In astronomy singular exceptions might occur, and in an approximate manner they do occur. We may point to the rings of Saturn as objects which, though undoubtedly obeying the law of gravity, are yet unique, as far as our observation of the universe has gone. They agree, indeed, with the other bodies of the planetary system in the stability of their movements, which never diverge far from the mean position. There seems to be little doubt that these rings are composed of swarms of small meteoric stones; formerly they were thought to be solid continuous rings, and mathematicians proved that if so constituted an entirely exceptional event might have happened under certain circumstances. Had the rings been exactly uniform all round, and with a centre of gravity coinciding for a moment with that of Saturn, a singular case of unstable equilibrium would have arisen, necessarily resulting in the sudden collapse of the rings, and the fall of their debris upon the surface of the planet. Thus in one single case the theory of gravity would give a result wholly unlike anything else known in the mechanism of the heavens.
It is possible that we might meet with singular exceptions in crystallography. If a crystal of the second or dimetric system, in which the third axis is usually unequal to either of the other two, happened to have the three axes equal, it might be mistaken for a crystal of the cubic system, but would exhibit different faces and dissimilar properties. There is, again, a possible class of diclinic crystals in which two axes are at right angles and the third axis inclined to the other two. This class is chiefly remarkable for its non-existence, since no crystals have yet been proved to have such axes. It seems likely that the class would constitute only a singular case of the more general triclinic system, in which all three axes are inclined to each other at various angles. Now if the diclinic form were merely accidental, and not produced by any general law of molecular constitution, its actual occurrence would be infinitely improbable, just as it is infinitely improbable that any star should indicate the North Pole with perfect exactness.
In the curves denoting the relation between the temperature and pressure of water there is, as shown by Professor J. Thomson, one very remarkable point entirely unique, at which alone water can remain in the three conditions of gas, liquid, and solid in the same vessel. It is the triple point at which three lines meet, namely (1) the steam line, which shows at what temperatures and pressures water is just upon the point of becoming gaseous; (2) the ice line, showing when ice is just about to melt; and (3) the hoar-frost line, which similarly indicates the pressures and temperatures at which ice is capable of passing directly into the state of gaseous vapour.[545]
Divergent Exceptions.
Closely analogous to singular exceptions are those divergent exceptions, in which a phenomenon manifests itself in unusual magnitude or character, without becoming subject to peculiar laws. Thus in throwing ten coins, it happened in four cases out of 2,048 throws, that all the coins fell with heads uppermost (p. [208]); these would usually be regarded as very singular events, and, according to the theory of probabilities, they would be rare; yet they proceed only from an unusual conjunction of accidental events, and from no really exceptional causes. In all classes of natural phenomena we may expect to meet with similar divergencies from the average, sometimes due merely to the principles of probability, sometimes to deeper reasons. Among every large collection of persons, we shall probably find some persons who are remarkably large or remarkably small, giants or dwarfs, whether in bodily or mental conformation. Such cases appear to be not mere lusus naturæ, since they occur with a frequency closely accordant with the law of error or divergence from an average, as shown by Quetelet and Mr. Galton.[546] The rise of genius, and the occurrence of extraordinary musical or mathematical faculties, are attributed by Mr. Galton to the same principle of divergence.
When several distinct forces happen to concur together, we may have surprising or alarming results. Great storms, floods, droughts, and other extreme deviations from the average condition of the atmosphere thus arise. They must be expected to happen from time to time, and will yet be very infrequent compared with minor disturbances. They are not anomalous but only extreme events, analogous to extreme runs of luck. There seems, indeed, to be a fallacious impression in the minds of many persons, that the theory of probabilities necessitates uniformity in the happening of events, so that in the same space of time there will always be nearly the same number of railway accidents and murders. Buckle has superficially remarked upon the constancy of such events as ascertained by Quetelet, and some of his readers acquire the false notion that there is a mysterious inexorable law producing uniformity in human affairs. But nothing can be more opposed to the teachings of the theory of probability, which always contemplates the occurrence of unusual runs of luck. That theory shows the great improbability that the number of railway accidents per month should be always equal, or nearly so. The public attention is strongly attracted to any unusual conjunction of events, and there is a fallacious tendency to suppose that such conjunction must be due to a peculiar new cause coming into operation. Unless it can be clearly shown that such unusual conjunctions occur more frequently than they should do according to the theory of probabilities, we should regard them as merely divergent exceptions.
Eclipses and remarkable conjunctions of the heavenly bodies may also be regarded as results of ordinary laws which nevertheless appear to break the regular course of nature, and never fail to excite surprise. Such events vary greatly in frequency. One or other of the satellites of Jupiter is eclipsed almost every day, but the simultaneous eclipse of three satellites can only take place, according to the calculations of Wargentin, after the lapse of 1,317,900 years. The relations of the four satellites are so remarkable, that it is actually impossible, according to the theory of gravity, that they should all suffer eclipse simultaneously. But it may happen that while some of the satellites are really eclipsed by entering Jupiter’s shadow, the others are either occulted or rendered invisible by passing over his disk. Thus on four occasions, in 1681, 1802, 1826, and 1843, Jupiter has been witnessed in the singular condition of being apparently deprived of satellites. A close conjunction of two planets always excites admiration, though such conjunctions must occur at intervals in the ordinary course of their motions. We cannot wonder that when three or four planets approach each other closely, the event is long remembered. A most remarkable conjunction of Mars, Jupiter, Saturn, and Mercury, which took place in the year 2446 B.C., was adopted by the Chinese Emperor, Chuen Hio, as a new epoch for the chronology of his Empire, though there is some doubt whether the conjunction was really observed, or was calculated from the supposed laws of motion of the planets. It is certain that on the 11th November, 1524, the planets Venus, Jupiter, Mars, and Saturn were seen very close together, while Mercury was only distant by about 16° or thirty apparent diameters of the sun, this conjunction being probably the most remarkable which has occurred in historical times.
Among the perturbations of the planets we find divergent exceptions arising from the peculiar accumulation of effects, as in the case of the long inequality of Jupiter and Saturn (p. [455]). Leverrier has shown that there is one place between the orbits of Mercury and Venus, and another between those of Mars and Jupiter, in either of which, if a small planet happened to exist, it would suffer comparatively immense disturbance in the elements of its orbit. Now between Mars and Jupiter there do occur the minor planets, the orbits of which are in many cases exceptionally divergent.[547]
Under divergent exceptions we might place all or nearly all the instances of substances possessing physical properties in a very high or low degree, which were described in the chapter on Generalisation (p. [607]). Quicksilver is divergent among metals as regards its melting point, and potassium and sodium as regards their specific gravities. Monstrous productions and variations, whether in the animal or vegetable kingdoms, should probably be assigned to this class of exceptions.
It is worthy of notice that even in such a subject as formal logic, divergent exceptions seem to occur, not of course due to chance, but exhibiting in an unusual degree a phenomenon which is more or less manifested in all other cases. I pointed out in p. [141] that propositions of the general type A = BC ꖌ bc are capable of expression in six equivalent logical forms, so that they manifest in a higher degree than any other proposition yet discovered the phenomenon of logical equivalence.