In 1661, Newton entered Cambridge University, where he was awarded a master’s degree in 1668. He was for the most part self-taught, learning his mathematics from books, especially from Descartes’s Géométrie and Wallis’s Arithmetica infinitorum. During the two years 1664-1665, when an outbreak of the Great Plague closed the university, Newton remained in seclusion at home. In these "wonderful years," he began to do his own original research. Beginning in 1664 he laid the foundations of the differential calculus, which he described as the "method of fluxions"; and, in 1665, he began investigating the "inverse method of fluxions," or the integral calculus. Newton formulated his principle of universal gravitation in the same period. This idea culminated in his masterwork, the Principia Mathematica (1687), which explains the motions of the heavenly bodies in the language of mathematics. In 1669 Newton’s former teacher resigned his professorship in favor of his pupil, who by that time was considered the most promising mathematician in England. Newton remained at Cambridge until 1696.

If Newton had overcome his "wariness to impart," there might never have been a controversy over who discovered the calculus. For many years his methods remained unknown, except to a few friends. He wrote De Analysi per Aequationes Infinitas in 1669 but did not publish it until 1711; while the Tractus de quadratura curvarum, composed in 1671, did not appear until 1704.

In Newton’s terminology, a variable quantity x, depending on time, is called a fluent; and its rate of change with time is said to be the fluxion of the fluent, denoted by (dx/dt in modern notation). He chose the letter o to represent an infinitely small quantity, with xo indicating the corresponding change in . For an illustration of his fluxional methods, Newton provides the equation xy - a = 0. He substitutes x + o for x, and y + o for y, then expands to get

After using the original equation xy - a = 0 and dividing by o, the equation is reduced to

The term involving o is neglected, since "o is supposed to be infinitely small," leaving

In 1665, Newton generalized the familiar binomial theorem for expanding expressions of the form (1 + a)ⁿ, n being a positive integer, to the case where n is a fractional exponent, positive or negative; the result is an infinite (binomial) series, rather than a polynomial. By means of the expansion of (1 - x²)^1/2, he arrived at what today would be written as