Every study of a generalisation or extension," De Morgan has well said, "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 où Bréte Bλérov 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 de

1 Syllabus of a Proposed System of Logic, p. 34.

pending 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.

Solids are in every way contrasted to gases. Each solid substance has its own peculiar degree of density, hardness, compressibility, transparency, tenacity, elasticity, power of conducting heat and electricity, magnetic properties, capability of producing frictional electricity, and so forth. Even different specimens of the same kind of substance will differ widely, according to the accidental treatment received. And not only has each substance its own specific properties, but, when crystallised, its properties vary in each direction with regard to the axes of crystallisation. The velocity of radiation, the rate of conduction of heat, the coefficients of expansibility and compressibility, the thermo-electric properties, all vary in different crystallographic directions.

It is probable that many apparent differences between liquids, and even between solids, will be explained when we learn to regard them under exactly corresponding circumstances. The extreme generality of the properties of gases is in reality only true at an infinitely high temperature, when they are all equally remote from their condensing points. Now, it is found that if we compare liquids for instance, different kinds of alcohols-not at equal temperatures, but at points equally distant from their respective boiling points, the laws and coefficients of expansion are nearly equal. The vapourtensions of liquids also are more nearly equal, when compared at corresponding points, and the boiling-points appear in many cases to be simply related to the chenical composition. No doubt the progress of investigation will enable us to discover generality, where at present we only see variety and puzzling complexity.

In some cases substances exhibit the same physical properties in the liquid as in the solid state. Lead has a high refractive power, whether in solution, or in solid salts, crystallised or vitreous. The magnetic power of iron is conspicuous, whatever be its chemical condition; indeed, the magnetic properties of substances, though varying with temperature, seem not to be greatly affected by other physical changes. Colour, absorptive power for heat or light rays, and a few other properties are also often the same in liquids and gases. Iodine and bromine possess a deep colour whenever they are chemically uncombined. Nevertheless, we can seldom argue safely from the properties of a substance in one condition to those in another condition. Ice is an insulator, water a conductor of electricity, and the same contrast exists in most other substances. The conducting power of a liquid for electricity increases with the temperature, while that of a solid decreases. By degrees we may learn to distinguish between those properties of matter which depend upon the intimate construction of the chemical molecule, and those which depend upon the contact, conflict, mutual attraction, or other relations of distinct molecules. The properties of a substance with respect to light seem generally to depend upon the molecule; thus, the power of certain substances to cause the plane of polarisation of a ray of light to rotate, is exactly the same whatever be its degree of density, or the diluteness of the solution in which it is contained. Taken as a whole, the physical properties of substances and their quantitative laws, present a problem of infinite complexity, and centuries must elapse before any moderately complete generalisations on the subject become possible.

Uniform Properties of all Matter.

Some laws are held to be true of all matter in the universe absolutely, without exception, no instance to the contrary having ever been noticed. This is the case with the laws of motion, as laid down by Galileo and Newton. It is also conspicuously true of the law of universal gravitation. The rise of modern physical science may perhaps be considered as beginning at the time when Galileo

showed, in opposition to the Aristotelians, that matter is equally affected by gravity, irrespective of its form, magnitude, or texture. All objects fall with equal rapidity, when disturbing causes, such as the resistance of the air, are removed or allowed for. That which was rudely demonstrated by Galileo from the leaning tower of Pisa, was proved by Newton to a high degree of approximation, in an experiment which has been mentioned (p. 443).

Newton formed two pendulums, as nearly as possible the same in outward shape and size by taking two equal round wooden boxes, and suspending them by equal threads, eleven feet long. The pendulums were therefore equally subject to the resistance of the air. He filled one box with wood, and in the centre of oscillation of the other he placed an equal weight of gold. The pendulums were then equal in weight as well as in size; and, on setting them simultaneously in motion, Newton found that they vibrated for a length of time with equal vibrations. He tried the same experiment with silver, lead, glass, sand, common salt, water, and wheat, in place of the gold, and ascertained that the motion of his pendulum was exactly the same whatever was the kind of matter inside.1 He considered that a difference of a thousandth part would have been apparent. The reader must observe that the pendulums were made of equal weight only in order that they might suffer equal retardation from the air. The meaning of the experiment is that all substances manifest exactly equal acceleration from the force of gravity, and that therefore the inertia or resistance of matter to force, which is the only independent measure of mass known to us, is always proportional to gravity.

These experiments of Newton were considered conclusive up to very recent times, when certain discordances between the theory and observations of the movements of planets led Nicolai, in 1826, to suggest that the equal gravitation of different kinds of matter might not be absolutely exact. It is perfectly philosophical thus to call in question, from time to time, some of the best accepted laws. On this occasion Bessel carefully repeated the experiments of Newton with pendulums composed of

Principia, bk. iii. Prop. VI. Motte's translation, vol. ii. p. 220