A Brachistochrone curve, or curve of fastest descent, is the curve between two points that is covered in the least time by a body that rolls down it under the action of constant gravity. It is the solution of the Brachistochrone problem.

Galileo incorrectly stated in 1638 in his Discourse on two new sciences that this curve was an arc of a circle. Johann Bernoulli solved the problem (by reference to the previously analysed tautochrone curve) before posing it to readers of Acta Eruditorum in June 1696. Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz and Guillaume Franšois Antoine de l'H˘pital. Four of the solutions (excluding l'H˘pital's) were published in the May 1697 edition of the same publication. The brachistochrone curve was proved to be a cycloid.

In an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called (in 1766) the calculus of variations. Joseph-Louis de Lagrange did further work that resulted in modern infinitesimal calculus.

Another rivalry, between Newton and Leibniz, also contributed to this development. Each claimed to have solved the brachistochrone problem before the other, and they continued to quarrel over their subsequent work on the calculus.

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