Indivisibles and the cycloid in the early 17th century

Abstract

We observe the application of Bonaventura Cavalieri’s (1598 - 1647) method of “indivisibles,” a mathematical method popular in the early 17th century for finding the area contained by curvilinear spaces, to the problem of finding the area under one arch of the so-called “cycloid” curve, that is, the curve traced by a point fixed upon the circumference of a circle which rolls along a horizontal line. We first briefly discuss the method itself, as well as what is understood by the notion of “indivisible.” Next, we explicate two different solutions to the stated problem of finding the area under one arch of the cycloid curve, one from Gilles Personne de Roberval (1602 - 1675), the other from Pierre de Fermat (1601 - 1665). Attention is paid to the ways in which these solutions utilize the method of “indivisibles.” Emphasis is placed throughout on the relationship between the notion of “indivisible” and the notion of the infinite.