Newton Invents Calculus

Around 1665, a young Isaac Newton began working out the mathematical methods now known as calculus, the study of change and accumulation. Leibniz would later arrive at the same ideas independently. Calculus gave scientists a precise language for motion, growth and curves, and it underpins physics, engineering, economics and much of modern science.

Newton was a student at Trinity College, Cambridge, when the university closed in 1665 because of an outbreak of bubonic plague. He returned to his family home at Woolsthorpe in Lincolnshire and spent much of the next two years there. By this time mathematicians had already built much of the groundwork. Descartes had joined algebra to geometry, Fermat had devised ways of finding tangents and maxima, and John Wallis and others had made progress on areas under curves. Newton's teacher Isaac Barrow had also noted the link between finding tangents and finding areas. What was missing was a general, systematic method.

Newton supplied much of it during this period, working largely alone. He developed what he called the method of fluxions, treating quantities as flowing and changing over time, with the rate of change being a "fluxion." He also worked on infinite series, including the generalized binomial theorem, and recognized that finding rates of change and finding areas are inverse operations, a result now called the fundamental theorem of calculus. Much of this survived in private manuscripts. Newton later described these years as remarkably productive, though the popular story of a single "miracle year" simplifies a process that continued for decades.

Newton was reluctant to publish. His methods circulated among a few colleagues in manuscript form, and a full account of his fluxions appeared in print only in the early 18th century. Meanwhile, the German philosopher and diplomat Gottfried Wilhelm Leibniz developed his own version in the mid-1670s and published it in 1684. His notation, including the integral sign and the dy/dx form, proved far more convenient and is still used today.

The result was a bitter priority dispute. Supporters of Newton accused Leibniz of plagiarism, and the Royal Society, with Newton influencing the proceedings, issued a report in his favor in 1712. Most modern historians conclude that the two men worked independently, though Newton clearly came first in time and Leibniz first in print. The quarrel divided British and continental mathematicians for generations, and British mathematics, tied to Newton's clumsier notation, fell somewhat behind until the 19th century.

Newton applied his methods in his Principia of 1687, which laid out the laws of motion and universal gravitation. Calculus has been central to science ever since.