Archimedes made many significant contributions to mathematics, though not all mathematicians would agree with the label “most significant.” But one of his contributions did advance the field of calculus by showing that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex (endpoint), and 2/3 the area of the circumscribed parallelogram.

To figure this out, he constructed an “infinite” sequence of triangles (or wedges), finding the area of segments composing the parabola. He began with the first area, A, then added more triangles between the existing ones and the parabola to get areas of:

Based on his iterations, he determined the following (the first time anyone had determined the summation of an infinite series; for more about infinite series, see below):

Archimedes also applied this method of exhaustion (not literally becoming tired, but close to it) to approximate the area of a circle, which, in turn, led to a better approximation of pi (π). Using such integrations, he also determined the volume and surface area of a sphere and cone, the surface area of an ellipse, and many others. His work is considered the first steps toward integration that would eventually lead to integral calculus. (For more information about Archimedes and his wedges, see “Geometry and Trigonometry.”)

Astronomer and mathematician Johannes Kepler used sectors to calculate the shape of the elliptical paths of the planets.