than those of the direct magnitudes. But the formation of these differential equations was not supposed to admit of any general and abstract rules. Now the Analysis of Variations, considered in the most philosophical point of view, may be regarded as essentially destined, by its nature, to bring within the reach of the calculus the actual establishment of the differential equations; for, in a great number of important and difficult questions, such is the general effect of the varied equations, which, still more indirect than the simple differential equations with respect to the special objects of the investigation, are also much more easy to form, and from which we may then, by invariable and complete analytical methods, the object of which is to eliminate the new order of auxiliary infinitesimals which have been introduced, deduce those ordinary differential equations which it would often have been impossible to establish directly. The method of variations forms, then, the most sublime part of that vast system of mathematical analysis, which, setting out from the most simple elements of algebra, organizes, by an uninterrupted succession of ideas, general methods more and more powerful, for the study of natural philosophy, and which, in its whole, presents the most incomparably imposing and unequivocal monument of the power of the human intellect.
We must, however, also admit that the conceptions which are habitually considered in the method of variations being, by their nature, more indirect, more general, and especially more abstract than all others, the employment of such a method exacts necessarily and continuously the highest known degree of intellectual exertion, in order never to lose sight of the precise object of the investigation, in following reasonings which offer to the mind such uncertain resting-places, and in which signs are of scarcely any assistance. We must undoubtedly attribute in a great degree to this difficulty the little real use which geometers, with the exception of Lagrange, have as yet made of such an admirable conception.