The Development of the Calculus

The Greeks made the first step in the inquiry of the infinitely small quantities by an attempt to determine the area of curves. The method of exhaustions they used for this purpose consisted of making the curve a limiting area, to which the circumscribed and inscribed polygons continually approached by increasing the number of their sides. The area obtained was considered to be the area of the curve. The method of integration is somewhat similar, to the extent that it involves finding the limits of sums. Zeno of Elea (c. 450 B.C.) was one of the first to work with problems that led to the consideration of infinitesimal magnitudes, and Leucippus (c. 440 B.C.) and Democrites (c. 400 B.C.) taught that magnitudes are composed of indivisible elements in infinite numbers. Archimedes' (c. 225 B.C.) work was the nearest approach to actual integration among the Greeks: his first noteworthy advance was to prove that the area of a parabolic segment is 4/3 of the triangle with the same base and vertex, or 2/3 of the circumscribed quadrilateral. He also anticipated many modern formulas in his treatment of solids bounded by curved surfaces.