Finally, although, in the summary exposition which was the object of this chapter, I have had to exhibit the condition of extreme imperfection which still belongs to the integral calculus, the student would have a false idea of the general resources of the transcendental analysis if he gave that consideration too great an importance. It is with it, indeed, as with ordinary analysis, in which a very small amount of fundamental knowledge respecting the resolution of equations has been employed with an immense degree of utility. Little advanced as geometers really are as yet in the science of integrations, they have nevertheless obtained, from their scanty abstract conceptions, the solution of a multitude of questions of the first importance in geometry, in mechanics, in thermology, &c. The philosophical explanation of this double general fact results from the necessarily preponderating importance and grasp of _abstract_ branches of knowledge, the least of which is naturally found to correspond to a crowd of _concrete_ researches, man having no other resource for the successive extension of his intellectual means than in the consideration of ideas more and more abstract, and still positive. * * * * * In order to finish the complete exposition of the philosophical character of the transcendental analysis, there remains to be considered a final conception, by which the immortal Lagrange has rendered this analysis still better adapted to facilitate the establishment of equations in the most difficult problems, by considering a class of equations still more _indirect_ than the ordinary differential equations. It is the _Calculus_, or, rather, the _Method of Variations_; the general appreciation of which will be our next subject. CHAPTER V. THE CALCULUS OF VARIATIONS. In order to grasp with more ease the philosophical character of the _Method of Variations_, it will be well to begin by considering in a summary manner the special nature of the problems, the general resolution of which has rendered necessary the formation of this hyper-transcendental analysis. It is still too near its origin, and its applications have been too few, to allow us to obtain a sufficiently clear general idea of it from a purely abstract exposition of its fundamental theory. PROBLEMS GIVING RISE TO IT. The mathematical questions which have given birth to the _Calculus of Variations_ consist generally in the investigation of the _maxima_ and _minima_ of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. Geometers for a long time designated all the questions of this character by the common name of _Isoperimetrical Problems_, which, however, is really suitable to only the smallest number of them.