CHAPTER XXVI.

CHARACTER OF THE EXPERIMENTALIST.

In the present age there seems to be a tendency to believe that the

importance of individual genius is less than it was--

“The individual withers, and the world is more and more.”

Society, it is supposed, has now assumed so highly developed a form,

that what was accomplished in past times by the solitary exertions of a

great intellect, may now be worked out by the united labours of an army

of investigators. Just as the well-organised power of a modern army

supersedes the single-handed bravery of the mediæval knights, so we are

to believe that the combination of intellectual labour has superseded

the genius of an Archimedes, a Newton, or a Laplace. So-called original

research is now regarded as a profession, adopted by hundreds of

men, and communicated by a system of training. All that we need to

secure additions to our knowledge of nature is the erection of great

laboratories, museums, and observatories, and the offering of pecuniary

rewards to those who can invent new chemical compounds, detect new

species, or discover new comets. Doubtless this is not the real meaning

of the eminent men who are now urging upon Government the endowment of

physical research. They can only mean that the greater the pecuniary

and material assistance given to men of science, the greater the result

which the available genius of the country may be expected to produce.

Money and opportunities of study can no more produce genius than

sunshine and moisture can generate living beings; the inexplicable

germ is wanting in both cases. But as, when the germ is present, the

plant will grow more or less vigorously according to the circumstances

in which it is placed, so it may be allowed that pecuniary assistance

may favour development of intellect. Public opinion however is not

discriminating, and is likely to interpret the agitation for the

endowment of science as meaning that science can be had for money.

All such notions are erroneous. In no branch of human affairs,

neither in politics, war, literature, industry, nor science, is the

influence of genius less considerable than it was. It is possible

that the extension and organisation of scientific study, assisted by

the printing-press and the accelerated means of communication, has

increased the rapidity with which new discoveries are made known, and

their details worked out by many heads and hands. A Darwin now no

sooner propounds original ideas concerning the evolution of living

creatures, than those ideas are discussed and illustrated, and

applied by naturalists in every part of the world. In former days his

discoveries would have been hidden for decades of years in scarce

manuscripts, and generations would have passed away before his theory

had enjoyed the same amount of criticism and corroboration as it has

already received. The result is that the genius of Darwin is more

valuable, not less valuable, than it would formerly have been. The

advance of military science and the organisation of enormous armies

has not decreased the value of a skilful general; on the contrary,

the rank and file are still more in need than they used to be of the

guiding power of a far-seeing intellect. The swift destruction of the

French military power was not due alone to the perfection of the German

army, nor to the genius of Moltke; it was due to the combination of a

well-disciplined multitude with a leader of the highest powers. So in

every branch of human affairs the influence of the individual is not

withering, but is growing with the extent of the material resources

which are at his command.

Turning to our own subject, it is a work of undiminished interest to

reflect upon those qualities of mind which lead to great advances in

natural knowledge. Nothing, indeed, is less amenable than genius to

scientific analysis and explanation. Even definition is out of the

question. Buffon said that “genius is patience,” and certainly patience

is one of its most requisite components. But no one can suppose that

patient labour alone will invariably lead to those conspicuous results

which we attribute to genius. In every branch of science, literature,

art, or industry, there are thousands of men and women who work with

unceasing patience, and thereby ensure moderate success; but it would

be absurd to suppose that equal amounts of intellectual labour yield

equal results. A Newton may modestly attribute his discoveries to

industry and patient thought, and there is reason to believe that

genius is unconscious and unable to account for its own peculiar

powers. As genius is essentially creative, and consists in divergence

from the ordinary grooves of thought and action, it must necessarily be

a phenomenon beyond the domain of the laws of nature. Nevertheless, it

is always an interesting and instructive work to trace out, as far as

possible, the characteristics of mind by which great discoveries have

been achieved, and we shall find in the analysis much to illustrate the

principles of scientific method.

*Error of the Baconian Method.*

Hundreds of investigators may be constantly engaged in experimental

inquiry; they may compile numberless note-books full of scientific

facts, and endless tables of numerical results; but, if the views of

induction here maintained be true, they can never by such work alone

rise to new and great discoveries. By a system of research they may

work out deductively the details of a previous discovery, but to arrive

at a new principle of nature is another matter. Francis Bacon spread

abroad the notion that to advance science we must begin by accumulating

facts, and then draw from them, by a process of digestion, successive

laws of higher and higher generality. In protesting against the false

method of the scholastic logicians, he exaggerated a partially true

philosophy, until it became as false as that which preceded it. His

notion of scientific method was a kind of scientific bookkeeping.

Facts were to be indiscriminately gathered from every source, and

posted in a ledger, from which would emerge in time a balance of

truth. It is difficult to imagine a less likely way of arriving at

great discoveries. The greater the array of facts, the less is the

probability that they will by any routine system of classification

disclose the laws of nature they embody. Exhaustive classification in

all possible orders is out of the question, because the possible orders

are practically infinite in number.

It is before the glance of the philosophic mind that facts must display

their meaning, and fall into logical order. The natural philosopher

must therefore have, in the first place, a mind of impressionable

character, which is affected by the slightest exceptional phenomenon.

His associating and identifying powers must be great, that is, a

strange fact must suggest to his mind whatever of like nature has

previously come within his experience. His imagination must be active,

and bring before his mind multitudes of relations in which the

unexplained facts may possibly stand with regard to each other, or to

more common facts. Sure and vigorous powers of deductive reasoning

must then come into play, and enable him to infer what will happen

under each supposed condition. Lastly, and above all, there must be the

love of certainty leading him diligently and with perfect candour, to

compare his speculations with the test of fact and experiment.

*Freedom of Theorising.*

It would be an error to suppose that the great discoverer seizes at

once upon the truth, or has any unerring method of divining it. In all

probability the errors of the great mind exceed in number those of the

less vigorous one. Fertility of imagination and abundance of guesses at

truth are among the first requisites of discovery; but the erroneous

guesses must be many times as numerous as those which prove well

founded. The weakest analogies, the most whimsical notions, the most

apparently absurd theories, may pass through the teeming brain, and no

record remain of more than the hundredth part. There is nothing really

absurd except that which proves contrary to logic and experience. The

truest theories involve suppositions which are inconceivable, and no

limit can really be placed to the freedom of hypothesis.

Kepler is an extraordinary instance to this effect. No minor laws

of nature are more firmly established than those which he detected

concerning the orbits and motions of planetary masses, and on these

empirical laws the theory of gravitation was founded. Did we not learn

from his own writings the multitude of errors into which he fell, we

might have imagined that he had some special faculty of seizing on the

truth. But, as is well known, he was full of chimerical notions; his

favourite and long-studied theory was founded on a fanciful analogy

between the planetary orbits and the regular solids. His celebrated

laws were the outcome of a lifetime of speculation, for the most part

vain and groundless. We know this because he had a curious pleasure

in dwelling upon erroneous and futile trains of reasoning, which

most persons consign to oblivion. But Kepler’s name was destined to

be immortal, on account of the patience with which he submitted his

hypotheses to comparison with observation, the candour with which he

acknowledged failure after failure, and the perseverance and ingenuity

with which he renewed his attack upon the riddles of nature.

Next after Kepler perhaps Faraday is the physical philosopher who has

given us the best insight into the progress of discovery, by recording

erroneous as well as successful speculations. The recorded notions,

indeed, are probably but a tithe of the fancies which arose in his

active brain. As Faraday himself said--“The world little knows how

many of the thoughts and theories which have passed through the mind

of a scientific investigator, have been crushed in silence and secrecy

by his own severe criticism and adverse examination; that in the most

successful instances not a tenth of the suggestions, the hopes, the

wishes, the preliminary conclusions have been realised.”

Nevertheless, in Faraday’s researches, published in the *Philosophical

Transactions*, in minor papers, in manuscript note-books, or in other

materials, made known in his interesting life by Dr. Bence Jones, we

find invaluable lessons for the experimentalist. These writings are

full of speculations which we must not judge by the light of subsequent

discovery. It may perhaps be said that Faraday committed to the

printing press crude ideas which a friend would have counselled him to

keep back. There was occasionally even a wildness and vagueness in his

notions, which in a less careful experimentalist would have been fatal

to the attainment of truth. This is especially apparent in a curious

paper concerning Ray-vibrations; but fortunately Faraday was aware of

the shadowy character of his speculations, and expressed the feeling in

words which must be quoted. “I think it likely,” he says,[478] “that

I have made many mistakes in the preceding pages, for even to myself

my ideas on this point appear only as the shadow of a speculation, or

as one of those impressions upon the mind, which are allowable for a

time as guides to thought and research. He who labours in experimental

inquiries knows how numerous these are, and how often their apparent

fitness and beauty vanish before the progress and development of real

natural truth.” If, then, the experimentalist has no royal road to the

discovery of the truth, it is an interesting matter to consider by what

logical procedure he attains the truth.

[478] *Experimental Researches in Chemistry and Physics*, p. 372.

*Philosophical Magazine*, 3rd Series, May 1846, vol. xxviii. p. 350.

If I have taken a correct view of logical method, there is really no

such thing as a distinct process of induction. The probability is

infinitely small that a collection of complicated facts will fall

into an arrangement capable of exhibiting directly the laws obeyed

by them. The mathematician might as well expect to integrate his

functions by a ballot-box, as the experimentalist to draw deep truths

from haphazard trials. All induction is but the inverse application

of deduction, and it is by the inexplicable action of a gifted mind

that a multitude of heterogeneous facts are ranged in luminous order

as the results of some uniformly acting law. So different, indeed,

are the qualities of mind required in different branches of science,

that it would be absurd to attempt to give an exhaustive description

of the character of mind which leads to discovery. The labours of

Newton could not have been accomplished except by a mind of the utmost

mathematical genius; Faraday, on the other hand, has made the most

extensive additions to human knowledge without passing beyond common

arithmetic. I do not remember meeting in Faraday’s writings with a

single algebraic formula or mathematical problem of any complexity.

Professor Clerk Maxwell, indeed, in the preface to his new *Treatise

on Electricity*, has strongly recommended the reading of Faraday’s

researches by all students of science, and has given his opinion

that though Faraday seldom or never employed mathematical formulæ,

his methods and conceptions were not the less mathematical in their

nature.[479] I have myself protested against the prevailing confusion

between a mathematical and an exact science,[480] yet I certainly think

that Faraday’s experiments were for the most part qualitative, and that

his mathematical ideas were of a rudimentary character. It is true that

he could not possibly investigate such a subject as magne-crystallic

action without involving himself in geometrical relations of some

complexity. Nevertheless I think that he was deficient in mathematical

deductive power, that power which is so highly developed by the modern

system of mathematical training at Cambridge.

[479] See also *Nature*, September 18, 1873; vol. viii. p. 398.

[480] *Theory of Political Economy*, pp. 3–14.

Faraday was acquainted with the forms of his celebrated lines of force,

but I am not aware that he ever entered into the algebraic nature

of those curves, and I feel sure that he could not have explained

their forms as depending on the resultant attractions of all the

magnetic particles. There are even occasional indications that he did

not understand some of the simpler mathematical doctrines of modern

physical science. Although he so clearly foresaw the correlation of the

physical forces, and laboured so hard with his own hands to connect

gravity with other forces, it is doubtful whether he understood the

doctrine of the conservation of energy as applied to gravitation.

Faraday was probably equal to Newton in experimental skill, and in that

peculiar kind of deductive power which leads to the invention of simple

qualitative experiments; but it must be allowed that he exhibited

little of that mathematical power which enabled Newton to follow out

intuitively the quantitative results of a complicated problem with such

wonderful facility. Two instances, Newton and Faraday, are sufficient

to show that minds of widely different conformation will meet with

suitable regions of research. Nevertheless, there are certain traits

which we may discover in all the highest scientific minds.

