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grated on his tomb. It was a drawing of the outre called 

 by mathematicians the logarithmic spiral, with the inscrip- 

 tion Eadem mutata resurgo : a double allusion, first, to his 

 hope of a resurrection, next, to the remarkable properties of 

 the curve, well known to mathematicians, which consist in 

 this, that many operations which, in most instances, convert 

 one curve into another, in the logarithmic spiral only repro- 

 duce the original. 



M. Fontenelle, his contemporary, says, ' M. Bernoulli 

 was of a bilious and melancholy temperament, a character 

 which, more than any other, gives the zeal and perseverance 



necessary for great things In all his researches his 



mareh was slow and sure ; neither his genius nor his habit 

 of success inspired him with confidence; he published 

 nothing without handling it over and over again ; and he 

 never ceased to fear the public which held him in so much 

 veneration. ' It is worth while to observe that the above 

 was written in the year of his death, and before the oppor- 

 tunity of reviewing his brother's career could furnish tempta- 

 tion to exaggerate points of contrast ; and before we quit 

 this subject, we may observe that the career of James Ber- 

 noulli is, on one point, a contradiction to a favourite theory, 

 a consequence of the generalising spirit in which biogra- 

 phies are frequently written. The qualities of the man 

 in question, be he who he may, are made the necessary 

 accompaniments of all who distinguish themselves in a 

 similar way. Thus, because several great mathematicians 

 have originated their best discoveries very young, it is laid 

 down as a sort of law of nature that they should always do 

 so : but James Bernoulli did nothing which would have 

 made him famous, even among contemporaries, till after he 

 was thirty years old, and then not from a principle of his 

 own, but from a hint thrown cut by Leibnitz, and which 

 r see BARROW] we might almost imagine his own genius 

 would have seized. Yet he is one of the most original ma- 

 thematicians that ever lived. 



He was married, and left a son and daughter. His ' Ars 

 Conjectandi,' one of the earliest works on the theory of pro- 

 babilities, and big treatise on series, were published posthu- 

 mously in 1713, under the care of Nicolas Bernoulli the 

 elder. Part of it was republished by Baron Maseres in 

 1795, in a volume of tracts. His complete works were pub- 

 lished at Geneva, 1744, in two voU. 4to. There is a letter 

 of his in the Journal de Physique, September, 1792, which 

 wHl be presently alluded to. He edited the Geometry of 

 Descartes, in 1695. 



(See elnge by Fontenelle, in the collection ; the memoir 

 by Lacroix in the Biographic Uiiieerselle ; Montucla, Hist, 

 den Math., throughout ; and the Preface to Lacroix, Calc. 

 Diff. et Int.) 



JOHN BKHNOHLLI I., brother of the preceding, was born 

 July 27th, 1667 (old style). He was the ninth child of his 

 father, who intended him for commercial pursuits, and sent 

 him to the University at Basle in 1682, where, like his 

 brother, he found his own vocation. He was made master 

 of arts in 1685, on which occasion he read a thesis in Greek 

 verse, in refutation, we suppose, of the divine right, &c., the 

 subject being, that the prince is made for his subjects. 



He then studied medicine, and in 1 690 published a dis- 

 sertation on effervescence and fermentation ; but he soon 

 began to apply himself to mathematics. In 1690 he tra- 

 velled to Geneva and into France, where he formed many 

 acquaintances, with such men as Malebranche, the Cassinis, 

 De rilopital. Sec. He returned to Basle in 1692, and from 

 that time dates his correspondence with Leibnitz. It is 

 well known how strenuously he defended the cause of the 

 latter in the dispute about the invention of fluxions, which 

 will appear in its proper place, and the vigorous war of pro- 

 blems which he maintained with the English school. In 

 1 693 (our authority the Hoge of the Berlin Academy, in 

 Formey's collection of 1757, says 1691, but this must be a 

 misprint) he was elected professor of mathematics at Wolf- 

 enbuttel ; but on his marriage with a lady of Basle, named 

 Dorothea Falckner, March 6th, 1694, he returned to his own 

 country, was received doctor of medicine, and kept a public 

 act on the Motion of the Muscles. 



In 1695 he accepted a professorship at Groningen, at 

 which place he remained till he succeeded his brother James 

 at Basle in 1705, where he died January 1st, 1748. We 

 shall have to speak of five of his descendants. He published 

 no separate works, but his memoirs are to be found in all 

 the scientific transactions of his day. They were collected 

 in four quarto volumes by Cramer, and published at Lau- 



sanne and Genera in 1 742. His correspondence with Leib- 

 nitz was published in two vols. 4to. at the same places in 

 1745. 



