XXVIII. ANALOGY…

XXVIII. ANALOGY

As we have seen in the previous chapter, generalisation passes insensibly into reasoning by analogy, and the difference is one of degree. We are said to generalise when we view many objects as agreeing in a few properties, so that the resemblance is extensive rather than deep. When we have only a few objects of thought, but are able to discover many points of resemblance, we argue by analogy that the correspondence will be even deeper than appears. It may not be true that the words are always used in such distinct senses, and there is great vagueness in the employment of these and many logical terms; but if any clear discrimination can be drawn between generalisation and analogy, it is as indicated above.

It has been said, indeed, that analogy denotes not a resemblance between things, but between the relations of things. A pilot is a very different man from a prime minister, but he bears the same relation to a ship that the minister does to the state, so that we may analogically describe the prime minister as the pilot of the state. A man differs still more from a horse, nevertheless four men bear to three men the same relation as four horses bear to three horses. There is a real analogy between the tones of the Monochord, the Sages of Greece, and the Gates of Thebes, but it does not extend beyond the fact that they were all seven in number. Between the most discrete notions, as, for instance, those of time and space, analogy may exist, arising from the fact that the mathematical conditions of the lapse of time and of motion along a line are similar. There is no identity of nature between a word and the thing it signifies; the substance *iron* is a heavy solid, the word *iron* is either a momentary disturbance of the air, or a film of black pigment on white paper; but there is analogy between words and their significates. The substance iron is to the substance iron-carbonate, as the name iron is to the name iron-carbonate, when these names are used according to their scientific definitions. The whole structure of language and the whole utility of signs, marks, symbols, pictures, and representations of various kinds, rest upon analogy. I may hope perhaps to enter more fully upon this important subject at some future time, and to attempt to show how the invention of signs enables us to express, guide, and register our thoughts. It will be sufficient to observe here that the use of words constantly involves analogies of a subtle kind; we should often be at a loss how to describe a notion, were we not at liberty to employ in a metaphorical sense the name of anything sufficiently resembling it. There would be no expression for the sweetness of a melody, or the brilliancy of an harangue, unless it were furnished by the taste of honey and the brightness of a torch.

A cursory examination of the way in which we popularly use the word analogy, shows that it includes all degrees of resemblance or similarity. The analogy may consist only in similarity of number or ratio, or in like relations of time and space. It may also consist in simple resemblance between physical properties. We should not be using the word inconsistently with custom, if we said that there was an analogy between iron, nickel, and cobalt, manifested in the strength of their magnetic powers. There is a still more perfect analogy between iodine and chlorine; not that every property of iodine is identical with the corresponding property of chlorine; for then they would be one and the same kind of substance, and not two substances; but every property of iodine resembles in all but degree some property of chlorine. For almost every substance in which iodine forms a component, a corresponding substance may be discovered containing chlorine, so that we may confidently infer from the compounds of the one to the compounds of the other substance. Potassium iodide crystallises in cubes; therefore it is to be expected that potassium chloride will also crystallise in cubes. The science of chemistry as now developed rests almost entirely upon a careful and extensive comparison of the properties of substances, bringing deep-lying analogies to light. When any new substance is encountered, the chemist is guided in his treatment of it by the analogies which it seems to present with previously known substances.

In this chapter I cannot hope to illustrate the all-pervading influence of analogy in human thought and science. All science, it has been said, at the outset, arises from the discovery of identity, and analogy is but one name by which we denote the deeper-lying cases of resemblance. I shall only try to point out at present how analogy between apparently diverse classes of phenomena often serves as a guide in discovery. We thus commonly gain the first insight into the nature of an apparently unique object, and thus, in the progress of a science, we often discover that we are treating over again, in a new form, phenomena which were well known to us in another form.

*Analogy as a Guide in Discovery.*

There can be no doubt that discovery is most frequently accomplished by following up hints received from analogy, as Jeremy Bentham remarked.[524] Whenever a phenomenon is perceived, the first impulse of the mind is to connect it with the most nearly similar phenomenon. If we could ever meet a thing wholly *sui generis*, presenting no analogy to anything else, we should be incapable of investigating its nature, except by purely haphazard trial. The probability of success by such a process is so slight, that it is preferable to follow up the faintest clue. As I have pointed out already (p. 418), the possible experiments are almost infinite in number, and very numerous also are the hypotheses upon which we may proceed. Now it is self-evident that, however slightly superior the probability of success by one course of procedure may be over another, the most probable one should always be adopted first.

[524] *Essay on Logic*, *Works*, vol. viii. p. 276.

The chemist having discovered what he believes to be a new element, will have before him an infinite variety of modes of treating and investigating it. If in any of its qualities the substance displays a resemblance to an alkaline metal, for instance, he will naturally proceed to try whether it possesses other properties of the alkaline metals. Even the simplest phenomenon presents so many points for notice that we have a choice from among many hypotheses.

It would be difficult to find a more instructive instance of the way in which the mind is guided by analogy than in the description by Sir John Herschel of the course of thought by which he was led to anticipate in theory one of Faraday’s greatest discoveries. Herschel noticed that a screw-like form, technically called helicoidal dissymmetry, was observed in three cases, namely, in electrical helices, plagihedral quartz crystals, and the rotation of the plane of polarisation of light. As he said,[525] “I reasoned thus: Here are three phenomena agreeing in a *very strange peculiarity*. Probably, this peculiarity is a connecting link, physically speaking, among them. Now, in the case of the crystals and the light, this probability has been turned into certainty by my own experiments. Therefore, induction led me to conclude that a similar connection exists, and must turn up, somehow or other, between the electric current and polarised light, and that the plane of polarisation would be deflected by magneto-electricity.” By this course of analogical thought Herschel had actually been led to anticipate Faraday’s great discovery of the influence of magnetic strain upon polarised light. He had tried in 1822–25 to discover the influence of electricity on light, by sending a ray of polarised light through a helix, or near a long wire conveying an electric current. Such a course of inquiry, followed up with the persistency of Faraday, and with his experimental resources, would doubtless have effected the discovery. Herschel also suggests that the plagihedral form of quartz crystals must be due to a screw-like strain during crystallisation; but the notion remains unverified by experiment.

[525] *Life of Faraday*, by Bence Jones, vol. ii. p. 206.

*Analogy in the Mathematical Sciences.*

Whoever wishes to acquire a deep acquaintance with Nature must observe that there are analogies which connect whole branches of science in a parallel manner, and enable us to infer of one class of phenomena what we know of another. It has thus happened on several occasions that the discovery of an unsuspected analogy between two branches of knowledge has been the starting-point for a rapid course of discovery. The truths readily observed in the one may be of a different character from those which present themselves in the other. The analogy, once pointed out, leads us to discover regions of one science yet undeveloped, to which the key is furnished by the corresponding truths in the other science. An interchange of aid most wonderful in its results may thus take place, and at the same time the mind rises to a higher generalisation, and a more comprehensive view of nature.

No two sciences might seem at first sight more different in their subject matter than geometry and algebra. The first deals with circles, squares, parallelograms, and other forms in space; the latter with mere symbols of number. Prior to the time of Descartes, the sciences were developed slowly and painfully in almost entire independence of each other. The Greek philosophers indeed could not avoid noticing occasional analogies, as when Plato in the Thæetetus describes a square number as *equally equal*, and a number produced by multiplying two unequal factors as *oblong*. Euclid, in the 7th and 8th books of his Elements, continually uses expressions displaying a consciousness of the same analogies, as when he calls a number of two factors a *plane number*, ἐπίπεδος ἀριθμός, and distinguishes a square number of which the two factors are equal as an equal-sided and plane number, ἰσόπλευρος καὶ ἐπίπεδος ἀριθμός. He also calls the root of a cubic number its side, πλευρά. In the Diophantine algebra many problems of a geometrical character were solved by algebraic or numerical processes; but there was no general system, so that the solutions were of an isolated character. In general the ancients were far more advanced in geometric than symbolic methods; thus Euclid in his 4th book gives the means of dividing a circle by purely geometric means into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he was totally unacquainted with the theory of the roots of unity exactly corresponding to this division of the circle.