*The Newtonian Method, the True Organum.*

Laplace was of opinion that the *Principia* and the *Opticks* of

Newton furnished the best models then available of the delicate art

of experimental and theoretical investigation. In these, as he says,

we meet with the most happy illustrations of the way in which, from

a series of inductions, we may rise to the causes of phenomena, and

thence descend again to all the resulting details.

The popular notion concerning Newton’s discoveries is that in early

life, when driven into the country by the Great Plague, a falling apple

accidentally suggested to him the existence of gravitation, and that,

availing himself of this hint, he was led to the discovery of the law

of gravitation, the explanation of which constitutes the *Principia*.

It is difficult to imagine a more ludicrous and inadequate picture of

Newton’s labours. No originality, or at least priority, was claimed

by Newton as regards the discovery of the law of the inverse square,

so closely associated with his name. In a well-known Scholium[481]

he acknowledges that Sir Christopher Wren, Hooke, and Halley, had

severally observed the accordance of Kepler’s third law of motion with

the principle of the inverse square.

[481] *Principia*, bk. i. Prop. iv.

Newton’s work was really that of developing the methods of deductive

reasoning and experimental verification, by which alone great

hypotheses can be brought to the touchstone of fact. Archimedes was the

greatest of ancient philosophers, for he showed how mathematical theory

could be wedded to physical experiments; and his works are the first

true Organum. Newton is the modern Archimedes, and the *Principia*

forms the true Novum Organum of scientific method. The laws which he

established are great, but his example of the manner of establishing

them is greater still. Excepting perhaps chemistry and electricity,

there is hardly a progressive branch of physical and mathematical

science, which has not been developed from the germs of true scientific

procedure which he disclosed in the *Principia* or the *Opticks*.

Overcome by the success of his theory of universal gravitation, we

are apt to forget that in his theory of sound he originated the

mathematical investigation of waves and the mutual action of particles;

that in his corpuscular theory of light, however mistaken, he first

ventured to apply mathematical calculation to molecular attractions

and repulsions; that in his prismatic experiments he showed how far

experimental verification could be pushed; that in his examination

of the coloured rings named after him, he accomplished the most

remarkable instance of minute measurement yet known, a mere practical

application of which by Fizeau was recently deemed worthy of a medal

by the Royal Society. We only learn by degrees how complete was his

scientific insight; a few words in his third law of motion display his

acquaintance with the fundamental principles of modern thermodynamics

and the conservation of energy, while manuscripts long overlooked prove

that in his inquiries concerning atmospheric refraction he had overcome

the main difficulties of applying theory to one of the most complex of

physical problems.

After all, it is only by examining the way in which he effected

discoveries, that we can rightly appreciate his greatness. The

*Principia* treats not of gravity so much as of forces in general, and

the methods of reasoning about them. He investigates not one hypothesis

only, but mechanical hypotheses in general. Nothing so much strikes

the reader of the work as the exhaustiveness of his treatment, and the

unbounded power of his insight. If he treats of central forces, it

is not one law of force which he discusses, but many, or almost all

imaginable laws, the results of each of which he sketches out in a few

pregnant words. If his subject is a resisting medium, it is not air or

water alone, but resisting media in general. We have a good example

of his method in the scholium to the twenty-second proposition of the

second book, in which he runs rapidly over many suppositions as to

the laws of the compressing forces which might conceivably act in an

atmosphere of gas, a consequence being drawn from each case, and that

one hypothesis ultimately selected which yields results agreeing with

experiments upon the pressure and density of the terrestrial atmosphere.

Newton said that he did not frame hypotheses, but, in reality, the

greater part of the *Principia* is purely hypothetical, endless

varieties of causes and laws being imagined which have no counterpart

in nature. The most grotesque hypotheses of Kepler or Descartes were

not more imaginary. But Newton’s comprehension of logical method was

perfect; no hypothesis was entertained unless it was definite in

conditions, and admitted of unquestionable deductive reasoning; and the

value of each hypothesis was entirely decided by the comparison of its

consequences with facts. I do not entertain a doubt that the general

course of his procedure is identical with that view of the nature of

induction, as the inverse application of deduction, which I advocate

throughout this book. Francis Bacon held that science should be founded

on experience, but he mistook the true mode of using experience,

and, in attempting to apply his method, ludicrously failed. Newton

did not less found his method on experience, but he seized the true

method of treating it, and applied it with a power and success never

since equalled. It is a great mistake to say that modern science is

the result of the Baconian philosophy; it is the Newtonian philosophy

and the Newtonian method which have led to all the great triumphs of

physical science, and I repeat that the *Principia* forms the true

“Novum Organum.”

In bringing his theories to a decisive experimental verification,

Newton showed, as a general rule, exquisite skill and ingenuity.

In his hands a few simple pieces of apparatus were made to give

results involving an unsuspected depth of meaning. His most beautiful

experimental inquiry was that by which he proved the differing

refrangibility of rays of light. To suppose that he originally

discovered the power of a prism to break up a beam of white light

would be a mistake, for he speaks of procuring a glass prism to

try the “celebrated phenomena of colours.” But we certainly owe to

him the theory that white light is a mixture of rays differing in

refrangibility, and that lights which differ in colour, differ also in

refrangibility. Other persons might have conceived this theory; in

fact, any person regarding refraction as a quantitative effect must see

that different parts of the spectrum have suffered different amounts

of refraction. But the power of Newton is shown in the tenacity with

which he followed his theory into every consequence, and tested each

result by a simple but conclusive experiment. He first shows that

different coloured spots are displaced by different amounts when viewed

through a prism, and that their images come to a focus at different

distances from the lense, as they should do, if the refrangibility

differed. After excluding by many experiments a variety of indifferent

circumstances, he fixes his attention upon the question whether the

rays are merely shattered, disturbed, and spread out in a chance

manner, as Grimaldi supposed, or whether there is a constant relation

between the colour and the refrangibility.

If Grimaldi was right, it might be expected that a part of the spectrum

taken separately, and subjected to a second refraction, would suffer a

new breaking up, and produce some new spectrum. Newton inferred from

his own theory that a particular ray of the spectrum would have a

constant refrangibility, so that a second prism would merely bend it

more or less, but not further disperse it in any considerable degree.

By simply cutting off most of the rays of the spectrum by a screen,

and allowing the remaining narrow ray to fall on a second prism, he

proved the truth of this conclusion; and then slowly turning the first

prism, so as to vary the colour of the ray falling on the second

one, he found that the spot of light formed by the twice-refracted

ray travelled up and down, a palpable proof that the amount of

refrangibility varies with the colour. For his further satisfaction,

he sometimes refracted the light a third or fourth time, and he found

that it might be refracted upwards or downwards or sideways, and yet

for each colour there was a definite amount of refraction through each

prism. He completed the proof by showing that the separated rays may

again be gathered together into white light by an inverted prism, so

that no number of refractions alters the character of the light. The

conclusion thus obtained serves to explain the confusion arising in the

use of a common lense; he shows that with homogeneous light there is

one distinct focus, with mixed light an infinite number of foci, which

prevent a clear view from being obtained at any point.

What astonishes the reader of the *Opticks* is the persistence with

which Newton follows out the consequences of a preconceived theory, and

tests the one notion by a wonderful variety of simple comparisons with

fact. The ease with which he invents new combinations, and foresees

the results, subsequently verified, produces an insuperable conviction

in the reader that he has possession of the truth. And it is certainly

the theory which leads him to the experiments, most of which could

hardly be devised by accident. Newton actually remarks that it was by

mathematically determining all kinds of phenomena of colours which

could be produced by refraction that he had “invented” almost all the

experiments in the book, and he promises that others who shall “argue

truly,” and try the experiments with care, will not be disappointed in

the results.[482]

[482] *Opticks*, bk. i. part ii. Prop. 3. 3rd ed. p. 115.

The philosophic method of Huyghens was the same as that of Newton,

and Huyghens’ investigation of double refraction furnishes almost

equally beautiful instances of theory guiding experiment. So far as

we know double refraction was first discovered by accident, and was

described by Erasmus Bartholinus in 1669. The phenomenon then appeared

to be entirely exceptional, and the laws governing the two paths of

the refracted rays were so unapparent and complicated, that Newton

altogether misunderstood the phenomenon, and it was only at the latter

end of the last century that scientific men began to comprehend its

laws.

Nevertheless, Huyghens had, with rare genius, arrived at the true

theory as early as 1678. He regarded light as an undulatory motion of

some medium, and in his *Traité de la Lumière* he pointed out that,

in ordinary refraction, the velocity of propagation of the wave is

equal in all directions, so that the front of an advancing wave is

spherical, and reaches equal distances in equal times. But in crystals,

as he supposed, the medium would be of unequal elasticity in different

directions, so that a disturbance would reach unequal distances in

equal times, and the wave produced would have a spheroidal form.

Huyghens was not satisfied with an unverified theory. He calculated

what might be expected to happen when a crystal of calc-spar was cut

in various directions, and he says: “I have examined in detail the

properties of the extraordinary refraction of this crystal, to see if

each phenomenon which is deduced from theory would agree with what is

really observed. And this being so, it is no slight proof of the truth

of our suppositions and principles; but what I am going to add here

confirms them still more wonderfully; that is, the different modes

of cutting this crystal, in which the surfaces produced give rise to

refraction exactly such as they ought to be, and as I had foreseen

them, according to the preceding theory.”

Newton’s mistaken corpuscular theory of light caused the theories and

experiments of Huyghens to be disregarded for more than a century; but

it is not easy to imagine a more beautiful or successful application of

the true method of inductive investigation, theory guiding experiment,

and yet wholly relying on experiment for confirmation.

*Candour and Courage of the Philosophic Mind.*

Perfect readiness to reject a theory inconsistent with fact is a

primary requisite of the philosophic mind. But it would be a mistake

to suppose that this candour has anything akin to fickleness; on the

contrary, readiness to reject a false theory may be combined with a

peculiar pertinacity and courage in maintaining an hypothesis as long

as its falsity is not actually apparent. There must, indeed, be no

prejudice or bias distorting the mind, and causing it to pass over the

unwelcome results of experiment. There must be that scrupulous honesty

and flexibility of mind, which assigns adequate value to all evidence;

indeed, the more a man loves his theory, the more scrupulous should be

his attention to its faults. It is common in life to meet with some

theorist, who, by long cogitation over a single theory, has allowed it

to mould his mind, and render him incapable of receiving anything but

as a contribution to the truth of his one theory. A narrow and intense

course of thought may sometimes lead to great results, but the adoption

of a wrong theory at the outset is in such a mind irretrievable.

The man of one idea has but a single chance of truth. The fertile

discoverer, on the contrary, chooses between many theories, and is

never wedded to any one, unless impartial and repeated comparison has

convinced him of its validity. He does not choose and then compare; but

he compares time after time, and then chooses.

Having once deliberately chosen, the philosopher may rightly entertain

his theory with the strongest fidelity. He will neglect no objection;

for he may chance at any time to meet a fatal one; but he will bear

in mind the inconsiderable powers of the human mind compared with the

tasks it has to undertake. He will see that no theory can at first be

reconciled with all objections, because there may be many interfering

causes, and the very consequences of the theory may have a complexity

which prolonged investigation by successive generations of men may not

exhaust. If, then, a theory exhibit a number of striking coincidences

with fact, it must not be thrown aside until at least one *conclusive

discordance* is proved, regard being had to possible error in

establishing that discordance. In science and philosophy something must

be risked. He who quails at the least difficulty will never establish a

new truth, and it was not unphilosophic in Leslie to remark concerning

his own inquiries into the nature of heat--

“In the course of investigation, I have found myself compelled to

relinquish some preconceived notions; but I have not abandoned them

hastily, nor, till after a warm and obstinate defence, I was driven

from every post.”[483]

[483] *Experimental Inquiry into the Nature of Heat.* Preface, p. xv.