The author of the cloge already cited says, that the qua- 

 lities of his heart were not less estimable than those of his 

 head, and that he was 'juste, droit, sincere, et pieux.' To 

 the last quality he has an undoubted right ; but his whole 

 history is an unfortunate example of impetuosity of temper 

 and narrowness of mind, which betrayed him into a want 

 of fairness, almost amounting to baseness. The assertion 

 of the eulogist is, as the reader will see, a tolerable specimen 

 of the extent to which such productions may be trusted as 

 to points of personal disposition and manners. The cele- 

 brated dispute with James Bernoulli is of a character 

 unique in history, and forms an episode so characteristic of 

 the state of science at the period, as well as of the disposi- 

 tions of the two celebrated brothers, that it is worth while to 

 dwell a little upon it. 



Before the mathematical sciences were possessed of ge- 

 neral methods of investigation, problems of which hundreds 

 are now soluble by one process were so many separate ques- 

 tions with separate difficulties. It had been the practice 

 of centuries for mathematicians who had found a particular 

 solution of any case, to propose the question as a challenge 

 to others. In the years preceding 1696 John Bernoulli 

 had showered new problems upon the world, which though 

 addressed to all, were generally considered as particularly 

 aimed at his elder brether, of whose established reputation 

 he seems to have been jealous. In 1696 John Bernoulli 

 proposed the well-known problem of Ui&brachistochron, or 

 ' to find the curve on which a material point will fall from one 

 given point to another in the least possible time.' This was 

 answered by Leibnitz, Newton, James Bernoulli, and De 

 1 H6pital ; but the third hit upon a method of solving more 

 general questions of the same kind ; and feeling perhaps 

 that it was time to assert the superiority which his age and 

 reputation might be supposed to give him, returned a 

 counter-challenge with his solution. It was a problem of a 

 much more general and abstruse character, one limited case 

 of which is the following ; 'Of all the curve lines which can 

 be described on a giv in rectilinear base, and of a given 

 length, to find that which contains the greatest area.' He 

 added another, which amounted to asking for the curve of 

 quickest descent, not from a point to a point, but from a 

 point to a given straight line: and ended by stating that a 

 person of his acquaintance (probably himself) would give 

 his brother due praise, and fifty florins besides, if he would 

 solve these problems within three months, and publish his 

 solutions within a year. John Bernoulli, in an answer 

 published immediately afterwards (for private correspond- 

 ence between the brothers had ceased), praises the solutions 

 which Newton, Leibnitz, and De 1'Hopital had given of 

 his problem, and admits the correctness of that of his bro- 

 ther, but reproaches him with the time he had employed 

 upon it. He goes on to say, that as to his brother's new 

 problems, they were in reality contained in his own ; that 

 difficult as they might appear, he had immediately over- 

 come them ; that instead of three months, it only took him 

 three minutes to penetrate the whole mystery. He sent 

 the results of his solutions accordingly, and required fulfil- 

 ment of the promise ; adding, that as it had cost him too 

 little trouble to gain the money, he should give it to the poor. 

 He had in fact solved the second problem, which as he truly 

 stated, is not of difficult deduction from his own ; but he 

 deceived himself as to the first. James Bernoulli quietly 

 answered, in the Journal des Savons for February, 1698, 

 that his brother's solution was wrong ; that if no one pub 

 lished any further solution, he would engage, 1. To find 

 out what his brother's method had been ; 2. Whatever it 

 was, to show that it was wrong ; 3. To give a true solution 

 of the problem. And he added, that whatever sum any one 

 would undertake to give him for succeeding in each of the 

 three undertakings, he would forfeit as much if he failed in 

 the first, twice as much if he failed in the second, and three 

 times as much if he failed in the third. The positive tone 

 of this announcement alarmed John Bernoulli, who well 

 knew that his brother was not a man to be much mistaken 

 when he spoke so strongly ; and he accordingly looked 

 again at his solution, corrected it as he thought, admitted 

 that he had been too precipitate, and again demanded the 

 reward. He proposed also another problem, for the solu- 

 tion of which he offered 200 florins, if done within the year. 

 James Bernoulli replied, ' I recommend my brother to look 