During the middle ages, on the contrary, algebra advanced beyond geometry, and modes of solving equations were gradually discovered by those who had no notion that at every step they were implicitly solving geometric problems. It is true that Regiomontanus, Tartaglia, Bombelli, and possibly other early algebraists, solved isolated geometrical problems by the aid of algebra, but particular numbers were always used, and no consciousness of a general method was displayed. Vieta in some degree anticipated the final discovery, and occasionally represented the roots of an equation geometrically, but it was reserved for Descartes to show, in the most general manner, that every equation may be represented by a curve or figure in space, and that every bend, point, cusp, or other peculiarity in the curve indicates some peculiarity in the equation. It is impossible to describe in any adequate manner the importance of this discovery. The advantage was two-fold: algebra aided geometry, and geometry gave reciprocal aid to algebra. Curves such as the well-known sections of the cone were found to correspond to quadratic equations; and it was impossible to manipulate the equations without discovering properties of those all-important curves. The way was thus opened for the algebraic treatment of motions and forces, without which Newton’s *Principia* could never have been worked out. Newton indeed was possessed by a strong infatuation in favour of the ancient geometrical methods; but it is well known that he employed symbolic methods to discover his theorems, and he now and then, by some accidental use of algebraic expression, confessed its greater power and generality.

Geometry, on the other hand, gave great assistance to algebra, by affording concrete representations of relations which would otherwise be too abstract for easy comprehension. A curve of no great complexity may give the whole history of the variations of value of a troublesome mathematical expression. As soon as we know, too, that every regular geometrical curve represents some algebraic equation, we are presented by observation of mechanical movements with abundant suggestions towards the discovery of mathematical problems. Every particle of a carriage-wheel when moving on a level road is constantly describing a cycloidal curve, the curious properties of which exercised the ingenuity of all the most skilful mathematicians of the seventeenth century, and led to important advancements in algebraic power. It may be held that the discovery of the Differential Calculus was mainly due to geometrical analogy, because mathematicians, in attempting to treat algebraically the tangent of a curve, were obliged to entertain the notion of infinitely small quantities.[526] There can be no doubt that Newton’s fluxional, that is, geometrical mode of stating the differential calculus, however much it subsequently retarded its progress in England, facilitated its apprehension at first, and I should think it almost certain that Newton discovered the principles of the calculus geometrically.

[526] Lacroix, *Traité Élémentaire de Calcul Différentiel et de Calcul Intégral*, 5^{me} édit. p. 699.

We may accordingly look upon this discovery of analogy, this happy alliance, as Bossut calls it,[527] between geometry and algebra, as the chief source of discoveries which have been made for three centuries past in mathematical methods. This is certainly the opinion of Lagrange, who says, “So long as algebra and geometry have been separate, their progress was slow, and their employment limited; but since these two sciences have been united, they have lent each other mutual strength, and have marched together with a rapid step towards perfection.”

[527] *Histoire des Mathématiques*, vol. i. p. 298.

The advancement of mechanical science has also been greatly aided by analogy. An abstract and intangible existence like force demands much power of conception, but it has a perfect concrete representative in a line, the end of which may denote the point of application, and the direction the line of action of the force, while the length can be made arbitrarily to denote the amount of the force. Nor does the analogy end here; for the moment of the force about any point, or its product into the perpendicular distance of its line of action from the point, is found to be represented by an area, namely twice the area of the triangle contained between the point and the ends of the line representing the force. Of late years a great generalisation has been effected; the Double Algebra of De Morgan is true not only of space relations, but of forces, so that the triangle of forces is reduced to a case of pure geometrical addition. Nay, the triangle of lines, the triangle of velocities, the triangle of forces, the triangle of couples, and perhaps other cognate theorems, are reduced by analogy to one simple theorem, which amounts to this, that there are two ways of getting from one angular point of a triangle to another, which ways, though different in length, are identical in their final results.[528] In the system of quaternions of the late Sir W. R. Hamilton, these analogies are embodied and carried out in the most general manner, so that whatever problem involves the threefold dimensions of space, or relations analogous to those of space, is treated by a symbolic method of the most comprehensive simplicity.

[528] See Goodwin, *Cambridge Philosophical Transactions* (1845), vol. viii. p. 269. O’Brien, “On Symbolical Statics,” *Philosophical Magazine*, 4th Series, vol. i. pp. 491, &c. See also Professor Clerk Maxwell’s delightful *Manual of Elementary Science*, called *Matter and Motion*, published by the Society for Promoting Christian Knowledge. In this admirable little work some of the most advanced results of mechanical and physical science are explained according to the method of quaternions, but with hardly any use of algebraic symbols.

It ought to be added that to the discovery of analogy between the forms of mathematical and logical expressions, we owe the greatest advance in logical science. Boole based his extension of logical processes upon the notion that logic is an algebra of two quantities 0 and 1. His profound genius for symbolic investigation led him to perceive by analogy that there must exist a general system of logical deduction, of which the old logicians had seized only a few fragments. Mistaken as he was in placing algebra as a higher science than logic, no one can deny that the development of the more complex and dependent science had advanced far beyond that of the simpler science, and that Boole, in drawing attention to the connection, made one of the most important discoveries in the history of science. As Descartes had wedded algebra and geometry, so did Boole accomplish the marriage of logic and algebra.

*Analogy in the Theory of Undulations.*

There is no class of phenomena which more thoroughly illustrates alike the power and weakness of analogy than the waves which agitate every kind of medium. All waves, whatsoever be the matter through which they pass, obey the principles of rhythmical or harmonic motion, and the subject therefore presents a fine field for mathematical generalisation. Each kind of medium may allow of waves peculiar in their conditions, so that it is a beautiful exercise in analogical reasoning to decide how, in making inferences from one kind of medium to another, we must make allowance for difference of circumstances. The waves of the ocean are large and visible, and there are the yet greater tidal waves which extend around the globe. From such palpable cases of rhythmical movement we pass to waves of sound, varying in length from about 32 feet to a small fraction of an inch. We have but to imagine, if we can, the fortieth octave of the middle C of a piano, and we reach the undulations of yellow light, the ultra-violet being about the forty-first octave. Thus we pass from the palpable and evident to that which is obscure, if not incomprehensible. Yet the same phenomena of reflection, interference, and refraction, which we find in some kinds of waves, may be expected to occur, *mutatis mutandis*, in other kinds.

From the great to the small, from the evident to the obscure, is not only the natural order of inference, but it is the historical order of discovery. The physical science of the Greek philosophers must have remained incomplete, and their theories groundless, because they did not understand the nature of undulations. Their systems were based upon the notion of movement of translation from place to place. Modern science tends to the opposite notion that all motion is alternating or rhythmical, energy flowing onwards but matter remaining comparatively fixed in position. Diogenes Laertius indeed correctly compared the propagation of sound with the spreading of waves on the surface of water when disturbed by a stone, and Vitruvius displayed a more complete comprehension of the same analogy. It remained for Newton to create the theory of undulatory motion in showing by mathematical deductive reasoning that the particles of an elastic fluid by vibrating backwards and forwards, might carry a pulse or wave moving from the source of disturbance, while the disturbed particles return to their place of rest. He was even able to make a first approximation by theoretical calculation to the velocity of sound-waves in the atmosphere. His theory of sound formed a hardly less important epoch in science than his far more celebrated theory of gravitation. It opened the way to all the subsequent applications of mechanical principles to the insensible motion of molecules. He seems to have been, too, upon the brink of another application of the same principles which would have advanced science by a century of progress, and made him the undisputed founder of all the theories of matter. He expressed opinions at various times that light might be due to undulatory movements of a medium occupying space, and in one intensely interesting sentence remarks[529] that colours are probably vibrations of different lengths, “much after the manner that, in the sense of hearing, nature makes use of aërial vibrations of several bignesses to generate sounds of divers tones, for the analogy of nature is to be observed.” He correctly foresaw that red and yellow light would consist of the longer undulations, and blue and violet of the shorter, while white light would be composed of an indiscriminate mixture of waves of various lengths. Newton almost overcame the strongest apparent difficulty of the undulatory theory of light, namely, the propagation of light in straight lines. For he observed that though waves of sound bend round an obstacle to some extent, they do not do so in the same degree as water-waves.[530] He had but to extend the analogy proportionally to light-waves, and not only would the difficulty have vanished, but the true theory of diffraction would have been open to him. Unfortunately he had a preconceived theory that rays of light are bent from and not towards the shadow of a body, a theory which for once he did not sufficiently compare with observation to detect its falsity. I am not aware, too, that Newton has, in any of his works, displayed an understanding of the phenomena of interference without which his notion of waves must have been imperfect.

