Definition:Method of Exhaustion

Definition

The method of exhaustion is a technique for calculating an approximation to the physical extent of a geometric figure whose extremities are curved lines.

To measure the area of a circle, for example, this is done by:

Circumscribing a polygon around the circle

Inscribing a polygon inside the circle

Measuring the areas of those polygons

Noting that the area of the circle is between those two

Increasing the number of sides on those polygons to make them closer and closer to the circle and each other.

As the number of sides of the polygons increases, their areas become ever closer to each other.

Historical Note

The method of exhaustion was invented by Eudoxus of Cnidus in around $370$ BCE.

It was later exploited to good effect by Archimedes, and is used throughout Euclid's The Elements during the course of calculating volumes of solid figures.

Some sources credit Archimedes with invention of the method, but this is incorrect.

Sources

• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $2$: The Logic of Shape: Taming irrationals