Wallis entered Cambridge University in 1632, studied theology, and received a master’s degree in 1640, the same year in which he took Holy Orders. He held a faculty position at Cambridge for about a year, but vacated it upon deciding to marry. During England’s Civil War of 1642-1648, Wallis aided the Puritan cause by deciphering captured coded Royalist dispatches. As a reward for this service (and although he was yet to show any mathematical promise), Wallis was appointed professor of geometry at Oxford in 1649. Because the position required him to give public lectures on theoretical mathematics, Wallis embarked at the age of 32 on a systematic and productive study of the subject. He retained his post at Oxford until his death, over 50 years later.

Wallis’s Tractus de sectionibus conicis of 1656 is the first elementary textbook to treat conics using Descartes’s new coordinate geometry. In it, the ellipse, hyperbola and parabola are each identified with an equation of second degree. In 1655, he had published the Arithmetica infinitorum, the work on which his reputation is grounded. The Arithmetica contains a formula equivalent to

for the area under the curve y = xⁿ. This is often regarded as the first general theorem to appear in the calculus. After giving a somewhat rigorous demonstration for several integral powers of x, Wallis inferred it to be true for every positive integer; then, relying on "permanence of form," he asserted that the formula held even when n is negative (but not equal to -1) or fractional. The result was not new, having been anticipated by Cavalieri. Where Cavalieri relied almost entirely on geometric reasoning, Wallis held to an arithmetic argument whenever possible. With the advent of his "arithmetic integration," the geometric method of indivisibles virtually ceased to appear in the calculus.

The familiar knot symbol for infinity makes its first appearance in print in the Arithmetica. As does Wallis’s famous infinite product expansion for p ,