Editing Jacob Bernoulli (section)

==Important works==
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 the fundamental result that <math>\sum{\frac{1}{n}}</math> diverges, which Bernoulli believed were new but they had actually been proved by [[Pietro Mengoli]] 40 years earlier and was proved by Nicole Oresme in the 14th century already.<ref>D. J. Struik (1986) A Source Book In Mathematics, 1200-1800, p. 320</ref> Bernoulli could not find a closed form for <math>\sum{\frac{1}{n^2}}</math>, but he did show that it converged to a finite limit less than 2. [[Euler]] was the first to find [[Basel problem|the limit of this series]] in 1737. Bernoulli also studied [[#Discovery of the mathematical constant e|the exponential series]] which came out of examining compound interest.

In May 1690, in a paper published in ''Acta Eruditorum'', Jacob Bernoulli showed that the problem of determining the [[Tautochrone curve|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 differential equation]],

:<math> y' = p(x)y + q(x)y^n. </math>

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.

[[File:Bernoulli - Ars conjectandi, 1713 - 058.tif|thumb|''Ars conjectandi'', 1713 (Milano, [[Fondazione Mansutti]]).]]
Bernoulli's most original work was ''[[Ars Conjectandi]]'', published in Basel in 1713, eight years after his death. The work was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. The book also covers other related subjects, including a review of [[combinatorics]], in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of [[Bernoulli numbers]] in a discussion of the exponential series. Inspired by Huygens' work, Bernoulli also gives many examples on how much one would expect to win playing various games of chance. The term [[Bernoulli trial]] resulted from this work.

In the last part of the book, Bernoulli sketches many areas of [[Probability theory|mathematical probability]], including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; and the law of large numbers.

Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity.