searches of Bunsen and Roscoe on the chemical action of light, we were absolutely 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 doubtless the very 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 is analogous thereto.

'Preliminary Discourse on the Study of Natural Philosophy,' p. 122.

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 (XIV) to consider the establishment of unitary magnitudes, in terms of which our results may be expressed. As every phenomenon is usually the sum of several distinct quantities proceeding from 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 millemetre into ten, or one hundred, or one thousand, or ten thousand parts, and mentally at any rate we can carry

on the process ad infinitum. Any finite space, then, must be conceived as made up of an infinite number of parts, each of which must consequently be infinitely small. We cannot entertain some of the simplest geometrical notions without allowing this. The conception of a square involves the conception of a side and diagonal, which, as Euclid admirably proves in the 117th proposition of his tenth book, have no common measuref, meaning, as I apprehend, 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 appreciate. 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 f See De Morgan, 'Study of Mathematics,' in U. K. S. Library, p. 81.

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 infinites.

One result which immediately follows from these considerations is, that we cannot possibly adjust any two quantities in absolute equality. The suspension of Mahomet's coffin between two precisely equal magnets, is theoretically conceivable but practically impossible. The story of the Merchant of Venice,' turns upon the infinite improbability, that an exact quantity of flesh could be cut. Unstable equilibrium cannot exist in nature, for it is that which is destroyed by an infinitely small displacement. It might be possible to balance an egg on its end practically, because no egg has a surface of perfect curvature. Suppose the egg shell to be perfectly smooth, and the feat would become impossible.

The Fallacious Indications of the Senses.

I may briefly remind the reader how little we can trust to our unassisted senses in estimating the degree, quantity, or magnitude of any phenomenon. The eye cannot correctly estimate the comparative brightness of two luminous bodies which differ much in brilliancy; for we know that the iris is constantly adjusting itself to the intensity

8 Lacroix, 'Essai sur l'Enseignement ou manière d'étudier les Mathématiques,' 2nd ed. Paris, 1816, pp. 292-294.

of the light received, and thus admits more or less light according to circumstances. The moon which shines with almost dazzling brightness by night, is pale and nearly imperceptible while the eye is yet affected by the vastly more powerful light of day. Much has been recorded concerning the comparative brightness of the zodiacal light at different times h, but it would be difficult to prove that these changes are not due to the varying darkness at the time, or the different acuteness of the observer's eye. For a like reason it is exceedingly difficult to establish the existence of any change in the form or comparative brightness of nebulæ; the appearance of a nebula greatly depends upon the keenness of sight of the observer, or the accidental condition of freshness or fatigue of his eye; the same is true of lunar observations; and even the use of the best telescope fails to remedy this difficulty. In judging of colours again, we must remember that light of any given colour tends to dull the sensibility of the eye for light of the same colour.

Nor is the eye when unassisted by instruments a much better judge of magnitude. Our estimates of the size of minute bright points, such as the fixed stars, are completely falsified by the effects of irradiation. Tycho calculated from the apparent size of the star-discs, that no one of the principal fixed stars could be contained within the area of the earth's orbit. Apart, however, from irradiation or other distinct causes of error, our visual estimates of sizes and shapes are often astonishingly incorrect. Artists almost invariably draw distant mountains or other objects in ludicrous disproportion to nearer objects, as a comparison of a sketch with a photograph at once shows, The extraordinary apparent difference of size of the sun

h 'Cosmos,' Translated by Otté, vol. i. pp. 131-134.

iReport of the British Association,' 1871, p. 84. Grant's History of Physical Astronomy,' pp. 568-9.

Y