A second process, to which the name of generalisation is often given, consists in passing from a fact or partial law to a multitude of unexamined cases, which we believe to be subject to the same conditions. Instead of merely recognising similarity as it is brought before us, we predict its existence before our senses can detect it, so that generalisation of this kind endows us with a prophetic power of more or less probability. Having observed that many substances assume, like water and mercury, the three states of solid, liquid, and gas, and having assured ourselves by frequent trial that the greater the means we possess of heating and cooling, the more substances we can vaporise and freeze, we pass confidently in advance of fact, and assume that all substances are capable of these three forms. Such a generalisation was accepted by Lavoisier and Laplace before many of the corroborative facts now in our possession were known. The reduction of a single comet beneath the sway of gravity was considered sufficient indication that all comets obey the same power. Few persons doubted that the law of gravity extended over the whole heavens; certainly the fact that a few stars out of many millions manifest the action of gravity, is now held to be sufficient evidence of its general extension over the visible universe.
Value of Generalisation.
It might seem that if we know particular facts, there can be little use in connecting them together by a general law. The particulars must be more full of useful information than an abstract general statement. If we know, for instance, the properties of an ellipse, a circle, a parabola, and hyperbola, what is the use of learning all these properties over again in the general theory of curves of the second degree? If we understand the phenomena of sound and light and water-waves separately, what is the need of erecting a general theory of waves, which, after all, is inapplicable to practice until resolved again into particular cases? But, in reality, we never do obtain an adequate knowledge of particulars until we regard them as cases of the general. Not only is there a singular delight in discovering the many in the one, and the one in the many, but there is a constant interchange of light and knowledge. Properties which are unapparent in the hyperbola may be readily observed in the ellipse. Most of the complex relations which old geometers discovered in the circle will be reproduced mutatis mutandis in the other conic sections. The undulatory theory of light might have been unknown at the present day, had not the theory of sound supplied hints by analogy. The study of light has made known many phenomena of interference and polarisation, the existence of which had hardly been suspected in the case of sound, but which may now be sought out, and perhaps found to possess unexpected interest. The careful study of water-waves shows how waves alter in form and velocity with varying depth of water. Analogous changes may some time be detected in sound waves. Thus there is mutual interchange of aid.
“Every study of a generalisation or extension,” De Morgan has well said,[492] “gives additional power over the particular form by which the generalisation is suggested. Nobody who has ever returned to quadratic equations after the study of equations of all degrees, or who has done the like, will deny my assertion that οὐ βλέπει βλέπων may be predicated of any one who studies a branch or a case, without afterwards making it part of a larger whole. Accordingly it is always worth while to generalise, were it only to give power over the particular. This principle, of daily familiarity to the mathematician, is almost unknown to the logician.”
Comparative Generality of Properties.
Much of the value of science depends upon the knowledge which we gradually acquire of the different degrees of generality of properties and phenomena of various kinds. The use of science consists in enabling us to act with confidence, because we can foresee the result. Now this foresight must rest upon the knowledge of the powers which will come into play. That knowledge, indeed, can never be certain, because it rests upon imperfect induction, and the most confident beliefs and predictions of the physicist may be falsified. Nevertheless, if we always estimate the probability of each belief according to the due teaching of the data, and bear in mind that probability when forming our anticipations, we shall ensure the minimum of disappointment. Even when he cannot exactly apply the theory of probabilities, the physicist may acquire the habit of making judgments in general agreement with its principles and results.
Such is the constitution of nature, that the physicist learns to distinguish those properties which have wide and uniform extension, from those which vary between case and case. Not only are certain laws distinctly laid down, with their extension carefully defined, but a scientific training gives a kind of tact in judging how far other laws are likely to apply under any particular circumstances. We learn by degrees that crystals exhibit phenomena depending upon the directions of the axes of elasticity, which we must not expect in uniform solids. Liquids, compared even with non-crystalline solids, exhibit laws of far less complexity and variety; and gases assume, in many respects, an aspect of nearly complete uniformity. To trace out the branches of science in which varying degrees of generality prevail, would be an inquiry of great interest and importance; but want of space, if there were no other reason, would forbid me to attempt it, except in a very slight manner.
Gases, so far as they are really gaseous, not only have exactly the same properties in all directions of space, but one gas exactly resembles other gases in many qualities. All gases expand by heat, according to the same law, and by nearly the same amount; the specific heats of equivalent weights are equal, and the densities are exactly proportional to the atomic weights. All such gases obey the general law, that the volume multiplied by the pressure, and divided by the absolute temperature, is constant or nearly so. The laws of diffusion and transpiration are the same in all cases, and, generally speaking, all physical laws, as distinguished from chemical laws, apply equally to all gases. Even when gases differ in chemical or physical properties, the differences are minor in degree. Thus the differences of viscosity are far less marked than in the liquid and solid states. Nearly all gases, again, are colourless, the exceptions being chlorine, the vapours of iodine, bromine, and a few other substances.
Only in one single point, so far as I am aware, do gases present distinguishing marks unknown or nearly so, in the solid and liquid states. I mean as regards the light given off when incandescent. Each gas when sufficiently heated, yields its own peculiar series of rays, arising from the free vibrations of the constituent parts of the molecules. Hence the possibility of distinguishing gases by the spectroscope. But the molecules of solids and liquids appear to be continually in conflict with each other, so that only a confused noise of atoms is produced, instead of a definite series of luminous chords. At the same temperature, accordingly, all solids and liquids give off nearly the same rays when strongly heated, and we have in this case an exception to the greater generality of properties in gases.
Liquids are in many ways intermediate in character between gases and solids. While incapable of possessing different elasticity in different directions, and thus denuded of the rich geometrical complexity of solids, they retain the variety of density, colour degrees of transparency, great diversity in surface tension, viscosity, coefficients of expansion, compressibility, and many other properties which we observe in solids, but not for the most part in gases. Though our knowledge of the physical properties of liquids is much wanting in generality at present, there is ground to hope that by degrees laws connecting and explaining the variations may be traced out.