[529] Birch, *History of the Royal Society*, vol. iii. p. 262, quoted by Young, *Works*, vol. i. p. 246.

[530] *Opticks*, Query 28, 3rd edit. p. 337.

While the general principles of undulatory motion will be the same in whatever medium the motion takes place, the circumstances may be excessively different. Between light travelling 186,000 miles per second and sound travelling in air only about 1,100 feet in the same time, or almost 900,000 times as slowly, we cannot expect a close outward resemblance. There are great differences, too, in the character of the vibrations. Gases scarcely admit of transverse vibration, so that sound travelling in air is a longitudinal wave, the particles of air moving backwards and forwards in the same line in which the wave moves onwards. Light, on the other hand, appears to consist entirely in the movement of points of force transversely to the direction of propagation of the ray. The light-wave is partially analogous to the bending of a rod or of a stretched cord agitated at one end. Now this bending motion may take place in any one of an infinite number of planes, and waves of which the planes are perpendicular to each other cannot interfere any more than two perpendicular forces can interfere. The complicated phenomena of polarised light arise out of this transverse character of the luminous wave, and we must not expect to meet analogous phenomena in atmospheric sound-waves. It is conceivable that in solids we might produce transverse sound undulations, in which phenomena of polarisation might be reproduced. But it would appear that even between transverse sound and light-waves the analogy holds true rather of the principles of harmonic motion than the circumstances of the vibrating medium; from experiment and theory it is inferred that the plane of polarisation in plane polarised light is perpendicular to instead of being coincident with the direction of vibration, as it would be in the case of transverse sound undulations. If so the laws of elastic forces are essentially different in application to the luminiferous ether and to ordinary solid bodies.[531]

[531] Rankine, *Philosophical Transactions* (1856), vol. cxlvi. p. 282.

*Analogy in Astronomy.*

We shall be much assisted in gaining a true appreciation of the value of analogy in its feebler degrees, by considering how much it has contributed to the progress of astronomical science. Our point of observation is so fixed with regard to the universe, and our means of examining distant bodies are so restricted, that we are necessarily guided by limited and apparently feeble resemblances. In many cases the result has been confirmed by subsequent direct evidence of the most forcible character.

While the scientific world was divided in opinion between the Copernican and Ptolemaic systems, it was analogy which furnished the most satisfactory argument. Galileo discovered, by the use of his new telescope, the four small satellites which circulate round Jupiter, and make a miniature planetary world. These four Medicean Stars, as they were called, were plainly seen to revolve round Jupiter in various periods, but approximately in one plane, and astronomers irresistibly inferred that what might happen on the smaller scale might also be found true of the greater planetary system. This discovery gave “the holding turn,” as Herschel expressed it, to the opinions of mankind. Even Francis Bacon, who, little to the credit of his scientific sagacity, had previously opposed the Copernican views, now became convinced, saying “We affirm the solisequium of Venus and Mercury; since it has been found by Galileo that Jupiter also has attendants.” Nor did Huyghens think it superfluous to adopt the analogy as a valid argument.[532] Even in an advanced stage of physical astronomy, the Jovian system has not lost its analogical interest; for the mutual perturbations of the four satellites pass through all their phases within a few centuries, and thus enable us to verify in a miniature case the principles of stability, which Laplace established for the great planetary system. Oscillations or disturbances which in the motions of the planets appear to be secular, because their periods extend over millions of years, can be watched, in the case of Jupiter’s satellites, through complete revolutions within the historical period of astronomy.[533]

[532] *Cosmotheoros* (1699), p. 16.

[533] Laplace, *System of the World*, vol. ii. p. 316.

In obtaining a knowledge of the stellar universe we must sometimes depend upon precarious analogies. We still hold upon this ground the opinion, entertained by Bruno as long ago as 1591, that the stars may be suns attended by planets like our earth. This is the most probable first assumption, and it is supported by spectrum observations, which show the similarity of light derived from many stars with that of the sun. But at the same time we learn by the prism that there are nebulæ and stars in conditions widely different from anything known in our system. In the course of time the analogy may perhaps be restored to comparative completeness by the discovery of suns in various stages of nebulous condensation. The history of the evolution of our own world may be traced back in bodies less developed, or traced forwards in systems more advanced towards the dissipation of energy, and the extinction of life. As in a great workshop, we may perhaps see the material work of Creation as it has progressed through thousands of millions of years.

In speculations concerning the physical condition of the planets and their satellites, we depend upon analogies of a weak character. We may be said to know that the moon has mountains and valleys, plains and ridges, volcanoes and streams of lava, and, in spite of the absence of air and water, the rocky surface of the moon presents so many familiar appearances that we do not hesitate to compare them with the features of our globe. We infer with high probability that Mars has polar snow and an atmosphere absorbing blue rays like our own; Jupiter undoubtedly possesses a cloudy atmosphere, possibly not unlike a magnified copy of that surrounding the earth, but our tendency to adopt analogies receives a salutary correction in the recently discovered fact that the atmosphere of Uranus contains hydrogen.

Philosophers have not stopped at these comparatively safe inferences, but have speculated on the existence of living creatures in other planets. Huyghens remarked that as we infer by analogy from the dissected body of a dog to that of a pig and ox or other animal of the same general form, and as we expect to find the same viscera, the heart, stomach, lungs, intestines, &c., in corresponding positions, so when we notice the similarity of the planets in many respects, we must expect to find them alike in other respects.[534] He even enters into an inquiry whether the inhabitants of other planets would possess reason and knowledge of the same sort as ours, concluding in the affirmative. Although the power of intellect might be different, he considers that they would have the same geometry if they had any at all, and that what is true with us would be true with them.[535] As regards the sun, he wisely observes that every conjecture fails. Laplace entertained a strong belief in the existence of inhabitants on other planets. The benign influence of the sun gives birth to animals and plants upon the surface of the earth, and analogy induces us to believe that his rays would tend to have a similar effect elsewhere. It is not probable that matter which is here so fruitful of life would be sterile upon so great a globe as Jupiter, which, like the earth, has its days and nights and years, and changes which indicate active forces. Man indeed is formed for the temperature and atmosphere in which he lives, and, so far as appears, could not live upon the other planets. But there might be an infinity of organisations relative to the diverse constitutions of the bodies of the universe. The most active imagination cannot form any idea of such various creatures, but their existence is not unlikely.[536]

[534] *Cosmotheoros* (1699), p. 17.

[535] Ibid. p. 36.

[536] *System of the World*, vol. ii. p. 326. *Essai Philosophique*, p. 87.

We now know that many metals and other elements never found in organic structures are yet capable of forming compounds with substances of vegetable or animal origin. It is therefore just possible that at different temperatures creatures formed of different yet analogous compounds might exist, but it would seem indispensable that carbon should form the basis of organic structures. We have no analogies to lead us to suppose that in the absence of that complex element life can exist. Could we find globes surrounded by atmospheres resembling our own in temperature and composition, we should be almost forced to believe them inhabited, but the probability of any analogical argument decreases rapidly as the condition of a globe diverges from that of our own. The Cardinal Nicholas de Cusa held long ago that the moon was inhabited, but the absence of any appreciable atmosphere renders the existence of inhabitants highly improbable. Speculations resting upon weak analogies hardly belong to the scope of true science, and can only be tolerated as an antidote to the far worse dogmas which assert that the thousand million of persons on earth, or rather a small fraction of them, are the sole objects of care of the Power which designed this limitless Universe.

*Failures of Analogy.*

So constant is the aid which we derive from the use of analogy in all attempts at discovery or explanation, that it is most important to observe in what cases it may lead us into difficulties. That which we expect by analogy to exist

(1) May be found to exist;

(2) May seem not to exist, but nevertheless may really exist;

(3) May actually be non-existent.

In the second case the failure is only apparent, and arises from our obtuseness of perception, the smallness of the phenomenon to be noticed, or the disguised character in which it appears. I have already pointed out that the analogy of sound and light seems to fail because light does not apparently bend round a corner, the fact being that it does so bend in the phenomena of diffraction, which present the effect, however, in such an unexpected and minute form, that even Newton was misled, and turned from the correct hypothesis of undulations which he had partially entertained.

