Properties which are unapparent in the hyperbola may readily be discovered in the ellipse. Most of the complex relations which the 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 polarization, 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 and importance. The careful study of water-waves shows how waves may alter in form and velocity with varying depth of water. Analogous changes may sometimes be detected in sound waves. Thus there is a mutual interchange of aid. par Every study of a generalization or extension,' as De Morgan has well said, 'gives additional power over the ticular form by which the generalization 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 generalize, 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 Physical 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 very 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 soon 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 found to 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 a great many qualities. All gases expand by heat, according to the one same law, and by nearly the same amount; the specific heats of equivalent weights are equal, and the densities, though not the same, 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, which apply to one gas apply equally to all other gases. Even when gases differ in chemical or physical properties, the differences are minor in degree or number. 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 some other sub stances. . 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 when pursuing separate paths. Hence the possibility of distinguishing gases by the spectroBut the molecules of solids and liquids appear scope. 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 a single exception to the general rule of 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 thus much wanting in generality at present, there is ground to hope that by degrees laws connecting and explaining the varieties of character may be traced out. Liquids ought to be compared together, not at uniform temperatures, but at points of temperature similarly related to the points of fusion and ebullition. Solids are in every way contrasted to gases. Each solid substance has its own peculiar density, hardness, compressibility, degree of 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 be widely different, according to the accidental treatment it has received. And not only has each substance its own specific properties, but, when crystallized, its own properties peculiar to each direction, regard being had to the axes of crystallization. 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 highly probable that many apparent differences among liquids, and even among solids, will be resolved and explained, when we learn to regard them under exactly corresponding circumstances. The extreme generality of the properties of gases is really 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 vapour-tensions of liquids also are much more nearly equal, when thus compared at corresponding points, and the boiling-points themselves appear to be simply related to the chemical composition in many cases. No doubt the progress of investigation will often enable us to discover generality, where we at present 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, crystallized, 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 physical changes. changes. Colour, absorptive power for heat or light rays, and a few other properties are also often the same both 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 that 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 gene |