History of Calculus - Pioneers of Modern Calculus

Pioneers of Modern Calculus

In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative. In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed an adequality method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation. Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."

On the integral side, Cavalieri developed his method of indivisibles in the 1630s and 40s, providing a more modern form of the ancient Greek method of exhaustion, and computing Cavalieri's quadrature formula, the area under the curves xn of higher degree, which had previously only been computed for the parabola, by Archimedes. Torricelli extended this work to other curves such as the cycloid, and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. James Gregory, influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus in the mid-17th century. The first full proof of the fundamental theorem of calculus was given by Isaac Barrow.

Newton and Leibniz, building on this work, independently developed the surrounding theory of infinitesimal calculus in the late 17th century. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton provided some of the most important applications to physics, especially of integral calculus.

The first proof of Rolle's theorem was given by Michel Rolle in 1691 using methods developed by the Dutch mathematician Johann van Waveren Hudde. The mean value theorem in its modern form was stated by Bernard Bolzano and Augustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Important contributions were also made by Barrow, Huygens, and many others.