. But this is a popular book, not a scholarly one, and Alexander pulls off the impressive feat of putting a subtle mathematical concept centre stage in a ripping historical narrative that I found even more satisfying in part two.

Perhaps it is because here there are just two key players, Hobbes and his nemesis Wallis. Wallis was England¡¯s leading mathematician in the years before Isaac Newton astonished the world with his formalisation of the algorithms of calculus and his revolutionary theory of gravity. The tale of Hobbes and Wallis, the geometer and the infinitesimalist, is told in the context of their well-documented (and decidedly uncivil) intellectual and political ¡°war¡± over infinitesimals.

I was initially uneasy about the adversarial approach Alexander takes in part one, pitting the Jesuit hierarchy against the infinitesimalists. For the purposes of a gripping narrative, transitional figures such as Nicole Oresme are left out, and the fact that most mathematicians, not only Jesuits, favoured geometry is glossed over. In part two, too, the narrative requires winners and losers, and here geometry is the loser. But Alexander does not mention, for example, that Newton himself was so concerned by the novelty and lack of rigour in his new calculus that he chose to present his theory of gravity mostly in terms of Euclidean geometry.

Nevertheless, this is a complex story told with skill and verve, and overall Alexander does an excellent job of presenting both sides of the debate. I particularly liked his treatment of the way innovators such as Wallis highlighted the difference between proving known facts rigorously and developing new knowledge with more heuristic methods. He thereby shows that mathematics is a far more flexible and radical tool than many lay readers realise.

There is much in this fascinating book, and it makes an interesting case for its intriguing conclusion: Renaissance Italy had led the world in art, science and mathematics, but it was England where ¡°the face of modernity¡± emerged, with unprecedented political and religious freedoms reflected in the freedom and flexibility offered by this new kind of mathematics.