973 



BERNOULLI. 



BERNOULLI. 



974 



JAMES BERNOULLI I , was born at Basel, December 27th 1654. His 

 father intended that he should be a divine, and had him taught the 

 classics and scholastic pbilo-ophy, but no mathematics. Accident 

 threw geometrical books in his way, and he studied them with ardour 

 in spite of the opposition of his father. He took for his device 

 Phaeton driving the chariot of the Sun, with the motto 'Invito patre 

 sidera verso.' At the age of twenty-two ho travelled to Geneva, and 

 from thence to France. It ia recorded of him that at the former 

 place he taught a blind girl to write, and that at Bordeaux he pre- 

 pared gnomonical tables. At his return, in 1680, he began to study 

 the philosophy of Descartes. 



The comet of 1680 drew from him hi3 'Conamen Novi Systematis,' 

 &c., an attempt to explain the phenomena of those bodies. He 

 imagined that they were satellites of a planet too distant to be visible, 

 and thence conjectured that their returns might be calculated. With 

 regard to the question of their predictive faculties, he supposes that 

 the head of the comet, being durable, denotes nothing, but that the 

 tail, being accidental, may be a symbol of the anger of heaven. M. 

 Fontenelle, as became the writer of an e'loge, calls this a " me"nage- 

 ment pour 1'opinion populaire ; " but we cannot follow him in viewing 

 it as such. In 1682 he published his treatise 'De Gravitate JEtheris,' 

 now of little note. His lasting fame dates from the year 1684, in 

 which Leibnitz published his first essays on the Differential Calculus 

 in the Leipzig Acts. From this time he and his brother John applied 

 themselves to the new science with a success and to an extent which 

 made Leibnitz declare that it was as much theirs as his. In 1687 he 

 was elected professor of mathematics at the University of Basel. His 

 celebrity attracted many foreigners to that place, and his researches 

 on the theory of series were investigations undertaken as official 

 exercises. 



The integral calculus was first inquired into by James Bernoulli, in 

 two essays published in 1691. His future labours were, in a great 

 measure, developments of the inexhaustible method of investigation 

 just named. Of that part which concerns his brother as well as him- 

 self we shall presently speak. He died at Basel of a slow fever, 

 August 16, 1705, in his fifty-first year. After the example of Archi- 

 medes, he ordered that one of his discoveries should be engraved on 

 his tomb. It was a drawing of the curve called by mathematicians 

 the logarithmic spiral, with the inscription ' 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 march 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 opportunity of reviewing his brother's 

 career could furnish temptation to exaggerate points of contrast ; and 

 before we quit this subject, we may observe that the career of James 

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

 consequence of the generalising spirit in which biographies are fre- 

 quently 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 mathema- 

 ticians have originated their best discoveries when 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 out by 

 Leibnitz, and which [BARROW] we might almost imagine his own 

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

 maticians that ever lived. 



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

 tandi,' one of the earliest works on the theory of probabilities, and his 

 treatise on series, were published posthumously 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 published at Geneva, in two vols. 4to, 1744. There is a letter 

 of his in the ' Journal de Physique,' September, 1792, which will be 

 presently alluded to. He edited the geometry of Descartes, in 1695. 



(See tloge by Fontenelle, in the collection ; the memoir by Lacroix 

 in the Biographie Universelle ; Montucla, Hist, des Math, throughout ; 

 and the Preface to Lacroix, Calc. Diff. ct Int.) 



JOHN BERNOULLI 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 Basel 

 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 1690 published a dissertation on 

 effervescence and fermentation ; but he soon began to apply himself 



to mathematics. In 1690 he travelled to Geneva and into France, 

 where ho formed many acquaintances, with such men as Malebranche, 

 the Cassinis, De l'H6pital, &c. He returned to Basel in 1692, and 

 from that titne 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, and the vigorous war of 

 problems which he maintained with the English school. In 1693 (our 

 authority the ' e'loge ' of the Berlin Academy, in Forme/a collection 

 of 1757, says 1691, but this must be a misprint) he waa elected pro- 

 fessor of mathematics at Wolfenbiittel ; but on his marriage with a 

 lady of Basel, named Dorothea Falckner, March 6th 1694, he returned 

 to hia-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 Basel 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 hia day. They were 

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

 sanne and Geneva in 1742. His correspondence with Leibnitz was 

 published in two vols. 4to, at the same places, in 1745. 



The author of the ' e'loge ' already cited says, that the qualities 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 per- 

 sonal disposition and manners. The celebrated dispute with James 

 Bernoulli is of a character unique in history, and forms an episode so 

 characteristic of ths state of science at the period, as well as of the 

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

 dwell a little upon it. 



Before the mathematical sciences were possessed of general methods 

 of investigation, problems of which hundreds are now soluble by one 

 process, were so many separate questions 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 

 now problems upon the world, which, though addressed to all, were 

 generally considered as particularly aimed at his elder brother, of 

 whose established reputation he seems to have been jealous. In 1696 

 John Bernoulli proposed the well-known problem of the 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'Hopital; 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 supe- 

 riority 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 given rectilinear base, and of a given length, to find that which con- 

 tains 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 

 correspondence 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 hia brother, but reproaches him 

 with the time he had employed upon it. He goes on. to aay, that as 

 to his brother's new problems, they were iu reality contained in his 

 own ; that difficult as they might appear, he had immediately over- 

 come them; and 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 fulfilment 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 Savans' for February 1698, that his brother's 

 solution was wrong ; that if no one published 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 solution of which he offered 200 florins, 



