∫dx√(1 + (f'(x) )2),
that one which renders the integral ∫f(x)dx, taken between the same limits, a maximum. It is evidently always so in other questions of this class.
Methods of the older Geometers. In the solutions which geometers before Lagrange gave of these problems, they proposed, in substance, to reduce them to the ordinary theory of maxima and minima. But the means employed to effect this transformation consisted in special simple artifices peculiar to each case, and the discovery of which did not admit of invariable and certain rules, so that every really new question constantly reproduced analogous difficulties, without the solutions previously obtained being really of any essential aid, otherwise than by their discipline and training of the mind. In a word, this branch of mathematics presented, then, the necessary imperfection which always exists when the part common to all questions of the same class has not yet been distinctly grasped in order to be treated in an abstract and thenceforth general manner.
METHOD OF LAGRANGE.
Lagrange, in endeavouring to bring all the different problems of isoperimeters to depend upon a common analysis, organized into a distinct calculus, was led to conceive a new kind of differentiation, to which he has applied the characteristic δ, reserving the characteristic d for the common differentials. These differentials of a new species, which he has designated under the name of Variations, consist of the infinitely small increments which the integrals receive, not by virtue of analogous increments on the part of the corresponding variables, as in the ordinary transcendental analysis, but by supposing that the form of the function placed under the sign of integration undergoes an infinitely small change. This distinction is easily conceived with reference to curves, in which we see the ordinate, or any other variable of the curve, admit of two sorts of differentials, evidently very different, according as we pass from one point to another infinitely near it on the same curve, or to the corresponding point of the infinitely near curve produced by a certain determinate modification of the first curve. It is moreover clear, that the relative variations of different magnitudes connected with each other by any laws whatever are calculated, all but the characteristic, almost exactly in the same manner as the differentials. Finally, from the general notion of variations are in like manner deduced the fundamental principles of the algorithm proper to this method, consisting simply in the evidently permissible liberty of transposing at will the characteristics specially appropriated to variations, before or after those which correspond to the ordinary differentials.
This abstract conception having been once formed, Lagrange was able to reduce with ease, and in the most general manner, all the problems of Isoperimeters to the simple ordinary theory of maxima and minima. To obtain a clear idea of this great and happy transformation, we must previously consider an essential distinction which arises in the different questions of isoperimeters.
Two Classes of Questions. These investigations must, in fact, be divided into two general classes, according as the maxima and minima demanded are absolute or relative, to employ the abridged expressions of geometers.
Questions of the first Class. The first case is that in which the indeterminate definite integrals, the maximum or minimum of which is sought, are not subjected, by the nature of the problem, to any condition; as happens, for example, in the problem of the brachystochrone, in which the choice is to be made between all imaginable curves. The second case takes place when, on the contrary, the variable integrals can vary only according to certain conditions, which usually consist in other definite integrals (which depend, in like manner,