priori_, no rational guarantee of their necessary suitableness to represent the mathematical law of every other class of phenomena. I will explain presently the general artifice, so profoundly ingenious, by which the human mind has succeeded in diminishing, in a remarkable degree, this fundamental difficulty which is presented by the relation of the concrete to the abstract in mathematics, without, however, its having been necessary to multiply the number of these analytical elements. THE TWO PRINCIPAL DIVISIONS OF THE CALCULUS. The preceding explanations determine with precision the true object and the real field of abstract mathematics. I must now pass to the examination of its principal divisions, for thus far we have considered the calculus as a whole. The first direct consideration to be presented on the composition of the science of the _calculus_ consists in dividing it, in the first place, into two principal branches, to which, for want of more suitable denominations, I will give the names of _Algebraic calculus_, or _Algebra_, and of _Arithmetical calculus_, or _Arithmetic_; but with the caution to take these two expressions in their most extended logical acceptation, in the place of the by far too restricted meaning which is usually attached to them. The complete solution of every question of the _calculus_, from the most elementary up to the most transcendental, is necessarily composed of two successive parts, whose nature is essentially distinct. In the first, the object is to transform the proposed equations, so as to make apparent the manner in which the unknown quantities are formed by the known ones: it is this which constitutes the _algebraic_ question. In the second, our object is to _find the values_ of the formulas thus obtained; that is, to determine directly the values of the numbers sought, which are already represented by certain explicit functions of given numbers: this is the _arithmetical_ question.[4] It is apparent that, in every solution which is truly rational, it necessarily follows the algebraical question, of which it forms the indispensable complement, since it is evidently necessary to know the mode of generation of the numbers sought for before determining their actual values for each particular case. Thus the stopping-place of the algebraic part of the solution becomes the starting point of the arithmetical part. [Footnote 4: Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes, _a_ and _b_, _x³_ + 3_ax_ = 2_b_,