In June 1696, Jean Bernoulli addressed a letter to the mathematicians of Europe challenging them to solve two problems: (1) to determine the brachistochrone between two given points not in the same vertical line, that is, the curve between two points that is covered in the least time by a point-like body that starts at the higher point with zero speed and is constrained to move along the curve to the lower point under the action of constant gravity and assuming no friction; and (2) to determine a curve such that, if a straight line drawn through a fixed point A meets it in two points P1, P2, then AP1m+AP2m will be constant. This challenge was first made in the Ada Lipsiensia for June 1696. Leibniz and Bernoulli were confident that only a person who knows calculus could solve this problem.Six months were allowed by Bernoulli for the solution of the problem, and in the event of none being sent to him he promised to publish his own. The six months elapsed without any solution being produced. However, he received a letter from Leibniz, stating that he had “cut the knot of the most beautiful of these problems,” and requesting that the period for their solution should be extended to Christmas next, so that the French and Italian mathematicians might have no reason to complain of the shortness of the period. Bernoulli adopted the suggestion, and publicly announced the postponement to notify those who might not see the Ada Lipsiensia about the contest.
On today’s date, Newton returned at 4:00 pm from working at the Royal Mint and found in his post the problems that Bernoulli had sent directly to him; two copies of the printed paper containing the problems. Newton stayed up to 4:00 am before arriving at the solutions; on the following day he sent a solution of them to Montague, then president of the Royal Society for anonymous publication. He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it. He also solved the second problem, and in so doing showed that by the same method other curves might be found which cut off three or more segments having similar properties. Solutions were also obtained from Leibniz and the Marquis de l’Hôpital. Although Newton’s solution was anonymous, he was recognized by Bernoulli as its author; tanquam ex ungue leonem (“we recognize the lion by his claw”).