## Jacob Bernoulli
Jacob Bernoulli is sometimes called Jacques Bernoulli. His father was Nicolaus Bernoulli, a mathematician. During the time that Jacob Bernoulli was taking his university degree in theology he was studying mathematics and astronomy against the wishes of his parents. In 1676, finishing his degree, Bernoulli moved to Geneva where he worked as a tutor. He then travelled to France spending 2 years studying with the followers of Descartes. In 1681 Bernoulli travelled to the Netherlands where he met many mathematicians. Continuing his studies with the leading mathematicians and scientists of Europe he went to England where, among others, he met Boyle and Hooke. As a result of his travels, Bernoulli began a correspondence with many mathematicians which he carried on over many years. Jacob Bernoulli returned to Switzerland and taught mechanics at the University in Basel from 1683, giving a series of important lectures on the mechanics of solids and liquids. Bernoulli studied the work of Descartes, Wallis, and Barrow and through these he became interested in infinitesimal geometry. Jacob began publishing in 1682. Jacob's younger brother, Johann Bernoulli, began to work on mathematical topics, and asked him to teach him mathematics. Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and the two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were the first to try to understand and apply Leibniz's theories. Although Jacob and Johann Bernoulli both worked on similar problems their relationship was soon to change from one of collaborators to one of rivals. The brothers were probably equally at fault in their quarrel. Whether the rivalry spurred them on to greater things or whether they might have achieved more had they continued their initial collaboration, it is impossible to say. Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as fundamental result that the sum of 1/n diverges, which Bernoulli believed were new but they had actually been proved 40 years earlier. Bernoulli could not find a closed form for the sum of 1/n2, but he did show that it converged to a finite limit less than 2. Euler was the first to find the sum of this series, pi2/6, in 1737. Bernoulli also studied the exponential series which came out of examining compound interest. In 1690 in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent, is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by Huygens in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. In 1696 Bernoulli solved the equation, now called the Bernoulli equation, y' = p(x)y + q(x)yn. Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694. In 1695 he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced. Jacob Bernoulli's most original work was published 8 years after his death. In the book Bernoulli reviewed work of others on probability, in particular work by van Schooten, Leibniz, and Prestet. The Bernoulli numbers appear in the book in a discussion of the exponential series. Many examples are given on how much one would expect to win playing various game of chance. Jacob Bernoulli continued to hold the chair of mathematics at Basel until his death in 1705 when the chair was filled by his brother Johann. Jacob had always found the properties of the logarithmic spiral to be almost magical and he had requested that it be carved on his tombstone. |