Bernoulli, sor, and provoked at the ingratitude which he exhi- 

 ;ames. 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 cycloids which a heavy body may describe in 

 its descent from a point to a line given in position, 

 that cycloid 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 

 thrae 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 Sga- 

 rana, 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 brother a 

 torrent of the coarsest invective. 



In order to put an end to this geometrical warfare, 

 which had now degenerated into personal abuse, 

 .Leibnitz, Newton, and the Marquis de L'Hospital, 

 were appointed arbiters ; but they do not seem to 

 have come to any decision on the subject. In 1700, 

 James Bernoulli published, in a letter to his brother, 

 the formula; of the isoperimetrical problem without a 

 demonstration, and invited him to make his own pub- 

 lic. John was still ignorant of the defect of his own 

 method ; and so much was he convinced of its accu- 

 racy, that he sent it under seal to the Academy of 

 Sciences at Paris, in February 1701, on condition that 

 it should not be opened till the appearance of his 

 brother's demonstration. In consequence of this, 

 James Bernoulli published his solution separately at 

 Basle, and also in the acts of Leipsic for May 1701, 

 i;nder the title of Analysis magni prohlematis hope- 

 nmetrici. The fame which was acquired by this ad- 

 mirable specimen of mathematical genius completely 

 VOL.111. I'MII in. 



BERNOULLI. 481 



silenced the pretensions of John Bernoulli for five 

 years ; but after the death of his brother, in 1705, he 

 published his solution in the memoirs of the academy 

 for 170G, as if he had thought his brother the only 

 person who could detect the false principle upon 

 which it was founded. After an interval of thirteen 

 years, John Bernoulli discovered the source of his er- 

 ror. He ingenuously confessed his mistake ; and 

 published a new solution in the memoirs of the aca- 

 demy for 1718, which did not differ much from that 

 of his brother. 



In the year 1699, James Bernoulli was elected a 

 foreign associate of the Academy of Sciences at Pa- 

 ris ; and, in 1701, the same honour was conferred up- 

 on him by the Royal Academy of Berlin. The se- 

 dentary life which he led, and his intense application 

 to study, brought upon him a severe attack of the 

 gout, accompanied with a slow fever, which put an. 

 end to his life on the 16th of August 1705, in the 

 51st year of his age. He was married in the year 

 1684, and left behind him one son and a daughter, 

 neither of whom seem to have inherited any portion oi 

 their father's genius. The son was bred to the pro- 

 fession of a painter. James Bernoulli was engaged, 

 at the time of his death, in a work entitled, Ars Con- 

 jectandi, or the art of forming conjectures concerning 

 contingent events. It was printed at Basle in 1713; 

 and contains a valuable treatise on infinite series, in 

 which its author has given an admirable demonstration 

 of the first case of the binomial theorem. This de- 

 monstration has been lately re-published in the third 

 volume of the Scriptores Logaril/imici, by Baron Ma- 

 seres. 



Bernoulli was of a bilious and melancholy tempe- 

 rament, and possessed great perseverance in surmount- 

 ing difficulties. His genius, though of the first or- 

 der, was not of that quick and versatile character 

 which seizes a subject with instinctive penetration, 

 and invents and discovers by a process almost intui- 

 tive. It was marked rather by an excess of caution. 

 He proceeded with slowness and suspicion, afraid of 

 error, yet resolved to avoid it ; and even after success 

 had taught him the extent of his own powers, and af- 

 ter the applause of all Europe had stamped immorta- 

 lity upon his name, he did not possess that confidence 

 in his talents which is generally the most prominent 

 qualities of flattered genius. When he challenged his 

 brother John to the solution of the isoperimetrical 

 problem, on which he had for a long time laboured, 

 he acted with more confidence than he usually dis- 

 played ; but his excessive caution gave him ample 

 security against the chance of error. His brother 

 John, on the contrary, whose genius was more acute 

 but less profound, obtained his solution of the pro- 

 blem almost instantaneously. Without even revising 

 his investigations, he gave them to the world, careless 

 about the mortification which he afterwards felt when 

 they were proved to be erroneous. In the keen dis- 

 pute which this circumstance occasioned, the charac- 

 ters of the two brothers appeared in their natural co- 

 lours. The cold defiance, the chastened severity, and 

 the temperate sarcasms of the one, form a striking 

 contrast witli the thoughtless ostentation, the rude 

 invectives, and the coarse raillery of the other. 

 The writings of James Bernoulli are vervnumereui, 

 3 p 



Bernoull i, 

 James. 