Faraday’s life, again, furnishes most interesting illustrations of this

tenacity of the philosophic mind. Though so candid in rejecting some

theories, there were others to which he clung through everything. One

of his favourite notions resulted in a brilliant discovery; another

remains in doubt to the present day.

*The Philosophic Character of Faraday.*

In Faraday’s researches concerning the connection of magnetism and

light, we find an excellent instance of the pertinacity with which a

favourite theory may be pursued, so long as the results of experiment

do not clearly negative the notions entertained. In purely quantitative

questions, as we have seen, the absence of apparent effect can seldom

be regarded as proving the absence of all effect. Now Faraday was

convinced that some mutual relation must exist between magnetism and

light. As early as 1822, he attempted to produce an effect upon a ray

of polarised light, by passing it through water placed between the

poles of a voltaic battery; but he was obliged to record that not the

slightest effect was observable. During many years the subject, we are

told,[484] rose again and again to his mind, and no failure could make

him relinquish his search after this unknown relation. It was in the

year 1845 that he gained the first success; on August 30th he began

to work with common electricity, vainly trying glass, quartz, Iceland

spar, &c. Several days of labour gave no result; yet he did not desist.

Heavy glass, a transparent medium of great refractive powers, composed

of borate of lead, was now tried, being placed between the poles of a

powerful electro-magnet while a ray of polarised light was transmitted

through it. When the poles of the electro-magnet were arranged in

certain positions with regard to the substance under trial, no effects

were apparent; but at last Faraday happened fortunately to place a

piece of heavy glass so that contrary magnetic poles were on the same

side, and now an effect was witnessed. The glass was found to have the

power of twisting the plane of polarisation of the ray of light.

[484] Bence Jones, *Life of Faraday*, vol. i. p. 362.

All Faraday’s recorded thoughts upon this great experiment are replete

with curious interest. He attributes his success to the opinion,

almost amounting to a conviction, that the various forms, under which

the forces of matter are made manifest, have one common origin,

and are so directly related and mutually dependent that they are

convertible. “This strong persuasion,” he says,[485] “extended to the

powers of light, and led to many exertions having for their object

the discovery of the direct relation of light and electricity. These

ineffectual exertions could not remove my strong persuasion, and I

have at last succeeded.” He describes the phenomenon in somewhat

figurative language as *the magnetisation of a ray of light*, and

also as *the illumination of a magnetic curve or line of force*. He

has no sooner got the effect in one case, than he proceeds, with his

characteristic comprehensiveness of research, to test the existence

of a like phenomenon in all the substances available. He finds that

not only heavy glass, but solids and liquids, acids and alkalis, oils,

water, alcohol, ether, all possess this power; but he was not able to

detect its existence in any gaseous substance. His thoughts cannot be

restrained from running into curious speculations as to the possible

results of the power in certain cases. “What effect,” he says, “does

this force have in the earth where the magnetic curves of the earth

traverse its substance? Also what effect in a magnet?” And then he

falls upon the strange notion that perhaps this force tends to make

iron and oxide of iron transparent, a phenomenon never observed. We can

meet with nothing more instructive as to the course of mind by which

great discoveries are made, than these records of Faraday’s patient

labours, and his varied success and failure. Nor are his unsuccessful

experiments upon the relation of gravity and electricity less

interesting, or less worthy of study.

[485] Ibid. vol. ii. p. 199.

Throughout a large part of his life, Faraday was possessed by the idea

that gravity cannot be unconnected with the other forces of nature. On

March 19th, 1849, he wrote in his laboratory book,--“Gravity. Surely

this force must be capable of an experimental relation to electricity,

magnetism, and the other forces, so as to bind it up with them in

reciprocal action and equivalent effect?”[486] He filled twenty

paragraphs or more with reflections and suggestions, as to the mode

of treating the subject by experiment. He anticipated that the mutual

approach of two bodies would develop electricity in them, or that a

body falling through a conducting helix would excite a current changing

in direction as the motion was reversed. “*All this is a dream*,”

he remarks; “still examine it by a few experiments. Nothing is too

wonderful to be true, if it be consistent with the laws of nature;

and in such things as these, experiment is the best test of such

consistency.”

[486] See also his more formal statement in the *Experimental

Researches in Electricity*, 24th Series, § 2702, vol. iii. p. 161.

He executed many difficult and tedious experiments, which are described

in the 24th Series of Experimental Researches. The result was *nil*,

and yet he concludes: “Here end my trials for the present. The results

are negative; they do not shake my strong feeling of the existence of

a relation between gravity and electricity, though they give no proof

that such a relation exists.”

He returned to the work when he was ten years older, and in 1858–9

recorded many remarkable reflections and experiments. He was much

struck by the fact that electricity is essentially a *dual force*,

and it had always been a conviction of Faraday that no body could be

electrified positively without some other body becoming electrified

negatively; some of his researches had been simple developments of

this relation. But observing that between two mutually gravitating

bodies there was no apparent circumstance to determine which should be

positive and which negative, he does not hesitate to call in question

an old opinion. “The evolution of *one* electricity would be a new

and very remarkable thing. The idea throws a doubt on the whole; but

still try, for who knows what is possible in dealing with gravity?”

We cannot but notice the candour with which he thus acknowledges in

his laboratory book the doubtfulness of the whole thing, and is yet

prepared as a forlorn hope to frame experiments in opposition to

all his previous experience of the course of nature. For a time his

thoughts flow on as if the strange detection were already made, and he

had only to trace out its consequences throughout the universe. “Let us

encourage ourselves by a little more imagination prior to experiment,”

he says; and then he reflects upon the infinity of actions in nature,

in which the mutual relations of electricity and gravity would come

into play; he pictures to himself the planets and the comets charging

themselves as they approach the sun; cascades, rain, rising vapour,

circulating currents of the atmosphere, the fumes of a volcano, the

smoke in a chimney become so many electrical machines. A multitude of

events and changes in the atmosphere seem to be at once elucidated by

such actions; for a moment his reveries have the vividness of fact.

“I think we have been dull and blind not to have suspected some such

results,” and he sums up rapidly the consequences of his great but

imaginary theory; an entirely new mode of exciting heat or electricity,

an entirely new relation of the natural forces, an analysis of

gravitation, and a justification of the conservation of force.

Such were Faraday’s fondest dreams of what might be, and to many a

philosopher they would have been sufficient basis for the writing of

a great book. But Faraday’s imagination was within his full control;

as he himself says, “Let the imagination go, guarding it by judgment

and principle, and holding it in and directing it by experiment.” His

dreams soon took a very practical form, and for many days he laboured

with ceaseless energy, on the staircase of the Royal Institution, in

the clock tower of the Houses of Parliament, or at the top of the Shot

Tower in Southwark, raising and lowering heavy weights, and combining

electrical helices and wires in every conceivable way. His skill and

long experience in experiment were severely taxed to eliminate the

effects of the earth’s magnetism, and time after time he saved himself

from accepting mistaken indications, which to another man might have

seemed conclusive verifications of his theory. When all was done there

remained absolutely no results. “The experiments,” he says, “were well

made, but the results are negative;” and yet, he adds, “I cannot accept

them as conclusive.” In this position the question remains to the

present day; it may be that the effect was too slight to be detected,

or it may be that the arrangements adopted were not suited to develop

the particular relation which exists, just as Oersted could not detect

electro-magnetism, so long as his wire was perpendicular to the plane

of motion of his needle. But these are not matters which concern us

further here. We have only to notice the profound conviction in the

unity of natural laws, the active powers of inference and imagination,

the unbounded licence of theorising, combined above all with the utmost

diligence in experimental verification which this remarkable research

exhibits.

*Reservation of Judgment.*

There is yet another characteristic needed in the philosophic mind; it

is that of suspending judgment when the data are insufficient. Many

people will express a confident opinion on almost any question which is

put before them, but they thereby manifest not strength, but narrowness

of mind. To see all sides of a complicated subject, and to weigh all

the different facts and probabilities correctly, require no ordinary

powers of comprehension. Hence it is most frequently the philosophic

mind which is in doubt, and the ignorant mind which is ready with a

positive decision. Faraday has himself said, in a very interesting

lecture:[487] “Occasionally and frequently the exercise of the judgment

ought to end in *absolute reservation*. It may be very distasteful, and

great fatigue, to suspend a conclusion; but as we are not infallible,

so we ought to be cautious; we shall eventually find our advantage, for

the man who rests in his position is not so far from right as he who,

proceeding in a wrong direction, is ever increasing his distance.”

[487] Printed in *Modern Culture*, edited by Youmans, p. 219.

Arago presented a conspicuous example of this high quality of mind, as

Faraday remarks; for when he made known his curious discovery of the

relation of a magnetic needle to a revolving copper plate, a number

of supposed men of science in different countries gave immediate and

confident explanations of it, which were all wrong. But Arago, who

had both discovered the phenomenon and personally investigated its

conditions, declined to put forward publicly any theory at all.

At the same time we must not suppose that the truly philosophic mind

can tolerate a state of doubt, while a chance of decision remains open.

In science nothing like compromise is possible, and truth must be one.

Hence, doubt is the confession of ignorance, and involves a painful

feeling of incapacity. But doubt lies between error and truth, so that

if we choose wrongly we are further away than ever from our goal.

Summing up, then, it would seem as if the mind of the great discoverer

must combine contradictory attributes. He must be fertile in

theories and hypotheses, and yet full of facts and precise results of

experience. He must entertain the feeblest analogies, and the merest

guesses at truth, and yet he must hold them as worthless till they are

verified in experiment. When there are any grounds of probability he

must hold tenaciously to an old opinion, and yet he must be prepared

at any moment to relinquish it when a clearly contradictory fact is

encountered. “The philosopher,” says Faraday,[488] “should be a man

willing to listen to every suggestion, but determined to judge for

himself. He should not be biased by appearances; have no favourite

hypothesis; be of no school; and in doctrine have no master. He should

not be a respecter of persons, but of things. Truth should be his

primary object. If to these qualities be added industry, he may indeed

hope to walk within the veil of the temple of nature.”

[488] *Life of Faraday*, vol. i. p. 225.

BOOK V.

GENERALISATION, ANALOGY, AND CLASSIFICATION.

CHAPTER XXVII.

GENERALISATION.

I have endeavoured to show in preceding chapters that all inductive

reasoning is an inverse application of deductive reasoning, and

consists in demonstrating that the consequences of certain assumed laws

agree with facts of nature gathered by active or passive observation.

The fundamental process of reasoning, as stated in the outset, consists

in inferring of a thing what we know of similar objects, and it is on

this principle that the whole of deductive reasoning, whether simply

logical or mathematico-logical, is founded. All inductive reasoning

must be founded on the same principle. It might seem that by a plain

use of this principle we could avoid the complicated processes of

induction and deduction, and argue directly from one particular case

to another, as Mill proposed. If the Earth, Venus, Mars, Jupiter,

and other planets move in elliptic orbits, cannot we dispense with

elaborate precautions, and assert that Neptune, Ceres, and the last

discovered planet must do so likewise? Do we not know that Mr.

Gladstone must die, because he is like other men? May we not argue

that because some men die therefore he must? Is it requisite to ascend

by induction to the general proposition “all men must die,” and then

descend by deduction from that general proposition to the case of Mr.

