Amir Alexander’s Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World is less a history of math (although there is quite a bit) than a history of the people and institutions who fought a protracted philosophical battle over something we now consider a trivial bit of precalculus. The idea of infinitesimals, at the time of their development called “indivisibles,” sparked vociferous opposition from the supposedly progressive Jesuits in the 1600s, becoming part of their vendetta against Galileo, leading to banishments and other sentences against Italian mathematicians, and eventually pushing the progress of math itself from Italy out to Germany, England, and the Netherlands.
If you’ve taken calculus at any point, then you’ve encountered infinitesimals, which first appeared in the work of the Greek mathematician Archimedes (the “eureka!” guy). These mathematical quantities are so small that they can’t be measured, but their size is still not quite zero, because you can add up a quantity (or an infinity) of infinitesimals and get a concrete nonzero result. Alexander’s book tells the history of infinitesimals from the ancient Greeks through the philosophical war in Italy between the Jesuits, who opposed the concept of indivisibles as heretical, and the Jesuats, a rival religious order founded in Siena that included several mathematicians of the era who published on the theory of indivisibles, including Bonaventura Cavalieri. When the Jesuits won this battle via politicking within the Catholic hierarchy, the Jesuats were forced to disband, and the work involved in infinitesimals shifted to England, where Alexander describes a second battle, between Thomas Hobbes (yep, the Leviathan guy) and John Wallis, the latter of whom used infinitesimals and some novel work with infinite series in pushing an inductive approach to mathematics and to disprove Hobbes’ assertion that he had solved the problem of squaring the circle.
Wallis’ work with infinitesimals extended beyond the controversy with Hobbes into the immediate precursors of the calculus developed by Isaac Newton and Gottfried Leibniz, including methods of calculating the area under a curve using these infinitesimals (which Wallis described as width-less parallelograms). Alexander stops short of that work, however, choosing instead to spend the book’s 300 pages on the two philosophical battles, first in Italy and then in England, that came before infinitesimals gained acceptance in the mathematical world and well before Newton or Leibniz entered the picture. Hobbes was wrong – the ancient problem of squaring the circle, which means drawing a square using only a straightedge and compass that has the same area as that of a given circle, is insoluble because the mathematical solution requires the square root of pi, and you can’t draw that. The impossibility of this solution wasn’t proven until 1882, two hundred years after Hobbes’ death, but the philosopher was convinced he’d solved it, which allowed Wallis to tear Hobbes apart in their back-and-forth and, along with some of his own politicking, gave Wallis and the infinitesimals the victory in mathematical circles as well.
Alexander tells a good story here, but doesn’t get far enough into the math for my tastes. The best passage in the book is the description of Hobbes’ work, including the summary of the political philosophy of Leviathan, a sort of utopian autocracy where the will of the sovereign is the will of all of the people, and the sovereign thus rules by acclamation of the populace rather than heredity or divine right. (I was supposed to read Leviathan in college but found the prose excruciating and gave up, so this was all rather new to me.) But Alexander skimps on the historical importance of infinitesimals, devoting just a six-page epilogue to what happened after Wallis won the debate. You can’t have integral calculus without infinitesimals, and calculus is kind of important, but none of its early history appears here, even though there’s a direct line from Wallis to Newton. That makes Infinitesimal a truncated read, great for what it covers, but missing the final chapter.
Next up: The Collected Stories of Katherine Anne Porter, winner of the Pulitzer Prize for Fiction in 1966.