structural:expansion
Direct method

In a letter to Henri Basnage, held at the University of Basel Public Library, dated 30 March 1697, Johann Bernoulli stated that he had found two methods (always referred to as "direct" and "indirect") to show that the Brachistochrone was the "common cycloid", also called the "roulette". Following advice from Leibniz, he included only the indirect method in the Acta Eruditorum Lipsidae of May 1697. He wrote that this was partly because he believed it was sufficient to convince anyone who doubted the conclusion, partly because it also resolved two famous problems in optics that "the late Mr. Huygens" had raised in his treatise on light. In the same letter he criticised Newton for concealing his method.

In addition to his indirect method he also published the five other replies to the problem that he received.

Johann Bernoulli's direct method is historically important as a proof that the brachistochrone is the cycloid. The method is to determine the curvature of the curve at each point. All the other proofs, including Newton's (which was not revealed at the time) are based on finding the gradient at each point.

In 1718, Bernoulli explained how he solved the brachistochrone problem by his direct method.

He explained that he had not published it in 1697, for reasons that no longer applied in 1718. This paper was largely ignored until 1904 when the depth of the method was first appreciated by Constantin Carathéodory, who stated that it shows that the cycloid is the only possible curve of quickest descent. According to him, the other solutions simply implied that the time of descent is stationary for the cycloid, but not necessarily the minimum possible.