ISOPERIMETRICAL PROBLEMS. 



, when (by the shift- 

 secutive position RD) 



Problem.. K'(J "auahir itsTonlceutive value W=dy + ddy. 



There are two ways in which this difficulty may be 



avoided. The fir,t is the one we have above taken, viz. 

 resolving the question of the maximum or minimum, as 

 a separate and independent problem, and then adapting 

 the conclusion so obtained to the particular case in ques- 

 tion- accordingly this is the course pursued at first by 

 the Bernoullis ; but, not to mention the want of ana- 

 lytical neatness in such a mode of proceeding, the ques- 

 tions of maxima and minima, which thus i.nse, would 

 in most cases be found of extreme difficulty, and their 

 subsequent adaptation to infinitely small quantities a 

 matter of uncommon delicacy. The other method con- 

 sists in looking on the difficulty (in its true light) as 

 merely one of notation, and obviating it by a refine- 

 ment in that point, viz.. by employing a new character- 

 istic 3 for that hypothesis of differentiation, by which 

 the point C shifts, as it were, from one curve, BCD, to 

 another infinitely near.it, along the fixed ordinate, and 

 a new name variation for this peculiar change in the 

 value of dy. We have here the origin of the calculus 

 of variations ; and the manner in which this simple arti- 

 fice enables us to combine the question of the mini- 

 mum, with the peculiar circumstances of the case where 

 it arises, cannot be better shewn than by the very in- 

 stance before us. 



It is, first of all, evident, that as S means nothing more 

 than differentiation, on another hypothesis, all the rules 

 of the differential calculus must be attended to in the 

 management of the new symbol, only making the va- 

 riations of such quantities as do not change on this hy- 

 pothesis, zero. Thus, 



hardly be said of any other branch of mathematical Isopcrirae- 

 science,) that in this the order of invention is precisely "jical 

 the one calculated to afford te most distinct and lumi- ^^ l ^, 

 nous view of it to one unacquainted with the subject, 

 and to give him a radical knowledge orits principles. 



James Bernoulli, having resolved his brother's pro- 

 blem, proposed, in his turn, the celebrated defiance, which 

 at once concentered the attention of the mathematical 

 world upon these researches, and which has imposed 

 the name of isoperimetrical problems on all which de- 

 pend on similar principles. In the Leipsic Acts, 1697, 

 appeared, accompanied with the promise of a pecuniary 

 reward to his brother, in case of a complete solution, a 

 programma requiring the nature of a curve, in which a 

 Certain integral expression (J y n dx orj\ n dx, y be- 

 ing the ordinate, and * the arc) shall be greater or less 

 than in any other curve nf l/ie same length between its 

 fixed extremities. We have here the first instance of a 

 question of what is called relative maxima and minima, 

 that is, where the curve in which a certain integral (A ) 

 is to be made a maximum or minimum, is to be selected, 

 not from among all curves whatever, but only from such 

 as have at least one property in common expressed by 

 some other integral (B), which, in the present instance. 



because the length of BF is given, and the velocities 

 through BF and CD are uniform. Again, 



3(rfy + dy)=0, or }dy'=z Idy, 

 because GD dy +- dy' is invariable. Silly, 

 ids t 



= 0: 



v v 



by the condition of the minimum. Now, 



dy 3 dy dy . 



= = ~idy, 

 u- ds y 



and, 



(a) 



dy' . . , dy' 



~ ds' ds' 



So that our equation (a) becomes 

 dy' 



v'ds' 



Idy 



.. 



v ds 



and here, as in all problems of maxima and minima, 

 the variation ^d y divides off, and leaves 



dy' dy = ^ 

 v'd s 1 v ds ~ 

 the same equation as before. 



By the aid of this principle of James Bernoulli, we 

 are already in a condition to resolve a variety of pro- 

 blems. In the present instance, it is easy to see in 

 what manner it supplies the place of the metaphysical 

 principle of his brother's solution: but, without at 

 present stopping to examine the farther cases to which 

 >t is applicable, we will continue to trace the history of 

 the subject; and it fortunately happens (which can 



is 



The first attempts of John Bernoulli towards the solu 

 tion of this question, although not destitute of inge- 

 nuity, fell short, it must be confessed, of what might 

 have been expected from his great abilities. He sup- 

 posed that the condition of equal length, and the maxi- 

 mum property, might universally be satisfied together, 

 by making both the ordinate QC and abscissa AQ vary 

 at once, two elements BC, CD, only being considered. 

 It would indeed be so were the quantities concerned fi- 

 nite ; but, even then, were a third condition expressed 

 by some other integral formula (C) superadded, the 

 method must at last be abandoned. When, however, 

 the elements of the curve are infinitely 'diminished, the 

 final equation (at least in the case whereyj" d x is to be 

 a maximum ) is reduced by effacing all that part of it 

 which is infinitely smaller than the rest, to a single 

 term, and to a form altogether illusory ; so that his ana- 

 lysis, which, even in this case, led him to a distinct con- 

 clusion, is positively erroneous, and offends against the 

 principles of the infinitesimal calculus. 



A different and legitimate view of the subject was 

 taken by James, who was thus enabled to detect and ex- 

 pose the fallacy of his brother's solutions; but, although 

 this was done with all possible forbearance and mode- 

 ration, it is painful to observe, that the mortification 

 thereby experienced by the latter, was productive of a 

 deep and lasting rancour, which obliterated in his-mind 

 the candour of a philosopher, and in his language the 

 decency of a gentleman ; which assailed his brother's 

 glory during the brief remainder of his life, and long 

 after his death continued to asperse his memory. 



James Bernoulli observes, that the variation of one 

 ordinate is indeed sufficient, when but one condition is 

 to be satisfied, but when two are concerned, it will be 

 necessary to make the same number of ordinates vary. 

 The principle of his solution, which became the foun- 

 dation of all subsequent researches, until the invention 

 of the calculus of variations, is as follows : The four 

 equidistant, and infinitely near ordinates PB, QC, RD, 



