[Footnote 6: This is less strictly true in the English system of numeration than in the French, since "twenty-one" is our more usual mode of expressing this number.]
To sum up as comprehensively as possible the preceding views, we may say, that to determine the _value_ of a number is nothing else than putting its primitive expression under the form
_a_ + _bz_ + _cz²_ + _dz³_ + _ez⁴_ . . . . . + _pz^m_,
_z_ being generally equal to 10, and the coefficients _a_, _b_, _c_, _d_, &c., being subjected to the conditions of being whole numbers less than _z_; capable of becoming equal to zero; but never negative. Every arithmetical question may thus be stated as consisting in putting under such a form any abstract function whatever of different quantities, which are supposed to have themselves a similar form already. We might then see in the different operations of arithmetic only simple particular cases of certain algebraic transformations, excepting the special difficulties belonging to conditions relating to the nature of the coefficients.
It clearly follows that abstract mathematics is essentially composed of the _Calculus of Functions_, which had been already seen to be its most important, most extended, and most difficult part. It will henceforth be the exclusive subject of our analytical investigations. I will therefore no longer delay on the _Calculus of Values_, but pass immediately to the examination of the fundamental division of the _Calculus of Functions_.
_Principle of its Fundamental Division._ We have determined, at the beginning of this chapter, wherein properly consists the difficulty which we experience in putting mathematical questions into _equations_. It is essentially because of the insufficiency of the very small number of analytical elements which we possess, that the relation of the concrete to the abstract is usually so difficult to establish. Let us endeavour now to appreciate in a philosophical manner the general process by which the human mind has succeeded, in so great a number of important cases, in overcoming this fundamental obstacle to _The establishment of Equations_.
1. _By the Creation of new Functions._ In looking at this important question from the most general point of view, we are led at once to the conception of one means of facilitating the establishment of the equations of phenomena. Since the principal obstacle in this matter comes from the too small number of our analytical elements, the whole question would seem to be reduced to creating new ones. But this means, though natural, is really illusory; and though it might be useful, it is certainly insufficient.