ISOPERIMETRY. 



tliem, lie prepofed to mathematicians in general, at the con- 

 clufion of his folution of liis brother's problem. 



The rivalry in glory that had long divided the Bernoiiillis, 

 \ra3 fully difpla)Ted on this occafion. At firll it was a little 

 moderated by their habits of feeing each other, at lead oc- 

 cafionally, and by the intervention of their common friends ; 

 but John haWng been appointed profL-fToi- of matiiematics at 

 Groningen in 169J, all private intercom fe between them 

 foon ccafed, and they no longer correfponded except through 

 the medium of periodical publications, for the purpofe of 

 propofing to each other the mod difficult problems ; and 

 here it was that James Bernouilli, defirous of avenging him- 

 felf of the ingratitude of his brother, to whom he had been 

 preceptor, challenged him by name to anfwcr the following 

 problero. 



Of all ifoperimctrieal curves dcfcribed on the fame com- 

 •mon bafe B N, to find B F N, fuch that anotlicr curve, 

 B Z N, (hall contain the grcattft fpace ; the ordinate of 

 which, P Z, is in any multiplicatc or fubmultiplicate ratio of 

 the ordinate P F, or of the arc B F. Or, as we ihould fay 

 now, the ordinate of which, P Z, is any fundion of P F, or 

 of the arc B F. Fi^. I ,-. 



To this leading propolition, he added another more analo- 

 gous to that of the line of (aiftLJi defcent, which was, to 

 frnd among all the cycloids, which a heavy body may de- 

 fcribe from a pirint to a line given in poiition, that cycloitl 

 which is defcribed in the leall poffible time, which propofi- 

 tions he concluded in nearly the following words. " A per- 

 fon for whom I pledge myfelf { Prodit Nox NE.\ro, pro qui 

 caveo) engages to give my brother, independently of the 

 praife he will deferve, a prize of 50 florins, on condition 

 that within three months, he engages to refolve thefe pro- 

 blems, and within a year publifhes legitimate folutions of 

 them."' Adding, " if at the expiration of this time, no 

 one (hail have refolved them, I will make public my fo- 

 lutions." 



Thefe propofitions, as we have before obferved, were fent 

 with the folution of the hrachyfiochrone (a term by which 

 John Bernouilli's problem was delignated); and as foon as he 

 had noticed .he fouitions to this, in doing which he bellow- 

 ed great praife on that of Newton and de I'Hopital, and 

 fome flight cenfure on his brother's, he undertook the folu- 

 tion of James's problems above-mentionL-J, and imagining 

 that his theory of the line of f.viftell defcent was alone fuffi- 

 cient to folve them, the following expreffions of ingenuous 

 vanity efcaped him. " Difficult," fays he, " as thefe problems 

 appear, I did not fail to a;<ply to them the inftant they came 

 to my hands, and hear with what fuccefs; inftead of three 

 months allowed me to found their depth, and the remainder 

 of the year to find their folutions, I have employed only 

 three minutes to examine, enter upon, and dive to tiie bot- 

 tom of this myftery." Thefe iiigh founding phrafes were ac- 

 companied with the coni^rudtion he gave of the problems, 

 awd the coofequent demand of the prize, which he faid l;e 

 Ihould give to the poor, a.s it coll him fo little trouble to 

 gain it. Bui the bufmefs was by no means fo far advanced as 

 he fuppofed, as his folution only anfwered for particular cafes, 

 in confequence of hi^ having made only two elements of the 

 curve enter therein, while the general folution required 

 three ; and he therefore thus laid himfelf open to the keen 

 reproaches of his brother, who foon perceived in what re- 

 fpect the folution was defective, and being at the fame time 

 perfectly fure of his own, he publifhed an advertifement in 

 1698, in which he affertedthat his brother's method was de- 

 feiiive. He Hill allowed geometricians time to find the folu- 

 tion, and if no one gave it, he pledged liimfelf for tl^ce 



things; ift. To divine with pritcifion the analyfis ofhi'g 

 brother : jdly Whatever it might be, to point out fallacies in 

 it : and, 3dly. To give the true folution of the problem in all 

 its parts. Adding, at the fame time, that if any perfon was 

 fufficiently interclKd in the progrefs of fcience to venture a 

 wager upon thefe articles, that he would engage to forfeit an 

 equal fum if he failed in the firil ; double the funi if he did 

 not fucceed in the lecond ; and triple the fum if he did not 

 accomplifti the third. 



