Failure of the Law of Continuity.
There are certain cautions which must be given as to the application of the principle of continuity. In the first place, where this principle really holds true, it may seem to fail owing to our imperfect means of observation. Though a physical law may not admit of perfectly abrupt change, there is no limit to the approach which it may make to abruptness. When we warm a piece of very cold ice, the absorption of heat, the temperature, and the dilatation of the ice vary according to apparently simple laws until we come to the zero of the Centigrade scale. Everything is then changed; an enormous absorption of heat takes place without any rise of temperature, and the volume of the ice decreases as it changes into water. Unless carefully investigated, this change appears to be perfectly abrupt; but accurate observation seems to show that there is a certain forewarning; the ice does not turn into water all at once, but through a small fraction of a degree the change is gradual. All the phenomena concerned, if measured very exactly, would be represented not by angular lines, but continuous curves, undergoing rapid flexures; and we may probably assert with safety that between whatever points of temperature we examine ice, there would be found some indication, though almost infinitesimally small, of the apparently abrupt change which was to occur at a higher temperature. It might also be pointed out that the important and apparently simple physical laws, such as those of Boyle and Mariotte, Dalton and Gay-Lussac, &c., are only approximately true, and the divergences from the simple laws are forewarnings of abrupt changes, which would otherwise break the law of continuity.
Secondly, it must be remembered that mathematical laws of some complexity will probably present singular cases or negative results, which may bear the appearance of discontinuity, as when the law of retraction suddenly yields us with perfect abruptness the phenomenon of total internal reflection. In the undulatory theory, however, there is no real change of law between refraction and reflection. Faraday in the earlier part of his career found so many substances possessing magnetic power, that he ventured on a great generalisation, and asserted that all bodies shared in the magnetic property of iron. His mistake, as he afterwards discovered, consisted in overlooking the fact that though magnetic in a certain sense, some substances have negative magnetism, and are repelled instead of being attracted by the magnet.
Thirdly, where we might expect to find a uniform mathematical law prevailing, the law may undergo abrupt change at singular points, and actual discontinuity may arise. We may sometimes be in danger of treating under one law phenomena which really belong to different laws. For instance, a spherical shell of uniform matter attracts an external particle of matter with a force varying inversely as the square of the distance from the centre of the sphere. But this law only holds true so long as the particle is external to the shell. Within the shell the law is wholly different, and the aggregate gravity of the sphere becomes zero, the force in every direction being neutralised by an exactly equal opposite force. If an infinitely small particle be in the superficies of a sphere, the law is again different, and the attractive power of the shell is half what it would be with regard to particles infinitely close to the surface of the shell. Thus in approaching the centre of a shell from a distance, the force of gravity shows double discontinuity in passing through the shell.25
It may admit of question, too, whether discontinuity is really unknown in nature. We perpetually do meet with events which are real breaks upon the previous law, though the discontinuity may be a sign that some independent cause has come into operation. If the ordinary course of the tides is interrupted by an enormous