Bernoulli's Challenge and the Brachistochrone

At the end of the seventeenth century, prominent mathematicians gravitated towards problems formulated in terms of maxima and minima. The interest germinated from a combination (of uncertain proportions) of aesthetic preferences, theological reasons, and the sheer allure of mathematical challenges. The most famous of such problems appears in a letter of June 9, 1696, from John Bernoulli to his friend Gottfried Leibniz: “Given two points A and B in a vertical plane, find the path AMB down which a moveable point M must, by virtue of its weight, fall from A to B in the shortest possible time” (Leibniz, 1962; Orio, 2009). In just a week, on June 16, Leibniz wrote back with the solution, adding that he solved the problem against his will, but that he was attracted to its beauty like Eve before the apple. He expressed his solution in the form of a differential equation, and proposed to name the curve tachystoptota (curve of quickest descent). Bernoulli responded a few days later taking up the biblical reference. He was “very happy about this comparison provided that he was not regarded as the snake that had offered the apple” (Knobloch, 2012). Bernoulli points out that Leibniz's solution corresponds to the cycloid (the curve traced out by a point on the rim of a circle rolling on a flat plane) and proposed to name the curve brachistochrone. The cycloid was admired by Galileo “as a very gracious curve to be adapted to the arches of a bridge” (Drake, 1978, p. 406) and was proven by Huygens (1673) to correspond to the isochronous pendulum. In the meantime, Bernoulli had already communicated the problem as a challenge to Rudolf Christian von Bodenhausen in Florence, in Switzerland to his brother Jacob Bernoulli, and in France to Pierre Varignon. In his response to Leibniz, John Bernoulli mentions two solutions. The first is similar to Leibniz's. The second uses a clever map from the swiftest path to the problem of finding the trajectory of a light ray propagating in a medium of continuously varying index of refraction. This mapping is a hallmark of the optical – mechanical analogy, which had a profound influence in later formulations of mechanics.