The philosophy of mathematics · Auguste Comte
METHOD OF LAGRANGE.
_Derived Functions._ The conception of Lagrange, in its admirable simplicity, consists in representing the transcendental analysis as a great algebraic artifice, by which, in order to facilitate the establishment of equations, we introduce, in the place of the primitive functions, or concurrently with them, their _derived_ functions; that is, according to the definition of Lagrange, the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable. The special calculus of indirect functions has for its constant object, here as well as in the conceptions of Leibnitz and of Newton, to eliminate these _derivatives_ which have been thus employed as auxiliaries, in order to deduce from their relations the corresponding equations between the primitive magnitudes. _An Extension of ordinary Analysis._ The transcendental analysis is, then, nothing but a simple though very considerable extension of ordinary analysis. Geometers have long been accustomed to introduce in analytical investigations, in the place of the magnitudes themselves which they wished to study, their different powers, or their logarithms, or their sines, &c., in order to simplify the equations, and even to obtain them more easily. This successive _derivation_ is an artifice of the same nature, only of greater extent, and procuring, in consequence, much more important resources for this common object. But, although we can readily conceive, _à priori_, that the auxiliary consideration of these derivatives _may_ facilitate the establishment of equations, it is not easy to explain why this _must_ necessarily follow from this mode of derivation rather than from any other transformation. Such is the weak point of the great idea of Lagrange. The precise advantages of this analysis cannot as yet be grasped in an abstract manner, but only shown by considering separately each principal question, so that the verification is often exceedingly laborious.
Text from Project Gutenberg, public domain in the United States.