Buonaventura Cavalieri

Introduction: a geometry of indivisibles

    Galileo's books became quite well known around Europe, at least as much for their notoriety as for their scientific content.  One of those who was greatly influenced by Galileo was Buonaventura Cavalieri (1598-1647), a Jesuat priest (the Jesuats, or Theatines, are not to be confused with the Jesuit priests, whose numbers were much greater), and university-trained mathematician.  Cavalieri taught briefly at the University of Pisa, but obtained the chair of mathematics at the University of Bologna in 1629, where he taught until his death. In 1635, he published Geometria indivisibilis continuorum nova quadam ratione promota (A Certain Method for the Development of a New Geometry of Continuous Indivisibles) elucidating methods for computing areas, volumes and centers of gravity of figures by means of indivisibles.  This method treated a plane region as being made up of infinitely many parallel lines, each considered to be an infinitesimally thin rectangle and an indivisible part of the region (since its width cannot be further subdivided); the area of the region therefore consisted of "all the lines", as Cavalieri would say.  Specifically, the area under a curve y = f(x) between x = a and x = b is divided into indivisibles, one at each point x on the base of the figure along the x-axis.  Each of the indivisibles is simultaneously viewed as a one-dimensional line segment and as an infinitesimally thin two-dimensional rectangle.  The indivisible at point x has height f(x) and width dx (using what would be Leibniz' notation from the late 1600s).  Therefore it had area f(x)dx.  The area of the entire region was thus the sum of these areas: f(x)dx.  Similarly, a solid figure was made up of indivisibles each of which was a parallel cross-sectional slice of the solid, and the volume consisted of "all the planes".

    It soon became apparent that the method of indivisibles was subject to some paradoxical behavior.  While it produced correct results in many cases, including the situation spelled out in the reading below, it could also be used in ways that led to obvious false results.  For instance, how was it possible to add up indivisible areas, each of which was smaller than any positive number and obtain a positive sum?  For another, consider a scalene triangle and, by dropping the altitude to the base of the triangle, partition it into two triangles of unequal area.  Both the left and right triangles consist of "all the lines" in each according to Cavalieri.  But we can easily see that each indivisible line in the left triangle corresponds to an equal indivisible line in the right one, implying that both triangles must have equal area!

    This sort of strange behavior worried a number of mathematicians, and Cavalieri was criticized for using a method that could produce such absurdities.  (In fact, these criticisms forced him to rewrite the Geometria indivisibilis and publish a second edition of the work, from which our selection comes.)  Still, he and others were successful in using the method to solve many geometrical problems, so there was clearly value in proceeding with such explorations.  It would be left for later thinkers to resolve these paradoxes.
 

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