But the reader must not for a moment suppose that, because we depend more and more upon mathematical methods, we leave logical methods behind us. Number, as I have endeavoured to show, is logical in its origin, and quantity is but a development of number, or analogous thereto.
Division of the Subject.
The general subject of quantitative investigation will have to be divided into several parts. We shall firstly consider the means at our disposal for measuring phenomena, and thus rendering them more or less amenable to mathematical treatment. This task will involve an analysis of the principles on which accurate methods of measurement are founded, forming the subject of the remainder of the present chapter. As measurement, however, only yields ratios, we have in the next chapter to consider the establishment of unit magnitudes, in terms of which our results may be expressed. As every phenomenon is usually the sum of several distinct quantities depending upon different causes, we have next to investigate in Chapter XV. the methods by which we may disentangle complicated effects, and refer each part of the joint effect to its separate cause.
It yet remains for us in subsequent chapters to treat of quantitative induction, properly so called. We must follow out the inverse logical method, as it presents itself in problems of a far higher degree of difficulty than those which treat of objects related in a simple logical manner, and incapable of merging into each other by addition and subtraction.
Continuous Quantity.
The phenomena of nature are for the most part manifested in quantities which increase or decrease continuously. When we inquire into the precise meaning of continuous quantity, we find that it can only be described as that which is divisible without limit. We can divide a millimetre into ten, or a hundred, or a thousand, or ten thousand parts, and mentally at any rate we can carry on the division ad infinitum. Any finite space, then, must be conceived as made up of an infinite number of parts each infinitely small. We cannot entertain the simplest geometrical notions without allowing this. The conception of a square involves the conception of a side and diagonal, which, as Euclid beautifully proves in the 117th proposition of his tenth book, have no common measure,3 meaning no finite common measure. Incommensurable quantities are, in fact, those which have for their only common measure an infinitely small quantity. It is somewhat startling to find, too, that in theory incommensurable quantities will be infinitely more frequent than commensurable. Let any two lines be drawn haphazard; it is infinitely unlikely that they will be commensurable, so that the commensurable quantities, which we are supposed to deal with in practice, are but singular cases among an infinitely greater number of incommensurable cases.
Practically, however, we treat all quantities as made up of the least quantities which our senses, assisted by the best measuring instruments, can perceive. So long as microscopes were uninvented, it was sufficient to regard an inch as made up of a thousand thousandths of an inch; now we must treat it as composed of a million millionths. We might apparently avoid all mention of infinitely small quantities, by never carrying our approximations beyond quantities which the senses can appreciate. In geometry, as thus treated, we should never assert two quantities to be equal, but only to be apparently equal. Legendre really adopts this mode of treatment in the twentieth proposition of the first book of his Geometry; and it is practically adopted throughout the physical sciences, as we shall afterwards see. But though our fingers, and senses, and