In the third class of cases analogy fails us altogether, and we expect that to exist which really does not exist. Thus we fail to discover the phenomena of polarisation in sound travelling through the atmosphere, since air is not capable of any appreciable transverse undulations. These failures of analogy are of peculiar interest, because they make the mind aware of its superior powers. There have been many philosophers who said that we can conceive nothing in the intellect which we have not previously received through the senses. This is true in the sense that we cannot *image* them to the mind in the concrete form of a shape or a colour; but we can speak of them and reason concerning them; in short, we often know them in everything but a sensuous manner. Accurate investigation shows that all material substances retard the motion of bodies through them by subtracting energy by impact. By the law of continuity we can frame the notion of a vacuous space in which there is no resistance whatever, nor need we stop there; for we have only to proceed by analogy to the case where a medium should accelerate the motion of bodies passing through it, somewhat in the mode which Aristotelians attributed falsely to the air. Thus we can frame the notion of *negative density*, and Newton could reason exactly concerning it, although no such thing exists.[537]

[537] *Principia*, bk. ii. Section ii. Prop. x.

In every direction of thought we may meet ultimately with similar failures of analogy. A moving point generates a line, a moving line generates a surface, a moving surface generates a solid, but what does a moving solid generate? When we compare a polyhedron, or many-sided solid, with a polygon, or plane figure of many sides, the volume of the first is analogous to the area of the second; the face of the solid answers to the side of the polygon; the edge of the solid to the point of the figure; but the corner, or junction of edges in the polyhedron, is left wholly unrepresented in the plane of the polygon. Even if we attempted to draw the analogies in some other manner, we should still find a geometrical notion embodied in the solid which has no representative in the figure of two dimensions.[538]

[538] De Morgan, *Cambridge Philosophical Transactions*, vol. xi. Part ii. p. 246.

Faraday was able to frame some notion of matter in a fourth condition, which should be to gas what gas is to liquid.[539] Such substance, he thought, would not fall far short of *radiant matter*, by which apparently he meant the supposed caloric or matter assumed to constitute heat, according to the corpuscular theory. Even if we could frame the notion, matter in such a state cannot be known to exist, and recent discoveries concerning the continuity of the solid, liquid, and gaseous states remove the basis of the speculation.

[539] *Life of Faraday*, vol. i. p. 216.Return to passage

From these and many other instances which might be adduced, we learn that analogical reasoning leads us to the conception of many things which, so far as we can ascertain, do not exist. In this way great perplexities have arisen in the use of language and mathematical symbols. All language depends upon analogy; for we join and arrange words so that they may represent the corresponding junctions or arrangements of things and their equalities. But in the use of language we are obviously capable of forming many combinations of words to which no corresponding meaning apparently exists. The same difficulty arises in the use of mathematical signs, and mathematicians have needlessly puzzled themselves about the square root of a negative quantity, which is, in many applications of algebraic calculation, simply a sign without any analogous meaning, there being a failure of analogy.

XXIX. EXCEPTIONAL PHENOMENA

If science consists in the detection of identity and the recognition of uniformity existing in many objects, it follows that the progress of science depends upon the study of exceptional phenomena. Such new phenomena are the raw material upon which we exert our faculties of observation and reasoning, in order to reduce the new facts beneath the sway of the laws of nature, either those laws already well known, or those to be discovered. Not only are strange and inexplicable facts those which are on the whole most likely to lead us to some novel and important discovery, but they are also best fitted to arouse our attention. So long as events happen in accordance with our anticipations, and the routine of every-day observation is unvaried, there is nothing to impress upon the mind the smallness of its knowledge, and the depth of mystery, which may be hidden in the commonest sights and objects. In early times the myriads of stars which remained in apparently fixed relative positions upon the heavenly sphere, received less notice from astronomers than those few planets whose wandering and inexplicable motions formed a riddle. Hipparchus was induced to prepare the first catalogue of stars, because a single new star had been added to those nightly visible; and in the middle ages two brilliant but temporary stars caused more popular interest in astronomy than any other events, and to one of them we owe all the observations of Tycho Brahe, the mediæval Hipparchus.

In other sciences, as well as in that of the heavens, exceptional events are commonly the points from which we start to explore new regions of knowledge. It has been beautifully said that Wonder is the daughter of Ignorance, but the mother of Invention; and though the most familiar and slight events, if fully examined, will afford endless food for wonder and for wisdom, yet it is the few peculiar and unlooked-for events which most often lead to a course of discovery. It is true, indeed, that it requires much philosophy to observe things which are too near to us.

The high scientific importance attaching, then, to exceptions, renders it desirable that we should carefully consider the various modes in which an exception may be disposed of; while some new facts will be found to confirm the very laws to which they seem at first sight clearly opposed, others will cause us to limit the generality of our previous statements. In some cases the exception may be proved to be no exception; occasionally it will prove fatal to our previous most confident speculations; and there are some new phenomena which, without really destroying any of our former theories, open to us wholly new fields of scientific investigation. The study of this subject is especially interesting and important, because, as I have before said (p. 587), no important theory can be built up complete and perfect all at once. When unexplained phenomena present themselves as objections to the theory, it will often demand the utmost judgment and sagacity to assign to them their proper place and force. The acceptance or rejection of a theory will depend upon discriminating the one insuperable contradictory fact from many, which, however singular and inexplicable at first sight, may afterwards be shown to be results of different causes, or possibly the most striking results of the very law with which they stand in apparent conflict.

I can enumerate at least eight classes or kinds of exceptional phenomena, to one or other of which any supposed exception to the known laws of nature can usually be referred; they may be briefly described as below, and will be sufficiently illustrated in the succeeding sections.

(1) Imaginary, or false exceptions, that is, facts, objects, or events which are not really what they are supposed to be.

(2) Apparent, but congruent exceptions, which, though apparently in conflict with a law of nature, are really in agreement with it.

(3) Singular exceptions, which really agree with a law of nature, but exhibit remarkable and unique results of it.

(4) Divergent exceptions, which really proceed from the ordinary action of known processes of nature, but which are excessive in amount or monstrous in character.

(5) Accidental exceptions, arising from the interference of some entirely distinct but known law of nature.

(6) Novel and unexplained exceptions, which lead to the discovery of a new series of laws and phenomena, modifying or disguising the effects of previously known laws, without being inconsistent with them.

(7) Limiting exceptions showing the falsity of a supposed law in some cases to which it had been extended, but not affecting its truth in other cases.

(8) Contradictory or real exceptions which lead us to the conclusion that a supposed hypothesis or theory is in opposition to the phenomena of nature, and must therefore be abandoned.

It ought to be clearly understood that in no case is a law of nature really thwarted or prevented from being fulfilled. The effects of a law may be disguised and hidden from our view in some instances: in others the law itself may be rendered inapplicable altogether; but if a law is applicable it must be carried out. Every law of nature must therefore be stated with the utmost generality of all the instances really coming under it. Babbage proposed to distinguish between *universal principles*, which do not admit of a single exception, such as that every number ending in 5 is divisible by five, and *general principles* which are more frequently obeyed than violated, as that “men will be governed by what they believe to be their interest.”[540] But in a scientific point of view general principles must be universal as regards some distinct class of objects, or they are not principles at all. If a law to which exceptions exist is stated without allusion to those exceptions, the statement is erroneous. I have no right to say that “All liquids expand by heat,” if I know that water below 4° C. does not; I ought to say, “All liquids, except water below 4° C., expand by heat;” and every new exception discovered will falsify the statement until inserted in it. To speak of some laws as being *generally* true, meaning not universally but in the majority of cases, is a hurtful abuse of the word, but is quite usual. *General* should mean that which is true of a whole *genus* or class, and every true statement must be true of some assigned or assignable class.

[540] Babbage, *The Exposition of 1851*, p. 1.

*Imaginary or False Exceptions.*

When a supposed exception to a law of nature is brought to our notice, the first inquiry ought properly to be--Is there any breach of the law at all? It may be that the supposed exceptional fact is not a fact at all, but a mere figment of the imagination. When King Charles requested the Royal Society to investigate the curious fact that a live fish put into a bucket of water does not increase the weight of the bucket and its contents, the Royal Society wisely commenced their deliberations by inquiring whether the fact was so or not. Every statement, however false, must have some cause or prior condition, and the real question for the Royal Society to investigate was, how the King came to think that the fact was so. Mental conditions, as we have seen, enter into all acts of observation, and are often a worthy subject of inquiry. But there are many instances in the history of science, in which trouble and error have been caused by false assertions carelessly made, and carelessly accepted without verification.

