(173.) In the conduct of this verification, we are to consider whether

the cause or law to which we are conducted be one already known and

recognised as a more general one, whose nature is well understood, and

of which the phenomenon in question is but one more case in addition

to those already known, or whether it be one less general, less known,

or altogether new. In the latter case, our verification will suffice,

if it merely shows that all the cases considered are plainly cases in

point. But in the former, the process of verification is of a much

more severe and definite kind. We must trace the action of our cause

with distinctness and precision, as modified by all the circumstances

of each case; we must estimate its effects, and show that nothing

unexplained remains behind; at least, in so far as the presence of

unknown modifying causes is not concerned.

(174.) Now, this is precisely the sort of process in which _residual

phenomena_ (such as spoken of in art. 158.) may be expected to occur.

If our induction be really a valid and a comprehensive one, _whatever_

remains unexplained in the comparison of its conclusion with particular

cases, under all their circumstances, _is_ such a phenomenon, and comes

in its turn to be a subject of inductive reasoning to discover its

cause or laws. It is thus that we may be said to witness facts with

the eyes of reason; and it is thus that we are continually attaining a

knowledge of new phenomena and new laws which lie beneath the surface

of things, and give rise to the creation of fresh branches of science

more and more remote from common observation.

(175.) Physical astronomy affords numerous and splendid instances of

this. The law, for example, which asserts that the planets are retained

in their orbits about the sun, and satellites about their primaries,

by an attractive force, decreasing as the square of the distances

increases, comes to be verified in each particular case by deducing

from it the exact motions which, under the circumstances, ought to take

place, and comparing them with fact. This comparison, while it verifies

in general the existence of the law of gravitation as supposed, and

its adequacy to explain all the principal motions of every body in the

system, yet leaves some small deviations in those of the planets, and

some very considerable ones in that of the moon and other satellites,

still unaccounted for; residual phenomena, which still remain to be

traced up to causes. By further examining these, their causes have at

length been ascertained, and found to consist in the mutual actions of

the planets on each other, and the disturbing influence of the sun on

the motions of the satellites.

(176.) But a law of nature has not that degree of generality which

fits it for a stepping-stone to greater inductions, unless it be

_universal_ in its application. We cannot rely on its enabling us to

extend our views beyond the circle of instances from which it was

obtained, unless we have already had experience of its power to do so;

unless it actually _has_ enabled us before trial to say what will take

place in cases analogous to those originally contemplated; unless, in

short, we have studiously placed ourselves in the situation of its

antagonists, and even perversely endeavoured to find exceptions to

it without success. It is in the precise proportion that a law once

obtained endures this extreme severity of trial, that its value and

importance are to be estimated; and our next step in the verification

of an induction must therefore consist in _extending_ its application

to cases not originally contemplated; in studiously varying the

circumstances under which our causes act, with a view to ascertain

whether their effect is general; and in pushing the application of our

laws to extreme cases.

(177.) For example, a fair induction from a great number of facts led

Galileo to conclude that the accelerating power of gravity is the same

on all sorts of bodies, and on great and small masses indifferently;

and this he exemplified by letting bodies of very different natures

and weights fall at the same instant from a high tower, when it was

observed that they struck the ground at the same moment, abating a

certain trifling difference, due, as he justly believed it to be, to

the greater proportional resistance of the air to light than to heavy

bodies. The experiment could not, at that time, be fairly tried with

extremely light substances, such as cork, feathers, cotton, &c. because

of the great resistance experienced by these in their fall; no means

being then known of removing this cause of disturbance. It was not,

therefore, till after the invention of the air-pump that this law could

be put to the severe test of an extreme case. A guinea and a downy

feather were let drop at once from the upper part of a tall exhausted

glass, and struck the bottom at the same moment. Let any one make the

trial _in the air_, and he will perceive the force of an _extreme case_.

(178.) In the verification of a law whose expression is _quantitative_,

not only must its generality be established by the trial of it in as

various circumstances as possible, but every such trial must be one of

precise measurement. And in such cases the means taken for subjecting

it to trial ought to be so devised as to repeat and multiply a great

number of times any deviation (if any exist); so that, let it be ever

so small, it shall at last become sensible.

(179.) For instance, let the law to be verified be, that _the gravity

of every material body is in the direct proportion of its mass_, which

is only another mode of expressing Galileo’s law above mentioned.

The time of falling from any moderate height cannot be measured with

precision enough for our purpose: but if it can be repeated a very

great multitude of times _without any loss or gain_ in the intervals,

and the whole amount of the times of fall so repeated measured by

a clock; and if at the same time the resistance of the air can be

rendered _exactly alike_ for all the bodies tried, we have here

Galileo’s trial in a much more refined state; and it is evident that

almost unlimited exactness may be obtained. Now, all this Newton

accomplished by the simple and elegant contrivance of enclosing in a

hollow pendulum the same weights of a great number of substances the

most different that could be found in all respects, as gold, glass,

wood, water, wheat, &c.[44], and ascertaining the time required for

the pendulum so charged to make a great number of oscillations; in

each of which it is clear the weights had to fall, and be raised again

successively, without loss of time, through the same _identical_

spaces. Thus any difference, however inconsiderable, that might

exist in the time of one such fall and rise would be multiplied and

accumulated till they became sensible. And none having been discovered

by so delicate a process in any case, the law was considered verified

both in respect of generality and exactness. This, however, is nothing

to the verifications afforded by astronomical phenomena, where the

deviations, if any, accumulate for thousands of years instead of a few

hours.

(180.) The surest and best characteristic of a well-founded and

extensive induction, however, is when verifications of it spring up,

as it were, spontaneously, into notice, from quarters where they might

be least expected, or even among instances of that very kind which

were at first considered hostile to them. Evidence of this kind is

irresistible, and compels assent with a weight which scarcely any other

possesses. To give an example: M. Mitscherlich had announced a law to

this effect--_that_ the chemical elements of which all bodies consist

are susceptible of being classified in distinct groups, which he termed

_isomorphous_ groups; and _that_ these groups are so related, that

when similar combinations are formed of individuals belonging to two,

three, or more of them, such combinations will crystallize in the same

geometrical forms. To this curious and important law there appeared

a remarkable exception. According to professor Mitscherlich, the

arsenic and phosphoric acids _are_ similar combinations coming under

the meaning of his law, and their combinations with soda and water,

forming the salts known to chemists under the names of arseniate and

phosphate of soda, ought, if the law were general, to crystallize in

identical shapes. The fact, however, was understood to be otherwise.

But lately, Mr. Clarke, a British chemist, having examined the two

salts attentively, ascertained the fact that their compositions

deviate essentially from that similarity which M. Mitscherlich’s law

requires; and that, therefore, the exception in question disappears.

This was something: but, pursuing the subject further, the same

ingenious enquirer happily succeeded in producing a _new_ phosphate of

soda, differing from that generally known in containing a different

proportion of water, and agreeing in composition exactly with the

arseniate. The crystals of this new salt, when examined, were found by

him to be precisely identical in form with those of the arseniate: thus

verifying, in a most striking and totally unexpected manner, the law in

question, or, as it is called, the law of isomorphism.

(181.) Unexpected and peculiarly striking confirmations of inductive

laws frequently occur in the form of residual phenomena, in the course

of investigations of a widely different nature from those which gave

rise to the inductions themselves. A very elegant example may be cited

in the unexpected confirmation of the law of the developement of heat

in elastic fluids by compression, which is afforded by the phenomena

of sound. The enquiry into the cause of sound had led to conclusions

respecting its mode of propagation, from which its velocity in the air

could be precisely calculated. The calculations were performed; but,

when compared with fact, though the agreement was quite sufficient

to show the general correctness of the cause and mode of propagation

assigned, _yet_ the _whole_ velocity could not be shown to arise from

this theory. There was still a _residual_ velocity to be accounted

for, which placed dynamical philosophers for a long time in a great

dilemma. At length Laplace struck on the happy idea, that this might

arise from the _heat_ developed in the act of that condensation which

necessarily takes place at every vibration by which sound is conveyed.

The matter was subjected to exact calculation, and the result was

at once the complete explanation of the residual phenomenon, and a

striking confirmation of the general law of the developement of heat by

compression, under circumstances beyond artificial imitation.

(182.) In extending our inductions to cases not originally

contemplated, there is one step which always strikes the mind with

peculiar force, and with such a sensation of novelty and surprise,

as often gives it a weight beyond its due philosophic value. It is

the transition from the little to the great, and _vice versâ_, but

especially the former. It is so beautiful to see, for instance, an

experiment performed in a watch-glass, or before a blowpipe, succeed,

in a great manufactory, on many tons of matter, or, in the bosom of

a volcano, upon millions of cubic fathoms of lava, that we almost

forget that these great masses are made up of watch-glassfuls, and

blowpipe-beads. We see the enormous intervals between the stars and

planets of the heavens, which afford room for innumerable processes

to be carried on, for light and heat to circulate, and for curious

and complicated motions to go forward among them: we look more

attentively, and we see sidereal systems, probably not less vast and

complicated than our own, crowded apparently into a small space (from

the effect of their distance from us), and forming groups resembling

bodies of a substantial appearance, having form and outline: yet we

recoil with incredulous surprise when we are asked _why_ we cannot

conceive the atoms of a grain of sand to be as remote from each other

(proportionally to their sizes) as the stars of the firmament; and

why there may not be going on, in that little microcosm, processes

as complicated and wonderful as those of the great world around us.

Yet the student who makes any progress in natural philosophy will

encounter numberless cases in which this transfer of ideas from the one

extreme of magnitude to the other will be called for: he will find,

for instance, the phenomena of the propagation of winds referred to

the same laws which regulate the propagation of motions through the

smallest masses of air; those of lightning assimilated to the mere

communication of an electric spark, and those of earthquakes to the

tremors of a stretched wire: in short, he must lay his account to

finding the distinction of great and little altogether annihilated in

nature: and it is well for man that such is the case, and that the same

laws, which he can discover and verify in his own circumscribed sphere

of power, should prove available to him when he comes to apply them on

the greatest scale; since it is thus only that he is enabled to become

an exciting cause in operations of any considerable magnitude, and to

vindicate his importance in creation.

(183.) But the business of induction does not end here: its final

result must be followed out into all its consequences, and applied to

all those cases which seem even remotely to bear upon the subject of

enquiry. Every new addition to our stock of causes becomes a means

of fresh attack with new vantage ground upon all those unexplained

parts of former phenomena which have resisted previous efforts. It

can hardly be pressed forcibly enough on the attention of the student

of nature, that there is scarcely any natural phenomenon which can be

fully and completely explained in all its circumstances, without a

union of several, perhaps of all, the sciences. The great phenomena of

astronomy, indeed, may be considered exceptions; but this is merely

because their scale is so vast that one only of the most widely

extending forces of nature takes the lead, and all those agents whose

sphere of action is limited to narrower bounds, and which determine

the production of phenomena nearer at hand, are thrown into the back

ground, and become merged and lost in comparative insignificance. But

in the more intimate phenomena which surround us it is far otherwise.

Into what a complication of different branches of science are we not

led by the consideration of such a phenomenon as rain, for instance,

or flame, or a thousand others, which are constantly going on before

our eyes? Hence, it is hardly possible to arrive at the knowledge of

a law of any degree of generality in any branch of science, but it

immediately furnishes us with a means of extending our knowledge of

innumerable others, the most remote from the point we set out from; so

that, when once embarked in any physical research, it is impossible for

any one to predict where it may ultimately lead him.

(184.) This remark rather belongs to the inverse or _deductive_

process, by which we pursue laws into their remote consequences.

But it is very important to observe, that the successful process of

scientific enquiry demands continually the alternate use of both

the _inductive_ and _deductive_ method. The path by which we rise to

knowledge must be made smooth and beaten in its lower steps, and often

ascended and descended, before we can scale our way to any eminence,

much less climb to the summit. The achievement is too great for a

single effort; stations must be established, and communications kept

open with all below. To quit metaphor; there is nothing so instructive,

or so likely to lead to the acquisition of general views, as this

pursuit of the consequences of a law once arrived at into every subject

where it may seem likely to have an influence. The discovery of a

new law of nature, a new ultimate fact, or one that even temporarily

puts on that appearance, is like the discovery of a new element in

chemistry. Thus, selenium was hardly discovered by Berzelius in the

vitriol works of Fahlun, when it presently made its appearance in the

sublimates of Stromboli, and the rare and curious products of the

Hungarian mines. And thus it is with every new law, or general fact. It

is hardly announced before its traces are found every where, and every

one is astonished at its having so long remained concealed. And hence

it happens that unexpected lights are shed at length over parts of

science that had been abandoned in despair, and given over to hopeless

obscurity.

