The philosophy of mathematics · Auguste Comte
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.
We thus see that the _algebraic_ calculus and the _arithmetical_ calculus differ essentially in their object. They differ no less in the point of view under which they regard quantities; which are considered in the first as to their _relations_, and in the second as to their _values_. The true spirit of the calculus, in general, requires this distinction to be maintained with the most severe exactitude, and the line of demarcation between the two periods of the solution to be rendered as clear and distinct as the proposed question permits. The attentive observation of this precept, which is too much neglected, may be of much assistance, in each particular question, in directing the efforts of our mind, at any moment of the solution, towards the real corresponding difficulty. In truth, the imperfection of the science of the calculus obliges us very often (as will be explained in the next chapter) to intermingle algebraic and arithmetical considerations in the solution of the same question. But, however impossible it may be to separate clearly the two parts of the labour, yet the preceding indications will always enable us to avoid confounding them. In endeavouring to sum up as succinctly as possible the distinction just established, we see that ALGEBRA may be defined, in general, as having for its object the _resolution of equations_; taking this expression in its full logical meaning, which signifies the transformation of _implicit_ functions into equivalent _explicit_ ones. In the same way, ARITHMETIC may be defined as destined to _the determination of the values of functions_. Henceforth, therefore, we will briefly say that ALGEBRA is the _Calculus of Functions_, and ARITHMETIC the _Calculus of Values_.
We can now perceive how insufficient and even erroneous are the ordinary definitions. Most generally, the exaggerated importance attributed to Signs has led to the distinguishing the two fundamental branches of the science of the Calculus by the manner of designating in each the subjects of discussion, an idea which is evidently absurd in principle and false in fact. Even the celebrated definition given by Newton, characterizing _Algebra_ as _Universal Arithmetic_, gives certainly a very false idea of the nature of algebra and of that of arithmetic.[5] Having thus established the fundamental division of the calculus into two principal branches, I have now to compare in general terms the extent, the importance, and the difficulty of these two sorts of calculus, so as to have hereafter to consider only the _Calculus of Functions_, which is to be the principal subject of our study.
Text from Project Gutenberg, public domain in the United States.