Jakob Bernoulli (1654 - 1705) taught himself the new Calculus
of Newton and Leibniz, as did his brother Johann (1667 - 1748).
Jakob invented polar coordinates and made significant contributions to
the theory of probability. Johann first studied medicine (taking a
doctor's degree at the University of Basel, Switzerland) with a thesis on
muscle contraction. He soon became fascinated with Calculus, applying
these new methods to a variety of problems in geometry, differential
equations and mechanics. Both brothers became professors of mathematics
at Basel (with Johann succeeding his older brother).There was a bitter rivalry between them which was aggravated when they worked on the same problem. In 1690, Johann posed the Brachistochrone Problem as a challenge to the mathematicians of the world:
A wire is bent in a curve which
joins two points.
A bead slides down the wire without friction.
What curve will give the minimum time of descent?
It is said that Newton, after a day of work at the Mint, learned of the problem and solved it the same afternoon. It was also solved by Jakob Bernoulli and Leibniz.
Johann's son, Daniel (1700 - 1782) also gave up medicine for mathematics and went on to become an outstanding mathematical physicist. In three generations the Bernoulli family produced eight mathematicians and scientists, the above three being the most outstanding.
Note: The Brachistochrone curve is the cycloid: if a circle rolls without slipping on a flat surface, a point on its circumference traces a cycloid.