(185.) The verification of _quantitative_ laws has been already spoken

of (178.); but their importance in physical science is so very great,

inasmuch as they alone afford a handle to strict mathematical deductive

application, that something ought to be said of the nature of the

inductions by which they are to be arrived at. In their simplest or

least general stages (of which alone we speak at present) they usually

express some numerical relation between two quantities dependent on

each other, either as collateral effects of a common cause, or as the

amount of its effect under given numerical circumstances or _data_.

For example, the law of refraction before noticed (§ 22.) expresses,

by a very simple relation, the amount of angular deviation of a ray

of light from its course, when the _angle_ at which it is inclined to

the refracting surface is known, viz. that the _sine_ of the angle

which the incident ray makes with a perpendicular to the surface is

always to that of the angle made by the refracted ray with the same

perpendicular, in a constant proportion, so long as the refracting

substance is the same. To arrive inductively at laws of this kind,

where one quantity _depends_ on or _varies with_ another, all that is

required is a series of careful and exact measures in every different

state of the _datum_ and _quæsitum_. Here, however, the mathematical

form of the law being of the highest importance, the greatest attention

must be given to the _extreme cases_ as well as to all those points

where the one quantity changes rapidly with a small change of the

other.[45] The results must be set down in a table in which the _datum_

gradually increases in magnitude from the lowest to the highest limit

of which it is susceptible. It will depend then entirely on our

habit of treating mathematical subjects, how far we may be able to

include such a table in the distinct statement of a mathematical law.

The discovery of such laws is often remarkably facilitated by the

contemplation of a class of phenomena to be noticed further on, under

the head of Collective Instances, (see § 194.) in which the nature of

the mathematical expression in which the law sought is comprehended, is

pointed out by the figure of some curve brought under inspection by a

proper mode of experimenting.

(186.) After all, unless our induction embraces a series of cases which

absolutely include the whole scale of variation of which the quantities

in question admit, the mathematical expression so obtained cannot be

depended upon as the true one, and if the scale actually embraced be

small, the extension of laws so derived to extreme cases will in all

probability be exceedingly fallacious. For example, air is an elastic

fluid, and as such, if enclosed in a confined space and squeezed, its

bulk diminishes: now, from a great number of trials made in cases where

the air has been compressed into a half, a third, &c. even as far

as a fiftieth of its bulk, or less, it has been concluded that “the

density of air is proportional to the compressing force,” or the bulk

it occupies _inversely_ as that force; and when the air is rarefied

by taking off part of its natural pressure, the same is found to be

the case, within very extensive limits. Yet it is impossible that this

should be, strictly or mathematically speaking, the true law; for, if

it were so, there could be no limit to the condensation of air, while

yet we have the strongest analogies to show that long before it had

reached any very enormous pitch the air would be reduced into a liquid,

and even, perhaps, if pressed yet more violently, into a solid form.

(187.) Laws thus derived, by the direct process of including in

mathematical formulæ the results of a greater or less number of

measurements, are called “empirical laws.” A good example of such a

law is that given by Dr. Young (Phil. Trans. 1826,) for the decrement

of life, or the law of mortality. Empirical laws in this state are

evidently _unverified inductions_, and are to be received and reasoned

on with the utmost reserve. No confidence can ever be placed in them

beyond the limits of the data from which they are derived; and even

within those limits they require a special and severe scrutiny to

examine _how nearly_ they do represent the observed facts; that is to

say, whether, in the comparison of their results with the observed

quantities, the differences are such as may fairly be attributed to

error of observation. When so carefully examined, they become, however,

most valuable; and frequently, when afterwards verified theoretically

by a deductive process (as will be explained in our next chapter),

turn out to be rigorous laws of nature, and afford the noblest and

most convincing supports of which theories themselves are susceptible.

The finest instances of this kind are the great laws of the planetary

motions deduced by Kepler, entirely from a comparison of observations

with each other, with no assistance from theory. These laws, viz. that

the planets move in ellipses round the sun; that each describes about

the sun’s centre equal areas in equal times; and that in the orbits of

different planets the squares of the periodical times are proportional

to the cubes of the distances; were the results of inconceivable

labour of calculation and comparison: but they amply repaid the

labour bestowed on them, by affording afterwards the most conclusive

and unanswerable proofs of the Newtonian system. On the other hand,

when empirical laws are unduly relied on beyond the limits of the

observations from which they were deduced, there is no more fertile

source of fatal mistakes. The formulæ which have been empirically

deduced for the elasticity of steam (till very recently), and those

for the resistance of fluids, and other similar subjects, have almost

invariably failed to support the theoretical structures which have been

erected on them.

(188.) It is a remarkable and happy fact, that the shortest and most

direct of all inductions should be that which has led at once, or by

very few steps, to the highest of all natural laws,--we mean those of

motion and force. Nothing can be more simple, precise, and general,

than the enunciation of these laws; and, as we have once before

observed, their application to particular facts in the descending or

deductive method is limited by nothing but the limited extent of our

mathematics. It would seem, then, that dynamical science were taken

thenceforward out of the pale of induction, and transformed into a

matter of absolute _à priori_ reasoning, as much as geometry; and so

it would be, were our mathematics perfect, and all the _data_ known.

Unhappily, the first is so far from being the case, that in many

of the most interesting branches of dynamical enquiry they leave

us completely at a loss. In what relates to the motions of fluids,

for instance, this is severely felt. We can include our problems,

it is true, in algebraical equations, and we can demonstrate that

they _contain_ the solutions; but the equations themselves are so

intractable, and present such insuperable difficulties, that they often

leave us quite as much in the dark as before. But even were these

difficulties overcome, recourse to experience must still be had, to

establish the _data_ on which particular applications are to depend;

and although mathematical analysis affords very powerful means of

_representing_ in general terms the data of any proposed case, and

_afterwards_, by comparison of its results with fact, determining

_what_ those data must be to explain the observed phenomena, still,

in any mode of considering the matter, an appeal to experience in

every particular instance of application is unavoidable, even when

the general principles are regarded as sufficiently established

without it. Now, in all such cases of difficulty we must recur to our

inductive processes, and regard the branches of dynamical science where

this takes place as purely experimental. By this we gain an immense

advantage, viz. that in all those points of them where the abstract

dynamical principles _do_ afford distinct conclusions, we obtain

verifications for our inductions of the highest and finest possible

kind. When we work our way up inductively to one of these results, we

cannot help feeling the strongest assurance of the validity of the

induction.

(189.) The necessity of this appeal to experiment in every thing

relating to the motions of fluids on the large scale has long been

felt. Newton himself, who laid the first foundations of hydrodynamical

science (so this branch of dynamics is called), distinctly perceived

it, and set the example of laborious and exact experiments on their

resistance to motion, and other particulars. Venturi, Bernoulli, and

many others, have applied the method of experiment to the motions

of fluids in pipes and canals; and recently the brothers Weber have

published an elaborate and excellent experimental enquiry into the

phenomena of waves. One of the greatest and most successful attempts,

however, to bring an important, and till then very obscure, branch

of dynamical enquiry back to the dominion of experiment, has been

made by Chladni and Savart in the case of sound and vibratory motion

in general; and it is greatly to be wished that the example may be

followed in many others hardly less abstruse and impracticable when

theoretically treated. In such cases the inductive and deductive

methods of enquiry may be said to go hand in hand, the one verifying

the conclusions deduced by the other; and the combination of experiment

and theory, which may thus be brought to bear in such cases, forms

an engine of discovery infinitely more powerful than either taken

separately. This state of any department of science is perhaps of

all others the most interesting, and that which promises the most to

research.

(190.) It can hardly be expected that we should terminate this division

of our subject without some mention of the “prerogatives of instances”

of Bacon, by which he understands characteristic phenomena, selected

from the great miscellaneous mass of facts which occur in nature,

and which, by their number, indistinctness, and complication, tend

rather to confuse than to direct the mind in its search for causes

and general heads of induction. Phenomena so selected on account of

some peculiarly forcible way in which they strike the reason, and

impress us with a kind of sense of causation, or a particular aptitude

for generalization, he considers, and justly, as holding a kind of

prerogative dignity, and claiming our first and especial attention in

physical enquiries.

(191.) We have already observed that, in forming inductions, it will

most commonly happen that we are led to our conclusions by the especial

force of some two or three strongly impressive facts, rather than by

affording the whole mass of cases a regular consideration; and hence

the need of cautious verification. Indeed, so strong is this propensity

of the human mind, that there is hardly a more common thing than to

find persons ready to assign a cause for every thing they see, and, in

so doing, to join things the most incongruous, by analogies the most

fanciful. This being the case, it is evidently of great importance

that these first ready impulses of the mind should be made on the

contemplation of the cases most likely to lead to good inductions.

The misfortune, however, is, in natural philosophy, that the choice

does not rest with us. We must take the instances as nature presents

them. Even if we are furnished with a list of them in tabular order,

we must understand and compare them with each other, before we can

tell which _are_ the instances thus deservedly entitled to the highest

consideration. And, after all, after much labour in vain, and groping

in the dark, accident or casual observation will present a case which

strikes us at once with a full insight into a subject, before we can

even have time to determine to what class its _prerogative_ belongs.

For example, the laws of crystallography were obscure, and its causes

still more so, till Haüy fortunately dropped a beautiful crystal of

calcareous spar on a stone pavement, and broke it. In piecing together

the fragments, he observed their facets not to correspond with those of

the crystal in its entire state, but to belong to another form; and,

following out the hint offered by a “_glaring instance_” thus casually

obtruded on his notice, he discovered the beautiful laws of the

cleavage, and the primitive forms of minerals.

(192.) It has always appeared to us, we must confess, that the help

which the classification of instances, under their different titles of

prerogative, affords to inductions, however just such classification

may be in itself, is yet more apparent than real. The force of the

instance must be felt in the mind, before it can be referred to

its place in the system; and, before it can be either referred or

appretiated, it must be known; and when it _is_ appretiated, we are

ready enough to interweave it in our web of induction, without greatly

troubling ourselves with enquiring whence it derives the weight we

acknowledge it to have in our decisions. However, since much importance

is usually attached to this part of Bacon’s work, we shall here give a

few examples to illustrate the nature of some of his principal cases.

One, of what he calls “glaring instances,” has just been mentioned.

In these, the _nature_ or cause enquired into, (which in this case

is the cause of the assumption of a peculiar external form, or the

internal _structure_ of a crystal,) “stands naked and alone, and this

in an eminent manner, or in the highest degree of its power.” No doubt,

such instances as these are highly instructive; but the difficulty in

physics is to find such, not to perceive their force when found.

(193.) The contrary of glaring are “clandestine instances,” where

“the nature sought is exhibited in its weakest and most imperfect

state.” Of this, Bacon himself has given an admirable example in the

cohesion of fluids, as a _clandestine instance_ of the “_nature_ or

quality of consistence, or solidity.” Yet here, again, the same acute

discrimination which enabled Bacon to perceive the analogy which

connects fluids with solids, through the common property of cohesive

attraction, would, at the same time, have enabled him to draw from it,

if properly supported, every consequence necessary to forming just

notions of the cohesive force; nor does its reference to the class of

clandestine instances at all assist in bringing forward and maturing

the final results. When, however, the final result is obtained,--when

our induction is complete, and we would verify it,--this class of

instances is of great use, being, in fact, frequently no other than

that of _extreme cases_, such as we have already spoken of (in §

177.); which, by placing our conclusions, as it were, in violent

circumstances, try their temper, and bring their vigour to the test.

