Calculus

Calculus is the branch of mathematics that deals with continuous change, built from two operations that turn out to be inverses of one another. Differentiation finds the instantaneous rate at which a quantity is changing, the exact speed of a car at one moment rather than its average speed over a trip, or the exact slope of a curve at a single point rather than the slope of a straight line connecting two distant points on it. Integration does the reverse, reconstructing a total quantity, such as the distance a car has traveled, from a complete record of how fast it was going at every moment along the way.

Both operations rest on the idea of a limit: what happens to a calculation as some quantity, an interval of time, a segment of a curve, is allowed to shrink toward zero without ever quite reaching it. Ancient mathematicians, including Archimedes, had used limiting arguments to calculate areas and volumes centuries earlier, but it took until the late seventeenth century for the general, systematic machinery of calculus, applicable to essentially any smooth curve or changing quantity, to be worked out, independently, by Isaac Newton and Gottfried Wilhelm Leibniz.

The fundamental theorem of calculus, which formally connects differentiation and integration as inverse operations, is the result that ties the whole subject together, and it is difficult to overstate how much of later physics, engineering, and applied mathematics depends on it. Newton's laws of motion are differential equations, describing how a planet's position changes because of the forces acting on it, or how a bridge bends under load, or how a population grows over time, all require exactly the machinery calculus provides.

Isaac Newton developed the mathematical methods behind calculus, which he called the method of fluxions, in the mid-1660s, while Cambridge University was closed because of plague, but he did not publish the work, it circulated only among a small circle of colleagues in manuscript form for decades. Gottfried Wilhelm Leibniz developed an independent version in the 1670s while working in Paris and Hanover, and published his results starting in 1684, using a system of notation, including the integral sign and the dy over dx notation for a derivative, that proved clearer and more flexible than Newton's, and remains the notation used almost everywhere calculus is taught today.

When Newton's supporters began accusing Leibniz of plagiarizing Newton's unpublished work in the early 1700s, what might have been a private academic disagreement became a public, nationalistic scandal. The Royal Society, of which Newton was president, appointed a committee in 1712 to investigate and rule on priority. The committee's report, Commercium Epistolicum, found in Newton's favor, a verdict that carried the Royal Society's institutional authority but was compromised from the start, since Newton himself wrote large portions of the supposedly neutral report anonymously and orchestrated much of its content.

The dispute poisoned relations between British and continental mathematicians for most of the eighteenth century, British mathematicians, out of loyalty to Newton, continued using his more awkward fluxion notation long after continental mathematicians, using Leibniz's system, had made faster progress, a self-inflicted setback historians of mathematics have long noted. The modern consensus, reached only once nationalist stakes had faded, is that Newton and Leibniz arrived at calculus independently, by different conceptual routes, at nearly the same historical moment, and that both deserve credit for the discovery.