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.