Before James Bernoulli entered upon this brilliant career of discovery, lie was elected, in 1687, to the professorship of mathematics at Basle, an office which he filled with distinguished reputation during the whole of his life. He succeeded Peter Meger lin, who is known -to astronomers as a zealous de fender of the Copernican system.
In 1690 James Bernoulli solved the problem of the isochronaus curve, of which Huygens and Leib nitz had already obtained a solution ; and on this oc casion he proposed the celebrated problem of the catenarian curve, which Galileo had tried in vain. Huygens, Leibnitz, and John Bernoulli soon obtain ed a solution ; but this solution was extended by James Bernoulli to cases, in which the weight of the chain varies in different parts of its length, according to a given law. This able mathematician determined also the curvature of a beaded bow, and that of an elas tic rod, fixed at one end, and loaded at the other with a given weight. He found likewise, that the form of a sail, swollen by the action of the wind, is the common catenarian curve when the wind does not escape ; but that it is one of the curves called Lintearice, when the sail is supposed perfectly flexible, and expanded with a fluid pressing in every direction. John Bernoulli published a solution of the same problem in the Jour nal des Scavans for 1692 ; but it appears unquestion able, that he had received hints from his brother, who communicated to him, by letter, his opinion upon that subject.
The theory of curves, produced by the revolution of one curve upon another, now occupied the atten tion of James ; and in this rich and untrodden field he made many interesting discoveries. He found, that the logarithmic spiral was its own evolute, anti evolute, caustic, and pericaustic ; and that the cy cloid had a property anala•ous to it. The discovery of this constant reproduction of the logarithmic spi ral was a source of such pleasure to Bernoulli, that, in imitation of Archimedes, he requested that a lo garithmic spiral should be engraven on his tomb, with the motto of Eadem mutata resurgo, a beauti ful and happy allusion to the future hopes of the Christian. Besides these discoveries, James Ber noulli solved the problem of the paracentric isochro nal curve, proposed by Leibnitz in 1689, and also the problem of the curve of quickest descent, which his brother John had proposed in 1697.
About this time began that famous dispute upon isoperimetrical problems, between James and John Bernoulli, in which their talents were displayed to greater advantage than their dispositions. " These illustrious characters," as the writer of this article has elsewhere observed, " connected by the strongest ties of affinity, were, at the commencement of their distinguished career, united by the warmest affection. John was initiated by his cider brother into the ma thematical sciences ; and a generous emulation, soft. cued by friendship in the one, and gratitude in the other, continued for some years to direct their stu dies, and accelerate their progress. There are few
men, however, who can support, at the same time, the character of a rival and a friend. The success of the one party is apt to awaken the envy of the other ; and success itself is often the parent of pre sumption. A foundation is thus laid for future dis sensions; and it is a melancholy fact in the history of learning, that the most ardent friendships have been sacrificed on the altar of literary ambition. Such was the case between the two Bernoullis. As soon as John was settled professor of mathematics at Gro ningen, all friendly intercourse between the two bro thers was at an end. Regarding John as the aggres sor, and provoked at the ingratitude which he exhi bited, his brother James challenged him, by name, to solve the following problems :" 1. To find among all the' isoperimetrical curves, between given limits, such a curve that, a second curve being constructed, having its ordinates any functions of the ordinates or arcs of the former, the area of the second curve shall be a maximum or a minimum. 2. To find, among all the cycloid; which a heavy body may describe in its descent from a point to a line given in position, . that cycloki which is described in the shortest time possible. A prize of 50 florins was offered by James to his brother John, if he should solve these problems in the space of three months, and produce legitimate solutions in the course of a year ; and if, at the ex piry of these intervals, no solutions appeared, he pro. mised to lay his own before the public. This chal lenge was willingly accepted by John, who began the investigation as soon as he received the subject, and soon completed the solution. Elated with suc cess, he ostentatiously declared, that, instead of three months, he had discovered the whole mystery in throe minutes. He demanded the prize, and offered to give to the poor what had cost him so little trou ble to gain. Unfortunately, however, for John, his solution of the isoperimetrical problem was erro neous. His•brother published a notice, in which he came under three engagements : 1. To point out the method employed by his brother ; 2. To expose its errors, whatever the method was ; 3. To give a true solution of the problem. The boldness of this notice induced John to revise his solution ; and, having found his mistake, which he ascribed to the hurry in which it was obtained, he sent a new solution, and again demand ed the prize. In reply to this demand, James Bernoulli requested his brother to examine his new solution, as the pretext of hurry would be unavailing after a second failure ; but John replied, that his solution was correct, and that his time would be better em ployed in making • new discoveries. In a letter to Varignon, which was inserted in the Journal des Sya vans, with an additional notice, James Bernoulli at tacked, with a good deal of ridicule and sarcasm, the solution of John, who read the letter with the utmost indignation, and lavished on his torrent of the coarsest invective.