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.

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.

CHAPTER 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.

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.

CHAPTER XXX.

CLASSIFICATION.

The extensive subject of Classification has been deferred to a late

part of this treatise, because it involves questions of difficulty, and

did not seem naturally to fall into an earlier place. But it must not

be supposed that, in now formally taking up the subject, we are for

the first time entertaining the notion of classification. All logical

inference involves classification, which is indeed the necessary

accompaniment of the action of judgment. It is impossible to detect

similarity between objects without thereby joining them together in

thought, and forming an incipient class. Nor can we bestow a common

name upon objects without implying the existence of a class. Every

common name is the name of a class, and every name of a class is a

common name. It is evident also that to speak of a general notion or

concept is but another way of speaking of a class. Usage leads us to

employ the word classification in some cases and not in others. We

are said to form the *general notion* parallelogram when we regard

an infinite number of possible four-sided rectilinear figures as

resembling each other in the common property of possessing parallel

sides. We should be said to form a *class*, Trilobite, when we place

together in a museum a number of specimens resembling each other in

certain defined characters. But the logical nature of the operation

is the same in both cases. We form a *class* of figures called

parallelograms and we form a *general notion* of trilobites.

Science, it was said at the outset, is the detection of identify, and

classification is the placing together, either in thought or in actual

proximity of space, those objects between which identity has been

detected. Accordingly, the value of classification is co-extensive with

the value of science and general reasoning. Whenever we form a class

we reduce multiplicity to unity, and detect, as Plato said, the one in

the many. The result of such classification is to yield generalised

knowledge, as distinguished from the direct and sensuous knowledge of

particular facts. Of every class, so far as it is correctly formed, the

principle of substitution is true, and whatever we know of one object

in a class we know of the other objects, so far as identity has been

detected between them. The facilitation and abbreviation of mental

labour is at the bottom of all mental progress. The reasoning faculties

of Newton were not different in nature from those of a ploughman; the

difference lay in the extent to which they were exerted, and the number

of facts which could be treated. Every thinking being generalises more

or less, but it is the depth and extent of his generalisations which

distinguish the philosopher. Now it is the exertion of the classifying

and generalising powers which enables the intellect of man to cope

in some degree with the infinite number of natural phenomena. In the

chapters upon combinations and permutations it was made evident, that

from a few elementary differences immense numbers of combinations

can be produced. The process of classification enables us to resolve

these combinations, and refer each one to its place according to one

or other of the elementary circumstances out of which it was produced.

We restore nature to the simple conditions out of which its endless

variety was developed. As Professor Bowen has said,[560] “The first

necessity which is imposed upon us by the constitution of the mind

itself, is to break up the infinite wealth of Nature into groups and

classes of things, with reference to their resemblances and affinities,

and thus to enlarge the grasp of our mental faculties, even at the

expense of sacrificing the minuteness of information which can be

acquired only by studying objects in detail. The first efforts in

the pursuit of knowledge, then, must be directed to the business

of classification. Perhaps it will be found in the sequel, that

classification is not only the beginning, but the culmination and the

end, of human knowledge.”

[560] *A Treatise on Logic, or, the Laws of Pure Thought*, by Francis

Bowen, Professor of Moral Philosophy in Harvard College, Cambridge,

United States, 1866, p. 315.

*Classification Involving Induction.*

The purpose of classification is the detection of the laws of nature.

However much the process may in some cases be disguised, classification

is not really distinct from the process of perfect induction, whereby

we endeavour to ascertain the connexions existing between properties of

the objects under treatment. There can be no use in placing an object

in a class unless something more than the fact of being in the class is

implied. If we arbitrarily formed a class of metals and placed therein

a selection from the list of known metals made by ballot, we should

have no reason to expect that the metals in question would resemble

each other in any points except that they are metals, and have been

selected by the ballot. But when chemists select from the list the five

metals, potassium, sodium, cæsium, rubidium, and lithium and call them

the Alkaline metals, a great deal is implied in this classification.

On comparing the qualities of these substances they are all found to

combine very energetically with oxygen, to decompose water at all

temperatures, and to form strongly basic oxides, which are highly

soluble in water, yielding powerfully caustic and alkaline hydrates

from which water cannot be expelled by heat. Their carbonates are also

soluble in water, and each metal forms only one chloride. It may also

be expected that each salt of one of the metals will correspond to a

salt of each other metal, there being a general analogy between the

compounds of these metals and their properties.

