The following may be stated as the general problem of the algebraical calculus: F being a certain function of a given number, to find what function F will be of any function of that number. For example, a binomial _a_ + _b_ is a function of its two parts _a_ and _b_, and the parts are, in their turn, functions of _a + b_: now _(a + b)^n_ is a certain function of the binomial; what function will this be of _a_ and _b_, the two parts? The answer to this question is the binomial theorem. The formula _(a + b)^n = a^n + (n / 1) a^(n - 1) b + ((n·(n - 1)) / (1·2)) a^(n - 2) b^2 + &c._, shows in what manner the number which is formed by multiplying _a + b_ into itself _n_ times, might be formed without that process, directly from _a_, _b_, and _n_. And of this nature are all the theorems of the science of number. They assert the identity of the result of different modes of formation. They affirm that some mode of formation from _x_, and some mode of formation from a certain function of _x_, produce the same number. Besides these general theorems of formulæ, what remains in the algebraical calculus is the resolution of equations. But the resolution of an equation is also a theorem. If the equation be _x^2 + ax = b_, the resolution of this equation, viz. _x = -(1/2) a ± [root of]((1/4) a^2 + b)_, is a general proposition, which may be regarded as an answer to the question, If _b_ is a certain function of _x_ and _a_ (namely _x^2 + ax_), what function is _x_ of _b_ and _a_? The resolution of equations is, therefore, a mere variety of the general problem as above stated. The problem is--Given a function, what function is it of some other function? And in the resolution of an equation, the question is, to find what function of one of its own functions the number itself is. Such as above described, is the aim and end of the calculus. As for its processes, every one knows that they are simply deductive. In demonstrating an algebraical theorem, or in resolving an equation, we travel from the _datum_ to the _quæsitum_ by pure ratiocination; in which the only premises introduced, besides the original hypotheses, are the fundamental axioms already mentioned--that things equal to the same thing are equal to one another, and that the sums of equal things are equal. At each step in the demonstration or in the calculation, we apply one or other of these truths, or truths deducible from them, as, that the differences, products, &c., of equal numbers are equal.