(194.) Bacon’s “collective instances” (_instantiæ unionis_), are no

other than general facts, or laws of some degree of generality, and

are themselves the results of induction. But there is a species of

collective instance which Bacon does not seem to have contemplated, of

a peculiarly instructive character; and that is, where particular cases

are offered to our observation in such numbers at once as to make the

induction of their law a matter of ocular inspection. For example, the

parabolic form assumed by a jet of water spouted from a round hole,

is a _collective instance_ of the velocities and directions of the

motions of all the particles which compose it _seen at once_, and which

thus leads us, without trouble, to recognize the law of the motion of

a projectile. Again, the beautiful figures exhibited by sand strewed

on regular plates of glass or metal set in vibration, are _collective

instances_ of an infinite number of points which remain at rest while

the remainder of the plate vibrates; and in consequence afford us,

as it were, a sight of the law which regulates their arrangement

and sequence throughout the whole surface. The beautifully coloured

lemniscates seen around the optic axes of crystals exposed to polarized

light afford a superb example of the same kind, pointing at once to

the general mathematical expression of the law which regulates their

production.[46] Of such collective instances as these, it is easy to

see the importance, and its reason. They lead us to a general law by an

induction which offers itself spontaneously, and thus furnish advanced

points in our enquiries; and when we start from these, already “a

thousand steps are lost.”

(195.) A fine example of a collective instance is that of the system

of Jupiter or Saturn with its satellites. We have here, in miniature,

and seen at one view, a system similar to that of the planets about

the sun; of which, from the circumstance of our being involved in it,

and unfavourably situated for seeing it otherwise than in detail, we

are incapacitated from forming a general idea but by slow progressive

efforts of reason. Accordingly, the contemplation of the _circumjovial

planets_ (as they were called) most materially assisted in securing the

admission of the Copernican system.

(196.) Of “Crucial instances” we have also already spoken, as affording

the readiest and securest means of eliminating extraneous causes, and

deciding between rival hypotheses. Owing to the disposition of the

mind to form hypotheses, and to prejudge cases, it constantly happens

that, among all the possible suppositions which may occur, two or three

principal ones occupy us, to the exclusion of the rest; or it may

be that, if we have been less precipitate, out of a great multitude

rejected for obvious inapplicability to some one or other case, two or

three of better claims remain for decision; and this such instances

enable us to do. One of the instances cited by Bacon in illustration of

his crucial class is very remarkable, being neither more nor less than

the proposal of a direct experiment to determine whether the tendency

of heavy bodies downwards is a result of some peculiar mechanism in

themselves, or of the attraction of the earth “by the corporeal mass

thereof, as by a collection of bodies of the same nature.” If it be so,

he says, “it will follow that the nearer all bodies approach to the

earth, the stronger and with the greater force and velocity they will

tend to it; but the farther they are, the weaker and slower:” and his

experiment consists in comparing the effect of a spring and a weight

in keeping up the motions of two “clocks,” regulated together, and

removed alternately to the tops of high buildings and into the deepest

mines. By _clocks_ he could not have meant pendulum clocks, which were

not then known, (the first made in England was in 1662,) _fly_-clocks,

so that the comparison, though too coarse, was not contrary to sound

mechanical principles. In short, its principle was the comparison of

the effect of a spring with that of a weight, in producing certain

motions in certain times, on heights and in mines. Now, this is the

very same thing that has really been done in the recent experiments

of professors Airy and Whewell in Dolcoath mine: a pendulum (a weight

moved by gravity) has been compared with a chronometer balance, moved

and regulated by a spring. In his 37th aphorism, Bacon also speaks of

gravity as an incorporeal power, acting at a distance, and _requiring

time for its transmission_; a consideration which occurred at a later

period to Laplace, in one of his most delicate investigations.

(197.) A well chosen and strongly marked crucial instance is,

sometimes, of the highest importance; when two theories, which run

parallel to each other (as is sometimes the case) in their explanation

of great classes of phenomena, at length come to be placed at issue

upon a single fact. A beautiful instance of this will be cited in the

next section. We may add to the examples above given of such instances,

that of the application of chemical tests, which are almost universally

crucial experiments.

(198.) Bacon’s “travelling instances” are those in which the _nature_

or quality under investigation “travels,” or varies in degree; and thus

(according to § 152.) afford an indication of a cause by a gradation

of intensity in the effect. One of his instances is very happy, being

that of “paper, which is white when dry, but proves less so when wet,

and comes nearer to the state of transparency upon the exclusion of the

air, and admission of water.” In reading this, and many other instances

in the Novum Organum, one would almost suppose (had it been written)

that its author had taken them from Newton’s Optics.

(199.) The travelling instances, as well as what Bacon terms “frontier

instances,” are cases in which we are enabled to trace that general

law which seems to pervade all nature--the law, as it is termed, of

continuity, and which is expressed in the well known sentence, “Natura

non agit per saltum.” The pursuit of this law into cases where its

application is not at first sight obvious, has proved a fertile source

of physical discovery, and led us to the knowledge of an analogy and

intimate connection of phenomena between which at first we should never

have expected to find any.

(200.) For example, the transparency of gold leaf, which permits a

bluish-green light to pass through it, is a frontier instance between

the transparency of pellucid bodies and the opacity of metals, and it

prevents a breach of the law of continuity between transparent and

opake bodies, by exhibiting a body of the class generally regarded

the most opake in nature, as still possessed of some slight degree

of transparency. It thus proves that the quality of opacity is not a

_contrary_ or _antagonist_ quality to that of transparency, but only

its extreme lowest degree.

CHAP. VII.

OF THE HIGHER DEGREES OF INDUCTIVE GENERALIZATION, AND OF THE

FORMATION AND VERIFICATION OF THEORIES.

(201.) As particular inductions and laws of the first degree of

generality are obtained from the consideration of individual facts,

so Theories result from a consideration of these laws, and of the

proximate causes brought into view in the previous process, regarded

all together as constituting a new set of phenomena, the creatures

of reason rather than of sense, and each representing under general

language innumerable particular facts. In raising these higher

inductions, therefore, more scope is given to the exercise of pure

reason than in slowly groping out our first results. The mind is more

disencumbered of matter, and moves as it were in its own element. What

is now before it, it perceives more intimately, and less through the

medium of sense, or at least not in the same manner as when actually at

work on the immediate objects of sense. But it must not be therefore

supposed that, in the formation of theories, we are abandoned to

the unrestrained exercise of imagination, or at liberty to lay down

arbitrary principles, or assume the existence of mere fanciful causes.

The liberty of speculation which we possess in the domains of theory is

not like the wild licence of the slave broke loose from his fetters,

but rather like that of the freeman who has learned the lessons of

self-restraint in the school of just subordination. The ultimate

objects we pursue in the highest theories are the same as those of the

lowest inductions; and the means by which we can most securely attain

them bear a close analogy to those which we have found successful in

such inferior cases.

(202.) The immediate object we propose to ourselves in physical

theories is the analysis of phenomena, and the knowledge of the hidden

processes of nature in their production, so far as they can be traced

by us. An important part of this knowledge consists in a discovery

of the actual structure or mechanism of the universe and its parts,

through which, and by which, those processes are executed; and of the

agents which are concerned in their performance. Now, the mechanism of

nature is for the most part either on too large or too small a scale

to be immediately cognizable by our senses; and her agents in like

manner elude direct observation, and become known to us only by their

effects. It is in vain therefore that we desire to become witnesses to

the processes carried on with such means, and to be admitted into the

secret recesses and laboratories where they are effected. Microscopes

have been constructed which magnify more than a thousand times in

_linear_ dimension, so that the smallest visible grain of sand may

be enlarged to the appearance of one a thousand million times more

bulky; yet the only impression we receive by viewing it through such

a magnifier is, that it reminds us of some vast fragment of a rock,

while the intimate structure on which depend its colour, its hardness,

and its chemical properties, remains still concealed: we do not seem

to have made even an approach to a closer analysis of it by any such

scrutiny.

(203.) On the other hand, the mechanism of the great system of which

our planet forms a part escapes immediate observation by the immensity

of its scale, nay, even by the slowness of its evolutions. The motion

of the minute hand of a watch can hardly be perceived without the

closest attention, and that of the hour hand not at all. But what are

these, in respect of the impression of slowness they produce in our

minds, compared with a revolving movement which takes a whole year, or

twelve, thirty, or eighty years to complete, as is the case with the

planets in their revolutions round the sun. Yet no sooner do we come

to reflect on the linear dimensions of these orbs, (which however we

do not _see_, nor can we measure them but by a long, circuitous, and

difficult process,) than we are lost in astonishment at the swiftness

of the very motions which before seemed so slow.[47] The motion of the

sails of a windmill offers (on a small scale) an illustrative case. At

a distance the rotation seems slow and steady--but when we stand close

to one of the sails in its sweep, we are surprised at the swiftness

with which it rushes by us.

(204.) Again, the agents employed by nature to act on material

structures are invisible, and only to be traced by the effects they

produce. Heat dilates matter with an irresistible force; but what heat

is, remains yet a problem. A current of electricity passing along a

wire moves a magnetized needle at a distance; but except from this

effect we perceive no difference between the condition of the wire

when it conveys and when it does not convey the stream: and we apply

the terms current, or stream, to the electricity only because in some

of its relations it reminds us of something we have observed in a

stream of air or water. In like manner we see that the moon circulates

about the earth; and because we believe it to be a solid mass, and

have never seen one solid substance revolve round another within our

reach to handle and examine unless retained by a force or united by a

tie, we conclude that there _is_ a force, and a mode of connection,

between the moon and the earth; though, what that mode can be, we have

no conception, nor can imagine _how_ such a force can be exerted at a

distance, and with empty space, or at most an invisible fluid, between.

(See § 148.)

(205.) Yet are we not to despair, since we see regular and beautiful

results brought about in human works by means which nobody would, at

first sight, think could have any thing to do with them. A sheet of

blank paper is placed upon a frame, and shoved forwards, and after

winding its way successively over and under half a dozen rollers, and

performing many other strange evolutions, comes out printed on both

sides. And, after all, the acting cause in this process is nothing

more than a few gallons of water boiled in an iron vessel, at a

distance from the scene of operations. But _why_ the water so boiled

should be capable of producing the active energy which sets the whole

apparatus in motion is, and will probably long remain, a secret to us.

(206.) This, however, does not at all prevent our having a very perfect

comprehension of the whole subsequent process. We might frequent

printing-houses, and form a theory of printing, and having worked our

way up to the point where the mechanical action commenced (the boiler

of the steam-engine), and verified it by taking to pieces, and putting

together again, the train of wheels and the presses, and by sound

theoretical examination of all the transfers of motion from one part to

another; we should, at length, pronounce our theory good, and declare

that we understood printing thoroughly. Nay, we might even go away and

apply the principles of mechanism we had learned in this enquiry to

other widely different purposes; construct other machines, and put them

in motion by the same moving power, and all without arriving at any

correct idea as to the ultimate source of the force employed. But, if

we were inclined to theorize farther, we might do so; and it is easy

to imagine how two theorists might form very different _hypotheses_ as

to the origin of the power which alternately raised and depressed the

piston-rod of the engine. One, for example, might maintain that the

boiler (whose contents we will suppose that neither theorist has been

permitted to examine) was the den of some powerful unknown animal, and

he would not be without plausible analogies in the warmth, the supply

of fuel and water, the breathing noises, the smoke, and above all, the

mechanical power exerted. He would say (not without a show of reason),

that where there is a positive and wonderful effect, and many strong

analogies, such as materials consumed, and all the usual signs of life

maintained, we are not to deny the existence of animal life because

we know no animal that consumes such food. Nay, he might observe with

truth, that the fuel actually consists of the chemical ingredients

which constitute the chief food of all animals, &c.; while, on the

other hand, his brother theorist, who caught a glimpse of the fire,

and detected the peculiar sounds of ebullition, might acquire a better

notion of the case, and form a theory more in consonance with fact.

(207.) Now, nothing is more common in physics than to find two, or even

many, _theories_ maintained as to the origin of a natural phenomenon.

