the number of the elementary abstract functions, the very limited list of which has been given above. The determination of the values of these ten functions necessarily gives that of all the functions, infinite in number, which are considered in the whole of mathematical analysis, such at least as it exists at present. There can be no new arithmetical operations without the creation of really new analytical elements, the number of which must always be extremely small. The field of arithmetic is, then, by its nature, exceedingly restricted, while that of algebra is rigorously indefinite.
It is, however, important to remark, that the domain of the calculus of values is, in reality, much more extensive than it is commonly represented; for several questions truly arithmetical, since they consist of determinations of values, are not ordinarily classed as such, because we are accustomed to treat them only as incidental in the midst of a body of analytical researches more or less elevated, the too high opinion commonly formed of the influence of signs being again the principal cause of this confusion of ideas. Thus not only the construction of a table of logarithms, but also the calculation of trigonometrical tables, are true arithmetical operations of a higher kind. We may also cite as being in the same class, although in a very distinct and more elevated order, all the methods by which we determine directly the value of any function for each particular system of values attributed to the quantities on which it depends, when we cannot express in general terms the explicit form of that function. In this point of view the numerical solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of arithmetic, in which we must necessarily comprise all that which has for its object the determination of the values of functions. The considerations relative to this object are, in fact, constantly homogeneous, whatever the determinations in question, and are always very distinct from truly algebraic considerations.
To complete a just idea of the real extent of the calculus of values, we must include in it likewise that part of the general science of the calculus which now bears the name of the Theory of Numbers, and which is yet so little advanced. This branch, very extensive by its nature, but whose importance in the general system of science is not very great, has for its object the discovery of the properties inherent in different numbers by virtue of their values, and independent of any particular system of numeration. It forms, then, a sort of transcendental arithmetic; and to it would really apply the definition proposed by Newton for algebra.
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