subject of Functions, we require a nomenclature enabling us to express any two numbers by names which, without specifying what particular numbers they are, shall show what function each is of the other; or, in other words, shall put in evidence their mode of formation from one another. The system of general language called algebraical notation does this. The expressions a and a2+3a denote, the one any number, the other the number formed from it in a particular manner. The expressions a, b, n, and (a+b)n, denote any three numbers, and a fourth which is formed from them in a certain mode.
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 [pg 433] 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 = an + n/1 an-1 b + n.n-1/1.2 an-2 b2, etc., 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.
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, etc., of equal numbers are equal.