The fingularity of this advertifement, and the reputation 

 of the writer as a geometrician, a little ilaggcred John Ber- 

 nouilli's confidence in liis method. He revifed his folution, 

 allowed that he had m.ide a trifling miftake, which he 

 afcribod to too great precipitancy, and fent a new refult , but 

 without alFuming a more modeft tone, and again demanded 

 the prize. 



To thefe pretenfions, James Bernouilli laconically an- 

 fwered, " I beg my brother to revife his lad folution anew, 

 to examine it carefully in every point, and then to let us know 

 whether it be all right ; as he mull be aware, tliat no atten- 

 tion can be paid to his excufes of precipitancy after I have 

 publiflied my folution." But John Bernouilli, who was not 

 aivare of the radicd defedl of the method that he employed, 

 felt an entire confidence in his lall refult, and faid in reply 

 there was no neccfhty to revife what he had done, ar.d that 

 his time would be much better fpent in making new difcove- 

 ries. To this confident affertion James ironically anfwered, 

 «' I never believed that ray brother was mailer of the true 

 folution of the ifoperimetrical problem, and I doubt it no.v 

 more than ever, from the difficulty he makes of the rcvifion 

 of his folution ; if it coll him but three minutes, as he 

 aflerts, " to examine, enter upon, and dive to the bottom of 

 the whole myitery," furely the revifal could not require 

 more ; but fuppofe he (pent double that time, how many 

 new difcoveries v/ould he be robbed of by the ^x-i. mmuU'S 

 thus employed." 



To this John again replied, and the matter Hill remained 

 undecided ; till, in 1700, James Bernouilli printed at Baiil a 

 letter addreifed to his brother, m which he invited him with 

 great moderation to publifh his method, and concluded by 

 giving tlie formuks of the problem, but without their de- 

 monilrations. John foon perceived how far he differed from 

 his brother, but not difcovering the principle of the true fo- 

 lution, nor the defecl of his own method, he at length gave 

 it in a paper which was fent under a ieal to the Academy of 

 Sciences at Paris, in the month of February 1701, on con- 

 dition that it ihould not be opened without liis confent, ;;nd 

 after liis brother had publllhedhis analyfis. 



As foon as James Bernouilli was informed of this, he had 

 no longer any reafon to keep his folution a fecret ; he 

 accordingly made it public, arid maintaiued it by way of 

 a thelis at Safil, in T.Iarch 1701, with a dedication to the 

 four illuilrious mathematicians, de I'Hopital, Leibnitz, New- 

 ton, and Fatiode Duillier. He hkewile printed it feparalely 

 under the following title : " Analyfis magni problcmatici 

 ifopcrimetricu" This was confidered as a prodigy of faga- 

 city and invention ; and indeed, if the time be ccnlidered, it 

 will not be too much to affert that a more difficult j.' cblem 

 was never refolved. The marquis de I'Hopital wrote to 

 Leibnitz, tlut he had read it with avidity, and that he had 

 found it very d'u-eft and accurate, which tellimony Leibnitz 

 tranfmitted to John Bernouilli himfelf, though he was much 

 prejudiced in his favour, having himfelf before examined and 

 approved of John Bernouilli's folution ; the latter having 

 fubmitted it to him for his opinion. 



After ihii publicaiioo John Bernouilli maintained a porfeA 



fJentc, 



