Jakob I was the first of the Bernoulli family of scientists to achieve fame as a mathematician. As with the two other particularly outstanding Bernoullis – his brother, Johann I and nephew, Daniel – Jakob I played an important role in the development and popularization of the then recently invented integral and differential calculus of Isaac Newton and Gottfried Leibniz. His particular contribution to the calculus consisted in showing how it could be applied to a wide variety of fields of applied mathematics.

Jakob I, who was born in Basel, Switzerland, began studying theology and in 1676 traveled through Europe where he met many of the important scientists of the day, such as Robert Boyle in England. He returned to Basel in 1682 where he began lecturing on mechanics and held a chair in mathematics at Basel University from 1687 until his death. Apart from his mathematical work he was an influential figure in the European scientific community through his voluminous correspondence.

His most important contributions to mathematics were in the fields of probability and in the calculus of variations. His work on probability is contained in his treatise the Ars conjectandi (1713; The Art of Conjecturing) in which he made numerous important contributions to the subject, among which was his discovery of what is now known as the ‘law of large numbers’. The law has a number of forms. In effect it says that for an event of probability P in a large number of trials n the number of actual events approaches n P as n increases. Ars conjectandi also contains Bernoulli's work on permutations and combinations.

The Bernoulli family were always prone to rivalry and Jakob I and his younger brother, Johann I, became involved in a controversy over the problem of finding the shortest path between two points of a particle moving solely under the influence of gravity. The result of this vigorous dispute was the creation of the calculus of variations, a field that Leonhard Euler was later to develop. In addition to this Jakob I did important and useful work in the study of the catenary, which he applied to the design of bridges.