The reception of the Copernican theory was much impeded by the objection, that if the earth were moving, a stone dropped from the top of a high tower should be left behind, and should appear to move towards the west, just as a stone dropped from the mast-head of a moving ship would fall behind, owing to the motion of the ship. The Copernicans attempted to meet this grave objection in every way but the true one, namely, showing by trial that the asserted facts are not correct. In the first place, if a stone had been dropped with suitable precautions from the mast-head of a moving ship, it would have fallen close to the foot of the mast, because, by the first law of motion, it would remain in the same state of horizontal motion communicated to it by the mast. As the anti-Copernicans had assumed the contrary result as certain to ensue, their argument would of course have fallen through. Had the Copernicans next proceeded to test with great care the other assertion involved, they would have become still better convinced of the truth of their own theory. A stone dropped from the top of a high tower, or into a deep well, would certainly not have been deflected from the vertical direction in the considerable degree required to agree with the supposed consequences of the Copernican views; but, with very accurate observation, they might have discovered, as Benzenberg subsequently did, a very small deflection towards the east, showing that the eastward velocity is greater at the top than the bottom. Had the Copernicans then been able to detect and interpret the meaning of the small divergence thus arising, they would have found in it corroboration of their own views.

Multitudes of cases might be cited in which laws of nature seem to be evidently broken, but in which the apparent breach arises from a misapprehension of the case. It is a general law, absolutely true of all crystals yet submitted to examination, that no crystal has a re-entrant angle, that is an angle which towards the axis of the crystal is greater than two right angles. Wherever the faces of a crystal meet they produce a projecting edge, and wherever edges meet they produce a corner. Many crystals, however, when carelessly examined, present exceptions to this law, but closer observation always shows that the apparently re-entrant angle really arises from the oblique union of two distinct crystals. Other crystals seem to possess faces contradicting all the principles of crystallography; but careful examination shows that the supposed faces are not true faces, but surfaces produced by the orderly junction of an immense number of distinct thin crystalline plates, each plate being in fact a separate crystal, in which the laws of crystallography are strictly observed. The roughness of the supposed face, the striæ detected by the microscope, or inference by continuity from other specimens where the true faces of the plates are clearly seen, prove the mistaken character of the supposed exceptions. Again, four of the faces of a regular octahedron may become so enlarged in the crystallisation of iron pyrites and some other substances, that the other four faces become imperceptible and a regular tetrahedron appears to be produced, contrary to the laws of crystallographic symmetry. Many other crystalline forms are similarly modified, so as to produce a series of what are called *hemihedral* forms.

In tracing out the isomorphic relations of the elements, great perplexity has often been caused by mistaking one substance for another. It was pointed out that though arsenic was supposed to be isomorphous with phosphorus, the arseniate of soda crystallised in a form distinct from that of the corresponding phosphate. Some chemists held this to be a fatal objection to the doctrine of isomorphism; but it was afterwards pointed out by Clarke, that the arseniate and phosphate in question were not corresponding compounds, as they differed in regard to the water of crystallisation.[541] Vanadium again appeared to be an exception to the laws of isomorphism, until it was proved by Professor Roscoe, that what Berzelius supposed to be metallic vanadium was really an oxide of vanadium.[542]

[541] Daubeny’s *Atomic Theory*, p. 76.

[542] *Bakerian Lecture, Philosophical Transactions* (1868), vol. clviii. p. 2.

*Apparent but Congruent Exceptions.*

Not unfrequently a law of nature will present results in certain circumstances which appear to be entirely in conflict with the law itself. Not only may the action of the law be much complicated and disguised, but it may in various ways be reversed or inverted, so that careless observers are misled. Ancient philosophers generally believed that while some bodies were heavy by nature, others, such as flame, smoke, bubbles, clouds, &c., were essentially light, or possessed a tendency to move upwards. So acute an inquirer as Aristotle failed to perceive the true nature of buoyancy, and the doctrine of intrinsic lightness, expounded in his works, became the accepted view for many centuries. It is true that Lucretius was aware why flame tends to rise, holding that--

“The flame has weight, though highly rare, Nor mounts but when compelled by heavier air.”

Archimedes also was so perfectly acquainted with the buoyancy of bodies immersed in water, that he could not fail to perceive the existence of a parallel effect in air. Yet throughout the early middle ages the light of true science could not contend with the glare of the Peripatetic doctrine. The genius of Galileo and Newton was required to convince people of the simple truth that all matter is heavy, but that the gravity of one substance may be overborne by that of another, as one scale of a balance is carried up by the preponderating weight in the opposite scale. It is curious to find Newton gravely explaining the difference of absolute and relative gravity, as if it were a new discovery proceeding from his theory.[543] More than a century elapsed before other apparent exceptions to the Newtonian philosophy were explained away.

[543] *Principia*, bk. ii. Prop. 20. Corollaries, 5 and 6.

Newton himself allowed that the motion of the apsides of the moon’s orbit appeared to be irreconcilable with the law of gravity, and it remained for Clairaut to remove the difficulty by more complete mathematical analysis. There must always remain, in the motions of the heavenly bodies, discrepancies of some amount between theory and observation; but such discrepancies have so often yielded in past times to prolonged investigation that physicists now regard them as merely apparent exceptions, which will afterwards be found to agree with the law of gravity.

The most beautiful instance of an apparent exception, is found in the total reflection of light, which occurs when a beam of light within a medium falls very obliquely upon the boundary separating it from a rarer medium. The general law is that when a ray strikes the limit between two media of different refractive indices, part of the light is reflected and part is refracted; but when the obliquity of the ray within the denser medium passes beyond a certain point, there is a sudden apparent breach of continuity, and the whole of the light is reflected. A clear reason can be given for this exceptional conduct of the light. According to the law of refraction, the sine of the angle of incidence bears a fixed ratio to the sine of the angle of refraction, so that the greater of the two angles, which is always that in the less dense medium, may increase up to a right angle; but when the media differ in refractive power, the less angle cannot become a right angle, as this would require the sine of an angle to be greater than the radius. It might seem that this is an exception of the kind described below as a limiting exception, by which a law is shown to be inapplicable beyond certain limits; but in the explanation of the exception according to the undulatory theory, we find that there is really no breach of the general law. When an undulation strikes a point in a bounding surface, spherical waves are produced and spread from the point. The refracted ray is the resultant of an infinite number of such spherical waves, and the bending of the ray at the common surface of two media depends upon the comparative velocities of propagation of the undulations in those media. But if a ray falls very obliquely upon the surface of a rarer medium, the waves proceeding from successive points of the surface spread so rapidly as never to intersect, and no resultant wave will then be produced. We thus perceive that from similar mathematical conditions arise distinct apparent effects.

There occur from time to time failures in our best grounded predictions. A comet, of which the orbit has been well determined, may fail, like Lexell’s Comet, to appear at the appointed time and place in the heavens. In the present day we should not allow such an exception to our successful predictions to weigh against our belief in the theory of gravitation, but should assume that some unknown body had through the action of gravitation deflected the comet. As Clairaut remarked, in publishing his calculations concerning the expected reappearance of Halley’s Comet, a body which passes into regions so remote, and which is hidden from our view during such long periods, might be exposed to the influence of forces totally unknown to us, such as the attraction of other comets, or of planets too far removed from the sun to be ever perceived by us. In the case of Lexell’s Comet it was afterwards shown, curiously enough, that its appearance was not one of a regular series of periodical returns within the sphere of our vision, but a single exceptional visit never to be repeated, and probably due to the perturbing powers of Jupiter. This solitary visit became a strong confirmation of the law of gravity with which it seemed to be in conflict.

*Singular Exceptions.*

Among the most interesting of apparent exceptions are those which I call *singular exceptions*, because they are more or less analogous to the singular cases or solutions which occur in mathematical science. A general mathematical law embraces an infinite multitude of cases which perfectly agree with each other in a certain respect. It may nevertheless happen that a single case, while really obeying the general law, stands out as apparently different from all the rest. The rotation of the earth upon its axis gives to all the stars an apparent motion of rotation from east to west; but while countless thousands obey the rule, the Pole Star alone seems to break it. Exact observations indeed show that it also revolves in a small circle, but a star might happen for a short time to exist so close to the pole that no appreciable change of place would be caused by the earth’s rotation. It would then constitute a perfect singular exception; while really obeying the law, it would break the terms in which it is usually stated. In the same way the poles of every revolving body are singular points.