For instance, in the case of heat itself, one considers it as a really

existing material fluid, of such exceeding subtlety as to penetrate

all bodies, and even to be capable of combining with them chemically;

while another regards it as nothing but a rapid vibratory or rotatory

motion in the ultimate particles of the bodies heated; and produces

a singularly ingenious train of mechanical reasoning to show, that

there is nothing contradictory to sound dynamical principles in such

a doctrine. Thus, again, with light: one considers it as consisting

in actual particles darted forth from luminous bodies, and acted

upon in their progress by forces of extreme intensity residing in the

substances on which they strike; another, in the vibratory motion of

the particles of luminous bodies, communicated to a peculiar subtle and

highly elastic ethereal medium, filling all space, and conveyed through

it into our eyes, as sounds are to our ears, by the undulations of the

air.

(208.) Now, are we to be deterred from framing hypotheses and

constructing theories, because we meet with such dilemmas, and

find ourselves frequently beyond our depth? Undoubtedly not. _Est

quodam prodire tenus si non datur ultra._ Hypotheses, with respect

to theories, are what presumed proximate causes are with respect to

particular inductions: they afford us motives for searching into

analogies; grounds of citation to bring before us all the cases which

seem to bear upon them, for examination. A well imagined hypothesis,

if it have been suggested by a fair inductive consideration of general

laws, can hardly fail at least of enabling us to generalize a step

farther, and group together several such laws under a more universal

expression. But this is taking a very limited view of the value and

importance of hypotheses: it may happen (and it has happened in the

case of the undulatory doctrine of light) that such a weight of analogy

and probability may become accumulated on the side of an hypothesis,

that we are compelled to admit one of two things; either that it is

an actual statement of what really passes in nature, or that the

reality, whatever it be, must run so close a parallel with it, as to

admit of some mode of expression common to both, at least in so far

as the phenomena actually known are concerned. Now, this is a very

great step, not only for its own sake, as leading us to a high point in

philosophical speculation, but for its applications; because whatever

conclusions we deduce from an hypothesis so supported must have at

least a strong presumption in their favour: and we may be thus led to

the trial of many curious experiments, and to the imagining of many

useful and important contrivances, which we should never otherwise have

thought of, and which, at all events, if verified in practice, are real

additions to our stock of knowledge and to the arts of life.

(209.) In framing a theory which shall render a rational account of

any natural phenomenon, we have _first_ to consider the agents on

which it depends, or the causes to which we regard it as ultimately

referable. These agents are not to be arbitrarily assumed; they

must be such as we have good inductive grounds to believe do exist

in nature, and do perform a part in phenomena analogous to those we

would render an account of; or such, whose presence in the actual

case can be demonstrated by unequivocal signs. They must be _veræ

causæ_, in short, which we can not only show to exist and to act,

but the laws of whose action we can derive independently, by direct

induction, from experiments purposely instituted; or at least make

such suppositions respecting them as shall not be contrary to our

experience, and which will remain to be verified by the coincidence of

the conclusions we shall deduce from them, with facts. For example,

in the theory of gravitation we suppose an agent,--_viz._ force, or

mechanical power,--to act on _any_ material body which is placed in

the presence of _any_ other, and to urge the two mutually towards each

other. This is a _vera causa_; for heavy bodies (that is, all bodies,

but some more, some less,) tend to, or endeavour to reach, the earth,

and require the exertion of force to counteract this endeavour, or

to keep them up. Now, that which opposes and neutralizes force _is_

force. And again, a plumb-line, which, when allowed to hang freely,

always hangs perpendicularly; is found to hang observably aside from

the perpendicular when in the neighbourhood of a considerable mountain;

thereby proving that a force is exerted upon it, which draws it towards

the mountain. Moreover, since it is a fact that the moon does circulate

about the earth, it must be drawn towards the earth by a force; for if

there were no force acting upon it, it would go on in a straight line

without turning aside to circulate in an orbit, and would, therefore,

soon go away and be lost in space. This force, then, which we call the

_force_ of gravity, is a real cause.

(210.) We have next to consider the laws which regulate the action of

these our primary agents; and these we can only arrive at in three

ways: 1st, By inductive reasoning; that is, by examining all the

cases in which we know them to be exercised, inferring, as well as

circumstances will permit, its amount or intensity in each particular

case, and then piecing together, as it were, these _disjecta membra_,

generalizing from them, and so arriving at the laws desired; 2dly,

By forming at once a bold hypothesis, particularizing the law,

and trying the truth of it by following out its consequences and

comparing them with facts; or, 3dly, By a process partaking of both

these, and combining the advantages of both without their defects,

viz. by assuming indeed the laws we would discover, but so generally

expressed, that they shall include an unlimited variety of particular

laws;--following out the consequences of this assumption, by the

application of such general principles as the case admits;--comparing

them in succession with all the particular cases within our knowledge;

and, lastly, _on this comparison_, so modifying and restricting the

general enunciation of our laws as to _make the results agree_.

(211.) All these three processes for the discovery of those general

elementary laws on which the higher theories are grounded are

applicable with different advantage in different circumstances.

We might exemplify their successive application to the case of

gravitation: but as this would rather lead into a disquisition too

particular for the objects of this discourse, and carry us too much

into the domain of technical mathematics, we shall content ourselves

with remarking, that the method last mentioned is that which

mathematicians (especially such as have a considerable command of

those general modes of representing and reasoning on quantity, which

constitute the higher analysis,) find the most universally applicable,

and the most efficacious; and that it is applicable with especial

advantage in cases where subordinate inductions of the kind described

in the last section have already led to laws of a certain generality

admitting of mathematical expression. Such a case, for instance,

is the elliptic motion of a planet, which is a general proposition

including the statement of an infinite number of particular _places_,

in which the laws of its motion allow it to be some time or other

found, and for which, of course, the law of force must be so assumed as

to account.

(212.) With regard to the first process of the three above enumerated,

it is in fact an induction of the kind described in § 185.; and all

the remarks we there made on that kind of induction apply to it in

this stage. The direct assumption of a particular hypothesis has been

occasionally practised very successfully. As examples, we may mention

Coulomb’s and Poisson’s theories of electricity and magnetism, in

both which, phenomena of a very complicated and interesting nature

are referred to the actions of attractive and repulsive forces,

following a law similar in its expression to the law of gravitation.

But the difficulty and labour, which, in the greater theories, always

attends the pursuit of a fundamental law into its remote consequences,

effectually precludes this method from being commonly resorted to as

a means of discovery, unless we have some good reason, from analogy

or otherwise, for believing that the attempt will prove successful,

or have been first led by partial inductions to particular laws which

naturally point it out for trial.

(213.) In this case the law assumes all the characters of a general

phenomenon resulting from an induction of particulars, but not yet

verified by comparison with _all_ the particulars, nor extended to all

that it is capable of including. (See § 171.) It is the verification

of such inductions which constitutes theory in its largest sense, and

which embraces an estimation of the influence of all such circumstances

as may modify the effect of the cause whose laws of action we have

arrived at and would verify. To return to our example: particular

inductions drawn from the motions of the several planets about the

sun, and of the satellites round their primaries, &c. having led us

to the general conception of an attractive force exerted by every

particle of matter in the universe on every other according to the law

to which we attach the name of gravitation; when we would verify this

induction, we must set out with assuming this law, considering the

whole system as subjected to its influence and implicitly obeying it,

and nothing interfering with its action; we then, for the first time,

perceive a train of modifying circumstances which had not occurred to

us when reasoning upwards from particulars to obtain the fundamental

law; we perceive that _all the planets_ must attract _each other_,

must therefore draw each other out of the orbits which they would

have if acted on only by the sun; and as this was never contemplated

in the inductive process, its validity becomes a question, which can

only be determined by ascertaining precisely how great a deviation

this new class of mutual actions will produce. To do this is no easy

task, or rather, it is the most difficult task which the genius of

man has ever yet accomplished: still, it _has_ been accomplished by

the mere application of the general laws of dynamics; and the result

(undoubtedly a most beautiful and satisfactory one) is, that all

those observed deviations in the motions of our system which stood

out as exceptions (§ 154.), or were noticed as residual phenomena and

reserved for further enquiry (§ 158.), in that imperfect view of the

subject which we got in the subordinate process by which we rose to

our general conclusion, prove to be the immediate consequences of the

above-mentioned mutual actions. As such, they are neither exceptions

nor residual facts, but fulfilments of general rules, and essential

features in the statement of the case, _without_ which our induction

would be invalid, and the law of gravitation positively untrue.

(214.) In the theory of gravitation, the law is all in all, applying

itself at once to the materials, and directly producing the result.

But in many other cases we have to consider not merely the laws

which regulate the actions of our ultimate causes, but a system of

mechanism, or a structure of parts, through the intervention of which

their effects become sensible to us. Thus, in the delicate and curious

electro-dynamic theory of Ampere, the mutual attraction or repulsion

of two magnets is referred to a more universal phenomenon, the mutual

action of electric currents, according to a certain fundamental law.

But, in order to bring the case of a magnet within the range of this

law, he is obliged to make a supposition of a peculiar structure or

mechanism, which constitutes a body a magnet, viz. that around each

particle of the body there shall be constantly circulating, in a

certain stated direction, a small current of electric fluid.

(215.) This, we may say, is too complex; it is artificial, and cannot

be granted: yet, if the admission of this or any other structure

tenfold more artificial and complicated will enable any one to present

in a general point of view a great number of particular facts,--to make

them a part of one system, and enable us to reason from the known to

the unknown, and actually to _predict facts before trial_,--we would

ask, why should it _not_ be granted? When we examine those instances

of nature’s workmanship which we can take to pieces and understand,

we find them in the highest degree artificial in our own sense of the

word. Take, for example, the structure of an eye, or of the skeleton of

an animal,--what complexity and what artifice! In the one, a _pellucid

muscle_; a lens formed with elliptical surfaces; a circular aperture

capable of enlargement or contraction without loss of form. In the

other, a framework of the most curious carpentry; in which occurs

not a single straight line, nor any known geometrical curve, yet all

evidently systematic, and constructed by rules which defy our research.

Or examine a crystallized mineral, which we can in some measure

dissect, and thus obtain direct evidence of an internal structure.

Neither artifice nor complication are here wanting; and though it

is easy to assert that these appearances are, after all, produced

by something which would be very simple, if we did but know it, it

is plain that the same might be _said_ of a steam-engine executing

the most complicated movements, previous to any investigation of its

nature, or any knowledge of the source of its power.

(216.) In estimating, however, the value of a theory, we are not to

look, _in the first instance_, to the question, whether it establishes

satisfactorily, or not, a particular process or mechanism; for of

this, after all, we can never obtain more than that indirect evidence

which consists in its leading to the same results. What, in the actual

state of science, is far more important for us to know, is whether our

theory truly represent _all_ the facts, and include _all_ the laws, to

which observation and induction lead. A theory which did this would,

no doubt, go a great way to establish any hypothesis of mechanism or

structure, which might form an essential part of it: but this is very

far from being the case, except in a few limited instances; and, till

it is so, to lay any great stress on hypotheses of the kind, except in

as much as they serve as a scaffold for the erection of general laws,

is to “quite mistake the scaffold for the pile.” Regarded in this

light, hypotheses have often an eminent use: and a facility in framing

them, if attended with an equal facility in laying them aside when

they have served their turn, is one of the most valuable qualities a

philosopher can possess; while, on the other hand, a bigoted adherence

to them, or indeed to peculiar views of any kind, in opposition to the

tenor of facts as they arise, is the bane of all philosophy.

(217.) There is no doubt, however, that the safest course, when it

can be followed, is to rise by inductions carried on among laws, as

among facts, from law to law, perceiving, as we go on, how laws which

we have looked upon as unconnected become particular cases, either

one of the other, or all of one still more general, and, at length,

blend altogether in the point of view from which we learn to regard

them. An example will illustrate what we mean. It is a general law,

that all hot bodies throw out or _radiate_ heat in all directions, (by

which we mean, not that heat is an actual substance darted out from hot

bodies, but only that the laws of the transmission of heat to distant

objects are similar to those which would regulate the distribution

of particles thrown forth in all directions,) and that other colder

bodies placed in their neighbourhood become hot, _as if_ they received

the heat so radiated. Again, all solid bodies which become heated in

one part _conduct_, or diffuse, the heat from that part through their

whole substance. Here we have two modes of communicating heat,--by

radiation, and by conduction; and both these have their peculiar,

and, to all appearance, very different laws. Now, let us bring a hot

and a cold body (of the same substance) gradually nearer and nearer

together,--as they approach, the heat will be communicated from the hot

to the cold one by the _laws of radiation_; and from the nearer to the

farther part of the colder one, as it gradually grows warm, by _those

of conduction_. Let their distance be diminished till they just lightly

touch. How does the heat _now_ pass from one to the other? Doubtless,

by radiation; for it may be proved, that in such a contact there is yet

an interval. Let them then be _forced_ together, and it will seem clear

that it must now be by _conduction_. Yet their _interval_ must diminish

gradually, as the force by which they are pressed together increases,

till they actually cohere, and form one. The law of continuity, then,

of which we have before spoken (§ 199.), forbids us to suppose that

the intimate nature of the process of communication is changed in this

transition from light to violent contact, and from that to actual

union. If so, we might ask, at what point does the change happen?

