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.1 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.2 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. 1 Maxwell's Theory of Heat, (1871), p. 175. 2 Galton, on the Height and Weight of Boys. Journal of the Anthropological Institute, 1875, p. 174 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.1 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 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 equivalence. 1 Grant's History of Physical Astronomy, p. 116 U U Accidental Exceptions. The third and largest class of exceptions contains those which arise from the casual interference of extraneous causes. A law may be in operation, and, if so, must be perfectly fulfilled; but, while we conceive that we are examining its results, we may have before us the effects of a different cause, possessing no connexion with the subject of our inquiry. The law is not really broken, but at the same time the supposed exception is not illusory. It may be a phenomenon which cannot occur but under the condition of the law in question, yet there has been such interference that there is an apparent failure of science. There is, for instance, no subject in which more rigorous and invariable laws have been established than in crystallography. As a general rule, each chemical substance possesses its own definite form, by which it can be infallibly recognised; but the mineralogist has to be on his guard against what are called pseudomorphic crystals. In some circumstances a substance, having assumed its proper crystalline form, may afterwards undergo chemical change; a new ingredient may be added, a former one removed, or one element may be substituted for another. In calcium carbonate the carbonic acid is sometimes replaced by sulphuric acid, so that we find gypsum in the form of calcite; other cases are known where the change is inverted and calcite is found in the form of gypsum. Mica, talc, steatite, hematite, are other minerals subject to these curious transmutations. Sometimes a crystal embedded in a matrix is entirely dissolved away, and a new mineral is subsequently deposited in the cavity as in a mould. Quartz is thus found cast in many forms wholly unnatural to it. A still more perplexing case sometimes occurs. Calcium carbonate is capable of assuming two distinct forms of crystallisation, in which it bears respectively the names of calcite and arragonite. Now arragonite, while retaining its outward form unchanged, may undergo an internal molecular change into calcite, as indicated by the altered cleavage. Thus we may come across crystals apparently of arragonite, which seem to break all the laws of crystallography, by possessing the cleavage of a different system of crystallisation. Some of the most invariable laws of nature are disguised by interference of unlooked-for causes. While the barometer was yet a new and curious subject of investigation, its theory, as stated by Torricelli and Pascal, seemed to be contradicted by the fact that in a well-constructed instrument the mercury would often stand far above 31 inches in height. Boyle showed that mercury could be made to stand as high as 75 inches in a perfectly cleansed tube, or about two and a half times as high as could be due to the pressure of the atmosphere. Many theories about the pressure of imaginary fluids were in consequence put forth, and the subject was involved in much confusion. until the adhesive or cohesive force between glass and mercury, when brought into perfect contact, was pointed out as the real interfering cause. It seems to me, however, that the phenomenon is not thoroughly understood as yet. Gay-Lussac observed that the temperature of boiling water was very different in some kinds of vessels from what it was in others. It is only when in contact with metallic surfaces or sharply broken edges that the temperature is fixed at 100° C. The suspended freezing of liquids is another case where the action of a law of nature appears to be interrupted. Spheroidal ebullition was at first sight a most anomalous phenomenon; it was almost incredible that water should not boil in a red-hot vessel, or that ice could actually be produced in a red-hot crucible. These paradoxical results are now fully explained as due to the interposition of a non-conducting film of vapour between the globule of liquid and the sides of the vessel. The feats of conjurors who handle liquid metals are accounted for in the same manner. At one time the passive state of steel was regarded as entirely anomalous. It may be assumed as a general law that when pieces of electro-negative and electro-positive metal are placed in nitric acid, and made to touch each other, the electro-negative metal will undergo rapid solution. But when iron is the electro-negative and platinum the electro-positive, the solution of the iron entirely and abruptly ceases. Faraday ingeniously proved 1 Discourse to the Royal Society, 28th May, 1684. |