A Short History of Calculus: From Archimedes to Weierstrass

The two basic questions of calculus are very old: how do you find the area of a curved shape, and what is the speed of a changing quantity at a single instant? Ancient Greece gave the first serious answer to the first question. In the method of exhaustion, developed by Eudoxus, a shape is filled with ever smaller pieces whose areas are known. Archimedes (3rd century BC) used it to find the area of a parabolic segment and the volume of a sphere; what he did was the idea we now call the integral, written without the word limit.

In the following centuries the idea advanced in different parts of the world. In the 11th century Ibn al-Haytham found formulas for sums of powers and used them to compute the volume of a paraboloid. In India, Madhava of the Kerala school arrived in the 14th century at infinite series for sine, cosine and arctangent; Europe would rediscover these series only three centuries later. In the first half of the 17th century the pieces began to come together: Descartes and Fermat described curves by equations, Fermat developed a method for finding tangents and maxima and minima, Cavalieri computed areas with "indivisibles", and Barrow saw in geometric form the relation between the tangent problem and the area problem.

Two people turned this accumulated work into a single method. Isaac Newton developed his method of "fluxions" in 1665-1666 but was very slow to publish it. Gottfried Wilhelm Leibniz reached the same ideas independently in the mid-1670s and published in 1684. The notations $\frac{dy}{dx}$ and $\int$ that we use today are Leibniz's. The real discovery of both was to make the fact that differentiation and integration are inverse operations, the fundamental theorem of calculus, into a general method of computation. The quarrel over who was first lasted for years and damaged both sides; today historians agree that the two worked independently.

The method worked extraordinarily well; in the 18th century the Bernoulli brothers and Euler used it to rebuild mechanics, astronomy and the theory of differential equations. But its foundation was not sound: an "infinitely small" quantity was divided by as if it were not zero and then discarded as if it were. In 1734 Bishop Berkeley mocked these quantities as "the ghosts of departed quantities", and his criticism was justified. The answer came a century later: in the 1820s Cauchy based calculus on the concept of limit, and Weierstrass made the limit precise with the epsilon-delta definition taught in courses today. Riemann defined the integral; Dedekind and Cantor put the real numbers themselves on firm ground.

This history says two things to a student. First, calculus is not something a genius discovered in one night; it is the product of two thousand years of shared work stretching from Greece to Iraq, from India to Europe. Second, it is entirely natural for the limit to feel difficult: after humanity started using the derivative, it took about a hundred and fifty years to define it correctly. When a calculus course starts with limits and moves on to derivatives and integrals, you are starting from the point history reached last.