There is a whole family of mathematical Bernoullis and it sometimes gets very confusing.

Daniel Bernoulli grew up with some of the most famous mathematicians of the age. His father Johann, brothers Nicolaus II and Johann II, and uncle Jacob were all accomplished mathematicians in their own right.

However, despite a keen interest in mathematics, Daniel read medicine at university at his father’s insistence, studying mathematics in his spare time.

After completing his doctorate, he was unable to work due to ill health - not good for business as a doctor! Not one for lazing around, he used the opportunity to publish his *Mathematical Exercises* with the help of Goldbach.

He also designed an hourglass for use at sea, one in which the stream of sand remained constant even when the ship was tossing and turning in the rough seas.

Daniel accompanied his brother to St Petersburg to teach mathematics to the Russian Tsar, but sadly Nicolaus II died only 5 months after they arrived.

Daniel was joined by Leonard Euler, one of his father’s most talented pupils. Daniel and Euler worked on vibrating systems, including studying the strings on a musical instruments. This led them to the idea of a fundamental vibration and its harmonics - vibrations whose frequencies were fixed multiples of the frequency of the fundamental vibration.

The mathematics they produced was fundamental and far reaching - any vibrating object behaved in a similar way to a musical instrument. The propagation of light, electricity and radio, even the ability of buildings to survive earthquakes depend on the same ideas.

Daniel began working on hydrodynamics, and went so far as to touch on the kinetic theory of gases discovered by Van der Waals a century later.

Daniel and his father fell out when they were jointly awarded the Grand Prize in 1734 His father was so outraged that his son’s mathematical could be compared to his own that he ceased all contact with him.

Daniel published his work on hydrodynamics, *Hydrodynamica* in 1738. A year later his father published Hydraulica, falsely pre-dating the publication date in order to take credit for Daniel’s work.

Daniel eventually won the Grand Prize of Paris an impressive total of 10 times for both nautical and astronomical topics.

He helped develop mathematical physics by combining the acceptance of Newton’s theories with the calculus of Leibniz, and in mechanics he used the principle of conservation of energy that gave a new interpretation (as an integral) of Newton’s basic equations.

Daniel’s father Jacob Bernoulli solved a long-standing problem and in so doing created a set of numbers - the Bernoulli numbers - that would have far reaching applications.

They appear, as if by magic, in the strangest of places; for example, they occur in the modern theory and solution of Fermat’s Last Theorem.

Jacob stumbled across the numbers when he investigated the sums of the powers of integers:

1+2+3+\cdots n={n(n+1)\over2}

1^2+2^2+3^2+\cdots n^2={n(2n+1)(n+2)\over6}

1^3+2^3+3^3+\cdots n^3={n^2(n+1)^2\over4}

Each of these has a formula as shown. There doesn’t seem to be any pattern to these but Jacob showed that they do follow a pattern - a very complicated pattern based on his numbers, 1,{1\over2},{1\over6},0,{-1\over30},0,{1\over42},0\cdots