Especially since it is also demonstrable, that the particles of the

most solid body are not, really, in contact. _Therefore_, the laws of

conduction and radiation have a mutual dependence, and the former are

only extreme cases of the latter. If, then, we would rightly understand

what passes, or what is the process of nature in the slow communication

of heat through the substance of a solid, we must ground our enquiries

upon what takes place at a distance, and then urge the laws to which we

have arrived, up to their extreme case.

(218.) When two theories run parallel to each other, and each explains

a great many facts in common with the other, any experiment which

affords a crucial instance to decide between them, or by which one or

other must fall, is of great importance. In thus verifying theories,

since they are grounded on general laws, we may appeal, not merely

to particular cases, but to whole classes of facts; and we therefore

have a great range among the individuals of these for the selection

of some particular effect which ought to take place oppositely in the

event of one of the two suppositions at issue being right and the other

wrong. A curious example is given by M. Fresnel, as decisive, in his

mind, of the question between the two great opinions on the nature

of light, which, since the time of Newton and Huyghens, have divided

philosophers. (See § 207.) When two very clean glasses are laid one

on the other, if they be not perfectly flat, but one or both in an

almost imperceptible degree convex or prominent, beautiful and vivid

colours will be seen between them; and if these be viewed through a

red glass, their appearance will be that of alternate dark and bright

stripes. These stripes are formed _between_ the two surfaces in

apparent contact, as any one may satisfy himself by using, instead of

a flat _plate_ of glass for the upper one, a triangular-shaped piece,

called a prism, like a three-cornered stick, and looking through the

inclined side of it next the eye, by which arrangement the reflection

of light from the upper surface is prevented from intermixing with that

from the surfaces in contact. Now, the coloured stripes thus produced

are explicable on both theories, and are appealed to by both as strong

confirmatory facts; but there is a difference in one circumstance

according as one or the other theory is employed to explain them.

In the case of the Huyghenian doctrine, the intervals between the

bright stripes ought to appear _absolutely black_; in the other,

_half bright_, when so viewed through a prism. This curious case of

difference was tried as soon as the opposing consequences of the two

theories were noted by M. Fresnel, and the result is stated by him to

be decisive in favour of that theory which makes light to consist in

the vibrations of an elastic medium.

(219.) Theories are best arrived at by the consideration of general

laws; but most securely verified by comparing them with particular

facts, because this serves as a verification of the whole train

of induction, from the lowest term to the highest. But then, the

comparison must be made with facts purposely selected so as to include

every variety of case, not omitting extreme ones, and in sufficient

number to afford every reasonable probability of detecting error. A

single numerical coincidence in a final conclusion, however striking

the coincidence or important the subject, is not sufficient. Newton’s

theory of sound, for example, leads to a numerical expression for the

actual velocity of sound, differing but little from that afforded by

the correct theory afterwards explained by Lagrange, and (when certain

considerations not contemplated by him are allowed for) agreeing with

fact; yet this coincidence is no verification of Newton’s view of the

general subject of sound, which is defective in an essential point,

as the great geometer last named has very satisfactorily shown. This

example is sufficient to inspire caution in resting the verification of

theories upon any thing but a very extensive comparison with a great

mass of observed facts.

(220.) But, on the other hand, when a theory will bear the test

of such extensive comparison, it matters little how it has been

originally framed. However strange and, at first sight, inadmissible

its postulates may appear, or however singular it may seem that

such postulates should have been fixed upon,--if they only lead

us, by legitimate reasonings, to conclusions in exact accordance

with numerous observations purposely made under such a variety of

circumstances as fairly to embrace the whole range of the phenomena

which the theory is intended to account for, we cannot refuse to admit

them; or if we still hesitate to regard them as demonstrated truths, we

cannot, at least, object to receive them as temporary substitutes for

such truths, until the latter shall become known. If they suffice to

explain all the phenomena known, it becomes highly improbable that they

will not explain more; and if all their conclusions we have tried have

proved correct, it is probable that others yet untried will be found so

too; so that _in rejecting them altogether, we should reject all the

discoveries to which they may lead_.

(221.) In all theories which profess to give a true account of the

process of nature in the production of any class of phenomena, by

referring them to general laws, or to the action of general causes,

through a train of modifying circumstances; before we can apply those

laws, or trace the action of those causes in any assigned case, we

require to know the circumstances: we must have data whereon to ground

their application. Now, these can be learned only from observation;

and it may seem to be arguing in a vicious circle to have recourse

to observation for any part of those theoretical conclusions, by

whose comparison with fact the theory itself is to be tried. The

consideration of an example will enable us to remove this difficulty.

The most general law which has yet been discovered in chemistry is

this, that all the elementary substances in nature are susceptible of

entering into combination with each other only in fixed or _definite

proportions_ by weight, and not arbitrarily; so that when any two

substances are put together with a view to unite them, if their weights

are not in some certain determinate proportion, a complete combination

will not take place, but some part of one or the other ingredient will

remain over and above, and uncombined. Suppose, now, we have found a

substance having all the outward characters of a homogeneous or unmixed

body, but which, on analysis, we discover to consist of sulphur, and

lead in the proportion of 20 parts of the former to 130 of the latter

ingredient; and we would know whether this is to be regarded as a

verification of the law of definite proportions or an exception to

it. The question is reduced to this, whether the proportion 20 to 130

be or be not _that_ fixed and definite proportion, (or one of them,

if there be more than one proportion possible,) in which, according

to the law in question, sulphur and lead can combine; now, this can

never be decided by merely looking at the law in all its generality.

It is clear, that when particularized by restricting its expression

to sulphur and lead, the law should state _what are_ those particular

fixed proportions in which these bodies can combine. That is to say,

there must be certain data or numbers, by which these are distinguished

from all other bodies in nature, and which require to be known before

we can apply the general law to the particular case. To determine such

data, observation must be consulted; and if we were to have recourse

to that of the combination of the two substances in question with each

other, no doubt there would be ground for the logical objection of

a vicious circle: but this is not done; the determination of these

numerical data is derived from experiments purposely made on a great

variety of different combinations, among which that under consideration

does not of necessity occur, and all these being found, independently

of each other, to agree in giving the same results, they are therefore

safely assumed as part of the system. Thus, the law of definite

proportions, when applied to the actual state of nature, requires two

separate statements, the one announcing the general law of combination,

the other particularizing the numbers appropriate to the several

elements of which natural bodies consist, or the data of nature. Among

these data, if arranged in a list, there will be found opposite to the

element sulphur the number 16, and opposite to lead, 104[48]; and since

20 is to 130 in the exact proportion of 16 to 104, it appears that the

combination in question affords a satisfactory verification of the law.

(222.) The great importance of physical data of this description,

and the advantage of having them well determined, will be obvious,

if we consider, that a list of them, when taken in combination with

the general law, affords the means of determining at once the exact

proportion of the ingredients of all natural compounds, if we only know

the place they hold in the system. In chemistry, the number of admitted

elements is between fifty and sixty, and new ones are added continually

as the science advances. Now, the moment the number corresponding to

any new substance added to the list is determined, we have, in fact,

ascertained all the proportions in which it can enter into combination

with all the others, so that a careful experiment made with the object

of determining this number is, in fact, equivalent to as many different

experiments as there are binary, ternary, or yet more complicated

combinations capable of existing, into which the new substance may

enter, as an ingredient.

(223.) The importance of obtaining exact physical data can scarcely

be too much insisted on, for without them the most elaborate theories

are little better than mere inapplicable forms of words. It would be

of little consequence to be informed, abstractedly, that the sun and

planets attract each other, with forces proportional to their masses,

and inversely as the squares of their distances: but, as soon as we

know the data of our system, as soon as we have an accurate statement

(no matter how obtained) of the distances, masses, and actual motions

of the several bodies which compose it, we need no more to enable us to

predict all the movements of its several parts, and the changes that

will happen in it for thousands of years to come; and even to extend

our views backwards into time, and recover from the past, phenomena,

which no observation has noted, and no history recorded, and which yet

(it is possible) may have left indelible traces of their existence in

their influence on the state of nature in our own globe, and those of

the other planets.

(224.) The proof, too, that our data _are_ correctly assumed, is

involved in the general verification of the whole theory, of which,

when once assumed, they form a part; and the same comparison with

observation which enables us to decide on the truth of the abstract

principle, enables us, at the same time, to ascertain whether we

have fixed the values of our data in accordance with the actual

state of nature. If not, it becomes an important question, whether

the assumed values can be corrected, so as to bring the results of

theory to agree with facts? Thus it happens, that as theories approach

to their perfection, a more and more exact determination of data

becomes requisite. Deviations from observed fact, which, in a first

or approximative verification, may be disregarded as trifling, become

important when a high degree of precision is attained. A difference

between the calculated and observed places of a planet, which would

have been disregarded by Kepler in his verification of the law of

elliptic motion, would now be considered fatal to the theory of

gravity, unless it could be shown to arise from an erroneous assumption

of some of the numerical data of our system.

(225.) The observations most appropriate for the ready and exact

determination of physical data are, therefore, those which it is most

necessary to have performed with exactness and perseverance. Hence

it is, that their performance, in many cases, becomes a national

concern, and observatories are erected and maintained, and expeditions

despatched to distant regions, at an expense which, to a superficial

view, would appear most disproportioned to their objects. But it

may very reasonably be asked why the direct assistance afforded by

governments to the execution of continued series of observations

adapted to this especial end should continue to be, as it has hitherto

almost exclusively been, confined to astronomy.

(226.) Physical data intended to be employed as elements of calculation

in extensive theories, require to be known with a much greater degree

of exactness than any single observation possesses, not only on account

of their dignity and importance, as affording the means of representing

an indefinite multitude of facts; but because, in the variety of

combinations that may arise, or in the changes that circumstances may

undergo, cases will occur when any trifling error in one of the data

may become enormously magnified in the final result to be compared

with observation. Thus, in the case of an eclipse of the sun, when

the moon enters very obliquely upon the sun’s disc, a trifling error

in the diameter of either the sun or moon may make a great one in the

time when the eclipse shall be announced to commence. It ought to

be remarked, that these are, of all others, the conjunctures where

observations are most available for the determination of data; for,

by the same rule that a small change in the data will, in such cases,

produce a great one in the thing to be observed; so, _vice versâ_, any

moderate amount of error, committed in an observation undertaken for

ascertaining its value, can produce but a very trifling one in the

_reverse_ calculation from which the data come to be determined by

observation. This remark extends to every description of physical data

in every department of science, and is never to be overlooked when the

object in view is the determination of data with the last degree of

precision.

(227.) But how, it may be asked, are we to ascertain _by_ observation,

data more precise than observation itself? How are we to conclude the

value of that which we do not see, with greater certainty than that

of quantities which we actually see and measure? It is the number of

observations which may be brought to bear on the determination of data

that enables us to do this. Whatever error we may commit in a single

determination, it is highly improbable that we should always err the

same way, so that, when we come to take an average of a great number of

determinations, (unless there be some constant cause which gives a bias

one way or the other,) we cannot fail, at length, to obtain a very near

approximation to the truth, and, even allowing a bias, to come much

nearer to it than can fairly be expected from any single observation,

liable to be influenced by the same bias.

(228.) This useful and valuable property of the average of a great

many observations, that it brings us nearer to the truth than any

single observation can be relied on as doing, renders it the most

constant resource in all physical enquiries where accuracy is desired.

