brachistochrone problem


The brachistochrone problem is a problem with which Johann Bernouilli challenged his contemporaries in Acta Eruditorum in June 1696:


Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise... Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.


Isaac Newton reportedly solved the problem between four in the evening and four in the morning after a hard day at the Royal Mint, later commenting: "I do not love to be dunned [pestered] and teased by foreigners about mathematical things..." Other correct solutions came in from Gottfried Leibniz, the Frenchman Guillaume de L'Hôpital, and Johann's brother Jakob. They, like Johann, realized that the solution to the brachistochrone problem, as it was also to the tautochrone problem, was a curve known as the cycloid. See also calculus of variations.