Gladstone? My answer undoubtedly is that we must ascend to general

propositions. The fundamental principle of the substitution of similars

gives us no warrant in affirming of Mr. Gladstone what we know of

other men, because we cannot be sure that Mr. Gladstone is exactly

similar to other men. Until his death we cannot be perfectly sure that

he possesses all the attributes of other men; it is a question of

probability, and I have endeavoured to explain the mode in which the

theory of probability is applied to calculate the probability that from

a series of similar events we may infer the recurrence of like events

under identical circumstances. There is then no such process as that

of inferring from particulars to particulars. A careful analysis of

the conditions under which such an inference appears to be made, shows

that the process is really a general one, and that what is inferred of

a particular case might be inferred of all similar cases. All reasoning

is essentially general, and all science implies generalisation. In

the very birth-time of philosophy this was held to be so: “Nulla

scientia est de individuis, sed de solis universalibus,” was the

doctrine of Plato, delivered by Porphyry. And Aristotle[489] held a

like opinion--Οὐδεμία δὲ τέχνη σκοπεȋ τὸ καθ’ ἕκαστον ... τὸ δὲ καθ’

ἕκαστον ἄπειρον καὶ οὐκ ἐπιστητόν. “No art treats of particular cases;

for particulars are infinite and cannot be known.” No one who holds the

doctrine that reasoning may be from particulars to particulars, can

be supposed to have the most rudimentary notion of what constitutes

reasoning and scíence.

[489] Aristotle’s *Rhetoric*, Liber I. 2. 11.

At the same time there can be no doubt that practically what we find

to be true of many similar objects will probably be true of the next

similar object. This is the result to which an analysis of the Inverse

Method of Probabilities leads us, and, in the absence of precise data

from which we may calculate probabilities, we are usually obliged to

make a rough assumption that similars in some respects are similars

in other respects. Thus it comes to pass that a large part of the

reasoning processes in which scientific men are engaged, consists in

detecting similarities between objects, and then rudely assuming that

the like similarities will be detected in other cases.

*Distinction of Generalisation and Analogy.*

There is no distinction but that of degree between what is known as

reasoning by *generalisation* and reasoning by *analogy*. In both

cases from certain observed resemblances we infer, with more or less

probability, the existence of other resemblances. In generalisation

the resemblances have great extension and usually little intension,

whereas in analogy we rely upon the great intension, the extension

being of small amount (p. 26). If we find that the qualities A and B

are associated together in a great many instances, and have never been

found separate, it is highly probable that on the next occasion when we

meet with A, B will also be present, and *vice versâ*. Thus wherever we

meet with an object possessing gravity, it is found to possess inertia

also, nor have we met with any material objects possessing inertia

without discovering that they also possess gravity. The probability

has therefore become very great, as indicated by the rules founded on

the Inverse Method of Probabilities (p. 257), that whenever in the

future we meet an object possessing either of the properties of gravity

and inertia, it will be found on examination to possess the other

of these properties. This is a clear instance of the employment of

generalisation.

In analogy, on the other hand, we reason from likeness in many points

to likeness in other points. The qualities or points of resemblance are

now numerous, not the objects. At the poles of Mars are two white spots

which resemble in many respects the white regions of ice and snow at

the poles of the earth. There probably exist no other similar objects

with which to compare these, yet the exactness of the resemblance

enables us to infer, with high probability, that the spots on Mars

consist of ice and snow. In short, many points of resemblance imply

many more. From the appearance and behaviour of those white spots we

infer that they have all the chemical and physical properties of

frozen water. The inference is of course only probable, and based upon

the improbability that aggregates of many qualities should be formed in

a like manner in two or more cases, without being due to some uniform

condition or cause.

In reasoning by analogy, then, we observe that two objects ABCDE....

and A′B′C′D′E′.... have many like qualities, as indicated by the

identity of the letters, and we infer that, since the first has another

quality, X, we shall discover this quality in the second case by

sufficiently close examination. As Laplace says,--“Analogy is founded

on the probability that similar things have causes of the same kind,

and produce the same effects. The more perfect this similarity, the

greater is this probability.”[490] The nature of analogical inference

is aptly described in the work on Logic attributed to Kant, where the

rule of ordinary induction is stated in the words, “*Eines in vielen,

also in allen*,” one quality in many things, therefore in all; and

the rule of analogy is “*Vieles in einem, also auch das übrige in

demselben*,”[491] many (qualities) in one, therefore also the remainder

in the same. It is evident that there may be intermediate cases in

which, from the identity of a moderate number of objects in several

properties, we may infer to other objects. Probability must rest either

upon the number of instances or the depth of resemblance, or upon the

occurrence of both in sufficient degrees. What there is wanting in

extension must be made up by intension, and *vice versâ*.

[490] *Essai Philosophique sur les Probabilités*, p. 86.

[491] Kant’s *Logik*, § 84, Königsberg, 1800, p. 207.

*Two Meanings of Generalisation.*

The term generalisation, as commonly used, includes two processes which

are of different character, but are often closely associated together.

In the first place, we generalise when we recognise even in two objects

a common nature. We cannot detect the slightest similarity without

opening the way to inference from one case to the other. If we compare

a cubical crystal with a regular octahedron, there is little apparent

similarity; but, as soon as we perceive that either can be produced

by the symmetrical modification of the other, we discover a groundwork

of similarity in the crystals, which enables us to infer many things

of one, because they are true of the other. Our knowledge of ozone

took its rise from the time when the similarity of smell, attending

electric sparks, strokes of lightning, and the slow combustion of

phosphorus, was noticed by Schönbein. There was a time when the rainbow

was an inexplicable phenomenon--a portent, like a comet, and a cause of

superstitious hopes and fears. But we find the true spirit of science

in Roger Bacon, who desires us to consider the objects which present

the same colours as the rainbow; he mentions hexagonal crystals from

Ireland and India, but he bids us not suppose that the hexagonal form

is essential, for similar colours may be detected in many transparent

stones. Drops of water scattered by the oar in the sun, the spray from

a water-wheel, the dewdrops lying on the grass in the summer morning,

all display a similar phenomenon. No sooner have we grouped together

these apparently diverse instances, than we have begun to generalise,

and have acquired a power of applying to one instance what we can

detect of others. Even when we do not apply the knowledge gained to

new objects, our comprehension of those already observed is greatly

strengthened and deepened by learning to view them as particular cases

of a more general property.

A second process, to which the name of generalisation is often

given, consists in passing from a fact or partial law to a multitude

of unexamined cases, which we believe to be subject to the same

conditions. Instead of merely recognising similarity as it is brought

before us, we predict its existence before our senses can detect it, so

that generalisation of this kind endows us with a prophetic power of

more or less probability. Having observed that many substances assume,

like water and mercury, the three states of solid, liquid, and gas, and

having assured ourselves by frequent trial that the greater the means

we possess of heating and cooling, the more substances we can vaporise

and freeze, we pass confidently in advance of fact, and assume that

all substances are capable of these three forms. Such a generalisation

was accepted by Lavoisier and Laplace before many of the corroborative

facts now in our possession were known. The reduction of a single

comet beneath the sway of gravity was considered sufficient indication

that all comets obey the same power. Few persons doubted that the law

of gravity extended over the whole heavens; certainly the fact that

a few stars out of many millions manifest the action of gravity, is

now held to be sufficient evidence of its general extension over the

visible universe.

*Value of Generalisation.*

It might seem that if we know particular facts, there can be little

use in connecting them together by a general law. The particulars

must be more full of useful information than an abstract general

statement. If we know, for instance, the properties of an ellipse, a

circle, a parabola, and hyperbola, what is the use of learning all

these properties over again in the general theory of curves of the

second degree? If we understand the phenomena of sound and light and

water-waves separately, what is the need of erecting a general theory

of waves, which, after all, is inapplicable to practice until resolved

again into particular cases? But, in reality, we never do obtain an

adequate knowledge of particulars until we regard them as cases of

the general. Not only is there a singular delight in discovering the

many in the one, and the one in the many, but there is a constant

interchange of light and knowledge. Properties which are unapparent

in the hyperbola may be readily observed in the ellipse. Most of the

complex relations which old geometers discovered in the circle will

be reproduced *mutatis mutandis* in the other conic sections. The

undulatory theory of light might have been unknown at the present day,

had not the theory of sound supplied hints by analogy. The study of

light has made known many phenomena of interference and polarisation,

the existence of which had hardly been suspected in the case of

sound, but which may now be sought out, and perhaps found to possess

unexpected interest. The careful study of water-waves shows how waves

alter in form and velocity with varying depth of water. Analogous

changes may some time be detected in sound waves. Thus there is mutual

interchange of aid.

“Every study of a generalisation or extension,” De Morgan has well

said,[492] “gives additional power over the particular form by which

the generalisation is suggested. Nobody who has ever returned to

quadratic equations after the study of equations of all degrees,

or who has done the like, will deny my assertion that οὐ βλέπει

βλέπων may be predicated of any one who studies a branch or a case,

without afterwards making it part of a larger whole. Accordingly

it is always worth while to generalise, were it only to give power

over the *particular*. This principle, of daily familiarity to the

mathematician, is almost unknown to the logician.”

[492] *Syllabus of a Proposed System of Logic*, p. 34.

*Comparative Generality of Properties.*

Much of the value of science depends upon the knowledge which we

gradually acquire of the different degrees of generality of properties

and phenomena of various kinds. The use of science consists in enabling

us to act with confidence, because we can foresee the result. Now this

foresight must rest upon the knowledge of the powers which will come

into play. That knowledge, indeed, can never be certain, because it

rests upon imperfect induction, and the most confident beliefs and

predictions of the physicist may be falsified. Nevertheless, if we

always estimate the probability of each belief according to the due

teaching of the data, and bear in mind that probability when forming

our anticipations, we shall ensure the minimum of disappointment. Even

when he cannot exactly apply the theory of probabilities, the physicist

may acquire the habit of making judgments in general agreement with its

principles and results.

Such is the constitution of nature, that the physicist learns to

distinguish those properties which have wide and uniform extension,

from those which vary between case and case. Not only are certain laws

distinctly laid down, with their extension carefully defined, but a

scientific training gives a kind of tact in judging how far other laws

are likely to apply under any particular circumstances. We learn by

degrees that crystals exhibit phenomena depending upon the directions

of the axes of elasticity, which we must not expect in uniform solids.

Liquids, compared even with non-crystalline solids, exhibit laws of

far less complexity and variety; and gases assume, in many respects,

an aspect of nearly complete uniformity. To trace out the branches of

science in which varying degrees of generality prevail, would be an

inquiry of great interest and importance; but want of space, if there

were no other reason, would forbid me to attempt it, except in a very

slight manner.

Gases, so far as they are really gaseous, not only have exactly the

same properties in all directions of space, but one gas exactly

resembles other gases in many qualities. All gases expand by heat,

according to the same law, and by nearly the same amount; the specific

heats of equivalent weights are equal, and the densities are exactly

proportional to the atomic weights. All such gases obey the general

law, that the volume multiplied by the pressure, and divided by the

absolute temperature, is constant or nearly so. The laws of diffusion

and transpiration are the same in all cases, and, generally speaking,

all physical laws, as distinguished from chemical laws, apply

equally to all gases. Even when gases differ in chemical or physical

properties, the differences are minor in degree. Thus the differences

of viscosity are far less marked than in the liquid and solid states.

Nearly all gases, again, are colourless, the exceptions being chlorine,

the vapours of iodine, bromine, and a few other substances.

Only in one single point, so far as I am aware, do gases present

distinguishing marks unknown or nearly so, in the solid and liquid

states. I mean as regards the light given off when incandescent.

Each gas when sufficiently heated, yields its own peculiar series

of rays, arising from the free vibrations of the constituent parts

of the molecules. Hence the possibility of distinguishing gases by

the spectroscope. But the molecules of solids and liquids appear to

be continually in conflict with each other, so that only a confused

*noise* of atoms is produced, instead of a definite series of luminous

chords. At the same temperature, accordingly, all solids and liquids

give off nearly the same rays when strongly heated, and we have in this

case an exception to the greater generality of properties in gases.

