On 16 August 1705, Jacob Bernoulli (1655–1705) passed away at Basel, Switzerland. Working with his younger brother Johann Bernoulli (1667–1748) at the University of Basel, he was known as a founder of the calculus of variations (a method for evaluating the action integral dl = √(dx²+dy²) for a small variation y→y+δy with l → l+ 𝒪(δy²)→l). He also contributed solutions to the Bernoulli differential equation, i.e. y’+P(x)y=Q(x)yⁿ, and outlined a solution to the ‘law of large numbers’ result in probability theory (related to the central limit theorem).
Jacob Bernoulli had initially studied theology and began work in ministry. Against the wishes of his father, Nicolaus Bernoulli (1623–1708), he sought out opportunities to learn mathematics and astronomy while travelling throughout Europe during 1676 to 1682. His theological background, nonetheless, was essential to his later work in mathematics.
“Jacob Bernoulli developed his art of conjecturing or doctrine of chances with the understanding that God has designed the universe to follow natural laws or regularities and that we only use ideas of chance where we lack knowledge of the underlying causes — not that these underlying causes do not in fact exist. To God everything is known and certain. In Bernoulli’s view, the law of large numbers shows that over the long run the underlying regularities of nature will manifest themselves. Finally, Bernoulli’s particular use of algebra and of the properties of binomial expansions to prove the lemmas that form the core of his demonstration of the law of large numbers fit with this ‘God’s eye’ view of the universe, in which everything is immediate and there is no scope for ordering into what is mathematically prior or posterior. Thus Jacob Bernoulli’s ideas about God and the world combine with his reliance on algebra in proving the law of large numbers to explain what has seemed so problematic to critics like Hacking about Bernoulli’s intended interpretation of his law of large numbers: why he ‘assumed’ the existence of a ratio of cases in his proof of the law of large numbers and nevertheless believed that the proof justified the use of observed frequencies to discover such ratios to a close approximation.”
―Sylla, Edith Dudley. “Jacob Bernoulli on Analysis, Synthesis, and the Law of Large Numbers.” in Analysis and Synthesis in Mathematics: History and Philosophy. Vol. 196. Eds. Michael Otte and Marco Panza.(New York, NY: Springer Science & Business Media, 1997), 80-81. Images: Painting by Niklaus Bernoulli (1662-1716); Chart online.