To grasp in a general manner the spirit of the different procedures which are employed in quadratures, we must observe that, by their nature, they can be primitively founded only on the differentiation of the ten simple functions. The results of this, conversely considered, establish as many direct theorems of the integral calculus, the only ones which can be directly known. All the art of integration afterwards consists, as has been said in the beginning of this chapter, in reducing all the other quadratures, so far as is possible, to this small number of elementary ones, which unhappily we are in most cases unable to effect. _Singular Solutions._ In this systematic enumeration of the various essential parts of the integral calculus, considered in their logical relations, I have designedly neglected (in order not to break the chain of sequence) to consider a very important theory, which forms implicitly a portion of the general theory of the integration of differential equations, but which I ought here to notice separately, as being, so to speak, outside of the integral calculus, and being nevertheless of the greatest interest, both by its logical perfection and by the extent of its applications. I refer to what are called _Singular Solutions_ of differential equations, called sometimes, but improperly, _particular_ solutions, which have been the subject of very remarkable investigations by Euler and Laplace, and of which Lagrange especially has presented such a beautiful and simple general theory. Clairaut, who first had occasion to remark their existence, saw in them a paradox of the integral calculus, since these solutions have the peculiarity of satisfying the differential equations without being comprised in the corresponding general integrals. Lagrange has since explained this paradox in the most ingenious and most satisfactory manner, by showing how such solutions are always derived from the general integral by the variation of the arbitrary constants. He was also the first to suitably appreciate the importance of this theory, and it is with good reason that he devoted to it so full a development in his "Calculus of Functions." In a logical point of view, this theory deserves all our attention by the character of perfect generality which it admits of, since Lagrange has given invariable and very simple procedures for finding the _singular_ solution of any differential equation which is susceptible of it; and, what is no less remarkable, these procedures require no integration, consisting only of differentiations, and are therefore always applicable. Differentiation has thus become, by a happy artifice, a means of compensating, in certain circumstances, for the imperfection of the integral calculus. Indeed, certain problems especially require, by their nature, the knowledge of these _singular_ solutions; such, for example, in geometry, are all the questions in which a curve is to be determined from any property of its tangent or