This problem, which afterwards suggested such a long series of analogous questions, consists in determining the curve which a heavy body must follow in order to descend from one point to another in the shortest possible time. Limiting the conditions to the simple fall in a vacuum, the only case which was at first considered, it is easily found that the required curve must be a reversed cycloid with a horizontal base, and with its origin at the highest point. But the question may become singularly complicated, either by taking into account the resistance of the medium, or the change in the intensity of gravity. _Isoperimeters._ Although this new class of problems was in the first place furnished by mechanics, it is in geometry that the principal investigations of this character were subsequently made. Thus it was proposed to discover which, among all the curves of the same contour traced between two given points, is that whose area is a maximum or minimum, whence has come the name of _Problem of Isoperimeters_; or it was required that the maximum or minimum should belong to the surface produced by the revolution of the required curve about an axis, or to the corresponding volume; in other cases, it was the vertical height of the center of gravity of the unknown curve, or of the surface and of the volume which it might generate, which was to become a maximum or minimum, &c. Finally, these problems were varied and complicated almost to infinity by the Bernouillis, by Taylor, and especially by Euler, before Lagrange reduced their solution to an abstract and entirely general method, the discovery of which has put a stop to the enthusiasm of geometers for such an order of inquiries. This is not the place for tracing the history of this subject. I have only enumerated some of the simplest principal questions, in order to render apparent the original general object of the method of variations. _Analytical Nature of these Problems._ We see that all these problems, considered in an analytical point of view, consist, by their nature, in determining what form a certain unknown function of one or more variables ought to have, in order that such or such an integral, dependent upon that function, shall have, within assigned limits, a value which is a maximum or a minimum with respect to all those which it would take if the required function had any other form whatever. Thus, for example, in the problem of the _brachystochrone_, it is well known that if _y_ = _f(z)_, _x_ = π(_z_), are the rectilinear equations of the required curve, supposing the axes of _x_ and of _y_ to be horizontal, and the axis of _z_ to be vertical, the time of the fall of a heavy body in that curve from the point whose ordinate is _z₁_, to that whose ordinate is _z₂_, is expressed in general terms by the definite integral ∫_{_z₂_}^{_z₁_}√(1 + (_f'(z))²_ + (π'(_z_))²/(2_gz_))_dz._