Method of Exhaustion

The method of exhaustion (methodus exhaustionibus, or méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The idea originated in the late 5th century BC with Antiphon, although it is not entirely clear how well he understood it. The theory was made rigorous a few decades later by Eudoxus of Cnidus. The first use of the term was in 1647 by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region (which can be “exhausted” so that its area becomes arbitrarily close to the true area). The proof involves assuming that the true area is greater than the second area, and then proving that assertion false, and then assuming that it is less than the second area, and proving that assertion false, too.

The method of exhaustion is seen as a precursor to the methods of calculus. The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries (in particular a rigorous definition of limit) subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. An important early intermediate step was Cavalieri's principle, also termed the "method of indivisibles", which was a bridge between the method of exhaustion and full-fledged integral calculus.