The entire domain of arithmetic is, then, much more extended than is commonly supposed; but this _calculus of values_ will still never be more than a point, so to speak, in comparison with the _calculus of functions_, of which mathematical science essentially consists. This comparative estimate will be still more apparent from some considerations which I have now to indicate respecting the true nature of arithmetical questions in general, when they are more profoundly examined.
_Its true Nature._ In seeking to determine with precision in what _determinations of values_ properly consist, we easily recognize that they are nothing else but veritable _transformations_ of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by analysis. In this point of view, the _calculus of values_ might be simply conceived as an appendix, and a particular application of the _calculus of functions_, so that _arithmetic_ would disappear, so to say, as a distinct section in the whole body of abstract mathematics.
In order thoroughly to comprehend this consideration, we must observe that, when we propose to determine the _value_ of an unknown number whose mode of formation is given, it is, by the mere enunciation of the arithmetical question, already defined and expressed under a certain form; and that in _determining its value_ we only put its expression under another determinate form, to which we are accustomed to refer the exact notion of each particular number by making it re-enter into the regular system of _numeration_. The determination of values consists so completely of a simple _transformation_, that when the primitive expression of the number is found to be already conformed to the regular system of numeration, there is no longer any determination of value, properly speaking, or, rather, the question is answered by the question itself. Let the question be to add the two numbers _one_ and _twenty_, we answer it by merely repeating the enunciation of the question,[6] and nevertheless we think that we have _determined the value_ of the sum. This signifies that in this case the first expression of the function had no need of being transformed, while it would not be thus in adding twenty-three and fourteen, for then the sum would not be immediately expressed in a manner conformed to the rank which it occupies in the fixed and general scale of numeration.