BERNOULLI. 



483 



Bernoulli, solutions For some time ; but before they were pub- 

 John lished, Sir Isaac Newton, the Marquis de L'Hospital, 

 * ""v ' an d James Bernoulli, succeeded in demonstrating, 

 that this curve, called the curve of quickest descent, is 

 a reversed cycloid. While employed on this subject, 

 James Bernoulli was led to the subject of isopei [me- 

 trical problems, which occasioned those differences 

 with his brother which we have already related in the 

 preceding article. 



The problem of orthogonal trajectories, whicli 

 Leibnitz had proposed to the English geometers, 

 was completely resolved by John Bernoulli in the 

 Leipsic Transactions for 1718. Sir Isaac Newton 

 had brought the problem to an equation, but did not 

 succeed in resolving the differential equation of the 

 trajectory. Two particular eases of it were solved 

 by the two Nicholas Bernoullis, the son and nephew 

 of John. A better solution, though defective in 

 point of generality, was given by Df Taylor in the 

 Philosophical Transactions of 1717; but it was left 

 for John Bernoulli to supply this radical defect. This 

 celebrated geometer succeeded, also, in the integra- 

 tion of several rational fractions, with which Taylor 

 had endeavoured to perplex him. 



The publication of Dr Taylor's method of incre- 

 ments gave rise to hostilities between him and John 

 Bernoulli more serious than the war of problems in 

 which they had been engaged* Taylor was charged 

 as a plagiarist in the Leipsic Transactions for 1716. 

 This anonymous attack from the pen of John Ber- 

 noulli was indignantly repelled by the English geo- 

 meter, who accused his antagonist of having only 

 altered and modified the solution of isoperimetrical 

 problems, which were given by his brother James. 

 Bernoulli again retorted under the concealed name of 

 Buscard ; but his reply was stained by a species of 

 angry invective, and insulting raillery, which was un- 

 worthy of a philosopher. 



In a dissertation on Orthogonal Trajectories, pub- 

 lished as the joint production of John Bernoulli and 

 his son Nicholas, they proposed the problem of re- 

 ciprocal trajectories, which was for a long time dis- 

 cussed between John Bernoulli and Dr Pemberton. 

 The friend of Newton carried on the controversy un- 

 der an anonymous disguise ; but he was unequal to a 

 contest with 6uch a formidable rival. Irritated at 

 the success of Bernoulli, the English geometers as- 

 sailed him m every quarter. Dr Keill challenged 

 him to determine the curve described by a body when 

 projected through a medium whose resistance varied 

 as the square of the velocity. In a short time the 

 exertions of Bernoulli were crowned with success; 

 and though Newtor. had solved only the case where 

 the resistance varied as the square of the velocity, 

 the Swiss geometer determined the path of the pro- 

 jectile, when the medium resisted, according to any 

 power of the velocity. Intoxicated with success, 

 Bernoulli demanded the solution obtained by Keill, 

 but when he found there was none to produce, he 

 attempted to punish the presumption of the English 

 philosopher, by the rudeness and severity of his wit. 

 The problem of Offenburgh, which consisted in 

 determining on the surface of a sphere, curves whose 

 perimeters could be expressed by algebraic quanti- 

 ties, had been tried in vain by Herman, ( Act. Petrop. 



1726) ; but John Bernoulli pointed out the error of Bernoulli, 

 Herman, and gave a general method for finding the 

 curves required. 



In the Memoirs of the Academy of Paris for 1730, 

 he published his determination of the isochronous 

 curve. In the same year, he carried off the prize 

 of the Academy of Sciences, on the spheroidal figure 

 of the planets, and on the motion of their aphelia ; 

 and in 1734, he shared the prize with his son Daniel, 

 for a dissertation on the change of inclination in the 

 planetary orbits ; an occasion, as will be seen in the 

 following article, which did not exhibit his character 

 to the greatest advantage. His work on the manage- 

 ment of ships, was published in 1718, and on this sub- 

 ject he was led into a controversy with Reuau. In 

 1713, he collected together the various works which 

 he had composed, and printed them at Lausanne, in 

 four volumes quarto. 



While our author held his professorship at Gron- 

 ingen, the university of Utrecht was solicitous to 

 rank him among its members. His salary and ap- 

 pointments, however, were increased, and he con- 

 tinued in his office at Groningen, where a violent fe- 

 ver had nearly terminated his labours in 1704, till 

 the pressing entreaties of his relations had almost in- 

 duced him to return to Basle. The rumour of his 

 departure incited the university of Utrecht to make 

 another effort to obtain the benefit of his talents j 

 and while he was hesitating what step to take, the 

 death of his brother put an end to his irresolution. 

 He returned to his native city, and succeeded his 

 brother in the professorship of mathematics on the 

 1 7th of November 1 705, where he delivered a discourse 

 De Fatis Nova Analyseos et Geometriw sublimis. 

 In this new situation he spent forty-two years of his 

 life, which were zealously devoted to the discharge 

 of his professional duties, and to the improvement 

 of the mathematical sciences. He took an active 

 share in promoting the objects of public instruction 

 in his native city, and he had the honour of being 

 twice rector, and nine times dean of the faculty of 

 philosophy in the university of Basle. These pro- 

 fessional labours he occasionally relieved by an epis- 

 tolary communication with the first philosophers of 

 the age; and he could number among his correspon- 

 dents the names of Newton, Leibnitz, Marquis De 

 L'Hospital, Euler, Maupertuis, Wolff, De Moivre, 

 Mairan, Montmort, Renau, Tschirnhausen, Miche- 

 lotti, Craig, Cheyne, Poleni, Cramer, Bulfinger, and 

 Gravesende. His correspondence with Leibnitz is 

 published in a work in two volumes quarto, which 

 appeared in 1745, under the title of Leilmtii ac Bet' 

 noulli Commercium Philosophicum ct Mathematicum, 

 and which contains much curious information re- 

 specting that campaign of problems, in which these 

 powerful combatants shone with such distinguished 

 lustre. 



Near the close of the year 1747, he was attacked 

 with a disorder in the bowels, which was not how- 

 ever sufficiently violent to interrupt his usual studies ; 

 but on the last night of the year, the disease reached 

 such an alarming height, that he expired on the 

 morning of the first of January 1748, in the eighty- 

 first year of his age. 



John Bernoulli had nine children, three of whemj 



