Method of Indivisibles

In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:

Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, which used limits but did not use infinitesimals.

Cavalieri's principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe. Archimedes was able to find the volume of a sphere given the volumes of a cone and cylinder using a method resembling Cavalieri's principle. In the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume. The transition from Cavalieri's indivisibles to Evangelista Torricelli's and John Wallis's infinitesimals was a major advance in the history of the calculus. The indivisibles were entities of codimension 1, so that a plane figure was thought as made out of an infinity of 1-dimensional lines. Meanwhile, infinitesimals were entities of the same dimension as the figure they make up; thus, a plane figure would be made out of "parallelograms" of infinitesimal width. Applying the formula for the sum of an arithmetic progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞.

If one knows that the volume of a cone is ${\displaystyle {\frac {1}{3}}\left({\text{base}}\times {\text{height}}\right)}$, then one can use Cavalieri's principle to derive the fact that the volume of a sphere is ${\displaystyle {\frac {4}{3}}\pi r^{3}}$, where ${\displaystyle r}$ is the radius.

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