Later Life of Isaac Newton - Bernoulli's Mathematical Challenge

Bernoulli's Mathematical Challenge

Newton's solution of the celebrated problems proposed by Johann Bernoulli and Leibniz deserves mention among his mathematical works. In June 1696 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, (2) to determine a curve such that, if a straight line drawn through a fixed point A meet it in two points P₁, P₂, then AP₁m+AP₂m will be constant. This challenge was first made in the Ada Lipsiensia for June 1696.

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; but 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; 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 for the information of those who might not see the Ada Lipsiensia.

On 29 January 1697 Newton returned at 4pm from working at the Royal Mint and found in his post the problems that Bernoulli had sent to him directly; two copies of the printed paper containing the problems. Newton stayed up to 4am 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; and, although Newton's solution was anonymous, he was recognized by Bernoulli as its author; "tanquam," says he, "ex ungue leonem" (we know the lion by his claw).

In 1699 Newton's position as a mathematician and natural philosopher was recognized by the French Academy of Sciences. In that year the Academy was remodelled, and eight foreign associates were created. Leibniz, Domenico Guglielmini (1655—1710), Hartsoeker, and E. W. Tschirnhaus were appointed on 4 February, James Bernoulli and John Bernoulli on 14 February, and Newton and Ole Rømer on 21 February.