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.Īlexander tells a good story here, but doesn’t get far enough into the math for my tastes. 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. 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. 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).
Infinitesimals series#
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. 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. 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. 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).
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. 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.
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