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. _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, upon the required functions) always retaining the same given value; as, for example, in all the geometrical questions relating to real _isoperimetrical_ figures, and in which, by the nature of the problem, the integral relating to the length of the curve, or to the area of the surface, must remain constant during the variation of that integral which is the object of the proposed investigation. The _Calculus of Variations_ gives immediately the general solution of questions of the former class; for it evidently follows, from the ordinary theory of maxima and minima, that the required relation must reduce to zero the _variation_ of the proposed integral with reference to each independent variable; which gives the condition common to both the maximum and the minimum: and, as a characteristic for distinguishing the one from the other, that the variation of the second order of the same integral must be negative for the maximum and positive for the minimum. Thus, for example, in the problem of the brachystochrone, we will have, in order to determine the nature of the curve sought, the equation of condition δ∫_{_z₂_}^{_z₁_}√([1 + (_f'(z)_)² + (π'(_z_))²]/(2_gz_))_dz_ = 0,