# History of Calculus: The Newton-Leibniz Controversy

The Newton-Leibniz Controversy was a dispute between the 17th-century mathematicians Isaac Newton and Gottfried Leibniz, over who had first invented calculus.

Newton had begun working on his form of calculus during his spare time in 1666, when he and many other students were sent home from Cambridge on account of the plague. He developed his method in order to solve physics problems, and he called it the “method of fluxions” (“fluxion” was his term for the derivative). However, he kept his findings to himself for nearly two decades, until publishing his Philosophae Naturalis Mathematica in 1687, which presented new (and now essential) theories of physical motion and used calculus to back them up.

Leibniz, on the other hand, only waited a decade to publish his results after finding the area under the curve in 1675. He published an explanation of differential calculus in 1684, and of integral calculus in 1686.

Since Leibniz published first, he received sole credit for discovering calculus. But Newton believed that Leibniz had plagiarized his own work, and had some circumstantial evidence in his favor: their network of colleagues overlapped, so Leibniz may have seen unpublished manuscripts, and some of Newton’s first descriptions of calculus (his “method of fluxions”) had appeared in a letter of correspondence to Leibniz.

Newton set out to prove this, and since his Philosophae had gained sufficient popularity to make him a scientific celebrity, he won the support of the British Royal Society. Leibniz, having few allies, was defenseless, and lived just long enough to hear the Royal Society proclaim Newton the sole discoverer of calculus a year before his death in 1715.

But nobody came out of the dispute well.

Leibniz had formulated calculus using a notation vastly more efficient that Newton’s, but because Newton claimed that the notation was meant to hide Leibniz’s treachery, England refused to use Leibniz’s notations as a matter of national pride (Leibniz was German). As a result of using outdated notation, British math fell behind the rest of the continent.

After reviewing Newton and Leibniz’s papers and correspondences, most modern historians have concluded that, although Leibniz likely saw some early manuscripts of Newton’s Philosophae, he had already come to his own conclusions by that time. Whereas Newton’s grounded his calculus in limits and concrete physical reality, Leibniz’s work was inspired by more abstract concepts and theory. Newton and Leibniz had invented calculus independently.

But even to say that Newton and Leibniz were the “sole inventors” of calculus, would be to discredit centuries of mathematical ideas leading up to the official “discoveries.” In the times of ancient Greece, Archimedes was the first to find the tangent line to a curve, and Antiphon developed the method of exhaustion for calculating area. In India, centuries before the European Enlightenment, Aryabhata expressed an astronomical problem as a differential equation, and Parameshvara developed an early version of the mean value theorem. Even during the Enlightenment, Fermat, Pascal, and Barrow had developed the concept of the derivative. Barrow even offered the first proof of the fundamental theorem of calculus, which links differentiation with integration. And Newton was Barrow’s own student.

Though calculus may often be said to be the creation of Newton and Leibniz, it was truly an invention of many mathematicians, over many centuries.

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