The idea that genius runs in families is supported by many examples in the arts and sciences. One striking case is the family of Johann Sebastian Bach, the most brilliant star in a constellation of talented musicians and composers.

In a similar vein, several generations of the Bernoulli family excelled in science and medicine. More that ten members of this Swiss family, over four generations, had distinguished careers in mathematics.

The fifth and tenth children of one generation, Jakob and Johann (aka James and John) Bernoulli were prominent enthusiasts and promoters of the calculus, which had been invented independently by Newton and Leibniz.

Jakob found equations for several “special curves” – the catenary, tractrix, cycloid and lemniscate. He introduced the term “integral calculus”, which was approved and adopted by Leibniz.

**Variational Calculus**

Johann devised a completely new branch of calculus, variational calculus, in finding the so-called *brachistochrone*, the curve along which a particle slides from one point to another in the shortest time. Jakob and Johann came into conflict over priority in solving this problem. Variational calculus is of fundamental importance in mechanics and quantum mechanics today.

Jakob published a volume, *Ars Conjecti*, on the theory of probability, which included the theorem called The Law of Large Numbers. While studying a problem in compound interest, he also discovered the fundamental number *e*, which is ubiquitous in mathematical analysis.

The result in calculus known as l’Hôpital’s Rule, was actually discovered by Johann Bernoulli. Johann, who was in the pay of l’Hôpital, remained silent until after l’Hôpital died, but then claimed priority.

Johann was also active in the Newton-Leibniz controversy over priority in inventing calculus. The historian of mathematics, Carl Boyer, called him Leibniz’s bulldog. When Johann’s son Daniel won a prize for which he himself had competed, Johann barred him from the family home.

**Hydrodynamics**

Daniel was the most celebrated of the next generation of Bernoullis. He applied probability theory to problems in commerce, medicine and astronomy. One of his papers demonstrated the advantages of universal inoculation against smallpox [see earlier *That’s Maths* post The End of Smallpox].

But his work in hydrodynamics had the most lasting impact, and “Bernoulli’s Principle”, relating pressure and velocity of a fluid, is named for him. This principle is critical in aerodynamics and in the theory of flight.

**A Star yet Brighter**

Several other members of the family in addition to Jakob, Johann and Daniel, made notable contributions to mathematics. But, bright as were the Bernoullis, their work was far surpassed by a famous pupil of Johann, the outstanding Swiss genius Leonhard Euler. Euler’s work dominated eighteenth century mathematics and continues to be of central importance today.