Liquids are in many ways intermediate in character between gases

and solids. While incapable of possessing different elasticity in

different directions, and thus denuded of the rich geometrical

complexity of solids, they retain the variety of density, colour

degrees of transparency, great diversity in surface tension, viscosity,

coefficients of expansion, compressibility, and many other properties

which we observe in solids, but not for the most part in gases. Though

our knowledge of the physical properties of liquids is much wanting in

generality at present, there is ground to hope that by degrees laws

connecting and explaining the variations may be traced out.

Solids are in every way contrasted to gases. Each solid substance

has its own peculiar degree of density, hardness, compressibility,

transparency, tenacity, elasticity, power of conducting heat and

electricity, magnetic properties, capability of producing frictional

electricity, and so forth. Even different specimens of the same kind

of substance will differ widely, according to the accidental treatment

received. And not only has each substance its own specific properties,

but, when crystallised, its properties vary in each direction with

regard to the axes of crystallisation. The velocity of radiation, the

rate of conduction of heat, the coefficients of expansibility and

compressibility, the thermo-electric properties, all vary in different

crystallographic directions.

It is probable that many apparent differences between liquids, and

even between solids, will be explained when we learn to regard them

under exactly corresponding circumstances. The extreme generality of

the properties of gases is in reality only true at an infinitely high

temperature, when they are all equally remote from their condensing

points. Now, it is found that if we compare liquids--for instance,

different kinds of alcohols--not at equal temperatures, but at points

equally distant from their respective boiling points, the laws and

coefficients of expansion are nearly equal. The vapour-tensions of

liquids also are more nearly equal, when compared at corresponding

points, and the boiling-points appear in many cases to be simply

related to the chemical composition. No doubt the progress of

investigation will enable us to discover generality, where at present

we only see variety and puzzling complexity.

In some cases substances exhibit the same physical properties in

the liquid as in the solid state. Lead has a high refractive power,

whether in solution, or in solid salts, crystallised or vitreous.

The magnetic power of iron is conspicuous, whatever be its chemical

condition; indeed, the magnetic properties of substances, though

varying with temperature, seem not to be greatly affected by other

physical changes. Colour, absorptive power for heat or light rays,

and a few other properties are also often the same in liquids and

gases. Iodine and bromine possess a deep colour whenever they are

chemically uncombined. Nevertheless, we can seldom argue safely from

the properties of a substance in one condition to those in another

condition. Ice is an insulator, water a conductor of electricity, and

the same contrast exists in most other substances. The conducting power

of a liquid for electricity increases with the temperature, while that

of a solid decreases. By degrees we may learn to distinguish between

those properties of matter which depend upon the intimate construction

of the chemical molecule, and those which depend upon the contact,

conflict, mutual attraction, or other relations of distinct molecules.

The properties of a substance with respect to light seem generally to

depend upon the molecule; thus, the power of certain substances to

cause the plane of polarisation of a ray of light to rotate, is exactly

the same whatever be its degree of density, or the diluteness of the

solution in which it is contained. Taken as a whole, the physical

properties of substances and their quantitative laws, present a problem

of infinite complexity, and centuries must elapse before any moderately

complete generalisations on the subject become possible.

*Uniform Properties of all Matter.*

Some laws are held to be true of all matter in the universe absolutely,

without exception, no instance to the contrary having ever been

noticed. This is the case with the laws of motion, as laid down

by Galileo and Newton. It is also conspicuously true of the law

of universal gravitation. The rise of modern physical science may

perhaps be considered as beginning at the time when Galileo showed,

in opposition to the Aristotelians, that matter is equally affected

by gravity, irrespective of its form, magnitude, or texture. All

objects fall with equal rapidity, when disturbing causes, such as the

resistance of the air, are removed or allowed for. That which was

rudely demonstrated by Galileo from the leaning tower of Pisa, was

proved by Newton to a high degree of approximation, in an experiment

which has been mentioned (p. 443).

Newton formed two pendulums, as nearly as possible the same in outward

shape and size by taking two equal round wooden boxes, and suspending

them by equal threads, eleven feet long. The pendulums were therefore

equally subject to the resistance of the air. He filled one box with

wood, and in the centre of oscillation of the other he placed an equal

weight of gold. The pendulums were then equal in weight as well as in

size; and, on setting them simultaneously in motion, Newton found that

they vibrated for a length of time with equal vibrations. He tried the

same experiment with silver, lead, glass, sand, common salt, water,

and wheat, in place of the gold, and ascertained that the motion of

his pendulum was exactly the same whatever was the kind of matter

inside.[493] He considered that a difference of a thousandth part

would have been apparent. The reader must observe that the pendulums

were made of equal weight only in order that they might suffer equal

retardation from the air. The meaning of the experiment is that all

substances manifest exactly equal acceleration from the force of

gravity, and that therefore the inertia or resistance of matter to

force, which is the only independent measure of mass known to us, is

always proportional to gravity.

[493] *Principia*, bk. iii. Prop. VI. Motte’s translation, vol. ii.

p. 220.

These experiments of Newton were considered conclusive up to very

recent times, when certain discordances between the theory and

observations of the movements of planets led Nicolai, in 1826, to

suggest that the equal gravitation of different kinds of matter might

not be absolutely exact. It is perfectly philosophical thus to call

in question, from time to time, some of the best accepted laws. On

this occasion Bessel carefully repeated the experiments of Newton with

pendulums composed of ivory, glass, marble, quartz, meteoric stones,

&c., but was unable to detect the least difference. This conclusion

is also confirmed by the ultimate agreement of all the calculations

of physical astronomy based upon it. Whether the mass of Jupiter be

calculated from the motion of its own satellites, from the effect

upon the small planets, Vesta, Juno, &c., or from the perturbation

of Encke’s Comet, the results are closely accordant, showing that

precisely the same law of gravity applies to the most different bodies

which we can observe. The gravity of a body, again, appears to be

entirely independent of its other physical conditions, being totally

unaffected by any alteration in the temperature, density, electric or

magnetic condition, or other physical properties of the substance.

One paradoxical result of the law of equal gravitation is the theorem

of Torricelli, to the effect that all liquids of whatever density

fall or flow with equal rapidity. If there be two equal cisterns

respectively filled with mercury and water, the mercury, though

thirteen times as heavy, would flow from an aperture neither more

rapidly nor more slowly than the water, and the same would be true of

ether, alcohol, and other liquids, allowance being made, however, for

the resistance of the air, and the differing viscosities of the liquids.

In its exact equality and its perfect independence of all

circumstances, except mass and distance, the force of gravity stands

apart from all the other forces and phenomena of nature, and has

not yet been brought into any relation with them except through the

general principle of the conservation of energy. Magnetic attraction,

as remarked by Newton, follows very different laws, depending upon the

chemical quality and molecular structure of each particular substance.

We must remember that in saying “all matter gravitates,” we exclude

from the term matter the basis of light-undulations, which is

immensely more extensive in amount, and obeys in many respects the

laws of mechanics. This adamantine substance appears, so far as can be

ascertained, to be perfectly uniform in its properties when existing

in space unoccupied by matter. Light and heat are conveyed by it with

equal velocity in all directions, and in all parts of space so far as

observation informs us. But the presence of gravitating matter modifies

the density and mechanical properties of the so-called ether in a way

which is yet quite unexplained.[494]

Leaving gravity, it is somewhat difficult to discover other laws

which are equally true of all matter. Boerhaave was considered to

have established that all bodies expand by heat; but not only is the

expansion very different in different substances, but we now know

positive exceptions. Many liquids and a few solids contract by heat

at certain temperatures. There are indeed other relations of heat to

matter which seem to be universal and uniform; all substances begin

to give off rays of light at the same temperature, according to the

law of Draper; and gases will not be an exception if sufficiently

condensed, as in the experiments of Frankland. Grove considers it to

be universally true that all bodies in combining produce heat; with

the doubtful exception of sulphur and selenium, all solids in becoming

liquids, and all liquids in becoming gases, absorb heat; but the

quantities of heat absorbed vary with the chemical qualities of the

matter. Carnot’s Thermodynamic Law is held to be exactly true of all

matter without distinction; it expresses the fact that the amount of

mechanical energy which might be theoretically obtained from a certain

amount of heat energy depends only upon the change of the temperatures,

so that whether an engine be worked by water, air, alcohol, ammonia, or

any other substance, the result would theoretically be the same, if the

boiler and condenser were maintained at similar temperatures.

[494] Professor Lovering has pointed out how obscure and uncertain

the ideas of scientific men about this ether are, in his interesting

Presidential Address before the American Association at Hartford,

1874. *Silliman’s Journal*, October 1874, p. 297. *Philosophical

Magazine*, vol. xlviii. p. 493.

*Variable Properties of Matter.*

I have enumerated some of the few properties of matter, which are

manifested in exactly the same manner by all substances, whatever be

their differences of chemical or physical constitution. But by far

the greater number of qualities vary in degree; substances are more

or less dense, more or less transparent, more or less compressible,

more or less magnetic, and so on. One common result of the progress of

science is to show that qualities supposed to be entirely absent from

many substances are present only in so low a degree of intensity that

the means of detection were insufficient. Newton believed that most

bodies were quite unaffected by the magnet; Faraday and Tyndall have

rendered it very doubtful whether any substance whatever is wholly

devoid of magnetism, including under that term diamagnetism. We are

rapidly learning to believe that there are no substances absolutely

opaque, or non-conducting, non-electric, non-elastic, non-viscous,

non-compressible, insoluble, infusible, or non-volatile. All tends

to become a matter of degree, or sometimes of direction. There may

be some substances oppositely affected to others, as ferro-magnetic

substances are oppositely affected to diamagnetics, or as substances

which contract by heat are opposed to those which expand; but the

tendency is certainly for every affection of one kind of matter to be

represented by something similar in other kinds. On this account one of

Newton’s rules of philosophising seems to lose all validity; he said,

“Those qualities of bodies which are not capable of being heightened,

and remitted, and which are found in all bodies on which experiment can

be made, must be considered as universal qualities of all bodies.” As

far as I can see, the contrary is more probable, namely, that qualities

variable in degree will be found in every substance in a greater or

less degree.

It is remarkable that Newton whose method of investigation was

logically perfect, seemed incapable of generalising and describing

his own procedure. His celebrated “Rules of Reasoning in Philosophy,”

described at the commencement of the third book of the *Principia*, are

of questionable truth, and still more questionable value.

*Extreme Instances of Properties.*

Although substances usually differ only in degree, great interest

may attach to particular substances which manifest a property in a

conspicuous and intense manner. Every branch of physical science

has usually been developed from the attention forcibly drawn to some

singular substance. Just as the loadstone disclosed magnetism and

amber frictional electricity, so did Iceland spar show the existence

of double refraction, and sulphate of quinine the phenomenon of

fluorescence. When one such startling instance has drawn the attention

of the scientific world, numerous less remarkable cases of the

phenomenon will be detected, and it will probably prove that the

property in question is actually universal to all matter. Nevertheless,

the extreme instances retain their interest, partly in a historical

point of view, partly because they furnish the most convenient

substances for experiment.

Francis Bacon was fully aware of the value of such examples, which he

called *Ostensive Instances* or Light-giving, Free and Predominant

Instances. “They are those,” he says,[495] “which show the nature

under investigation naked, in an exalted condition, or in the highest

degree of power; freed from impediments, or at least by its strength

predominating over and suppressing them.” He mentions quicksilver as

an ostensive instance of weight or density, thinking it not much less

dense than gold, and more remarkable than gold as joining density

to liquidity. The magnet is mentioned as an ostensive instance of

attraction. It would not be easy to distinguish clearly between these

ostensive instances and those which he calls *Instantiae Monodicae*,

or *Irregulares*, or *Heteroclitae*, under which he places whatever

is extravagant in its properties or magnitude, or exhibits least

similarity to other things, such as the sun and moon among the heavenly

bodies, the elephant among animals, the letter *s* among letters, or

the magnet among stones.[496]

[495] *Novum Organum*, bk. ii. Aphorisms, 24, 25.

