Bernoulli, Jakob (1654–1705)

Jakob (James, Jacques) Bernoulli, from Switzerland, was the first of eight members of the mathematically talented Bernoulli family. As directed by his parents, he was trained as a philosopher (master's degree, 1671) and theologian (licentiate, 1676) at the University of Basel.

His career interests, however, were in mathematics and, at least initially, in its application to astronomy. He studied mathematics during his extensive travels, subsequent to graduation, with such luminaries as Nicolas Malebranche (1638–1715) for two years in France, Johann van Waveren Hudde (1628–1704) in the Netherlands, and briefly with Robert Boyle (1627–1691) and Robert Hooke (1635–1703) in England. He then resettled in Basel, and while awaiting a more lucrative offer (and publishing a flawed theory pertaining to comets), he opened a private school of mathematics in 1682. The following year, he was appointed to a teaching post in mechanics at his alma mater, and in 1687, he obtained a professorship and chair in mathematics, which he held until his death.

In 1682, he became a correspondent disciple of Gottfried Wilhelm Leibniz (1646–1716), who coinvented the calculus along with Sir Isaac Newton (1642–1727). This primarily distance-learning arrangement led to Bernoulli's instrumental role in the development of elementary differential and integral calculus and ordinary differential equations.

Among his many discoveries were the system of polar coordinates, the isochrone (the path that an object falls with uniform velocity), and the logarithmic spiral (r = aebθ). He extended trigonometric functions to complex variables, which led analysis (the study of infinite series) into the study of algebra. Although the ∫ symbol was invented by his younger brother and student, Johann (1667–1748), the term integral was coined by Jakob in an article published in Acta Eruditorum in 1690.

His magnum opus, Ars Conjectandi, was published posthumously in 1713. Bernoulli spent 20 years writing [Page 87]the book but never brought it to fruition. It was one of the earliest rigorous treatises on probability theory. In the second of four parts, he proves by induction the binomial theorem. The fourth part contains the theorem of Bernoulli. Siméon-Denis Poisson (1781–1840), a descendent in Bernoulli's academic genealogy, renamed the theorem the law of large numbers. The modern Monte Carlo method, a technique of repeated sampling, is also known as Bernoulli trials.