It would be interesting again to compare the scrupulous accuracy of a modern trigonometrical survey with Eratosthenes’ rude but ingenious guess at the difference of latitude between Alexandria and Syene—or with Norwood’s measurement of a degree of latitude in 1635. “Sometimes I measured, sometimes I paced,” said Norwood; “and I believe I am within a scantling of the truth.” Such was the germ of those elaborate geodesical measurements which have made the dimensions of the globe known to us within a few hundred yards.
In other branches of science, the invention of an instrument has usually marked, if it has not made, an epoch. The science of heat might be said to commence with the construction of the thermometer, and it has recently been advanced by the introduction of the thermo-electric pile. Chemistry has been created chiefly by the careful use of the balance, which forms a unique instance of an instrument remaining substantially in the form in which it was first applied to scientific purposes by Archimedes. The balance never has been and probably never can be improved, except in details of construction. The torsion balance, introduced by Coulomb towards the end of last century, has rapidly become essential in many branches of investigation. In the hands of Cavendish and Baily, it gave a determination of the earth’s density; applied in the galvanometer, it gave a delicate measure of electrical forces, and is indispensable in the thermo-electric pile. This balance is made by simply suspending any light rod by a thin wire or thread attached to the middle point. And we owe to it almost all the more delicate investigations in the theories of heat, electricity, and magnetism.
Though we can now take note of the millionth of an inch in space, and the millionth of a second in time, we must not overlook the fact that in other operations of science we are yet in the position of the Chaldæans. Not many years have elapsed since the magnitudes of the stars, meaning the amounts of light they send to the observer’s eye, were guessed at in the rudest manner, and the astronomer adjudged a star to this or that order of magnitude by a rough comparison with other stars of the same order. To Sir John Herschel we owe an attempt to introduce a uniform method of measurement and expression, bearing some relation to the real photometric magnitudes of the stars.[181] Previous to the researches of Bunsen and Roscoe on the chemical action of light, we were devoid of any mode of measuring the energy of light; even now the methods are tedious, and it is not clear that they give the energy of light so much as one of its special effects. Many natural phenomena have hardly yet been made the subject of measurement at all, such as the intensity of sound, the phenomena of taste and smell, the magnitude of atoms, the temperature of the electric spark or of the sun’s photosphere.
To suppose, then, that quantitative science treats only of exactly measurable quantities, is a gross if it be a common mistake. Whenever we are treating of an event which either happens altogether or does not happen at all, we are engaged with a non-quantitative phenomenon, a matter of fact, not of degree; but whenever a thing may be greater or less, or twice or thrice as great as another, whenever, in short, ratio enters even in the rudest manner, there science will have a quantitative character. There can be little doubt, indeed, that every science as it progresses will become gradually more and more quantitative. Numerical precision is the soul of science, as Herschel said, and as all natural objects exist in space, and involve molecular movements, measurable in velocity and extent, there is no apparent limit to the ultimate extension of quantitative science. 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,[182] 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 instruments must stop somewhere, there is no reason why the mind should not go on. We can see that a proof which is only carried through a few steps in fact, might be carried on without limit, and it is this consciousness of no stopping-place, which renders Euclid’s proof of his 117th proposition so impressive. Try how we will to circumvent the matter, we cannot really avoid the consideration of the infinitely small and the infinitely great. The same methods of approximation which seem confined to the finite, mentally extend themselves to the infinite.