[496] Ibid. Aph. 28.

In optical science great use has been made of the high dispersive

power of the transparent compounds of lead, that is, the power of

giving a long spectrum (p. 432). Dollond, having noticed this peculiar

dispersive power in lenses made of flint glass, employed them to

produce an achromatic arrangement. The element strontium presents a

contrast to lead in this respect, being characterised by a remarkably

low dispersive power; but I am not aware that this property has yet

been turned to account.

Compounds of lead have both a high dispersive and a high refractive

index, and in the latter respect they proved very useful to Faraday.

Having spent much labour in preparing various kinds of optical glass,

Faraday happened to form a compound of lead, silica, and boracic

acid, now known as *heavy glass*, which possessed an intensely high

refracting power. Many years afterwards in attempting to discover the

action of magnetism upon light he failed to detect any effect, as has

been already mentioned, (p. 588), until he happened to test a piece of

the heavy glass. The peculiar refractive power of this medium caused

the magnetic strain to be apparent, and the rotation of the plane of

polarisation was discovered.

In almost every part of physical science there is some substance

of powers pre-eminent for the special purpose to which it is put.

Rock-salt is invaluable for its extreme diathermancy or transparency to

the least refrangible rays of the spectrum. Quartz is equally valuable

for its transparency, as regards the ultra-violet or most refrangible

rays. Diamond is the most highly refracting substance which is at the

same time transparent; were it more abundant and easily worked it would

be of great optical importance. Cinnabar is distinguished by possessing

a power of rotating the plane of polarisation of light, from 15 to 17

times as much as quartz. In electric experiments copper is employed for

its high conducting powers and exceedingly low magnetic properties;

iron is of course indispensable for its enormous magnetic powers; while

bismuth holds a like place as regards its diamagnetic powers, and was

of much importance in Tyndall’s decisive researches upon the polar

character of the diamagnetic force.[497] In regard to magne-crystallic

action the mineral cyanite is highly remarkable, being so powerfully

affected by the earth’s magnetism, that, when delicately suspended, it

assumes a constant position with regard to the magnetic meridian, and

may almost be used like the compass needle. Sodium is distinguished

by its unique light-giving powers, which are so extraordinary that

probably one half of the whole number of stars in the heavens have a

yellow tinge in consequence.

[497] *Philosophical Transactions* (1856) vol. cxlvi. p. 246.

It is remarkable that water, though the most common of all fluids,

is distinguished in almost every respect by extreme qualities. Of

all known substances water has the highest specific heat, being thus

peculiarly fitted for the purpose of warming and cooling, to which

it is often put. It rises by capillary attraction to a height more

than twice that of any other liquid. In the state of ice it is nearly

twice as dilatable by heat as any other known solid substance.[498] In

proportion to its density it has a far higher surface tension than any

other substance, being surpassed in absolute tension only by mercury;

and it would not be difficult to extend considerably the list of its

remarkable and useful properties.

[498] *Philosophical Magazine*, 4th Series, January 1870, vol. xxxix.

p. 2.

Under extreme instances we may include cases of remarkably low powers

or qualities. Such cases seem to correspond to what Bacon calls

*Clandestine Instances*, which exhibit a given nature in the least

intensity, and as it were in a rudimentary state.[499] They may often

be important, he thinks, as allowing the detection of the cause of

the property by difference. I may add that in some cases they may be

of use in experiments. Thus hydrogen is the least dense of all known

substances, and has the least atomic weight. Liquefied nitrous oxide

has the lowest refractive index of all known fluids.[500] The compounds

of strontium have the lowest dispersive power. It is obvious that

a property of very low degree may prove as curious and valuable a

phenomenon as a property of very high degree.

[499] *Novum Organum*, bk. ii. Aphorism 25.

[500] Faraday’s *Experimental Researches in Chemistry and Physics*,

p. 93.

*The Detection of Continuity.*

We should bear in mind that phenomena which are in reality of a

closely similar or even identical nature, may present to the senses

very different appearances. Without a careful analysis of the changes

which take place, we may often be in danger of widely separating facts

and processes, which are actually instances of the same law. Extreme

difference of degree or magnitude is a frequent cause of error. It is

truly difficult at the first moment to recognise any similarity between

the gradual rusting of a piece of iron, and the rapid combustion of a

heap of straw. Yet Lavoisier’s chemical theory was founded upon the

similarity of the oxydising process in one case and the other. We have

only to divide the iron into excessively small particles to discover

that it is really the more combustible of the two, and that it actually

takes fire spontaneously and burns like tinder. It is the excessive

slowness of the process in the case of a massive piece of iron which

disguises its real character.

If Xenophon reports truly, Socrates was misled by not making sufficient

allowance for extreme differences of degree and quantity. Anaxagoras

held that the sun is a fire, but Socrates rejected this opinion, on the

ground that we can look at a fire, but not at the sun, and that plants

grow by sunshine while they are killed by fire. He also pointed out

that a stone heated in a fire is not luminous, and soon cools, whereas

the sun ever remains equally luminous and hot.[501] All such mistakes

evidently arise from not perceiving that difference of quantity may be

so extreme as to assume the appearance of difference of quality. It is

the least creditable thing we know of Socrates, that after pointing

out these supposed mistakes of earlier philosophers, he advised his

followers not to study astronomy.

[501] *Memorabilia*, iv. 7.

Masses of matter of very different size may be expected to exhibit

apparent differences of conduct, arising from the various intensity of

the forces brought into play. Many persons have thought it requisite

to imagine occult forces producing the suspension of the clouds, and

there have even been absurd theories representing cloud particles as

minute water-balloons buoyed up by the warm air within them. But we

have only to take proper account of the enormous comparative resistance

which the air opposes to the fall of minute particles, to see that

all cloud particles are probably constantly falling through the air,

but so slowly that there is no apparent effect. Mineral matter again

is always regarded as inert and incapable of spontaneous movement. We

are struck by astonishment on observing in a powerful microscope, that

every kind of solid matter suspended in extremely minute particles

in pure water, acquires an oscillatory movement, often so marked as

to resemble dancing or skipping. I conceive that this movement is due

to the comparatively vast intensity of chemical action when exerted

upon minute particles, the effect being 5,000 or 10,000 greater in

proportion to the mass than in fragments of an inch diameter (p. 406).

Much that was formerly obscure in the science of electricity arose from

the extreme differences of intensity and quantity in which this form of

energy manifests itself. Between the brilliant explosive discharge of a

thunder-cloud and the gentle continuous current produced by two pieces

of metal and some dilute acid, there is no apparent analogy whatever.

It was therefore a work of great importance when Faraday demonstrated

the identity of the forces in action, showing that common frictional

electricity would decompose water like that from the voltaic battery.

The relation of the phenomena became plain when he succeeded in showing

that it would require 800,000 discharges of his large Leyden battery

to decompose one single grain of water. Lightning was now seen to be

electricity of excessively high tension, but extremely small quantity,

the difference being somewhat analogous to that between the force of

one million gallons of water falling through one foot, and one gallon

of water falling through one million feet. Faraday estimated that one

grain of water acting on four grains of zinc, would yield electricity

enough for a great thunderstorm.

It was long believed that electrical conductors and insulators belonged

to two opposed classes of substances. Between the inconceivable

rapidity with which the current passes through pure copper wire,

and the apparently complete manner in which it is stopped by a

thin partition of gutta-percha or gum-lac, there seemed to be no

resemblance. Faraday again laboured successfully to show that these

were but the extreme cases of a chain of substances varying in all

degrees in their powers of conduction. Even the best conductors, such

as pure copper or silver, offer resistance to the electric current. The

other metals have considerably higher powers of resistance, and we pass

gradually down through oxides and sulphides. The best insulators, on

the other hand, allow of an atomic induction which is the necessary

antecedent of conduction. Hence Faraday inferred that whether we can

measure the effect or not, all substances discharge electricity more

or less.[502] One consequence of this doctrine must be, that every

discharge of electricity produces an induced current. In the case of

the common galvanic current we can readily detect the induced current

in any parallel wire or other neighbouring conductor, and can separate

the opposite currents which arise at the moments when the original

current begins and ends. But a discharge of high tension electricity

like lightning, though it certainly occupies time and has a beginning

and an end, yet lasts so minute a fraction of a second, that it would

be hopeless to attempt to detect and separate the two opposite induced

currents, which are nearly simultaneous and exactly neutralise each

other. Thus an apparent failure of analogy is explained away, and

we are furnished with another instance of a phenomenon incapable of

observation and yet theoretically known to exist.[503]

[502] *Experimental Researches in Electricity*, Series xii. vol. i.

p. 420.

[503] *Life of Faraday*, vol. ii. p. 7.

Perhaps the most extraordinary case of the detection of unsuspected

continuity is found in the discovery of Cagniard de la Tour and

Professor Andrews, that the liquid and gaseous conditions of matter

are only remote points in a continuous course of change. Nothing is

at first sight more apparently distinct than the physical condition

of water and aqueous vapour. At the boiling-point there is an

entire breach of continuity, and the gas produced is subject to

laws incomparably more simple than the liquid from which it arose.

But Cagniard de la Tour showed that if we maintain a liquid under

sufficient pressure its boiling point may be indefinitely raised, and

yet the liquid will ultimately assume the gaseous condition with but

a small increase of volume. Professor Andrews, recently following out

this course of inquiry, has shown that liquid carbonic acid may, at

a particular temperature (30°·92 C.), and under the pressure of 74

atmospheres, be at the same time in a state indistinguishable from

that of liquid and gas. At higher pressures carbonic acid may be

made to pass from a palpably liquid state to a truly gaseous state

without any abrupt change whatever. As the pressure is greater the

abruptness of the change from liquid to gas gradually decreases, and

finally vanishes. Similar phenomena or an approximation to them have

been observed in other liquids, and there is little doubt that we may

make a wide generalisation, and assert that, under adequate pressure,

every liquid might be made to pass into a gas without breach of

continuity.[504] The liquid state, moreover, is considered by Professor

Andrews to be but an intermediate step between the solid and gaseous

conditions. There are various indications that the process of melting

is not perfectly abrupt; and could experiments be made under adequate

pressures, it is believed that every solid could be made to pass by

insensible degrees into the state of liquid, and subsequently into that

of gas.

[504] *Nature*, vol. ii. p. 278.

These discoveries appear to open the way to most important and

fundamental generalisations, but it is probable that in many other

cases phenomena now regarded as discrete may be shown to be different

degrees of the same process. Graham was of opinion that chemical

affinity differs but in degree from the ordinary attraction which

holds different particles of a body together. He found that sulphuric

acid continued to evolve heat when mixed even with the fiftieth

equivalent of water, so that there seemed to be no distinct limit to

chemical affinity. He concludes, “There is reason to believe that

chemical affinity passes in its lowest degree into the attraction of

aggregation.”[505]

[505] *Journal of the Chemical Society*, vol. viii. p. 51.