Now in forming this class of alkaline metals, we have done more than

merely select a convenient order of statement. We have arrived at a

discovery of certain empirical laws of nature, the probability being

very considerable that a metal which exhibits some of the properties

of alkaline metals will also possess the others. If we discovered

another metal whose carbonate was soluble in water, and which

energetically combined with water at all temperatures, producing a

strongly basic oxide, we should infer that it would form only a single

chloride, and that generally speaking, it would enter into a series

of compounds corresponding to the salts of the other alkaline metals.

The formation of this class of alkaline metals then, is no mere matter

of convenience; it is an important and successful act of inductive

discovery, enabling us to register many undoubted propositions as

results of perfect induction, and to make a great number of inferences

depending upon the principles of imperfect induction.

An excellent instance as to what classification can do, is found in

Mr. Lockyer’s researches on the sun.[561] Wanting some guide as to

what more elements to look for in the sun’s photosphere, he prepared

a classification of the elements according as they had or had not

been traced in the sun, together with a detailed statement of the

chief chemical characters of each element. He was then able to observe

that the elements found in the sun were for the most part those

forming stable compounds with oxygen. He then inferred that other

elements forming stable oxides would probably exist in the sun, and

he was rewarded by the discovery of five such metals. Here we have

empirical and tentative classification leading to the detection of the

correlation between existence in the sun, and the power of forming

stable oxides and then leading by imperfect induction to the discovery

of more coincidences between these properties.

[561] *Proceedings of the Royal Society*, November, 1873, vol. xxi.

p. 512.

Professor Huxley has defined the process of classification in the

following terms.[562] “By the classification of any series of objects,

is meant the actual or ideal arrangement together of those which are

like and the separation of those which are unlike; the purpose of this

arrangement being to facilitate the operations of the mind in clearly

conceiving and retaining in the memory the characters of the objects in

question.”

[562] *Lectures on the Elements of Comparative Anatomy*, 1864, p. 1.

This statement is doubtless correct, so far as it goes, but it does

not include all that Professor Huxley himself implicitly treats under

classification. He is fully aware that deep correlations, or in

other terms deep uniformities or laws of nature, will be disclosed

by any well chosen and profound system of classification. I should

therefore propose to modify the above statement, as follows:--“By

the classification of any series of objects, is meant the actual or

ideal arrangement together of those which are like and the separation

of those which are unlike, the purpose of this arrangement being,

primarily, to disclose the correlations or laws of union of properties

and circumstances, and, secondarily, to facilitate the operations

of the mind in clearly conceiving and retaining in the memory the

characters of the objects in question.”

*Multiplicity of Modes of Classification.*

In approaching the question how any given group of objects may be

best classified, let it be remarked that there must generally be an

unlimited number of modes of classifying a group of objects. Misled, as

we shall see, by the problem of classification in the natural sciences,

philosophers seem to think that in each subject there must be one

essentially natural system of classification which is to be selected,

to the exclusion of all others. This erroneous notion probably arises

also in part from the limited powers of thought and the inconvenient

mechanical conditions under which we labour. If we arrange the books

in a library catalogue, we must arrange them in some one order; if we

compose a treatise on mineralogy, the minerals must be successively

described in some one arrangement; if we treat such simple things as

geometrical figures, they must be taken in some fixed order. We shall

naturally select that arrangement which appears to be most convenient

and instructive for our principal purpose. But it does not follow

that this method of arrangement possesses any exclusive excellence,

and there will be usually many other possible arrangements, each

valuable in its own way. A perfect intellect would not confine itself

to one order of thought, but would simultaneously regard a group of

objects as classified in all the ways of which they are capable. Thus

the elements may be classified according to their atomicity into the

groups of monads, dyads, triads, tetrads, pentads, and hexads, and

this is probably the most instructive classification; but it does not

prevent us from also classifying them according as they are metallic

or non-metallic, solid, liquid or gaseous at ordinary temperatures,

useful or useless, abundant or scarce, ferro-magnetic or diamagnetic,

and so on.

