Cards on the table: I approached Amir Alexander’s book warily. The title places it in a suspect genre; according to an Amazon search, our overdetermined world has been decisively shaped by, among many other things, codfish, cotton, Islam, the Scots, air conditioning, the Lunar Society of Birmingham, and, if we include the testimony of documentary films, beer. And the book’s introduction suggests a morality play that flatters all the prejudices of the right-thinking: “On the one side were the advocates of intellectual freedom, scientific progress, and political reform; on the other, the champions of authority, universal and unchanging knowledge, and fixed political hierarchy.”

Much more was at stake than mathematical technicalities.

Well and clearly written, Infinitesimal tells two stories about a seventeenth-century mathematical dispute. The protagonists of each are developing and applying a technique (“the method of indivisibles”) that promises novel methods of discovery and proof. That technique is also rife with paradoxes and contradictions, and leads easily to errors; its advocates can give no rigorous account of what they’re doing, and some dismiss the need for one. Their antagonists uphold the rigorous ideals of classical mathematics, modeled on Euclid. For them, Alexander says, these ideals are a bulwark against chaos and disorder, not only in mathematics but in thought and politics. As a result, he argues, much more was at stake than mathematical technicalities.

The antagonist of the first story is the Jesuit order, trying to restore the unity and authority of the Roman Church