And it is surprising what a rapid effect, in equalizing fluctuations

and destroying deviations, a moderate multiplication of individual

observations has. A better example can hardly be taken than the average

height of the quicksilver in the common barometer, which measures

the pressure of the air, and whose fluctuations are proverbial.

Nevertheless, if we only observe it regularly every day, and, at the

end of each month, take an average of the observed heights, we shall

find the fluctuations surprisingly diminished in amount; and if we

go on for a whole year, or for many years in succession, the annual

averages will be found to agree with still greater exactness. This

equalizing power of averages, by destroying all such fluctuations as

are irregular or accidental, frequently enables us to obtain evidence

of fluctuations really regular, periodic in their recurrence, and

so much smaller in their amount than the accidental ones, that, but

for this mode of proceeding, they never would have become apparent.

Thus, if the height of the barometer be observed four times a day,

constantly, for a few months, and the averages taken, it will be seen

that a regular _daily_ fluctuation, of very small amount, takes place,

the quicksilver rising and falling twice in the four-and-twenty hours.

It is by such observations that we are enabled to ascertain--what no

single measure (unless by a fortunate coincidence), could give us any

idea, and never any certain knowledge of--the true _sea level_ at any

part of the coast, or the height at which the water of the ocean would

stand, if perfectly undisturbed by winds, waves, or tides: a subject of

very great importance, and upon which it would be highly desirable to

possess an extensive series of observations, at a great many points on

the coasts of the principal continents and islands over the whole globe.

(229.) In all cases where there is a direct and simple relation between

the phenomenon observed and a single _datum_ on which it depends,

every single observation will give a value of this quantity, and the

average of all (under certain restrictions) will be its exact value. We

say, under certain restrictions; for, if the circumstances under which

the observations are made be not alike, they may not all be equally

favourable to exactness, and it would be doing injustice to those most

advantageous, to class them with the rest. In such cases as these,

as well as in cases where the _data_ are numerous and complicated

together, so as not to admit of single, separate determination (a

thing of continual occurrence), we have to enter into very nice, and

often not a little intricate, considerations respecting the _probable_

accuracy of our results, or the limits of error within which it is

_probable_ they lie. In so doing we are obliged to have recourse to

a refined and curious branch of mathematical enquiry, called the

doctrine of probabilities, the object of which (as its name imports)

is to reduce our estimation of the probability of any conclusion to

calculation, so as to be able to give more than a mere guess at the

degree of reliance which ought to be placed in it.

(230.) To give some general idea of the considerations which such

computations involve, let us imagine a person firing with a pistol

at a wafer on a wall ten yards distant: we might, in a general way,

take it for granted, that he would hit the wall, but not the wafer,

at the first shot; but if we would form any thing like a probable

conjecture of _how near_ he would come to it, we must first have an

idea of his skill. No better way of judging could be devised than

by letting him fire a hundred shots at it, and marking where they

all struck. Suppose this done,--suppose the wafer has been hit once

or twice, that a certain number of balls have hit the wall within an

inch of it, a certain number between one and two inches, and so on,

and that one or two have been some feet wide of the mark. Still the

question arises, what estimate are we thence to form of his skill? how

_near_ (or nearer) may we, after this experience, safely, or at least

not unfairly, bet that he will come to the mark the next subsequent

shot? This the laws of probability enable us on such data to say.

Again, suppose, _before_ we were allowed to measure the distances,

the wafer were to have been taken away, and we were called upon, on

the mere evidence of the marks on the wall, to say where it had been

placed; it is clear that no reasoning would enable any one to say with

certainty; yet there is assuredly one place which we may fix on with

greater probability of being right than any other. Now, this is a very

similar case to that of an observer--an astronomer for example--who

would determine the exact place of a heavenly body. He points to it

his telescope, and obtains a series of results disagreeing among

themselves, but yet all agreeing within certain limits, and only a

comparatively small number of them deviating considerably from the mean

of all; and from these he is called upon to say, definitively, what he

shall consider to have been the most probable place of his star at the

moment. Just so in the calculation of physical _data_; where no two

results agree exactly, and where all come within limits, some wide,

some close, what have we to guide us when we would make up our minds

what to conclude respecting them? It is evident that any system of

calculation that can be shown to lead of necessity to the most probable

conclusion where certainty is not to be had must be valuable. However,

as this doctrine is one of the most difficult and delicate among the

applications of mathematics to natural philosophy, this slight mention

of it must suffice at present.

(231.) In the foregoing pages we have endeavoured to explain the spirit

of the methods to which, since the revival of philosophy, natural

science has been indebted for the great and splendid advances it has

made. What we have all along most earnestly desired to impress on the

student is, that natural philosophy is essentially united in all its

departments, through all which one spirit reigns and one method of

enquiry applies. It cannot, however, be studied as a whole, without

subdivision into parts; and, in the remainder of this discourse, we

shall therefore take a summary view of the progress which has been made

in the different branches into which it may be most advantageously

so subdivided, and endeavour to give a general idea of the nature of

each, and of its relations to the rest. In the course of this, we shall

have frequent opportunity to point out the influence of those general

principles we have above endeavoured to explain, on the progress of

discovery. But this we shall only do as cases arise, without entering

into any regular analysis of the history of each department with that

view. Such an analysis would, indeed, be a most useful and valuable

work, but would far exceed our present limits. We are not, however,

without a hope that this great desideratum in science will, ere long,

be supplied from a quarter every way calculated to do it justice.

PART III.

OF THE SUBDIVISION OF PHYSICS INTO DISTINCT BRANCHES,

AND THEIR MUTUAL RELATIONS.

CHAPTER I.

OF THE PHENOMENA OF FORCE, AND OF THE CONSTITUTION OF NATURAL BODIES.

(232.) Natural History may be considered in two very different lights:

either, 1st, as a collection of facts and objects presented by nature,

from the examination, analysis, and combination of which we acquire

whatever knowledge we are capable of attaining both of the order of

nature, and of the agents she employs for producing her ends, and from

which, therefore, all sciences arise; or, 2dly, as an assemblage of

phenomena to be explained; of effects to be deduced from causes; and of

materials prepared to our hands, for the application of our principles

to useful purposes. Natural history, therefore, considered in the

one or the other of these points of view, is either the beginning or

the end of physical science. As it offers to us, in a confused and

interwoven mass, the elements of all our knowledge, our business is to

disentangle, to arrange, and to present them in a separate and distinct

state: and to this end we are called upon to resolve the important but

complicated problem,--Given the effect, or assemblage of effects, to

find the causes. The principles on which this enquiry relies are those

which constitute the relation of cause and effect, as it exists with

reference to our minds; and their rules and mode of application have

been attempted to be sketched out, (though in far less detail than the

intrinsic interest of the subject, both in a logical and practical

point of view, would demand,) in the foregoing pages. It remains

now to bring together, in a summary statement, the results of the

general examination of nature, so far as it has been prosecuted to the

discovery of natural agents, and the mode in which they act.

(233.) The first great agent which the analysis of natural phenomena

offers to our consideration, more frequently and prominently than

any other, is force. Its effects are either, 1st, to counteract the

exertion of opposing force, and thereby to maintain _equilibrium_; or,

2dly, to produce _motion_ in matter.

(234.) Matter, or that, whatever it be, of which all the objects in

nature which manifest themselves directly to our senses consist,

presents us with two general qualities, which at first sight appear

to stand in contradiction to each other--activity and inertness. Its

activity is proved by its power of spontaneously setting other matter

in motion, and of itself obeying their mutual impulse, and moving

under the influence of its own and other force; inertness, in refusing

to move unless obliged to do so by a force impressed externally, or

mutually exerted between itself and other matter, and by persisting in

its state of motion or rest unless disturbed by some external cause.

Yet in reality this contradiction is only apparent. Force being the

cause, and motion the effect produced by it on matter, to say that

matter is inert, or has _inertia_, as it is termed, is only to say

that the cause is expended in producing its effect, and that the

same cause cannot (without renewal) produce double or triple its own

proper effect. In this point of view, equilibrium may be conceived as

a continual production of two opposite effects, each undoing at every

instant what the other has done.

(235.) However, if this should appear too metaphysical, at all events

this difference of effects gives rise to two great divisions of the

science of force, which are commonly known by the names of STATICS and

DYNAMICS; the latter term, which is general, and has been used by us

before in its general sense, being usually confined to the doctrine

of motion, as produced and modified by force. Each of these great

divisions again branches out into distinct subdivisions, according as

we consider the equilibrium or motion of matter in the three distinct

states in which it is presented to us in nature, the solid, liquid, and

aëriform state, to which, perhaps, ought to be added the _viscous_,

as a state intermediate between that of solidity and fluidity, the

consideration of which, though very obscure and difficult, offers a

high degree of interest on a variety of accounts.

_Statics and Dynamics._

(236.) The principles have been definitively fixed by Galileo and his

successors, down to Newton, on a basis of sound induction; and as they

are perfectly general, and apply to every case, they are competent,

as we have already before observed, to the solution of every problem

that can occur in the deductive processes, by which phenomena are

to be explained, or effects calculated. Hence, they include every

question that can arise respecting the motions and rest of the smallest

particles of matter, as well as of the largest masses. But the mode of

reasoning from these general principles differs materially, whether

we consider them as applied to masses of matter of a sensible size,

or to those excessively minute, and perhaps indivisible, molecules of

which such masses are composed. The investigations which relate to the

latter subject are extremely intricate, as they necessarily involve the

consideration of the hypotheses we may form respecting the intimate

constitution of the several sorts of bodies above enumerated.

(237.) On the other hand, those which respect the equilibrium and

motions of sensible masses of matter are happily capable of being

so managed as to render unnecessary the adoption of any particular

hypothesis of structure. Thus, in reasoning respecting the application

of forces to a solid mass, we suppose its parts indissolubly and

unalterably connected; it matters not by what tie, provided this

condition be satisfied, that one point of it cannot be moved without

setting all the rest in motion, so that the relative situation of the

parts one among another be not changed. This is the abstract notion

of a solid which the mechanician employs in his reasonings. And their

conclusions will apply to natural bodies, of course, only so far as

they conform to such a definition. In strictness of speaking, however,

there are no bodies which absolutely conform to it. No substance is

known whose parts are absolutely incapable of yielding one among

another; but the amount by which they do yield is so excessively

small as to be demonstrably incapable, in most cases, of having any

influence on the results: and in those where it has such influence, an

especial investigation of its amount can always be made. This gives

rise to two subdivisions of the application of mechanical reasonings

to solid masses. Those which refer to the action of forces on flexible

or elastic, and on inflexible or rigid, bodies, comprehending under

the latter all such whose resistance to flexure or fracture is so

very great as to permit our adoption of the language and ideas of the

extreme case without fear of material error.

(238.) In like manner, when we reason respecting the action of forces

on a fluid mass, all we have occasion to assume is, that its parts

are freely moveable one among the other. If, besides this, we choose

to regard a fluid as incompressible, and deduce conclusions on this

supposition, they will hold good only so far as there may be found

such fluids in nature. Now, in strictness, there are none such; but,

practically speaking, in the greater number of cases their resistance

to compression is so very great that the result of the reasoning so

carried on is not sensibly vitiated; and, in the remaining cases,

the same general principles enable us to enter on a special enquiry

directed to this point: and hence the division of fluids, in mechanical

language, into compressible and incompressible, the latter being only

the extreme or limiting case of the former.

(239.) As we propose here, however, only to consider what is the actual

constitution of nature, we shall regard all bodies, as they really

are, more or less flexible and yielding. We know for certain, that

the space which any material body appears to occupy is not entirely

filled by it; because there is none which by the application of a

sufficient force may not be _compressed_ or forced into a smaller

space, and which, either wholly, as in air or liquids, or in part, as

in the greater number of solids, will not recover its former dimensions

when the force is taken off. In the case of air, this condensation

may be urged to almost any extent; and not only does a mass of air

so condensed completely recover its original bulk, when the applied

pressure is removed, but if that ordinary pressure under which it

exists at the earth’s surface (and which arises from the weight of

the atmosphere) be also removed by an air-pump, it will still further

dilate itself without limit so far as we have yet been able to try

it. Hence we are led to the conclusion that the particles of air are

mutually elastic, and have a _tendency to recede from one another_,

which can only be counteracted by _force_, and therefore is itself a

force of the repulsive kind. Nevertheless, as air is heavy, and as

gravitation is a universal property of matter, there is no doubt that

this repulsive tendency must have a limit, and that there is a distance

to which, if the particles of the air could be removed from each other,

their mutual repulsion would cease, and an attraction take its place.