Mineralogists have spent a great deal of labour in trying to discover

the supposed natural system of classification for minerals. They have

constantly encountered the difficulty that the chemical composition

does not run together with the crystallographic form, and the various

physical properties of the mineral. Substances identical in the

forms of their crystals, especially those belonging to the first or

cubical system of crystals, are often found to have no resemblance

in chemical composition. The same substance, again, is occasionally

found crystallised in two essentially different crystallographic

forms; calcium carbonate, for instance, appearing as calc-spar and

arragonite. The simple truth is that if we are unable to discover

any correspondence, or, as we may call it, any *correlation* between

the properties of minerals, we cannot make any one arrangement which

will enable us to treat all these properties in a single system of

classification. We must classify minerals in as many different ways

as there are different groups of unrelated properties of sufficient

importance. Even if, for the purpose of describing minerals

successively in a treatise, we select one chief system, that, for

instance, having regard to chemical composition, we ought mentally to

regard the minerals as classified in all other useful modes.

Exactly the same may be said of the classification of plants. An

immense number of different modes of classifying plants have been

proposed at one time or other, an exhaustive account of which will be

found in the article on classification in Rees’s “Cyclopædia,” or in

the introduction to Lindley’s “Vegetable Kingdom.” There have been the

Fructists, such as Cæsalpinus, Morison, Hermann, Boerhaave or Gaertner,

who arranged plants according to the form of the fruit. The Corollists,

Rivinus, Ludwig, and Tournefort, paid attention chiefly to the number

and arrangement of the parts of the corolla. Magnol selected the calyx

as the critical part, while Sauvage arranged plants according to their

leaves; nor are these instances more than a small selection from the

actual variety of modes of classification which have been tried. Of

such attempts it may be said that every system will probably yield some

information concerning the relations of plants, and it is only after

trying many modes that it is possible to approximate to the best.

*Natural and Artificial Systems of Classification.*

It has been usual to distinguish systems of classification as natural

and artificial, those being called natural which seemed to express the

order of existing things as determined by nature. Artificial methods of

classification, on the other hand, included those formed for the mere

convenience of men in remembering or treating natural objects.

The difference, as it is commonly regarded, has been well described

by Ampére,[563] as follows: “We can distinguish two kinds of

classifications, the natural and the artificial. In the latter kind,

some characters, arbitrarily chosen, serve to determine the place of

each object; we abstract all other characters, and the objects are thus

found to be brought near to or to be separated from each other, often

in the most bizarre manner. In natural systems of classification, on

the contrary, we employ concurrently all the characters essential to

the objects with which we are occupied, discussing the importance of

each of them; and the results of this labour are not adopted unless

the objects which present the closest analogy are brought most near

together, and the groups of the several orders which are formed from

them are also approximated in proportion as they offer more similar

characters. In this way it arises that there is always a kind of

connexion, more or less marked, between each group and the group which

follows it.”

[563] *Essai sur la Philosophie des Sciences*, p. 9.

There is much, however, that is vague and logically false in this

and other definitions which have been proposed by naturalists to

express their notion of a natural system. We are not informed how the

*importance* of a resemblance is to be determined, nor what is the

measure of the *closeness* of analogy. Until all the words employed

in a definition are made clear in meaning, the definition itself is

worse than useless. Now if the views concerning classification here

upheld are true, there can be no sharp and precise distinction between

natural and artificial systems. All arrangements which serve any

purpose at all must be more or less natural, because, if closely enough

scrutinised, they will involve more resemblances than those whereby the

class was defined.

It is true that in the biological sciences there would be one

arrangement of plants or animals which would be conspicuously

instructive, and in a certain sense natural, if it could be attained,

and it is that after which naturalists have been in reality striving

for nearly two centuries, namely, that *arrangement which would display

the genealogical descent of every form from the original life germ*.

Those morphological resemblances upon which the classification of

living beings is almost always based are inherited resemblances, and

it is evident that descendants will usually resemble their parents and

each other in a great many points.

I have said that a natural is distinguished from an arbitrary or

artificial system only in degree. It will be found almost impossible

to arrange objects according to any circumstance without finding that

some correlation of other circumstances is thus made apparent. No

arrangement could seem more arbitrary than the common alphabetical

arrangement according to the initial letter of the name. But we cannot

scrutinise a list of names of persons without noticing a predominance

of Evans’s and Jones’s, under the letters E and J, and of names

beginning with Mac under the letter M. The predominance is so great

that we could not attribute it to chance, and inquiry would of course

show that it arose from important facts concerning the nationality

of the persons. It would appear that the Evans’s and Jones’s were of

Welsh descent, and those whose names bear the prefix Mac of Keltic

descent. With the nationality would be more or less strictly correlated

many peculiarities of physical constitution, language, habits, or

mental character. In other cases I have been interested in noticing

the empirical inferences which are displayed in the most arbitrary

arrangements. If a large register of the names of ships be examined

it will often be found that a number of ships bearing the same name

were built about the same time, a correlation due to the occurrence of

some striking incident shortly previous to the building of the ships.