The atomic theory is well established, but its limits are not marked

out. As Grove points out, we may by selecting sufficiently high

multipliers express any combination or mixture of elements in terms

of their equivalent weights.[506] Sir W. Thomson has suggested that

the power which vegetable fibre, oatmeal, and other substances possess

of attracting and condensing aqueous vapour is probably continuous,

or, in fact, identical with capillary attraction, which is capable

of interfering with the pressure of aqueous vapour and aiding its

condensation.[507] There are many cases of so-called catalytic or

surface action, such as the extraordinary power of animal charcoal

for attracting organic matter, or of spongy platinum for condensing

hydrogen, which can only be considered as exalted cases of a more

general power of attraction. The number of substances which are

decomposed by light in a striking manner is very limited; but many

other substances, such as vegetable colours, are affected by long

exposure; on the principle of continuity we might expect to find that

all kinds of matter are more or less susceptible of change by the

incidence of light rays.[508] It is the opinion of Grove that wherever

an electric current passes there is a tendency to decomposition, a

strain on the molecules, which when sufficiently intense leads to

disruption. Even a metallic conducting wire may be regarded as tending

to decomposition. Davy was probably correct in describing electricity

as chemical affinity acting on masses, or rather, as Grove suggests,

creating a disturbance through a chain of particles.[509] Laplace went

so far as to suggest that all chemical phenomena may be results of

the Newtonian law of attraction, applied to atoms of various mass and

position; but the time is probably far distant when the progress of

molecular philosophy and of mathematical methods will enable such a

generalisation to be verified or refuted.

[506] *Correlation of Physical Forces*, 3rd edit. p. 184.

[507] *Philosophical Magazine*, 4th Series, vol. xlii. p. 451.

[508] Grove, *Correlation of Physical Forces*, 3rd edit. p. 118.

[509] Ibid. pp. 166, 199, &c.

*The Law of Continuity.*

Under the title of the Law of Continuity we may place many applications

of the general principle of reasoning, that what is true of one case

will be true of similar cases, and probably true of what are probably

similar. Whenever we find that a law or similarity is rigorously

fulfilled up to a certain point in time or space, we expect with a high

degree of probability that it will continue to be fulfilled at least a

little further. If we see part only of a circle, we naturally expect

that the circular form will be continued in the part hidden from us. If

a body has moved uniformly over a certain space, we expect that it will

continue to move uniformly. The ground of such inferences is doubtless

identical with that of other inductive inferences. In continuous

motion every infinitely small space passed over constitutes a separate

constituent fact, and had we perfect powers of observation the smallest

finite motion would include an infinity of information, which, by the

principles of the inverse method of probabilities, would enable us to

infer with certainty to the next infinitely small portion of the path.

But when we attempt to infer from one finite portion of a path to

another finite portion, inference will be only more or less probable,

according to the comparative lengths of the portions and the accuracy

of observation; the longer our experience is, the more probable our

inference will be; the greater the length of time or space over which

the inference extends, the less probable.

This principle of continuity presents itself in nature in a great

variety of forms and cases. It is familiarly expressed in the dictum

*Natura non agit per saltum*. As Graham expressed the maxim, there are

in nature no abrupt transitions, and the distinctions of class are

never absolute.[510] There is always some notice--some forewarning

of every phenomenon, and every change begins by insensible degrees,

could we observe it with perfect accuracy. The cannon ball, indeed, is

forced from the cannon in an inappreciable portion of time; the trigger

is pulled, the fuze fired, the powder inflamed, the ball expelled,

all simultaneously to our senses. But there is no doubt that time is

occupied by every part of the process, and that the ball begins to

move at first with infinite slowness. Captain Noble is able to measure

by his chronoscope the progress of the shot in a 300-pounder gun, and

finds that the whole motion within the barrel takes place in something

less than one 200th part of a second. It is certain that no finite

force can produce motion, except in a finite space of time. The amount

of momentum communicated to a body is proportional to the accelerating

force multiplied by the time during which it acts uniformly. Thus a

slight force produces a great velocity only by long-continued action.

In a powerful shock, like that of a railway collision, the stroke of a

hammer on an anvil, or the discharge of a gun, the time is very short,

and therefore the accelerating forces brought into play are exceedingly

great, but never infinite. In the case of a large gun the powder in

exploding is said to exert for a moment a force equivalent to at least

2,800,000 horses.

[510] *Philosophical Transactions*, 1861. *Chemical and Physical

Researches*, p. 598.

Our belief in some of the fundamental laws of nature rests upon the

principle of continuity. Galileo is held to be the first philosopher

who consciously employed this principle in his arguments concerning

the nature of motion, and it is certain that we can never by mere

experience assure ourselves of the truth even of the first law of

motion. *A material particle*, we are told, *when not acted on by

extraneous forces will continue in the same state of rest or motion.*

This may be true, but as we can find no body which is free from the

action of extraneous causes, how are we to prove it? Only by observing

that the less the amount of those forces the more nearly is the law

found to be true. A ball rolled along rough ground is soon stopped;

along a smooth pavement it continues longer in movement. A delicately

suspended pendulum is almost free from friction against its supports,

but it is gradually stopped by the resistance of the air; place it

in the vacuous receiver of an air-pump and we find the motion much

prolonged. A large planet like Jupiter experiences almost infinitely

less friction, in comparison to its vast momentum, than we can produce

experimentally, and we find in such a case that there is not the least

evidence of the falsity of the law. Experience, then, informs us that

we may approximate indefinitely to a uniform motion by sufficiently

decreasing the disturbing forces. It is an act of inference which

enables us to travel on beyond experience, and assert that, in the

total absence of any extraneous force, motion would be absolutely

uniform. The state of rest, again, is a limiting case in which motion

is infinitely small or zero, to which we may attain, on the principle

of continuity, by successively considering cases of slower and slower

motion. There are many classes of phenomena, in which, by gradually

passing from the apparent to the obscure, we can assure ourselves of

the nature of phenomena which would otherwise be a matter of great

doubt. Thus we can sufficiently prove in the manner of Galileo, that

a musical sound consists of rapid uniform pulses, by causing strokes

to be made at intervals which we gradually diminish until the separate

strokes coalesce into a uniform hum or note. With great advantage we

approach, as Tyndall says, the sonorous through the grossly mechanical.

In listening to a great organ we cannot fail to perceive that the

longest pipes, or their partial tones, produce a tremor and fluttering

of the building. At the other extremity of the scale, there is no fixed

limit to the acuteness of sounds which we can hear; some individuals

can hear sounds too shrill for other ears, and as there is nothing in

the nature of the atmosphere to prevent the existence of undulations

far more rapid than any of which we are conscious, we may infer, by the

principle of continuity, that such undulations probably exist.

There are many habitual actions which we perform we know not how. So

rapidly are acts of minds accomplished that analysis seems impossible.

We can only investigate them when in process of formation, observing

that the best formed habit is slowly and continuously acquired, and

it is in the early stages that we can perceive the rationale of the

process.

Let it be observed that this principle of continuity must be held of

much weight only in exact physical laws, those which doubtless repose

ultimately upon the simple laws of motion. If we fearlessly apply

the principle to all kinds of phenomena, we may often be right in

our inferences, but also often wrong. Thus, before the development

of spectrum analysis, astronomers had observed that the more they

increased the powers of their telescopes the more nebulæ they could

resolve into distinct stars. This result had been so often found

true that they almost irresistibly assumed that all nebulæ would be

ultimately resolved by telescopes of sufficient power; yet Huggins has

in recent years proved by the spectroscope, that certain nebulæ are

actually gaseous, and in a truly nebulous state.

The principle of continuity must have been continually employed in the

inquiries of Galileo, Newton, and other experimental philosophers,

but it appears to have been distinctly formulated for the first time

by Leibnitz. He at least claims to have first spoken of “the law of

continuity” in a letter to Bayle, printed in the *Nouvelles de la

République des Lettres*, an extract from which is given in Erdmann’s

edition of Leibnitz’s works, p. 104, under the title “Sur un Principe

Général utile à l’explication des Lois de la Nature.”[511] It has

indeed been asserted that the doctrine of the *latens processus* of

Francis Bacon involves the principle of continuity,[512] but I think

that this doctrine, like that of the *natures* of substances, is merely

a vague statement of the principle of causation.

[511] *Life of Sir W. Hamilton*, p. 439.

[512] Powell’s *History of Natural Philosophy*, p. 201. *Novum

Organum*, bk. ii. Aphorisms 5–7.

*Failure of the Law of Continuity.*

There are certain cautions which must be given as to the application of

the principle of continuity. In the first place, where this principle

really holds true, it may seem to fail owing to our imperfect means

of observation. Though a physical law may not admit of perfectly

abrupt change, there is no limit to the approach which it may make to

abruptness. When we warm a piece of very cold ice, the absorption of

heat, the temperature, and the dilatation of the ice vary according

to apparently simple laws until we come to the zero of the Centigrade

scale. Everything is then changed; an enormous absorption of heat

takes place without any rise of temperature, and the volume of the ice

decreases as it changes into water. Unless carefully investigated, this

change appears to be perfectly abrupt; but accurate observation seems

to show that there is a certain forewarning; the ice does not turn into

water all at once, but through a small fraction of a degree the change

is gradual. All the phenomena concerned, if measured very exactly,

would be represented not by angular lines, but continuous curves,

undergoing rapid flexures; and we may probably assert with safety that

between whatever points of temperature we examine ice, there would be

found some indication, though almost infinitesimally small, of the

apparently abrupt change which was to occur at a higher temperature.

It might also be pointed out that the important and apparently simple

physical laws, such as those of Boyle and Mariotte, Dalton and

Gay-Lussac, &c., are only approximately true, and the divergences

from the simple laws are forewarnings of abrupt changes, which would

otherwise break the law of continuity.

Secondly, it must be remembered that mathematical laws of some

complexity will probably present singular cases or negative results,

which may bear the appearance of discontinuity, as when the law of

retraction suddenly yields us with perfect abruptness the phenomenon of

total internal reflection. In the undulatory theory, however, there is

no real change of law between refraction and reflection. Faraday in the

earlier part of his career found so many substances possessing magnetic

power, that he ventured on a great generalisation, and asserted that

all bodies shared in the magnetic property of iron. His mistake, as he

afterwards discovered, consisted in overlooking the fact that though

magnetic in a certain sense, some substances have negative magnetism,

and are repelled instead of being attracted by the magnet.

Thirdly, where we might expect to find a uniform mathematical law

prevailing, the law may undergo abrupt change at singular points,

and actual discontinuity may arise. We may sometimes be in danger of

treating under one law phenomena which really belong to different laws.

For instance, a spherical shell of uniform matter attracts an external

particle of matter with a force varying inversely as the square of the

distance from the centre of the sphere. But this law only holds true

so long as the particle is external to the shell. Within the shell

the law is wholly different, and the aggregate gravity of the sphere

becomes zero, the force in every direction being neutralised by an

exactly equal opposite force. If an infinitely small particle be in the

superficies of a sphere, the law is again different, and the attractive

power of the shell is half what it would be with regard to particles

infinitely close to the surface of the shell. Thus in approaching the

centre of a shell from a distance, the force of gravity shows double

discontinuity in passing through the shell.[513]

[513] Thomson and Tait, *Treatise on Natural Philosophy*, vol. i. pp.

346–351.

It may admit of question, too, whether discontinuity is really unknown

in nature. We perpetually do meet with events which are real breaks

upon the previous law, though the discontinuity may be a sign that

some independent cause has come into operation. If the ordinary

course of the tides is interrupted by an enormous irregular wave, we

attribute it to an earthquake, or some gigantic natural disturbance.

If a meteoric stone falls upon a person and kills him, it is clearly a

discontinuity in his life, of which he could have had no anticipation.

A sudden sound may pass through the air neither preceded nor followed

by any continuous effect. Although, then, we may regard the Law of

Continuity as a principle of nature holding rigorously true in many of

the relations of natural forces, it seems to be a matter of difficulty

to assign the limits within which the law is verified. Much caution is

required in its application.