Whenever the laws of nature are reduced to a mathematical form we may expect to meet with singular cases, and, as all the physical sciences will meet in the mathematical principles of mechanics, there is no part of nature where we may not encounter them. In mechanical science the motion of rotation may be considered an exception to the motion of translation. It is a general law that any number of parallel forces, whether acting in the same or opposite directions, will have a resultant which may be substituted for them with like effect. This resultant will be equal to the algebraic sum of the forces, or the difference of those acting in one direction and the other; it will pass through a point which is determined by a simple formula, and which may be described as the mean point of all the points of application of the parallel forces (p. 364). Thus we readily determine the resultant of parallel forces except in one peculiar case, namely, when two forces are equal and opposite but not in the same straight line. Being equal and opposite the amount of the resultant is nothing, yet, as the forces are not in the same straight line, they do not balance each other. Examining the formula for the point of application of the resultant, we find that it gives an infinitely great magnitude, so that the resultant is nothing at all, and acts at an infinite distance, which is practically the same as to say that there is no resultant. Two such forces constitute what is known in mechanical science as a *couple*, which occasions rotatory instead of rectilinear motion, and can only be neutralised by an equal and opposite couple of forces.

The best instances of singular exceptions are furnished by the science of optics. It is a general law that in passing through transparent media the plane of vibration of polarised light remains unchanged. But in certain liquids, some peculiar crystals of quartz, and transparent solid media subjected to a magnetic strain, as in Faraday’s experiment (pp. 588, 630), the plane of polarisation is rotated in a screw-like manner. This effect is so entirely *sui generis*, so unlike any other phenomena in nature, as to appear truly exceptional; yet mathematical analysis shows it to be only a single case of much more general laws. As stated by Thomson and Tait,[544] it arises from the composition of two uniform circular motions. If while a point is moving round a circle, the centre of that circle move upon another circle, a great variety of curious curves will be produced according as we vary the dimensions of the circles, the rapidity or the direction of the motions. When the two circles are exactly equal, the rapidities nearly so, and the directions opposite, the point will be found to move gradually round the centre of the stationary circle, and describe a curious star-like figure connected with the molecular motions out of which the rotational power of the media rises. Among other singular exceptions in optics may be placed the conical refraction of light, already noticed (p. 540), arising from the peculiar form assumed by a wave of light when 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 law of ordinary cases, that they afford the best opportunities for verification.

[544] *Treatise on Natural Philosophy*, vol. i. p. 50.

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]

[545] Maxwell’s *Theory of Heat*, (1871), p. 175.

*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.

[546] Galton, on the Height and Weight of Boys. *Journal of the Anthropological Institute*, 1875, p. 174.

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]

[547] Grant’s *History of Physical Astronomy*, p. 116.

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.

*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[548] 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,[549] 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.

[548] *Discourse to the Royal Society*, 28th May, 1684.

[549] Robert Hooke’s *Posthumous Works*, p. 365.

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 that this effect is due to a thin film of oxide of iron, which forms upon the surface of the iron and protects it.[550]

[550] *Experimental Researches in Electricity*, vol. ii. pp. 240–245.

The law of gravity is so simple, and disconnected from the other laws of nature, that it never suffers any disturbance, and is in no way disguised, but by the complication of its own effects. It is otherwise with those secondary laws of the planetary system which have only an empirical basis. The fact that all the long known planets and satellites have a similar motion from west to east is not necessitated by any principles of mechanics, but points to some common condition existing in the nebulous mass from which our system has been evolved. The retrograde motions of the satellites of Uranus constituted a distinct breach in this law of uniform direction, which became all the more interesting when the single satellite of Neptune was also found to be retrograde. It now became probable, as Baden Powell well observed, that the anomaly would cease to be singular, and become a case of another law, pointing to some general interference which has taken place on the bounds of the planetary system. Not only have the satellites suffered from this perturbance, but Uranus is also anomalous in having an axis of rotation lying nearly in the ecliptic; and Neptune constitutes a partial exception to the empirical law of Bode concerning the distances of the planets, which circumstance may possibly be due to the same disturbance.

Geology is a science in which accidental exceptions are likely to occur. Only when we find strata in their original relative positions can we surely infer that the order of succession is the order of time. But it not uncommonly happens that strata are inverted by the bending and doubling action of extreme pressure. Landslips may carry one body of rock into proximity with an unrelated series, and produce results apparently inexplicable.[551] Floods, streams, icebergs, and other casual agents, may lodge remains in places where they would be wholly unexpected. Though such interfering causes have been sometimes wrongly supposed to explain important discoveries, the geologist must bear the possibility of interference in mind. Scarcely more than a century ago it was held that fossils were accidental productions of nature, mere forms into which minerals had been shaped by no peculiar cause. Voltaire appears not to have accepted such an explanation; but fearing that the occurrence of fossil fishes on the Alps would support the Mosaic account of the deluge, he did not hesitate to attribute them to the remains of fishes accidentally brought there by pilgrims. In archæological investigations the greatest caution is requisite in allowing for secondary burials in ancient tombs and tumuli, for imitations, forgeries, casual coincidences, disturbance by subsequent races or by other archæologists. In common life extraordinary events will happen from time to time, as when a shepherdess in France was astonished at an iron chain falling out of the sky close to her, the fact being that Gay-Lussac had thrown it out of his balloon, which was passing over her head at the time.

[551] Murchison’s *Silurian System*, vol. ii. p. 733, &c.

*Novel and Unexplained Exceptions.*

When a law of nature appears to fail because some other law has interfered with its action, two cases may present themselves;--the interfering law may be a known one, or it may have been previously undetected. In the first case, which we have sufficiently considered in the preceding section, we have nothing to do but calculate as exactly as possible the amount of interference, and make allowance for it; the apparent failure of the law under examination should then disappear. But in the second case the results may be much more important. A phenomenon which cannot be explained by any known laws may indicate the interference of undiscovered natural forces. The ancients could not help perceiving that the general tendency of bodies downwards failed in the case of the loadstone, nor would the doctrine of essential lightness explain the exception, since the substance drawn upwards by the loadstone is a heavy metal. We now see that there was no breach in the perfect generality of the law of gravity, but that a new form of energy manifested itself in the loadstone for the first time.

Other sciences show us that laws of nature, rigorously true and exact, may be developed by those who are ignorant of more complex phenomena involved in their application. Newton’s comprehension of geometrical optics was sufficient to explain all the ordinary refractions and reflections of light. The simple laws of the bending of rays apply to all rays, whatever the character of the undulations composing them. Newton suspected the existence of other classes of phenomena when he spoke of rays as *having sides*; but it remained for later experimentalists to show that light is a transverse undulation, like the bending of a rod or cord.

Dalton’s atomic theory is doubtless true of all chemical compounds, and the essence of it is that the same compound will always be found to contain the same elements in the same definite proportions. Pure calcium carbonate contains 48 parts by weight of oxygen to 40 of calcium and 12 of carbon. But when careful analyses were made of a great many minerals, this law appeared to fail. What was unquestionably the same mineral, judging by its crystalline form and physical properties, would give varying proportions of its components, and would sometimes contain unusual elements which yet could not be set down as mere impurities. Dolomite, for instance, is a compound of the carbonates of magnesia and lime, but specimens from different places do not exhibit any fixed ratio between the lime and magnesia. Such facts could be reconciled with the laws of Dalton only by supposing the interference of a new law, that of Isomorphism.

It is now established that certain elements are related to each other, so that they can, as it were, step into each other’s places without apparently altering the shapes of the crystals which they constitute. The carbonates of iron, calcium, and magnesium, are nearly identical in their crystalline forms, hence they may crystallise together in harmony, producing mixed minerals of considerable complexity, which nevertheless perfectly verify the laws of equivalent proportions. This principle of isomorphism once established, not only explains what was formerly a stumbling-block, but gives valuable aid to chemists in deciding upon the constitution of new salts, since compounds of isomorphous elements which have identical crystalline forms must possess corresponding chemical formulæ.