This limit is probably attained at some very great height above the

earth’s surface, beyond which, of course, its atmosphere cannot extend.

(240.) What, however, we can only conclude by this or similar reasoning

respecting air, we see distinctly in liquids. They are all, though

in a small degree, compressible, and recover their former dimensions

completely when the pressure is removed; but they cannot be dilated (by

mechanical means), and have no tendency, while they remain liquids, to

enlarge themselves beyond a certain limit, and therefore they assume a

determinate _surface_ while at rest, and their parts actually resist

further separation with a considerable force, thus giving rise to the

phenomenon of the _cohesion of liquids_.

(241.) Both in air and in liquids, however, the most perfect freedom

of motion of the parts among each other subsists, which could hardly

be the case if they were not separate and independent of each other.

And from this, combined with the foregoing considerations, it has been

concluded that they do not actually touch, but are kept asunder at

determinate distances from each other, by the constant action of the

two forces of attraction and repulsion, which are supposed to balance

and counteract each other at the ordinary distances of the particles,

but to prevail, the one, or the other, according as they are forcibly

urged together or pulled asunder.

(242.) In solids, however, the case is very different. The mutual free

motion of their parts _inter se_ is powerfully impeded, and in some

almost destroyed. In some, a slow and gradual change of figure may be

produced to a great extent, by pressure or blows, as for instance in

the metals, clay, butter, &c.; in others, fracture is the consequence

of any attempt to change the figure by violence beyond a certain very

small limit. In solids, then, it is evident, that the consideration of

their intimate structure has a very great influence in modifying the

general results of the action of such attractive and repulsive forces

as may be assumed to account for the phenomena they present; yet the

general facts that their parts _cohere_ with a certain energy, and that

they resist displacement or intrusion on the part of other bodies,

are sufficient to demonstrate at least the existence of such forces,

whatever obscurity may subsist as to their mode of action.

(243.) This division of bodies into airs, liquids, and solids, gives

rise, then, to three distinct branches of mechanical science, in each

of which the general principles of equilibrium and motion have their

peculiar mode of application; viz. pneumatics, hydrostatics, and what

might, without impropriety, be termed stereostatics.

_Pneumatics._

(244.) Pneumatics relates to the equilibrium or movements of aërial

fluids under all circumstances of pressure, density, and elasticity.

The weight of the air, and its pressure on all the bodies on the

earth’s surface, were quite unknown to the ancients, and only first

perceived by Galileo, on the occasion of a sucking-pump refusing to

draw water above a certain height. Before his time it had always been

supposed that water rose by suction in a pipe, in consequence of a

certain natural _abhorrence of a vacuum_ or empty space, which obliged

the water to enter by way of supplying the place of the air sucked out.

But if any such abhorrence existed, and had the force of an _acting

cause_, which could urge water a single foot into a pipe, there is no

reason why the same principle should not carry it up two, three, or

any number of feet; none why it should suddenly stop short at a certain

height, and refuse to rise higher, however violent the suction might

be, nay, even fall back, if purposely forced up too high.

(245.) Galileo, however, at first contented himself with the

conclusion, that the natural abhorrence of a vacuum was not strong

enough to sustain the water more than about thirty-two feet above

its level; and, although the true cause of the phenomenon at length

occurred to him, in the pressure of the air on the general surface,

it was not satisfactorily demonstrated till his pupil, Torricelli,

conceived the happy idea of instituting an experiment on a small

scale by the use of a much heavier liquid, mercury, instead of water,

and, in place of sucking out the air from above, employing the much

more effectual method of filling a long glass tube with mercury, and

inverting it into a basin of the same metal. It was then at once seen,

as by a _glaring instance_, that the maintenance of the mercury in the

tube (which is nothing else than the common barometer) was the effect

of a perfectly definite external cause, while its fluctuations from day

to day, with the varying state of the atmosphere, strongly corroborated

the notion of its being due to the pressure of the external air on the

surface of the mercury in the reservoir.

(246.) The discovery of Torricelli was, however, at first much

misconceived, and even disputed, till the question was finally decided

by appeal to a _crucial instance_, one of the first, if not the very

first on record in physics, and for which we are indebted to the

celebrated Pascal. His acuteness perceived that if the weight of the

incumbent air be the direct cause of the elevation of the mercury,

it must be measured by the amount of that elevation, and therefore

that, by carrying a barometer up a high mountain, and so ascending

into the atmosphere _above_ a large portion of the incumbent air, the

pressure, as well as the length of the column sustained by it, must

be diminished; while, on the other hand, if the phenomenon were due

to the cause originally assigned, no difference could be expected to

take place, whether the observation were made on a mountain or on the

plain. Perhaps the decisive effect of the experiment which he caused

to be instituted for the purpose, on the Puy de Dôme, a high mountain

in Auvergne, while it convinced every one of the truth of Torricelli’s

views, tended more powerfully than any thing which had previously been

done in science to confirm, in the minds of men, that disposition to

experimental verification which had scarcely yet taken full and secure

root.

(247.) Immediately on this discovery followed that of the air-pump,

by Otto von Guericke of Magdeburgh, whose aim seems to have been to

decide the question, whether a vacuum could or could not exist, by

endeavouring to make one. The imperfection of his mechanism enabled

him only to diminish the aërial contents of his receivers, not

entirely to empty them; but the curious effects produced by even a

partial exhaustion of air speedily excited attention, and induced our

illustrious countryman, Robert Boyle, to the prosecution of those

experiments which terminated in his hands, and in those of Hauksbee,

Hooke, Mariotte, and others, in a satisfactory knowledge of the

general law of the equilibrium of the air under the influence of

greater or less pressures. These discoveries have since been extended

to all the various descriptions of aërial fluids which chemistry has

shown to exist, and to maintain their aëriform state under artificial

pressure, and even to those which may be produced from liquids reduced

to a state of vapour by heat, so long as they retain that state.

(248.) The manner in which the observed law of equilibrium of an

elastic fluid, like air, may be considered to originate in the mutual

repulsion of its particles, has been investigated by Newton, and

the actual statement of the law itself, as announced by Mariotte,

“that the density of the air, or the quantity of it contained in the

same space, is, _cæteris paribus_, proportional to the pressure it

supports,” has recently been verified within very extensive limits by

direct experiment, by a committee of the Royal Academy of Paris. This

law contains the principle of solution of every dynamical question

that can occur relative to the equilibrium of elastic fluids, and is

therefore to be regarded as one of the highest _axioms_ in the science

of pneumatics.

_Hydrostatics._

(249.) The principles of the equilibrium of liquids, understanding

by this word such fluids as do not, though quite at liberty, attempt

to dilate themselves beyond a certain point, are at once few and

simple. The first steps towards a knowledge of them were made by

Archimedes, who established the general fact, that a solid immersed in

a liquid loses a portion of its weight equal to that of the liquid it

displaces. It seems very astonishing, after this, that it should not

have been at once concluded that the weight thus said to be _lost_ is

only _counteracted_ by the upward pressure of the liquid, and that,

therefore, a portion of any liquid, surrounded on all sides by a liquid

of the same kind, does really exert its weight in keeping its place.

Yet the prejudice that “liquids do not gravitate in their natural

place” kept its ground, and was only dispelled with the mass of error

and absurdity which the introduction of a rational and experimental

philosophy by Galileo swept away.

(250.) The hydrostatical law of _the equal pressure of liquids in all

directions_, with its train of curious and important consequences, is

an immediate conclusion from the perfect mobility of their parts among

one another, in consequence of which each of them tends to recede from

an excess of pressure on one side, and thus bears upon the rest, and

distributes the pressure among its neighbours. In this form it was

laid down by Newton, and has proved one of the most useful and fertile

principles of physico-mathematical reasoning on the equilibrium of

fluid masses, as affording a means of tracing the action of a force

applied at any point of a liquid through its whole extent. It applies,

too, without any modification, to expansible fluids as well as to

liquids; and, in the applications of geometry to this subject, enables

us to dispense with any minute and intricate enquiries as to the mode

in which individual particles act on each other.

(251.) In a practical point of view, this law is remarkable for the

directness of its application to useful purposes. The immediate and

perfect distribution of a pressure applied on any one part, however

small, of a fluid surface through the whole mass, enables us to

communicate _at one instant_ the same pressure to any number of such

parts by merely increasing the surface of the fluid, which may be done

by enlarging the containing vessel; and if the vessel be so constructed

that a large portion of its surface shall be moveable together, the

pressures on all the similar parts of this portion will be united into

one consentient force, which may thus be increased to any extent we

please. The hydraulic press, invented by Bramah, (or rather applied by

him after a much more ancient inventor, Stevin,) is constructed on this

principle. A small quantity of water is driven by sufficient pressure

into a vessel _already full_, and provided with a moveable surface or

piston of great size. Under such circumstances something must give way;

the great surface of the piston accumulates the pressure on it to such

an extent that nothing can resist its violence. Thus, trees are torn

up by the roots; piles extracted from the earth; woollen and cotton

goods compressed into the most portable dimensions; and even hay, for

military service, reduced to such a state of coercion as to be easily

packed on board transports.

(252.) Liquids differ from aëriform fluids by their _cohesion_, which

may be regarded as a kind of approach to a solid state, and was so

regarded by Bacon (193.). Indeed, there can be little doubt that

the solid, liquid, and aëriform states of bodies are merely stages

in a progress of gradual transition from one extreme to the other;

and that, however strongly marked the distinctions between them may

appear, they will ultimately turn out to be separated by no sudden or

violent line of demarcation, but shade into each other by insensible

gradations. The late experiments of Baron Cagnard de la Tour may be

regarded as a first step towards the full demonstration of this (199.).

But the cohesion of liquids is not, like that of solids, so modified

by their structure in other respects as to destroy the mobility of

their parts one among another (unless in those cases of nearer approach

to the solid state which obtain in viscid or gummy liquids). On the

contrary, the two qualities co-exist, and give rise to a number of

curious and intricate phenomena.

(253.) One of the most remarkable of these is capillary attraction,

or capillarity as it is sometimes called. Every body has remarked the

adhesion of water to glass. The elevation of the general surface of

the liquid where it is in contact with the containing vessel; the form

of a drop suspended at the under side of a solid: these are instances

of capillary attraction. If a small glass tube with a bore as fine as

a hair be immersed in water, the water will be observed to rise in

it to a certain height, and to assume a concave surface at its upper

extremity. The attraction of the glass on the water, and the cohesion

of the parts of the water to each other, are no doubt the joint causes

of this curious effect; but the mode of action is at once obscure and

complex; and although the researches of Laplace and Young have thrown

great light on it, further investigation seems necessary before we can

be said distinctly to understand it.

(254.) As the capillarity and cohesion of the parts of liquids shows

them to possess the power of mutual attraction, so their elasticity

demonstrates that they also possess that of repulsion when forcibly

brought nearer than their natural state. From the extremely small

extent to which the compression of liquids can be carried by any force

we can employ, compared with that of air, we must conclude that this

repulsion is much more violent in the former than in the latter, but

counteracted also by a more powerful force of attraction. So much more

powerful, indeed, is the resistance of liquids to compression, that

they were usually regarded as incompressible; an opinion corroborated

by a celebrated experiment made at Florence, in which water was forced

through the pores (as it was said) of a golden ball. More recent

experiments by Canton, and since by Perkins, Oërsted, and others,

have demonstrated however the contrary, and assigned the amount of

compression.

(255.) The consideration of the motions of fluids, whether liquid

or expansible, is infinitely more complicated than that of their

equilibrium. When their motions are slow, it is reasonable to suppose

that the law of the equable distribution of pressure obtains; but in

very rapid displacements of their parts one among the other, it is not

easy to see how such an equable distribution can be accomplished, and

some phenomena exist which seem to indicate a contrary conclusion.

