Page:System of Logic.djvu/439

tion 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 $(a+b)^{n}=a^{n}+{\frac {n}{1}}a^{n-1}b+{\frac {n.n-1}{1.2}}a^{n-2}b^{2}+$, 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.

It would be inconsistent with the scale of this work, and not necessary to its design, to carry the analysis of the truths and processes of algebra any further; which is also the less needful, as the task has been, to a very great extent, performed by other writers. Peacock's Algebra, and Dr. Whewell's Doctrine of Limits, are full of instruction on the subject. The profound treatises of a truly philosophical mathematician, Professor De Morgan, should be studied by every one who desires to comprehend the evidence of mathematical truths, and the meaning of the obscurer processes of the calculus, and the speculations of M. Comte, in his Cours de Philosophie Positive, on the philosophy of the higher branches of mathematics, are among the many valuable gifts for which philosophy is indebted to that eminent thinker.

 

§ 7. If the extreme generality, and remoteness not so much from sense as from the visual and tactual imagination, of the laws of number, renders it a somewhat difficult effort of abstraction to conceive those laws as being in reality physical truths obtained by observation; the same difficulty does not exist with regard to the laws of extension. The facts of which those