ISOPEHIMETRY. 



filence, neither puMft'mg his own folution, nor ciMtrcifm(T 

 that of his brother's. At length, in 1 705, James Bevnoiiilli 

 died, and a (hort time after John BernoniUi pubhfhed his 

 fohition in the Memoires of the academy for 1706. This, 

 however, poflelFed the fame radical defei\ that has been before 

 Hated, namely, that the author had confidered only two ele- 

 ments of the curve, iiiftead of which it is requifite to have 

 three enter, or to employ an equivalent condition. In pro- 

 blems of the fame kind as that of the line of fwifteft defcent, 

 where it is fimply required to fulfil the conditions of the 

 maximum or minimum, the applying of this condition to 

 two elements is fufRcient to find the fiuxional equation of 

 tlie curve ; but when, befide the maximum or minimum, the 

 curve mull poffefsa farther property of being ifoperimctrical 

 to another, this new condition requires that a tliird element 

 of the curve (hall have a certain inclination with refpeft to 

 the other two ; and every determination, founded fimply on 

 the firft confideration, will give falfe refults ; except in thofe 

 cafes where a curve cannot fatisfyone of tlie two conditions, 

 ■without at the fame time fulfiUing the other ; and of this 

 John Bernouilli was at length fo convinced, that he made it 

 the bafis of a new folution, more than 13 years after his 

 brother's death, confeffing himfelf decL-ived in his firll. "J'ai 

 donner ici," fays he, " pour reparer cette inadvertence une 

 noilvelle maniere de refoudre," &c. This was a tardy 

 avowal, but it would ftill have done him honour, had he at 

 the fame time acknowledged that his new folution was in 

 fubftance the fame as his brother's ; but giyen in a form which 

 confiderably abridged the calculation ; in (lead of which, he 

 even in this feeks every occafion to afperfe his brother's me- 

 thod, and this after a lapfe of fo many years, when, as Mr. 

 Woodhoufe obfcrves, "the rccoUeftioii of his brother's kind- 

 nefs, or zeal for a brother's fame, ought to have afifuaged 

 and laid to fleep all angry paffions ;'' but witli regard to the 

 folution itfelf, it mult be acknowledged, confidering the 

 flate of analytical fcience at tliattime, to poffefs very dillin- 

 guifhing marks of a great mafter, and fairly merits the eulo- 

 gium wliich the author himfelf has bellowed upon it ; that 

 of being equally exempt from the tcdioufnefs of his brother's, 

 and tiie obfcurity of Taylor's calculation, alluding here to 

 the folution of the celebrated Dr. Brook Taylor, wiiich ap- 

 peared, in 1715, in his " Methodus Incrementorum " At 

 the period at which we have now arrived, James Bernouilli 

 had been dead feveral years, and the above paper was tlie 

 laft that John Bernouilh wrote on the fubjeft of ifoperiinetry, 

 but the theory was purfued by many other eminent mathe- 

 maticians, and introduced into feveral of their works. Simp- 

 fon, in his Trafts, has a chapter, entitled " An Invelligation 

 of a General Rule for the Refohition of Ifoperimetiical Pro- 

 blems of all Orders.'' He has alfo given the folution of 

 feveral ifoperimctrical problems, in his " Doftrine of Flux- 

 ions " Maclaurin has hkewife a chapter on the fame fubjecl 

 in his "Treatife of Fluxions." To thefe may alfo be added 

 Emerfon, Le Seur, Boflut, and Lacroix ; each of which 

 authors has introduced this doftrine into their refpeClive 

 works ; but the two writers who have mod contributed in 

 bringing to perfeftion die theory of ifoperimetry, are Eulcr 

 and Lagrange, the former having, befide feveral memoirs 

 in the Ada Petro, a traft on this fubjeft, entitled " Me- 

 thodus inveniendi Lincas Cui vas Proprietate Maxiini Mini- 

 mive gaiidentes," which, with a very few exceptions, is what 

 i". was intended to be, a complete treatife, containing efien- 

 tiaily all the requifite methods of folutions, with great variety 

 and abundanccof examples aiid ilhiltrations : there were Rill, 

 howi-ver, fome -defefts in this work, for want of a better 

 atgoi ihm, or more compendious procefs of elUblilhing the 



theorems ; and certain ritpplemetital formula; ; which defefts 

 have been finally removed by I^agrange, in his admirable and 

 refined " Calculus of Variations ;" and a very interelliiig 

 treatife on the fame fubjecl has lately been publilhed by 

 Mr. Woodhoufe, in which are combined tlie hiitory and pro- 

 grefs of the fcience, with fuch obfervations and remarks, as 

 feem moft calculated to render it inftruftive and familiar to 

 the Englifii Itudent. 



Having thus given a brief (Icetch of the fuccefilve improve- 

 ments that have been made in the theory or ifopiTimetry, 

 from tlie time of its firft introduction by John Bcrnouiili, 

 to its completion by Lagrange ; we fliall conclude tliis 

 article with the folution of a few problems which will illuf- 

 trate many of the reraaiks that have been made in the fore- 

 going pages, referrmg the reader wI;o wifiies for farther 

 information to the woi-ks above quoted. 



The refearches of Euler and Lagrange have i-endered the 

 folution of ifoperimctrical problems extremely fimple, but 

 the invclligations by which they arrived at their formulas of 

 folution are very profound, long, and embarraClng, which 

 nothing lefs than tlie genius of thefe celebrated men would 

 have been able to have reduced to that fimple form at which 

 tliey at length arrived, an invcftigation of which, at lead of 

 Lagrange's, will be given und.er the article Variation. 

 In this place we fiiall barely Itate the refults, and (hew their 

 application to the folution of a few problems of this kind. 



Let V rcprefent any fundticn of the variable quantities 

 x-^ndy, and let there be afliimed 



V rz M ,v + Nj +- P/ + Q j, &c. [A] 



where 6 =. -—, a =; — , r =: — , 5:c. 

 '^ X X X 



Then the folution of all ifoperimetrical problems, into 

 which no integral quantity enters, are folvible by the fel- 

 lowing general formula, 



N- 



R 



cc. = o [B] 



which however, for the fake of a more ready application, 

 may be divided into the following cafes. 



Cafe I. — If M, Q, andall the co-efficients except N and P, 



in the above value of V, be equal to zero, then we have 



p 5 



V = N p -f P /) ; and fince crcjierally N -t- , .- — 



■* a i XX- 



P 



See. = o, it bee 



N 



omes in this cal 



P = o ; or, fii 



feN 



= o;3 equently, 

 , it reduces to N v — 



/> P = o ; whence N >' = /P; and fubftituting this laft 

 value of Nj, WT have V =^P -|- P /> ; and taking the 

 fluents V = P/ + r, c being the correftion. 



If M be r.ot equal to o, we muft add the fluent of M a ; 

 then V —/yii + V p -\- c. 



Cafe 2. — Let M = o, N = o, and all the co-efficients after 



Q, then V =z P^ + Q ?> and the general formula in this 

 cafe becomes 



p (V 



- — + -rrr = °5 



wlicnce P :=-■*" , and the fluents give P = -'^ -f r; 



multiply 



