The Calculus

Abstract

Abstract With an absurd oversimplification, the “invention” of the calculus is sometimes ascribed to two men, Newton and Leibniz. In reality, the calculus is the product of a long evolution that was neither initiated nor terminated by Newton and Leibniz, but in which both played a decisive part. Scattered over seventeenth century Europe, for the most part outside the schools, was a group of spirited scientists who strove to continue the mathematical work of Galileo and Kepler. By correspondence and travel these men maintained close contact. Two central problems held their attention. First, the problem of tangents: to determine the tangent lines to a given curve, the fundamental problem of the differential calculus. Second, the problem of quadrature: to determine the area within a given curve, the fundamental problem of the integral calculus. Newton’s and Leibniz’ great merit is to have clearly recognized the intimate connection between these two problems. In their hands the new unified methods became powerful instruments of science. Much of the success was due to the marvelous symbolic notation invented by Leibniz. His achievement is in no way diminished by the fact that it was linked with hazy and untenable ideas which are apt to perpetuate a lack of precise understanding in minds that prefer mysticism to clarity. Newton, by far the greater scientist, appears to have been mainly inspired by Barrow (1630-1677), his teacher and predecessor at Cambridge. Leibniz was more of an outsider. A brilliant lawyer, diplomat, and philosopher, one of the most active and versatile minds of his century, he learned the new mathematics in an incredibly short time from the physicist Huygens while visiting Paris on a diplomatic mission. Soon afterwards he published results that contain the nucleus of the modern calculus. Newton, whose discoveries had been made much earlier, was averse to publication.