*Negative Arguments on the Principle of Continuity.*

Upon the principle of continuity we may sometimes found arguments of

great force which prove an hypothesis to be impossible, because it

would involve a continual repetition of a process *ad infinitum*, or

else a purely arbitrary breach at some point. Bonnet’s famous theory

of reproduction represented every living creature as containing

germs which were perfect representatives of the next generation, so

that on the same principle they necessarily included germs of the

next generation, and so on indefinitely. The theory was sufficiently

refuted when once clearly stated, as in the following poem called the

Universe,[514] by Henry Baker:--

“Each seed includes a plant: that plant, again,

Has other seeds, which other plants contain:

Those other plants have all their seeds, and those

More plants again, successively inclose.

“Thus, ev’ry single berry that we find,

Has, really, in itself whole forests of its kind,

Empire and wealth one acorn may dispense,

By fleets to sail a thousand ages hence.”

[514] *Philosophical Transactions* (1740), vol. xli. p. 454.

The general principle of inference, that what we know of one case must

be true of similar cases, so far as they are similar, prevents our

asserting anything which we cannot apply time after time under the same

circumstances. On this principle Stevinus beautifully demonstrated

that weights resting on two inclined planes and balancing each other

must be proportional to the lengths of the planes between their apex

and a horizontal plane. He imagined a uniform endless chain to be hung

over the planes, and to hang below in a symmetrical festoon. If the

chain were ever to move by gravity, there would be the same reason

for its moving on for ever, and thus producing a perpetual motion. As

this is absurd, the portions of the chain lying on the planes, and

equal in length to the planes, must balance each other. On similar

grounds we may disprove the existence of any *self-moving machine*;

for if it could once alter its own state of motion or rest, in however

small a degree, there is no reason why it should not do the like

time after time *ad infinitum*. Newton’s proof of his third law of

motion, in the case of gravity, is of this character. For he remarks

that if two gravitating bodies do not exert exactly equal forces in

opposite directions, the one exerting the strongest pull will carry

both away, and the two bodies will move off into space together with

velocity increasing *ad infinitum*. But though the argument might

seem sufficiently convincing, Newton in his characteristic way made

an experiment with a loadstone and iron floated upon the surface of

water.[515] In recent years the very foundation of the principle of

conservation of energy has been placed on the assumption that it is

impossible by any combination of natural bodies to produce force

continually from nothing.[516] The principle admits of application in

various subtle forms.

[515] *Principia*, bk. i. Law iii. Corollary 6.

[516] Helmholtz, Taylor’s *Scientific Memoirs* (1853), vol. vi.

p. 118.

Lucretius attempted to prove, by a most ingenious argument of this

kind, that matter must be indestructible. For if a finite quantity,

however small, were to fall out of existence in any finite time, an

equal quantity might be supposed to lapse in every equal interval of

time, so that in the infinity of past time the universe must have

ceased to exist.[517] But the argument, however ingenious, seems to

fail at several points. If past time be infinite, why may not matter

have been created infinite also? It would be most reasonable, again,

to suppose the matter destroyed in any time to be proportional to the

matter then remaining, and not to the original quantity; under this

hypothesis even a finite quantity of original matter could never wholly

disappear from the universe. For like reasons we cannot hold that the

doctrine of the conservation of energy is really proved, or can ever be

proved to be absolutely true, however probable it may be regarded.

[517] *Lucretius*, bk. i. lines 232–264.

*Tendency to Hasty Generalisation.*

In spite of all the powers and advantages of generalisation, men

require no incitement to generalise; they are too apt to draw hasty and

ill-considered inferences. As Francis Bacon said, our intellects want

not wings, but rather weights of lead to moderate their course.[518]

The process is inevitable to the human mind; it begins with childhood

and lasts through the second childhood. The child that has once been

hurt fears the like result on all similar occasions, and can with

difficulty be made to distinguish between case and case. It is caution

and discrimination in the adoption of conclusions that we have chiefly

to learn, and the whole experience of life is one continued lesson

to this effect. Baden Powell has excellently described this strong

natural propensity to hasty inference, and the fondness of the human

mind for tracing resemblances real or fanciful. “Our first inductions,”

he says,[519] “are always imperfect and inconclusive; we advance

towards real evidence by successive approximations; and accordingly we

find false generalisation the besetting error of most first attempts

at scientific research. The faculty to generalise accurately and

philosophically requires large caution and long training, and is not

fully attained, especially in reference to more general views, even

by some who may properly claim the title of very accurate scientific

observers in a more limited field. It is an intellectual habit which

acquires immense and accumulating force from the contemplation of wider

analogies.”

[518] *Novum Organum*, bk. 1 Aphorism 104.

[519] *The Unity of Worlds and of Nature*, 2nd edit. p. 116.

Hasty and superficial generalisations have always been the bane

of science, and there would be no difficulty in finding endless

illustrations. Between things which are the same in number there is a

certain resemblance, namely in number; but in the infancy of science

men could not be persuaded that there was not a deeper resemblance

implied in that of number. Pythagoras was not the inventor of a

mystical science of number. In the ancient Oriental religions the seven

metals were connected with the seven planets, and in the seven days

of the week we still have, and probably always shall have, a relic of

the septiform system ascribed by Dio Cassius to the ancient Egyptians.

The disciples of Pythagoras carried the doctrine of the number seven

into great detail. Seven days are mentioned in Genesis; infants acquire

their teeth at the end of seven months; they change them at the end

of seven years; seven feet was the limit of man’s height; every

seventh year was a climacteric or critical year, at which a change

of disposition took place. Then again there were the seven sages of

Greece, the seven wonders of the world, the seven rites of the Grecian

games, the seven gates of Thebes, and the seven generals destined to

conquer that city.

In natural science there were not only the seven planets, and the seven

metals, but also the seven primitive colours, and the seven tones

of music. So deep a hold did this doctrine take that we still have

its results in many customs, not only in the seven days of the week,

but the seven years’ apprenticeship, puberty at fourteen years, the

second climacteric, and legal majority at twenty-one years, the third

climacteric. The idea was reproduced in the seven sacraments of the

Roman Catholic Church, and the seven year periods of Comte’s grotesque

system of domestic worship. Even in scientific matters the loftiest

intellects have occasionally yielded, as when Newton was misled by the

analogy between the seven tones of music and the seven colours of his

spectrum. Other numerical analogies, though rejected by Galileo, held

Kepler in thraldom; no small part of Kepler’s labours during seventeen

years was spent upon numerical and geometrical analogies of the most

baseless character; and he gravely held that there could not be more

than six planets, because there were not more than five regular solids.

Even the genius of Huyghens did not prevent him from inferring that

but one satellite could belong to Saturn, because, with those of

Jupiter and the Earth, it completed the perfect number of six. A whole

series of other superstitions and fallacies attach to the numbers six

and nine.

It is by false generalisation, again, that the laws of nature have

been supposed to possess that perfection which we attribute to simple

forms and relations. The heavenly bodies, it was held, must move in

circles, for the circle was the perfect figure. Newton seemed to adopt

the questionable axiom that nature always proceeds in the simplest way;

in stating his first rule of philosophising, he adds:[520] “To this

purpose the philosophers say, that nature does nothing in vain, when

less will serve; for nature is pleased with simplicity, and affects

not the pomp of superfluous causes.” Keill lays down[521] as an axiom

that “The causes of natural things are such, as are the most simple,

and are sufficient to explain the phenomena: for nature always proceeds

in the simplest and most expeditious method; because by this manner of

operating the Divine Wisdom displays itself the more.” If this axiom

had any clear grounds of truth, it would not apply to proximate laws;

for even when the ultimate law is simple the results may be infinitely

diverse, as in the various elliptic, hyperbolic, parabolic, or circular

orbits of the heavenly bodies. Simplicity is naturally agreeable to a

mind of limited powers, but to an infinite mind all things are simple.

[520] *Principia*, bk. iii, *ad initium*.

[521] Keill, *Introduction to Natural Philosophy*, p. 89.

Every great advance in science consists in a great generalisation,

pointing out deep and subtle resemblances. The Copernican system was

a generalisation, in that it classed the earth among the planets; it

was, as Bishop Wilkins expressed it, “the discovery of a new planet,”

but it was opposed by a more shallow generalisation. Those who argued

from the condition of things upon the earth’s surface, thought that

every object must be attached to and rest upon something else. Shall

the earth, they said, alone be free? Accustomed to certain special

results of gravity they could not conceive its action under widely

different circumstances.[522] No hasty thinker could seize the deep

analogy pointed out by Horrocks between a pendulum and a planet,

true in substance though mistaken in some details. All the advances

of modern science rise from the conception of Galileo, that in the

heavenly bodies, however apparently different their condition, we shall

ultimately recognise the same fundamental principles of mechanical

science which are true on earth.

[522] Jeremiæ Horroccii *Opera Posthuma* (1673), pp. 26, 27.

Generalisation is the great prerogative of the intellect, but it

is a power only to be exercised safely with much caution and after

long training. Every mind must generalise, but there are the widest

differences in the depth of the resemblances discovered and the care

with which the discovery is verified. There seems to be an innate

power of insight which a few men have possessed pre-eminently, and

which enabled them, with no exemption indeed from labour or temporary

error, to discover the one in the many. Minds of excessive acuteness

may exist, which have yet only the powers of minute discrimination,

and of storing up, in the treasure-house of memory, vast accumulations

of words and incidents. But the power of discovery belongs to a

more restricted class of minds. Laplace said that, of all inventors

who had contributed the most to the advancement of human knowledge,

Newton and Lagrange appeared to possess in the highest degree the

happy tact of distinguishing general principles among a multitude of

objects enveloping them, and this tact he conceived to be the true

characteristic of scientific genius.[523]

[523] Young’s *Works*, vol. ii. p. 564.

CHAPTER XXVIII.

ANALOGY.

As we have seen in the previous chapter, generalisation passes

insensibly into reasoning by analogy, and the difference is one of

degree. We are said to generalise when we view many objects as agreeing

in a few properties, so that the resemblance is extensive rather

than deep. When we have only a few objects of thought, but are able

to discover many points of resemblance, we argue by analogy that the

correspondence will be even deeper than appears. It may not be true

that the words are always used in such distinct senses, and there is

great vagueness in the employment of these and many logical terms; but

if any clear discrimination can be drawn between generalisation and

analogy, it is as indicated above.

It has been said, indeed, that analogy denotes not a resemblance

between things, but between the relations of things. A pilot is a very

different man from a prime minister, but he bears the same relation to

a ship that the minister does to the state, so that we may analogically

describe the prime minister as the pilot of the state. A man differs

still more from a horse, nevertheless four men bear to three men the

same relation as four horses bear to three horses. There is a real

analogy between the tones of the Monochord, the Sages of Greece, and

the Gates of Thebes, but it does not extend beyond the fact that they

were all seven in number. Between the most discrete notions, as, for

instance, those of time and space, analogy may exist, arising from

the fact that the mathematical conditions of the lapse of time and

of motion along a line are similar. There is no identity of nature

between a word and the thing it signifies; the substance *iron* is

a heavy solid, the word *iron* is either a momentary disturbance of

the air, or a film of black pigment on white paper; but there is

analogy between words and their significates. The substance iron is

to the substance iron-carbonate, as the name iron is to the name

iron-carbonate, when these names are used according to their scientific

definitions. The whole structure of language and the whole utility

of signs, marks, symbols, pictures, and representations of various

kinds, rest upon analogy. I may hope perhaps to enter more fully upon

this important subject at some future time, and to attempt to show how

the invention of signs enables us to express, guide, and register our

thoughts. It will be sufficient to observe here that the use of words

constantly involves analogies of a subtle kind; we should often be at

a loss how to describe a notion, were we not at liberty to employ in

a metaphorical sense the name of anything sufficiently resembling it.

There would be no expression for the sweetness of a melody, or the

brilliancy of an harangue, unless it were furnished by the taste of

honey and the brightness of a torch.