Method of indivisibles

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In geometry, the method of indivisibles has its roots in the atomist theories of Democritus as well as the work of Archimedes. In the 17th century Bonaventura Cavalieri coined the term as a name for his method of calculating areas of plane figures and volumes of solids. Cavalieri discussed the method as early as 1621 with his mentor Galileo. Related techniques exploiting the infinitely small were exploited by Johannes Kepler in calculating volumes of wine barrels and other problems. The ancient doctrine of continuum made out of indivisible "atoms" was contrary to Aristotelian dogma that held the continuum to be divisible indefinitely though only potentially. The Jesuit order's attitude to mathematics was shaped by Clavius already in the 16th century. They adhered to the geometry of Euclid and the philosophy of Aristotle as their benchmarks of mathematical rigor. This led them to oppose the method of indivisibles which they viewed as contrary to philosophy as well as to common sense. Some of the famous Jesuit critics of the method of indivisibles were Guldin, Mario Bettini, and Andre Tacquet, the latter holding that the method "makes war upon geometry to such an extent, that if it is not to destroy it, it must itself be destroyed." In addition to Cavalieri, the greatest Italian practitioners of the method were Evangelista Torricelli and Stefano degli Angeli. The Jesuits proceeded to stifle the Italian school of indivisibilists, issuing a series of decrees against indivisibles, notably in 1632, when the method was banned from being taught at the vast Jesuit network of colleges. In 1651 the method was placed on a list of over 60 permanently prohibited subjects. In the 1650s and 60s Angeli put up a spirited defense of the method, refuting the critiques of Bettini and Tacquet, but in 1668 the rival Jesuat order, that numbered among its illustrious members both Cavalieri and Angeli, was suppressed by Papal brief, signaling an end to a flourishing Italian school of mathematics that will not re-establish itself for centuries as a leading research school. James Gregory studied under Angeli from 1664 until 1668. From then on the centers of activity in what is soon to become the infinitesimal calculus shifts north of the Alps, to England and France.


  • Amir Alexander (2014). Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Scientific American / Farrar, Straus and Giroux. ISBN 978-0374176815. 
  • Kirsti Andersen. Cavalieri's method of indivisibles. Arch. Hist. Exact Sci. 31 (1985), no. 4, 291–367.

See also[edit]

Cavalieri's principle