We may expect that from time to time extraordinary phenomena will be discovered, and will lead to new views of nature. The recent observation, for instance, that the resistance of a bar of selenium to a current of electricity is affected in an extraordinary degree by rays of light falling upon the selenium, points to a new relation between light and electricity. The allotropic changes which sulphur, selenium, and phosphorus undergo by an alteration in the amount of latent heat which they contain, will probably lead at some future time to important inferences concerning the molecular constitution of solids and liquids. The curious substance ozone has perplexed many chemists, and Andrews and Tait thought that it afforded evidence of the decomposition of oxygen by the electric discharge. The researches of Sir B. C. Brodie negative this notion, and afford evidence of the real constitution of the substance,[552] which still, however, remains exceptional in its properties and relations, and affords a hope of important discoveries in chemical theory.

[552] *Philosophical Transactions* (1872), vol. clxii. No. 23.

*Limiting Exceptions.*

We pass to cases where exceptional phenomena are actually irreconcilable with a law of nature previously regarded as true. Error must now be allowed to have been committed, but the error may be more or less extensive. It may happen that a law holding rigorously true of the facts actually under notice had been extended by generalisation to other series of facts then unexamined. Subsequent investigation may show the falsity of this generalisation, and the result must be to limit the law for the future to those objects of which it is really true. The contradiction to our previous opinions is partial and not total.

Newton laid down as a result of experiment that every ray of homogeneous light has a definite refrangibility, which it preserves throughout its course until extinguished. This is one case of the general principle of undulatory movement, which Herschel stated under the title “Principle of Forced Vibrations” (p. 451), and asserted to be absolutely without exception. But Herschel himself described in the *Philosophical Transactions* for 1845 a curious appearance in a solution of quinine; as viewed by transmitted light the solution appeared colourless, but in certain aspects it exhibited a beautiful celestial blue tint. Curiously enough the colour is seen only in the first portion of liquid which the light enters. Similar phenomena in fluor-spar had been described by Brewster in 1838. Professor Stokes, having minutely investigated the phenomena, discovered that they were more or less present in almost all vegetable infusions, and in a number of mineral substances. He came to the conclusion that this phenomenon, called by him Fluorescence, could only be explained by an alteration in the refrangibility of the rays of light; he asserts that light-rays of very short length of vibration in falling upon certain atoms excite undulations of greater length, in opposition to the principle of forced vibrations. No complete explanation of the mode of change is yet possible, because it depends upon the intimate constitution of the atoms of the substances concerned; but Professor Stokes believes that the principle of forced vibrations is true only so long as the excursions of an atom are very small compared with the magnitude of the complex molecules.[553]

[553] *Philosophical Transactions* (1852), vol. cxlii. pp. 465, 548, &c.

It is well known that in Calorescence the refrangibility of rays is increased and the wave-length diminished. Rays of obscure heat and low refrangibility may be concentrated so as to heat a solid substance, and make it give out rays belonging to any part of the spectrum, and it seems probable that this effect arises from the impact of distinct but conflicting atoms. Nor is it in light only that we discover limiting exceptions to the law of forced vibrations; for if we notice gentle waves lapping upon the stones at the edge of a lake we shall see that each larger wave in breaking upon a stone gives rise to a series of smaller waves. Thus there is constantly in progress a degradation in the magnitude of water-waves. The principle of forced vibrations seems then to be too generally stated by Herschel, but it must be a difficult question of mechanical theory to discriminate the circumstances in which it does and does not hold true.

We sometimes foresee the possible existence of exceptions yet unknown by experience, and limit the statement of our discoveries accordingly. Extensive inquiries have shown that all substances yet examined fall into one of two classes; they are all either ferro-magnetic, that is, magnetic in the same way as iron, or they are diamagnetic like bismuth. But it does not follow that every substance must be ferro-magnetic or diamagnetic. The magnetic properties are shown by Sir W. Thomson[554] to depend upon the specific inductive capacities of the substance in three rectangular directions. If these inductive capacities are all positive, we have a ferro-magnetic substance; if negative, a diamagnetic substance; but if the specific inductive capacity were positive in one direction and negative in the others, we should have an exception to previous experience, and could not place the substance under either of the present recognised classes.

[554] *Philosophical Magazine*, 4th Series, vol. i. p. 182.

So many gases have been reduced to the liquid state, and so many solids fused, that scientific men rather hastily adopted the generalisation that all substances could exist in all three states. A certain number of gases, such as oxygen, hydrogen, and nitrogen, have resisted all efforts to liquefy them, and it now seems probable from the experiments of Dr. Andrews that they are limiting exceptions. He finds that above 31° C. carbonic acid cannot be liquefied by any pressure he could apply, whereas below this temperature liquefaction is always possible. By analogy it becomes probable that even hydrogen might be liquefied if cooled to a very low temperature. We must modify our previous views, and either assert that *below a certain critical temperature* every gas may be liquefied, or else we must assume that a highly condensed gas is, when above the critical temperature, undistinguishable from a liquid. At the same time we have an explanation of a remarkable exception presented by liquid carbonic acid to the general rule that gases expand more by heat than liquids. Liquid carbonic acid was found by Thilorier in 1835 to expand more than four times as much as air; but by the light of Andrews’ experiments we learn to regard the liquid as rather a highly condensed gas than an ordinary liquid, and it is actually possible to reduce the gas to the apparently liquid condition without any abrupt condensation.[555]

[555] Maxwell, *Theory of Heat*, p. 123.

Limiting exceptions occur most frequently in the natural sciences of Botany, Zoology, Geology, &c., the laws of which are empirical. In innumerable instances the confident belief of one generation has been falsified by the wider observation of a succeeding one. Aristotle confidently held that all swans are white,[556] and the proposition seemed true until not a hundred years ago black swans were discovered in Western Australia. In zoology and physiology we may expect a fundamental identity to exist in the vital processes, but continual discoveries show that there is no limit to the apparently anomalous expedients by which life is reproduced. Alternate generation, fertilisation for several successive generations, hermaphroditism, are opposed to all we should expect from induction founded upon the higher animals. But such phenomena are only limiting exceptions showing that what is true of one class is not true of another. In certain of the cephalopoda we meet the extraordinary fact that an arm of the male is cast off and lives independently until it encounters the female.

[556] *Prior Analytics*, ii. 2, 8, and elsewhere.

*Real Exceptions to Supposed Laws.*

The exceptions which we have lastly to consider are the most important of all, since they lead to the entire rejection of a law or theory before accepted. No law of nature can fail; there are no such things as real exceptions to real laws. Where contradiction exists it must be in the mind of the experimentalist. Either the law is imaginary or the phenomena which conflict with it; if, then, by our senses we satisfy ourselves of the actual occurrence of the phenomena, the law must be rejected as illusory. The followers of Aristotle held that nature abhors a vacuum, and thus accounted for the rise of water in a pump. When Torricelli pointed out the visible fact that water would not rise more than 33 feet in a pump, nor mercury more than about 30 inches in a glass tube, they attempted to represent these facts as limiting exceptions, saying that nature abhorred a vacuum to a certain extent and no further. But the Academicians del Cimento completed their discomfiture by showing that if we remove the pressure of the surrounding air, and in proportion as we remove it, nature’s feelings of abhorrence decrease and finally disappear altogether. Even Aristotelian doctrines could not stand such direct contradiction.

Lavoisier’s ideas concerning the constitution of acids received complete refutation. He named oxygen the *acid generator*, because he believed that all acids were compounds of oxygen, a generalisation based on insufficient data. Berthollet, as early as 1789, proved by analysis that hydrogen sulphide and prussic acid, both clearly acting the part of acids, were devoid of oxygen; the former might perhaps have been interpreted as a limiting exception, but when so powerful an acid as hydrogen chloride (muriatic acid) was found to contain no oxygen the theory had to be relinquished. Berzelius’ theory of the dual formation of chemical compounds met a similar fate.

It is obvious that all conclusive *experimenta crucis* constitute real exceptions to the supposed laws of the theory which is overthrown. Newton’s corpuscular theory of light was not rejected on account of its absurdity or inconceivability, for in these respects it is, as we have seen, far superior to the undulatory theory. It was rejected because certain small fringes of colour did not appear in the exact place and of the exact size in which calculation showed that they ought to appear according to the theory (pp. 516–521). One single fact clearly irreconcilable with a theory involves its rejection. In the greater number of cases, what appears to be a fatal exception may be afterwards explained away as a singular or disguised result of the laws with which it seems to conflict, or as due to the interference of extraneous causes; but if we fail thus to reduce the fact to congruity, it remains more powerful than any theories or any dogmas.

