Most of the book is devoted to the interrelated stories of many leading mathematical/religious figures during these tumultuous early modern times. They include the priest Christopher Clavius and his mathematical work on the new Gregorian calendar, and the relentless Jesuit censors and the dead weight they finally succeeded in placing on Italian mathematics by proscribing infinitesimals.

Along the way there are striking geometric insights that buttress the case for infinitesimals. Cavalieri’s principle, for example, states that if two solids are included between parallel planes and cross-sections of these two solids at every height have equal areas, then the solids have the same volume.

Consider, for example, two towers of coins, one straight, the other of equal height but with the coins sticking haphazardly out the sides. It’s intuitive that the towers have equal volume, and would even if the coins were “infinitely thin” and the towers leaned and twisted every which way. Formulas for the areas and volumes of geometric figures were surprisingly easy to obtain using this principle, which was a precursor of integral calculus.

No one talks of infinitesimals any more: The modern notion of limits accomplishes everything they did, but much more rigorously. One exception is a recent reconstruction of infinitesimals — positive “numbers” smaller than every real number — devised by the logician Abraham Robinson and developed further by H. Jerome Keisler, my adviser at the University of Wisconsin.

Since the Jesuits succeeded in banning infinitesimals in Italy, the last part of Dr. Alexander’s finely detailed, dramatic story traces their subsequent history north to England. There one of the key figures is Thomas Hobbes, the 17th-century philosopher of authoritarianism, a strong advocate of law, order — and, like the Jesuits, of the top-down hierarchical nature of Euclidean geometry.

Hobbes’s hated antagonist, the mathematician John Wallis, used infinitesimals freely, along with any other ideas he thought might further mathematical insight. And further it they did, leading over time to calculus, differential equations, and science and technology that have truly shaped the modern world.