The Swiss Bernoulli brothers, James and John, were the first to achieve a full understanding of Leibnizs presentation of the calculus. Their subsequent publications did much to make the subject widely known to the rest of the continent.
James Bernoulli, the elder of the two, entered the University of Basel in 1671, receiving a masters degree in theology two years later and a licentiate (a degree just below the doctorate) in theology in 1676. Meanwhile, he was teaching himself mathematics, much against the wishes of his merchant father. Bernoulli spent two years in France familiarizing himself with Descartes Géométrie and the work of his followers. By 1687, he had sufficient mathematical reputation to be appointed to a vacant post at Basel. He also wrote to Leibniz in the same year, asking to be shown his new methods. This proved difficult because Leibnizs abbreviated explanations were full of errors. Still, Bernoulli mastered the material within several years and went on to make contributions to the calculus equal to those of Leibniz himself.
The Bernoulli brothers used the techniques of Leibnizs calculus as a means for handling a wide range of astronomical and physical problems, sometimes working independently to solve the same problem. In 1690, James Bernoulli challenged the mathematicians of Europe to determine the shape (that is, to find the equation) of a hanging flexible cable suspended in equilibrium at two points. The correct solution was presented a year later by his brother John in his first published paper. The desired curve was not a parabola, as some expected, but a curve known as the catenary -- from the Latin word catena, chain.
Bernoulli was more adapt at treating infinite series than most mathematicians of the day. He showed that
diverges, and that
1/12 + 1/22 + 1/32 + 1/42 + . . .
converges; but he confessed his inability to find the sum of the latter series. (Euler succeeded in finding its sum.) In 1690 he established what is known as the "Bernoullian inequality,"
(1 + x)n > 1 + nx, x > -1, n > 1, n an integer.
We also owe to him the word "integral" in its technical sense.