to exact any great intellectual efforts, seeing that we can hardly conceive of any precise relation which is not immediately a certain relation of equality, or which cannot be readily brought thereto by some very easy transformations.
Thus, when we admit every species of functions into the definition of equations, we do not at all account for the extreme difficulty which we almost always experience in putting a problem into an equation, and which so often may be compared to the efforts required by the analytical elaboration of the equation when once obtained. In a word, the ordinary abstract and general idea of an equation does not at all correspond to the real meaning which geometers attach to that expression in the actual development of the science. Here, then, is a logical fault, a defect of correlation, which it is very important to rectify.
Division of Functions into Abstract and Concrete. To succeed in doing so, I begin by distinguishing two sorts of functions, abstract or analytical functions, and concrete functions. The first alone can enter into veritable equations. We may, therefore, henceforth define every equation, in an exact and sufficiently profound manner, as a relation of equality between two abstract functions of the magnitudes under consideration. In order not to have to return again to this fundamental definition, I must add here, as an indispensable complement, without which the idea would not be sufficiently general, that these abstract functions may refer not only to the magnitudes which the problem presents of itself, but also to all the other auxiliary magnitudes which are connected with it, and which we will often be able to introduce, simply as a mathematical artifice, with the sole object of facilitating the discovery of the equations of the phenomena. I here anticipate summarily the result of a general discussion of the highest importance, which will be found at the end of this chapter. We will now return to the essential distinction of functions as abstract and concrete.
This distinction may be established in two ways, essentially different, but complementary of each other, à priori and à posteriori; that is to say, by characterizing in a general manner the peculiar nature of each species of functions, and then by making the actual enumeration of all the abstract functions at present known, at least so far as relates to the elements of which they are composed.
À priori, the functions which I call abstract are those which express a manner of dependence between magnitudes, which can be conceived between numbers alone, without there being need of indicating any phenomenon whatever in which it is realized. I name, on the other hand, concrete functions, those for which the mode of dependence expressed cannot be defined or conceived except by assigning a determinate case of physics, geometry, mechanics, &c., in which it actually exists.
Most functions in their origin, even those which are at present the most purely abstract, have begun by being concrete; so that it is easy to make the preceding distinction understood, by citing only the successive different points of view under which, in proportion as the science has become formed, geometers have considered the most simple analytical functions. I will indicate powers, for example, which have in general become abstract functions only since the labours of Vieta and Descartes. The functions x2, x3, which in our present analysis are so well conceived as simply abstract, were, for the geometers of antiquity, perfectly concrete functions, expressing the relation of the superficies of a square, or the volume of a cube to the length of their side. These had in their eyes such a character so exclusively, that it was only by means of the geometrical definitions that they discovered the elementary algebraic properties of these functions, relating to the decomposition of the variable into two parts, properties which were at that epoch only real theorems of geometry, to which a numerical meaning was not attached until long afterward.