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 refraction 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.

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

1 Thomson and Tait, Treatise on Natural Philosophy, vol. i. pp. 346-351.

the tides is interrupted by an enormous irregular wave, we attribute it to an earthquake, or some gigantic natural disturbance. If a meteoric stone falls upon a person and kills him, it is clearly a discontinuity in his life, of which he could have had no anticipation. A sudden sound may pass through the air neither preceded nor followed by any continuous effect. Although, then, we may regard the Law of Continuity as a principle of nature holding rigorously true in many of the relations of natural forces, it seems to be a matter of difficulty to assign the limits within which the law is verified. Much caution is required in its applica

tion.

Negative Arguments on the Principle of Continuity.

Upon the principle of continuity we may sometimes found arguments of great force which prove an hypothesis to be impossible, because it would involve a continual repetition of a process ad infinitum, or else a purely arbitrary breach at some point. Bonnet's famous theory of reproduction represented every living creature as containing germs which were perfect representatives of the next generation, so that on the same principle they necessarily included germs of the next generation, and so on indefinitely. The theory was sufficiently refuted when once clearly stated, as in the following poem called the Universe,1 by Henry Baker:

"Each seed includes a plant: that plant, again,
Has other seeds, which other plants contain :
Those other plants have all their seeds, and those
More plants again, successively inclose.

"Thus, ev'ry single berry that we find,
Has, really, in itself whole forests of its kind,
Empire and wealth one acorn may dispense,
By fleets to sail a thousand ages hence."

The general principle of inference, that what we know of one case must be true of similar cases, so far as they are similar, prevents our asserting anything which we cannot apply time after time under the same circumstances.

1 Philosophical Transactions (1740), vol. xli. p. 454.

On this principle Stevinus beautifully demonstrated that weights resting on two inclined planes and balancing each other must be proportional to the lengths of the planes between their apex and a horizontal plane. He imagined a uniform endless chain to be hung over the planes, and to hang below in a symmetrical festoon. If the chain were ever to move by gravity, there would be the same reason for its moving on for ever, and thus producing a perpetual motion. As this is absurd, the portions of the chain lying on the planes, and equal in length to the planes, must balance each other. On similar grounds we may disprove the existence of any self-moving machine; for if it could once alter its own state of motion or rest, in however small a degree, there is no reason why it should not do the like time after time ad infinitum. Newton's proof of his third law of motion, in the case of gravity, is of this character. For he remarks that if two gravitating bodies do not exert exactly equal forces in opposite directions, the one exerting the strongest pull will carry both away, and the two bodies will move off into space together with velocity increasing ad infinitum. But though the argument might seem sufficiently convincing, Newton in his characteristic way made an experiment with a loadstone and iron floated upon the surface of water. In recent years the very foundation of the principle of conservation of energy has been placed on the assumption that it is impossible by any combination of natural bodies to produce force continually from nothing. The principle admits of application in various subtle forms.

Lucretius attempted to prove, by a most ingenious argument of this kind, that matter must be indestructible. For if a finite quantity, however small, were to fall out of existence in any finite time, an equal quantity might be supposed to lapse in every equal interval of time, so that in the infinity of past time the universe must have ceased to exist. But the argument, however ingenious, seems to fail at several points. If past time be infinite, why may not matter have been created infinite also? It would be most reasonable, again, to suppose the matter

1 Principia, bk. i. Law iii. Corollary 6.

2 Helmholtz, Taylor's Scientific Memoirs (1853), vol. vi. p. 118. 3 Lucretius, bk. i. lines 232-264.

destroyed in any time to be proportional to the matter then remaining, and not to the original quantity; under this hypothesis even a finite quantity of original matter could never wholly disappear from the universe. For like reasons we cannot hold that the doctrine of the conservation of energy is really proved, or can ever be proved to be absolutely true, however probable it may be regarded.

Tendency to Hasty Generalisation.

In spite of all the powers and advantages of generalisation, men require no incitement to generalise; they are too apt to draw hasty and ill-considered inferences. As Francis Bacon said, our intellects want not wings, but rather weights of lead to moderate their course.1 The process is inevitable to the human mind; it begins with childhood and lasts through the second childhood. The child that has once been hurt fears the like result on all similar occasions, and can with difficulty be made to distinguish between case and case. It is caution and discrimination in the adoption of conclusions that we have chiefly to learn, and the whole experience of life is one continued lesson to this effect. Baden Powell has excellently described this strong natural propensity to hasty inference, and the fondness of the human mind for tracing resemblances real or fanciful. "Our first inductions," he says,2 "are always imperfect and inconclusive; we advance towards real evidence by successive approximations; and accordingly we find false generalisation the besetting error of most first attempts at scientific research. The faculty to generalise accurately and philosophically requires large caution and long training, and is not fully attained, especially in reference to more general views, even by some who may properly claim the title of very accurate scientific. observers in a more limited field. It is an intellectual habit which acquires immense and accumulating force from the contemplation of wider analogies."

Hasty and superficial generalisations have always been the bane of science, and there would be no difficulty in

1 Novum Organum, bk. I Aphorism 104.

2 The Unity of Worlds and of Nature, 2nd edit. p. 116.

finding endless illustrations. Between things which are the same in number there is a certain resemblance, namely in number; but in the infancy of science men could not be persuaded that there was not a deeper resemblance implied in that of number. Pythagoras was not the inventor of a mystical science of number. In the ancient Oriental religions the seven metals were connected with the seven planets, and in the seven days of the week we still have, and probably always shall have, a relic of the septiform system ascribed by Dio Cassius to the ancient Egyptians. The disciples of Pythagoras carried the doctrine of the number seven into great detail. Seven days are mentioned in Genesis; infants acquire their teeth at the end of seven months; they change them at the end of seven years; seven feet was the limit of man's height; every seventh year was a climacteric or critical year, at which a change of disposition took place. Then again there were the seven sages of Greece, the seven wonders of the world, the seven rites of the Grecian games, the seven gates of Thebes, and the seven generals destined to conquer that city.

In natural science there were not only the seven planets, and the seven metals, but also the seven primitive colours, and the seven tones of music. So deep a hold did this doctrine take that we still have its results in many customs, not only in the seven days of the week, but the seven years' apprenticeship, puberty at fourteen years, the second climacteric, and legal majority at twentyone years, the third climacteric. The idea was reproduced in the seven sacraments of the Roman Catholic Church, and the seven year periods of Comte's grotesque system of domestic worship. Even in scientific matters the loftiest intellects have occasionally yielded, as when Newton was misled by the analogy between the seven tones of music and the seven colours of his spectrum. Other numerical analogies, though rejected by Galileo, held Kepler in thraldom; no small part of Kepler's labours during seventeen years was spent upon numerical and geometrical analogies of the most baseless character; and he gravely held that there could not be more than six planets, because there were not more than five regular solids. Even the genius of Huyghens did not prevent him from inferring that but