So far as we can infer the results likely to be obtained by variations beyond the limits of experiment, we must proceed upon the same principles. If possible we must detect the exact laws in action, and then trust to them as a guide when we have no experience. If not, an empirical formula of the same character as those employed in interpolation is our only resource. But to extend our inference far beyond the limits of experience is exceedingly unsafe. Our knowledge is at the best only approximate, and takes no account of small tendencies. Now it usually happens that tendencies small within our limits of observation become perceptible or great under extreme circumstances. When the variable in our empirical formula is small, we are justified in overlooking the higher powers, and taking only two or three lower powers. But as the variable increases, the higher powers gain in importance, and in time yield the principal part of the value of the function.
This is no mere theoretical inference. Excepting the few primary laws of nature, such as the law of gravity, of the conservation of energy, &c., there is hardly any natural law which we can trust in circumstances widely different from those with which we are practically acquainted. From the expansion or contraction, fusion or vaporisation of substances by heat at the surface of the earth, we can form a most imperfect notion of what would happen near the centre of the earth, where the pressure almost infinitely exceeds anything possible in our experiments. The physics of the earth give us a feeble, and probably a misleading, notion of a body like the sun, in which an inconceivably high temperature is united with an inconceivably high pressure. If there are in the realms of space nebulæ consisting of incandescent and unoxidised vapours of metals and other elements, so highly heated perhaps that chemical composition is out of the question, we are hardly able to treat them as subjects of scientific inference. Hence arises the great importance of experiments in which we investigate the properties of substances under extreme circumstances of cold or heat, density or rarity, intense electric excitation, &c. This insecurity in extending our inferences arises from the approximate character of our measurements. Had we the power of appreciating infinitely small quantities, we should by the principle of continuity discover some trace of every change which a substance could undergo under unattainable circumstances. By observing, for instance, the tension of aqueous vapour between 0° and 100° C., we ought theoretically to be able to infer its tension at every other temperature; but this is out of the question practically because we cannot really ascertain the law precisely between those temperatures.
Many instances might be given to show that laws which appear to represent correctly the results of experiments within certain limits altogether fail beyond those limits. The experiments of Roscoe and Dittmar, on the absorption of gases in water16 afford interesting illustrations, especially in the case of hydrochloric acid, the quantity of which dissolved in water under different pressures follows very closely a linear law of variation, from which however it diverges widely at low pressures.17 Herschel, having deduced from observations of the double star γ Virginis an elliptic orbit for the motion of one component round the centre of gravity of both, found that for a time the motion of the star agreed very well with this orbit. Nevertheless divergence began to appear and after a time became so great that an entirely new orbit, of more than double the dimensions of the old one, had ultimately to be adopted.18
Illustrations of Empirical Quantitative Laws.
Although our object in quantitative inquiry is to discover the exact or rational formulæ, expressing the laws which apply to the subject, it is instructive to observe in how many important branches of science, no precise laws have yet been detected. The tension of aqueous vapour at different temperatures has been determined by a succession of eminent experimentalists—Dalton, Kaëmtz, Dulong, Arago, Magnus, and Regnault—and by the last mentioned the measurements were conducted with extraordinary care. Yet no incontestable