Born: 21 October 1687 in Basel, Switzerland
Died: 29 November 1759 in Basel, Switzerland

Nicolaus(I) Bernoulli was a nephew of Jacob Bernoulli and Johann Bernoulli. His early education involved studying mathematics with his uncles. In fact it was Jacob Bernoulli who supervised Nicolaus's Master's degree at the University of Basel which he was awarded in 1704. Five years later he was received a doctorate for a dissertation which studied the application of probability theory to certain legal questions.

In 1712 Nicolaus Bernoulli toured Europe visiting Holland, England and France. It was in France that he met Montmort and the two mathematicians became close friends and collaborated on mathematical questions in a long correspondence.

Nicolaus Bernoulli was appointed to Galileo's chair at Padua in 1716 which Hermann had filled immediately prior to Nicolaus's appointment. There he worked on geometry and differential equations. In 1722 he left Italy and returned to his home town to take up the chair of logic at the University of Basel. After nine years, remaining at the University of Basel, he was appointed to the chair of law. In addition to these academic appointments, he did four periods as rector of the university.

J O Fleckenstein, writing in [1], describes Nicolaus Bernoulli's contribution to mathematics:-

Nicolaus was a gifted but not very productive mathematician. As a result, his most important achievements are hidden throughout his correspondence, which comprises about 560 items. The most important part of his correspondence with Montmort (1710-1712) was published in the latter's "Essai d'analyse sur les jeux de hazard" (Paris, 1713).

From Montmort's work we can see that Nicolaus formulated certain problems in the theory of probability, in particular the problem which today is known as the St Petersburg problem. Nicolaus also corresponded with Leibniz during the years 1712 to 1716. In these letters Nicolaus discussed questions of convergence, and showed that (1+x)ⁿ diverges for x > 0.

Nicolaus also corresponded with Euler. Again quoting [1]:-

In his letters to Euler (1742-43) he criticises Euler's indiscriminate use of divergent series. In this correspondence he also solved the problem of the sum of the reciprocal squares ∑ (1/n²) = π²/6, which had confounded Leibniz and Jacob Bernoulli.

Nicolaus Bernoulli assisted in the publication of Jacob Bernoulli's Ars conjectandi. Later Nicolaus edited Jacob Bernoulli's complete works and supplemented it with results taken from Jacob's diary. Other problems he worked on involved differential equations. He studied the problem of orthogonal trajectories, making important contributions by the construction of orthogonal trajectories to families of curves, and he proved the equality of mixed second-order partial derivatives. He also made significant contributions in studying the Riccati equation.

One of the great controversies of the time was the Newton Leibniz argument. As might be expected Nicolaus supported Leibniz but he did produce some good arguments in his favour such as observing that Newton failed to understand higher derivatives properly which had led him into errors in the problem of inverse central force in a resisting medium.

1. J O Fleckenstein, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/doc/1G2-2830900410.html

Books:

1. H Bernhard, The Bernoulli family, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).

Articles:

1. E A Fellmann, Partielle Differentiation im Briefwechsel Eulers mit Niklaus I. Bernoulli - eine Miszelle, in History of mathematics (San Diego, CA, 1996), 223-236.
2. M Hürlimann (ed.), Die Mathematikerfamilie Bernoulli, Grosse Schweizer (Zürich, 1942), 112-119.