passing through certain double-refracting crystals. The laws obeyed by the wave are exactly the same as in other cases, yet the results are entirely sui generis. So far are such cases from contradicting the theory of ordinary cases, that they afford the supreme opportunities for verification.

In astronomy singular exceptions might occur, and in an approximate manner they do occur. We might point to the rings of Saturn as objects which, though undoubtedly obeying the law of gravity, are yet entirely 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. But a truly singular event might happen, or might have happened, under slightly different 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 at first sight for a crystal of the cubic system, but would in many ways 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 nonexistence in a material point of view, 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 necessitated 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 one very remarkable point entirely single and unique, at which alone water can remain in the three conditions of gas, liquid, and solid in the same vessel. It is the point at which three curves intersect, namely, the steam line showing at what temperatures and pressures water is just upon the point of becoming gaseous, and other similar lines which show when ice is just on the point of melting, and when ice is just about to assume the gaseous state directly.

Divergent Exceptions.

Closely analogous to singular exceptions are those divergent exceptions, in which a phenomenon manifests itself in very unusual magnitude or character, without however any degree becoming subject to peculiar laws. Thus #rowing ten coins, it happened in four cases out of throws, that all the coins fell with heads uppermost

238); these would usually be regarded as very events, and, according to the theory of probathey would be comparatively very rare; yet they

from an unusual conjunction of accidental from no really exceptional causes. In all classes anomena we may expect to meet with similar

on 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 usually occur with a frequency closely accordant with the law of error or divergence from an average, as shown by M. Quetelet and Mr. Galton (vol. i. p. 446). The rise of genius, or the occurrence of extraordinary musical or mathematical faculties, are attributed by M. Galton to the same principle of divergence.

Under this class of exceptions I am inclined to place all kinds of remarkable events arising from an unusual conjunction of many ordinary tendencies. 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 unfrequent compared with minor disturbances. They are not anomalous but only extreme events, exactly 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 closely the same number, for instance, of railway accidents and murders. Buckle has superficially remarked upon the comparative constancy of many such events as ascertained by Quetelet, and some of his readers acquire the false notion that there is a kind of mysterious inexorable law producing uniformity in natural and human affairs. But nothing can be more opposed to the teachings of the theory of probability, which always contemplates the occurrence of extreme and 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, but there is a fallacious tendency to suppose that every 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 or even fear. Such conjunctions of bodies 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 occasionally 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, as seen by us. 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 surprise and admiration, though conjunctions must naturally occur at intervals in the ordinary course of their motions. We cannot wonder, then, that when three or four planets approach each other closely, the event is long remembered. A most exceptional 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 that 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 his torical times.

Among the perturbations of the planetary motions we may find divergent exceptions arising from the peculiar accumulation or intensification of effects, as in the case of the long inequality of Jupiter and Saturn (vol. ii. p. 70). 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 divergents.

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. 162 of the first volume, that propositions of the general type ABC + be 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 equivalency.

Under the head of divergent exceptions we might doubtless place all or nearly all of the instances of substances possessing physical properties in a very high or low degree, which were described in the chapter on g Grant's History of Physical Astronomy,' p. 116.