(256.) Independent of this, there are difficulties of an almost

insuperable nature to the regular deductive application of the general

principles of mechanics to this subject, which arise from the excessive

intricacy of the pure mathematical enquiries to which its investigation

leads. It was Newton who set the example of a first attempt to draw

any conclusions respecting the motion of fluid masses by direct

reasoning from dynamical principles, and thus laid the foundation of

HYDRODYNAMICS; but it was not till the time of D’Alembert that the

method of reducing any question respecting the motions of fluids under

the action of forces to strict mathematical investigation could be said

to be completely understood. But the cases even now in which this mode

of treating such questions can be applied with full satisfaction are

few in comparison of those in which the experimental method of enquiry

as already observed (189.) is preferable. Such, for example, is that

of the resistance of fluids to bodies moving through them; a knowledge

of which is of great importance in naval architecture and in gunnery,

where the resistance of the air acts to an enormous extent. Such,

too, among the practical subjects which depend mainly on this branch

of science, are the use of sails in navigation; the construction of

windmills, and water-wheels; the transmission of water through pipes

and channels; the construction of docks and harbours, &c.

_Nature of Solids in general._

(257.) The intimate constitution of solids is, in all probability, very

complicated, and we cannot be said to know much of it. By some recent

delicate experiments on the dimensions of wires violently strained, it

has been shown that they are to a certain small extent capable of being

dilated by tension, as they are also of being compressed by pressure,

but within limits even narrower than those of liquids. Usually, when

strained too far, they break, and refuse to re-unite; or, if compressed

too forcibly, take a permanent contraction of dimension. Thus, wood

may be indented by a blow, and metals rendered denser and heavier by

hammering or rolling. There is a certain degree of confusion prevalent

in ordinary language about the hardness, elasticity, and other similar

qualities, of solids, which it may be well to remove. Hardness is that

disposition of a solid which renders it difficult to displace its

parts among themselves. Thus, steel is harder than iron; and diamond

almost infinitely harder than any other substance in nature: but the

compressibility of steel, or the extent to which it will yield to a

given pressure and recover itself, is not much less than that of soft

iron, and that of ice is very nearly the same with that of water.

(258.) Again, we call Indian rubber a very elastic body, and so it is;

but in a different sense from steel. Its parts admit of great mutual

displacement without permanent dislocation; however distorted, it

recovers its figure readily, but with a small force. Yet, if Indian

rubber were to be enclosed in a space that it just filled, so as not to

permit its parts to yield laterally, doubtless it would resist actual

compression with great violence. Here, then, we have an instance

of two kinds of elasticity in one substance; a feebler effort of

recovery from distorted figure, and a more violent one from a state

of altered dimension. Both, however, originate in the same causes,

and are referable to the same principles; the former being in fact

only a modified case of the latter, as the effort of a steel spring,

when bent, to recover its former shape, is referable to the same

forces which give to steel its hardness and strength to resist actual

compression and fracture.

(259.) The toughness of a solid, or that quality by which it will

endure heavy blows without breaking, is again distinct from hardness

though often confounded with it. It consists in a certain yielding of

parts with a powerful general cohesion, and is compatible with various

degrees of elasticity. Malleability is again another quality of solids,

especially metals, quite distinct from toughness, and depends on their

capability of being deprived of their figure without an effort to

recover it and without fracture.

(260.) Tenacity, again, is a property of solids more directly depending

on the cohesion of their parts than toughness. It consists in their

power of resisting separation by a strain steadily applied, while the

quality of toughness is materially influenced by their disposition

to communicate through their substance the jarring effect of a

blow. Accordingly, the tenacity of a solid is a direct measure of

the cohesive attraction of its parts, and is the best proof of the

existence of such a power.

_Crystallography._

(261.) It cannot be supposed that these and many other tangible

qualities, as they may be called, should subsist in solids without

a corresponding mechanism in their internal structure. That they

have such a mechanism, and that a very curious and intricate one,

the phenomena of crystallography sufficiently show. This interesting

and beautiful department of natural science is of comparatively very

modern date. That many natural substances affected certain forms must

have been known from the earliest times. Pliny appears to have been

acquainted with this fact, at least in some instances, as he describes

the forms of quartz and diamond. But till the time of Linnæus no

material attention seems to have been bestowed on the subject. He,

however, observed, and described with care, the crystalline forms

of a variety of substances, and even regarded them as so definite a

character of the solids which assumed them, that he supposed every

particular form to be generated by a particular salt. Romé de l’Isle

pursued the study of the crystalline forms of bodies yet farther. He

first ascertained the important fact of the constancy of the angles

at which their faces meet; and observing further that many of them

appear in several different shapes, first conceived the idea that these

shapes might be reducible to one, appropriated in a peculiar manner to

each _substance_, and modified by strict geometrical laws. Bergmann,

reasoning on a fact imparted to him by his pupil Gahn, made a yet

greater step, and showed how at least one species of crystal might

be built up of thin laminæ ranged in a certain order, and following

certain rules of superposition. He failed, however, in deducing just

and general conclusions from this remark, which, correctly viewed, is

the foundation of the most important law of crystallography, that which

connects the primitive form with other forms capable of being exhibited

by the same substance, by a certain fixed relation. An idea may be

formed of what is meant by this sort of connection of one form with

another, by considering a pointed pyramid built up of cubic stones,

disposed in layers, each of which separately is a square plate of the

thickness of a single stone. These layers, laid horizontally one on the

other, and decreasing regularly in size from the bottom to the top,

produce a pyramidal form with a rough or channeled surface; and if the

layers are so extremely thin that the channels cease to be visible to

the eye, the pyramid will seem smooth and perfect.

(262.) Very shortly after this, and without knowledge of what had been

done by Gahn and Bergmann, the Abbé Haüy, instructed by the accidental

fracture of a fine group of crystals, made the remark noticed already

(in 67.), and reasoning on it with more caution and success, and

pursuing it into all its detail, developed the general laws which

regulate the superposition of the layers of particles of which he

supposes all crystals to be built up, and which enable us, from knowing

their primitive forms, to discover, previous to trial, what other

forms they are capable of assuming; and which, according to this idea,

are called derivative or secondary forms. Mohs and others have since

imagined processes and systems by which the derivation of forms from

each other is facilitated, and have corrected some errors of over-hasty

generalization into which their predecessors had fallen, as well as

advanced, by an extraordinary diligence of research, our knowledge of

the forms which the various substances which occur in nature and art

actually do assume.

(263.) In what manner a variety in point of external form may originate

in a variety of figures in the ultimate particles of which a solid

is composed, may very readily be imagined by considering what would

happen if the bricks of which an edifice is constructed had all a

certain _leaning_ or bias in one direction out of the perpendicular.

Suppose every brick, for instance, when laid flat on its face, with

its longer edges north and south, had its eastern and western faces

upright, but its northern and southern ones leaning southwards at a

certain inclination the same for each brick; a house built of such

bricks would lean the same way, if the bricks fitted well together.

If, _besides this_, the eastern and western faces of the bricks,

instead of being truly upright, had an inclination eastward, the house

would have a similar one, and all its four corners, instead of being

upright, would lean to the south-east. Suppose, instead of a house, a

pyramid were built of such oblique bricks, with the sides of its base

directed to the four points of the compass; then its point, instead

of being situated vertically over the centre of its base, would stand

perpendicularly over some point to the south-east of that centre, and

the pyramid itself would have its sides facing the south and the east,

more highly inclined to the horizon than those towards the north and

west.

(264.) Whatever conception we may form of the manner in which the

particles of a crystal cohere and form masses, it is next to impossible

to divest ourselves of the idea of a determinate figure common to them

all. Any other supposition, indeed, would be incompatible with that

exact similarity in all other respects which the phenomena of chemistry

may be considered as having demonstrated. However, it must be borne in

mind that this idea, plausible as it may appear, is yet in some degree

hypothetical, and that the laws of crystallography, as determined from

inductive observation, are quite independent of any supposition of the

kind, or even of the existence of such things as ultimate particles or

atoms at all.

(265.) Still, that peculiar internal constitution of solid bodies,

whatever it be, which is indicated by the assumption of determinate

figures, by their splitting easier in some directions than in others,

and by their presenting glittering plane surfaces when broken into

fragments, cannot but have an important influence on all their

relations to external agents, as well as to their internal movements

and the mutual actions of their parts on one another. Accordingly, the

division of bodies into crystallized and uncrystallized, or imperfectly

crystallized, is one of the most universal importance; and almost all

the phenomena produced by those more intimate natural causes which

act within small limits, and as it were on the immediate mechanism

of solid substances, are remarkably modified by their crystalline

structure. Thus, in transparent solids, the course taken by the rays

of light, in traversing them, as well as the properties impressed upon

them in so doing, are intimately connected with this structure. The

recent experiments of M. Savart, too, have proved that this is also the

case with their power of resistance to external force, on which depends

their elasticity. Crystallized substances, according to the results of

these experiments, resist compression with different degrees of elastic

force, according to the direction in which it is attempted to compress

them; and all the phenomena dependent on their elasticity are affected

by this cause, especially those which relate to their vibratory

movements and their conveyance of sound.

(266.) There can be little doubt that modifications, similarly

depending on the internal structure of crystals, will be traced through

every department of physics. In that interesting one which relates

to the action of heat in expanding the dimensions of substances, a

beginning has already been made by Professor Mitscherlich. It had long

been known that all substances are dilated by heat, and no exception

to this law has been found, so long as we regard the _bulk_ of the

heated body. Thus, an iron rod when hot is both longer and thicker than

when cold; and the difference of dimension, though but trifling in

itself, is yet capable of being made sensible, and is of considerable

consequence in engineering. Thus, too, the quicksilver in a common

thermometer occupies a larger space when hot than when cold; and being

confined by the glass ball, (which also expands, but _not so much in

proportion_,) it is forced to rise in the tube. These and similar facts

had long been known; and accurate measures of the total amount of

dilatation of a variety of different bodies, under similar accessions

of heat, had been obtained and registered in tables. But no one had

suspected the important fact, that this expansion in crystallized

bodies takes place under totally different circumstances from what

obtains in uncrystallized ones. M. Mitscherlich has lately shown that

such substances expand differently in different directions, and has

even produced a case in which expansion in one direction is actually

accompanied with contraction in another. This step, the most important

beyond a doubt which has yet been made in pyrometry, can however only

be regarded as the first in a series of researches which will occupy

the next generation, and which promises to afford an abundant harvest

of new facts, as well as the elucidation of some of the most obscure

and interesting points in the doctrine of heat.

(267.) From what has been said, it is clear that if we look upon solid

bodies as collections of particles or atoms, held together and kept

in their places by the perpetual action of attractive and repulsive

forces, we cannot suppose these forces, at least in crystallized

substances, to act alike in all directions. Hence arises the conception

of _polarity_, of which we see an instance, on a great scale, in the

magnetic needle, but which, under modified forms, there is nothing to

prevent us from conceiving to act among the ultimate atoms of solid

or even fluid bodies, and to produce all the phenomena which they

exhibit in their crystallized state, either when acting on each other,

or on light, heat, &c. It is not difficult, if we give the reins to

imagination, to conceive how attractive and repulsive atoms, bound

together by some unknown tie, may form little machines or compound

particles, which shall have many of the properties which we refer to

polarity; and accordingly many ingenious suppositions have been made to

that effect: but in the actual state of science it is certainly safest

to wave these hypotheses, without however absolutely rejecting them,

and regard the _polarity of matter_ as one of the ultimate phenomena to

which the analysis of nature leads us, and of which it is our business

fully to investigate the laws, before we endeavour to ascertain its

causes, or trace the mechanism by which it is produced.

(268.) The mutual attractions and repulsions of the particles of

matter, then, and their polarity, whether regarded as an original or a

derivative property, are the forces which, acting with great energy,

and within very confined limits, we must look to as the principles on

which the intimate constitution of all bodies and many of their mutual

actions depend. These are what are understood by the general term of

_molecular forces_. Molecular attraction has been attempted to be

confounded by some with the general attraction of gravity, which all

matter exerts on all other matter; but this idea is refuted by the

plainest facts.

CHAP. II.

OF THE COMMUNICATION OF MOTION THROUGH BODIES.--OF SOUND AND LIGHT.