John Bernoulli earned a master’s degree in philosophy and, in 1690, a medical licentiate from the University of Basel, where his brother James was teaching. At the same time, he was secretly studying the publications of Leibniz with James’s help. Shortly thereafter, Bernoulli visited Paris where he contracted to teach the material to the young marquis de l’Hopital. Many of his own discoveries in calculus appeared in l’Hopital’s textbook. In 1695, supported by a recommendation from l’Hopital, Bernoulli obtained a position at Gröningen in Holland. Upon his brother’s death in 1705, he succeeded him as professor of mathematics at Basel, to remain there for 43 years. Bernoulli was a zealous defender of Leibniz against charges that he had plagiarized Newton’s discovery of the calculus.

In 1696, John Bernoulli published a mathematical challenge, a popular device in the early days of the calculus. The problem he posed was to determine the shape of the curve down which a bead will slide, from one point to another not directly beneath it, in the shortest possible time. This is the famous brachistochrone problem, which Bernoulli named from the Greek words for "quickest time." Five prominent mathematicians found a solution; namely, the two Bernoullis, Leibniz, l’Hopital and Newton. When Newton’s solution arrived, unsigned, Bernoulli is said to have exclaimed, "I recognize the lion by his paw." Not surprisingly, the sought-after curve is not a straight line, but an upside-down cycloid.

One of Bernoulli’s more notable achievements is the expansion of a function in series through repeated integration by parts:

This leads to interesting identities such as