The age of ships or other structures is usually correlated with their

general form, nature of materials, &c., so that ships of the same name

will often resemble each other in many points.

It is impossible to examine the details of some of the so-called

artificial systems of classification of plants, without finding that

many of the classes are natural in character. Thus in Tournefort’s

arrangement, depending almost entirely on the formation of the corolla,

we find the natural orders of the Labiatæ, Cruciferæ, Rosaceæ,

Umbelliferæ, Liliaceæ, and Papilionaceæ, recognised in his 4th, 5th,

6th, 7th, 9th, and 10th classes. Many of the classes in Linnæus’

celebrated sexual system also approximate to natural classes.

*Correlation of Properties.*

Habits and usages of language are apt to lead us into the error of

imagining that when we employ different words we always mean different

things. In introducing the subject of classification nominally I was

careful to draw the reader’s attention to the fact that all reasoning

and all operations of scientific method really involve classification,

though we are accustomed to use the name in some cases and not in

others. The name *correlation* requires to be used with the same

qualification. Things are correlated (*con*, *relata*) when they are

so related or bound to each other that *where one is the other is, and

where one is not the other is not*. Throughout this work we have then

been dealing with correlations. In geometry the occurrence of three

equal angles in a triangle is correlated with the existence of three

equal sides; in physics gravity is correlated with inertia; in botany

exogenous growth is correlated with the possession of two cotyledons,

or the production of flowers with that of spiral vessels. Wherever a

proposition of the form A = B is true there correlation exists. But it

is in the classificatory sciences especially that the word correlation

has been employed.

We find it stated that in the class Mammalia the possession of two

occipital condyles, with a well-ossified basi-occipital, is correlated

with the possession of mandibles, each ramus of which is composed of a

single piece of bone, articulated with the squamosal element of the

skull, and also with the possession of mammæ and non-nucleated red

blood-corpuscles. Professor Huxley remarks[564] that this statement of

the character of the class mammalia is something more than an arbitrary

definition; it is a statement of a law of correlation or co-existence

of animal structures, from which most important conclusions are

deducible. It involves a generalisation to the effect that in nature

the structures mentioned are always found associated together. This

amounts to saying that the formation of the class mammalia involves an

act of inductive discovery, and results in the establishment of certain

empirical laws of nature. Professor Huxley has excellently expressed

the mode in which discoveries of this kind enable naturalists to make

deductions or predictions with considerable confidence, but he has also

pointed out that such inferences are likely from time to time to prove

mistaken. I will quote his own words:

[564] *Lectures on the Elements of Comparative Anatomy, and on the

Classification of Animals*, 1864, p. 3.

“If a fragmentary fossil be discovered, consisting of no more than

a ramus of a mandible, and that part of the skull with which it

articulated, a knowledge of this law may enable the palæontologist to

affirm, with great confidence, that the animal of which it formed a

part suckled its young, and had non-nucleated red blood-corpuscles; and

to predict that should the back part of that skull be discovered, it

will exhibit two occipital condyles and a well-ossified basi-occipital

bone.

“Deductions of this kind, such as that made by Cuvier in the famous

case of the fossil opossum of Montmartre, have often been verified,

and are well calculated to impress the vulgar imagination; so that

they have taken rank as the triumphs of the anatomist. But it should

carefully be borne in mind, that, like all merely empirical laws, which

rest upon a comparatively narrow observational basis, the reasoning

from them may at any time break down. If Cuvier, for example, had had

to do with a fossil Thylacinus instead of a fossil Opossum, he would

not have found the marsupial bones, though the inflected angle of the

jaw would have been obvious enough. And so, though, practically,

any one who met with a characteristically mammalian jaw would be

justified in expecting to find the characteristically mammalian occiput

associated with it; yet, he would be a bold man indeed, who should

strictly assert the belief which is implied in this expectation, viz.,

that at no period of the world’s history did animals exist which

combined a mammalian occiput with a reptilian jaw, or *vice versâ*.”