Of late years not a few of the favourite doctrines of geologists have been rudely destroyed. It was the general belief that human remains were to be found only in those deposits which are actually in progress at the present day, so that the creation of man appeared to have taken place in this geological age. The discovery of a single worked flint in older strata and in connexion with the remains of extinct mammals was sufficient to explode such a doctrine. Similarly, the opinions of geologists have been altered by the discovery of the Eozoön in the Laurentian rocks of Canada; it was previously held that no remains of life occurred in any older strata than those of the Cambrian system. As the examination of the strata of the globe becomes more complete, our views of the origin and succession of life upon the globe must undergo many changes.

*Unclassed Exceptions.*

At every period of scientific progress there will exist a multitude of unexplained phenomena which we know not how to regard. They are the outstanding facts upon which the labours of investigators must be exerted,--the ore from which the gold of future discovery is to be extracted. It might be thought that, as our knowledge of the laws of nature increases, the number of such exceptions should decrease; but, on the contrary, the more we know the more there is yet to explain. This arises from several reasons; in the first place, the principal laws and forces in nature are numerous, so that he who bears in mind the wonderfully large numbers developed in the doctrine of combinations, will anticipate the existence of immensely numerous relations of one law to another. When we are once in possession of a law, we are potentially in possession of all its consequences; but it does not follow that the mind of man, so limited in its powers and capacities, can actually work them all out in detail. Just as the aberration of light was discovered empirically, though it should have been foreseen, so there are multitudes of unexplained facts, the connexion of which with laws of nature already known to us, we should perceive, were we not hindered by the imperfection of our deductive powers. But, in the second place, as will be more fully pointed out, it is not to be supposed that we have approximated to an exhaustive knowledge of nature’s powers. The most familiar facts may teem with indications of forces, now secrets hidden from us, because we have not mind-directed eyes to discriminate them. The progress of science will consist in the discovery from time to time of new exceptional phenomena, and their assignment by degrees to one or other of the heads already described. When a new fact proves to be merely a false, apparent, singular, divergent, or accidental exception, we gain a more minute and accurate acquaintance with the effects of laws already known to exist. We have indeed no addition to what was implicitly in our possession, but there is much difference between knowing the laws of nature and perceiving all their complicated effects. Should a new fact prove to be a limiting or real exception, we have to alter, in part or in whole, our views of nature, and are saved from errors into which we had fallen. Lastly, the new fact may come under the sixth class, and may eventually prove to be a novel phenomenon, indicating the existence of new laws and forces, complicating but not otherwise interfering with the effects of laws and forces previously known.

The best instance which I can find of an unresolved exceptional phenomenon, consists in the anomalous vapour-densities of phosphorus, arsenic, mercury, and cadmium. It is one of the most important laws of chemistry, discovered by Gay-Lussac, that equal volumes of gases exactly correspond to equivalent weights of the substances. Nevertheless phosphorus and arsenic give vapours exactly twice as dense as they should do by analogy, and mercury and cadmium diverge in the other direction, giving vapours half as dense as we should expect. We cannot treat these anomalies as limiting exceptions, and say that the law holds true of substances generally but not of these; for the properties of gases (p. 601), usually admit of the widest generalisations. Besides, the preciseness of the ratio of divergence points to the real observance of the law in a modified manner. We might endeavour to reduce the exceptions by doubling the atomic weights of phosphorus and arsenic, and halving those of mercury and cadmium. But this step has been maturely considered by chemists, and is found to conflict with all the other analogies of the substances and with the principle of isomorphism. One of the most probable explanations is, that phosphorus and arsenic produce vapour in an allotropic condition, which might perhaps by intense heat be resolved into a simpler gas of half the density; but facts are wanting to support this hypothesis, and it cannot be applied to the other two exceptions without supposing that gases and vapours generally are capable of resolution into something simpler. In short, chemists can at present make nothing of these anomalies. As Hofmann says, “Their philosophical interpretation belongs to the future.... They may turn out to be typical facts, round which many others of the like kind may come hereafter to be grouped; and they may prove to be allied with special properties, or dependent on particular conditions as yet unsuspected.”[557]

[557] Hofmann’s *Introduction to Chemistry*, p. 198.

It would be easy to point out a great number of other unexplained anomalies. Physicists assert, as an absolutely universal law, that in liquefaction heat is absorbed;[558] yet sulphur is at least an apparent exception. The two substances, sulphur and selenium, are, in fact, very anomalous in their relations to heat. Sulphur may be said to have two melting points, for, though liquid like water at 120° C., it becomes quite thick and tenacious between 221° and 249°, and melts again at a higher temperature. Both sulphur and selenium may be thrown into several curious states, which chemists conveniently dispose of by calling them *allotropic*, a term freely used when they are puzzled to know what has happened. The chemical and physical history of iron, again, is full of anomalies; not only does it undergo inexplicable changes of hardness and texture in its alloys with carbon and other elements, but it is almost the only substance which conveys sound with greater velocity at a higher than at a lower temperature, the velocity increasing from 20° to 100° C., and then decreasing. Silver also is anomalous in regard to sound. These are instances of inexplicable exceptions, the bearing of which must be ascertained in the future progress of science.

[558] Stewart’s *Elementary Treatise on Heat*, p. 80.

When the discovery of new and peculiar phenomena conflicting with our theories of the constitution of nature is reported to us, it becomes no easy task to steer a philosophically correct course between credulity and scepticism. We are not to assume, on the one hand, that there is any limit to the wonders which nature can present to us. Nothing except the contradictory is really impossible, and many things which we now regard as common-place were considered as little short of the miraculous when first perceived. The electric telegraph was a visionary dream among mediæval physicists;[559] it has hardly yet ceased to excite our wonder; to our descendants centuries hence it will probably appear inferior in ingenuity to some inventions which they will possess. Now every strange phenomenon may be a secret spring which, if rightly touched, will open the door to new chambers in the palace of nature. To refuse to believe in the occurrence of anything strange would be to neglect the most precious chances of discovery. We may say with Hooke, that “the believing strange things possible may perhaps be an occasion of taking notice of such things as another would pass by without regard as useless.” We are not, therefore, to shut our ears even to such apparently absurd stories as those concerning second-sight, clairvoyance, animal magnetism, ode force, table-turning, or any of the popular delusions which from time to time are current. The facts recorded concerning these matters are facts in some sense or other, and they demand explanation, either as new natural phenomena, or as the results of credulity and imposture. Most of the supposed phenomena referred to have been, or by careful investigation would doubtless be, referred to the latter head, and the absence of scientific ability in many of those who describe them is sufficient to cast a doubt upon their value.

[559] Jevons, *Proceedings of the Manchester Literary and Philosophical Society*, 6th March, 1877, vol. xvi. p. 164. See also Mr. W. E. A. Axon’s note on the same subject, ibid. p. 166.Return to passage

It is to be remembered that according to the principle of the inverse method of probability, the probability of any hypothetical explanation is affected by the probability of each other possible explanation. If no other reasonable explanation could be suggested, we should be forced to look upon spiritualist manifestations as indicating mysterious causes. But as soon as it is shown that fraud has been committed in several important cases, and that in other cases persons in a credulous and excited state of mind have deceived themselves, the probability becomes very considerable that similar explanations may apply to most like manifestations. The performances of conjurors sufficiently prove that it requires no very great skill to perform tricks the *modus operandi* of which shall entirely escape the notice of spectators. It is on these grounds of probability that we should reject the so-called spiritualist stories, and not simply because they are strange.

Certainly in the obscure phenomena of mind, those relating to memory, dreams, somnambulism, and other peculiar states of the nervous system, there are many inexplicable and almost incredible facts, and it is equally unphilosophical to believe or to disbelieve without clear evidence. There are many facts, too, concerning the instincts of animals, and the mode in which they find their way from place to place, which are at present quite inexplicable. No doubt there are many strange things not dreamt of in our philosophy, but this is no reason why we should believe in every strange thing which is reported to have happened.