Calculus is the mathematical study of continuous change, and is split into two distinct but related fields – differential calculus and integral calculus. Differentiation allows us to find out how much something changes (e.g. the acceleration of a car), while integration allows us to work out how far the car has travelled if we know its rate of acceleration.
Calculus makes it possible to take very complex problems and divide them into a series of much smaller sections, allowing us to reach an exact answer. This is useful when dealing with quantities that change in a non-linear way (from the speed of a rocket as it soars through space to the flow of electricity around a circuit). If you imagine a mathematics problem illustrated by curves on a graph, differential calculus concerns the slopes of the curves and integral calculus looks at those areas between and beneath the curves.
Elements of calculus were developed in antiquity by mathematicians from traditions as disparate as the Greek, Indian, Chinese, Japanese and Islamic. It was in Europe in the 17th century, that the Italian, Bonaventura Cavalieri (1598–1647), showed how volumes and areas may be computed as the total of the volumes and areas of infinitesimally small cross-sections. René Descartes, Pierre de Fermat, Blaise Pascal and John Wallis also made significant strides.
But who ‘invented’ it?
Credit for ‘inventing’ calculus became the subject of a ferocious dispute between Isaac Newton on one side and Gottfried Leibniz on the other. Leibniz introduced much of what became the standard theory and notation, while Newton was the first to apply calculus practically, to explain, for instance, planetary motion in his Principia (1687). Newton accused Leibniz of plagiarism, accusations that bedevilled the German until his death. Remarkably, the likely truth is that two great minds independently hit upon similar groundbreaking insights contemporaneously and divided by several countries. Both took what had hitherto been rather ragged and inconsistent ideas and theories, and synthesized them into a fully functioning system of mathematics.
Today, calculus has applications in an extraordinary number of fields – from physics and astronomy to mechanical and electrical engineering, and even in the social sciences.