There is one peculiarity. however, in the question of ordinary maxima and minima, to which the problem is here reduced, viz. that the quantity to be made a maxi mum or minimum is itself a differential expression, or in finitely small magnitude. This does not at all affect the truth of the conclusion, which is independent of the ab solute magnitude of the lines and velocities concerned, but it does the manner of treating it. In problems of maxima and minima, the quantities concerned are supposed to vary by increments infinitely smaller than the quantities them selves. Now, the variation of the length of FC, or the By the aid of this principle of James Bernoulli, we are already in a condition to resolve a variety of problems. 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 stuppinK to examine the farther cases to which it is applicable, we will continue to trace the history of the subject ; and it for tunately happens (which can hardly be said of any other branch of mathematical science,) that in this the order of invention is precisely the one calculated to afford the m ,st distinct and luminous view of it to one unacquainted with the subject, and to give him a Iaclical knowledge of its principles.
James Bernoulli, having resolved his brother's problem, 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 isolterimeirical problems on all w Inch depend on similar principles. In the Leipsic Acts, 1697, appeared, accom panied with the promise of a pecuniary reward to his brother, in case of a complete solution, a firog-ramma re quiring the nature of a curve, in which a certain integral expression (fy"d x snd x, y being the ordinate, and s the arc) shall be greater or less than in any other curve of the 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 mon' 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, is The first attempts of John Bernoulli towards the solution of this question, although not destitute of ingenuity, fell short, it must be confessed, of what might have been ex pected from his great abilities. lie supposed that the con
dition of equal length, and the maximum property, might universally be satisfied together, by making both the ordi nate QC and abscissa AQ vary at once, two elements BC, CD, only being considered. It would indeed be so, wet e the quantities concerned finite ; 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 di minished, the final equation (at least in the case where rs" d x is to be a maximum) is reduced by effacing all 'that part of it which is infinitely smaller than the rest, to 5 single term, and to a form altogether illusory ; so that his analysis, which, even in this case, led him to a distinct con clusion, is positively erroneous, and offends against the prin ciples of the infinitesimal calculus.
A diffeicnt and legitimate view of the subject was taken by James, who was thus enabled to detect and expose the fallacy of his brother's solutions ; but, although this was done with all possible forbearance and moderation, it is painful to observe, that the mortification thereby ex perienced by the latter, was productive of a deep and last ing rancour, which obliterated in his mind the candour of a philosopher, and in his language the decency of a gentle ; which assailed his brother's glory during the brief remainder of his life, and long after hisodeath continued to asperse his me mory.
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 equidis!ant, and infinitely near ordinates l'B, QC, RD, remain fixed, while C, D, vary along the lines QC, RD, by the infinitely small elements or variations C c, D d. Then, to determine their positions, or the relation between the differentials d y, d x, the first condition is, that