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.[11] 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.
[Footnote 11: Leibnitz had already considered the comparison of one curve with an other infinitely near to it, calling it "_Differentiatio de curva in curvam_." But this comparison had no analogy with the conception of Lagrange, the curves of Leibnitz being embraced in the same general equation, from which they were deduced by the simple change of an arbitrary constant.]
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.