The Method of Exhaustion

Eudoxus of Cnidus (390–337 BCE) and later Archimedes used the method of exhaustion: approximate a curved area with polygons of increasing number of sides. The area is 'exhausted' by the polygons from inside and outside simultaneously. This is the ancestor of integration — slice an area into infinitely many thin strips, each simple to measure, and add them up. Archimedes used it to prove that the area of a parabolic segment equals 4/3 the area of the inscribed triangle. His method anticipated Newton and Leibniz by nearly 2,000 years.