REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.

Before concluding a work on the Principles of Science, it will not be

inappropriate to add some remarks upon the limits and ultimate bearings

of the knowledge which we may acquire by the employment of scientific

method. All science consists, it has several times been stated, in the

detection of identities in the action of natural agents. The purpose of

inductive inquiry is to ascertain the apparent existence of necessary

connection between causes and effects, expressed in the form of natural

laws. Now so far as we thus learn the invariable course of nature, the

future becomes the necessary sequel of the present, and we are brought

beneath the sway of powers with which nothing can interfere.

By degrees it is found, too, that the chemistry of organised substances

is not entirely separated from, but is continuous with, that of earth

and stones. Life seems to be nothing but a special form of energy

which is manifested in heat and electricity and mechanical force. The

time may come, it almost seems, when the tender mechanism of the brain

will be traced out, and every thought reduced to the expenditure of

a determinate weight of nitrogen and phosphorus. No apparent limit

exists to the success of scientific method in weighing and measuring,

and reducing beneath the sway of law, the phenomena both of matter

and of mind. And if mental phenomena be thus capable of treatment by

the balance and the micrometer, can we any longer hold that mind is

distinct from matter? Must not the same inexorable reign of law which

is apparent in the motions of brute matter be extended to the subtle

feelings of the human heart? Are not plants and animals, and ultimately

man himself, merely crystals, as it were, of a complicated form? If

so, our boasted free will becomes a delusion, moral responsibility a

fiction, spirit a mere name for the more curious manifestations of

material energy. All that happens, whether right or wrong, pleasurable

or painful, is but the outcome of the necessary relations of time and

space and force.

Materialism seems, then, to be the coming religion, and resignation

to the nonentity of human will the only duty. Such may not generally

be the reflections of men of science, but I believe that we may thus

describe the secret feelings of fear which the constant advance of

scientific investigation excites in the minds of many. Is science,

then, essentially atheistic and materialistic in its tendency?

Does the uniform action of material causes, which we learn with an

ever-increasing approximation to certainty, preclude the hypothesis of

a benevolent Creator, who has not only designed the existing universe,

but who still retains the power to alter its course from time to time?

To enter upon actual theological discussions would be evidently beyond

the scope of this work. It is with the scientific method common to all

the sciences, and not with any of the separate sciences, that we are

concerned. Theology therefore would be at least as much beyond my scope

as chemistry or geology. But I believe that grave misapprehensions

exist as regards the very nature of scientific method. There are

scientific men who assert that the interposition of Providence is

impossible, and prayer an absurdity, because the laws of nature are

inductively proved to be invariable. Inferences are drawn not so

much from particular sciences as from the logical nature of science

itself, to negative the impulses and hopes of men. Now I may state

that my own studies in logic lead me to call in question such negative

inferences. Laws of nature are uniformities observed to exist in the

action of certain material agents, but it is logically impossible to

show that all other agents must behave as these do. The too exclusive

study of particular branches of physical science seems to generate an

over-confident and dogmatic spirit. Rejoicing in the success with which

a few groups of facts are brought beneath the apparent sway of laws,

the investigator hastily assumes that he is close upon the ultimate

springs of being. A particle of gelatinous matter is found to obey

the ordinary laws of chemistry; yet it moves and lives. The world is

therefore asked to believe that chemistry can resolve the mysteries of

existence.

*The Meaning of Natural Law.*

Pindar speaks of Law as the Ruler of the Mortals and the Immortals, and

it seems to be commonly supposed that the so-called Laws of Nature, in

like manner, rule man and his Creator. The course of nature is regarded

as being determined by invariable principles of mechanics which have

acted since the world began, and will act for evermore. Even if the

origin of all things is attributed to an intelligent creative mind,

that Being is regarded as having yielded up arbitrary power, and as

being subject like a human legislator to the laws which he has himself

enacted. Such notions I should describe as superficial and erroneous,

being derived, as I think, from false views of the nature of scientific

inference, and the degree of certainty of the knowledge which we

acquire by inductive investigation.

A law of nature, as I regard the meaning of the expression, is not a

uniformity which must be obeyed by all objects, but merely a uniformity

which is as a matter of fact obeyed by those objects which have come

beneath our observation. There is nothing whatever incompatible with

logic in the discovery of objects which should prove exceptions to any

law of nature. Perhaps the best established law is that which asserts

an invariable correlation to exist between gravity and inertia, so that

all gravitating bodies are found to possess inertia, and all bodies

possessing inertia are found to gravitate. But it would be no reproach

to our scientific method, if something were ultimately discovered

to possess gravity without inertia. Strictly defined and correctly

interpreted, the law itself would acknowledge the possibility; for with

the statement of every law we ought properly to join an estimate of the

number of instances in which it has been observed to hold true, and the

probability thence calculated, that it will hold true in the next case.

Now, as we found (p. 259), no finite number of instances can warrant

us in expecting with certainty that the next instance will be of like

nature; in the formulas yielded by the inverse method of probabilities

a unit always appears to represent the probability that our inference

will be mistaken. I demur to the assumption that there is any necessary

truth even in such fundamental laws of nature as the Indestructibility

of Matter, the Conservation of Energy, or the Laws of Motion. Certain

it is that men of science have recognised the conceivability of

other laws, and even investigated their mathematical consequences.

Airy investigated the mathematical conditions of a perpetual motion

(p. 223), and Laplace and Newton discussed imaginary laws of forces

inconsistent with those observed to operate in the universe (pp. 642,

706).

The laws of nature, as I venture to regard them, are simply general

propositions concerning the correlation of properties which have been

observed to hold true of bodies hitherto observed. On the assumption

that our experience is of adequate extent, and that no arbitrary

interference takes place, we are then able to assign the probability,

always less than certainty, that the next object of the same apparent

nature will conform to the same laws.

*Infiniteness of the Universe.*

We may safely accept as a satisfactory scientific hypothesis the

doctrine so grandly put forth by Laplace, who asserted that a perfect

knowledge of the universe, as it existed at any given moment, would

give a perfect knowledge of what was to happen thenceforth and for

ever after. Scientific inference is impossible, unless we may regard

the present as the outcome of what is past, and the cause of what is

to come. To the view of perfect intelligence nothing is uncertain. The

astronomer can calculate the positions of the heavenly bodies when

thousands of generations of men shall have passed away, and in this

fact we have some illustration, as Laplace remarks, of the power which

scientific prescience may attain. Doubtless, too, all efforts in the

investigation of nature tend to bring us nearer to the possession of

that ideally perfect power of intelligence. Nevertheless, as Laplace

with profound wisdom adds,[603] we must ever remain at an infinite

distance from the goal of our aspirations.

[603] *Théorie Analytique des Probabilités*, quoted by Babbage,

*Ninth Bridgewater Treatise*, p. 173.

Let us assume, for a time at least, as a highly probable hypothesis,

that whatever is to happen must be the outcome of what is; there then

arises the question, What is? Now our knowledge of what exists must

ever remain imperfect and fallible in two respects. Firstly, we do

not know all the matter that has been created, nor the exact manner

in which it has been distributed through space. Secondly, assuming

that we had that knowledge, we should still be wanting in a perfect

knowledge of the way in which the particles of matter will act upon

each other. The power of scientific prediction extends at the most

to the limits of the data employed. Every conclusion is purely

hypothetical and conditional upon the non-interference of agencies

previously undetected. The law of gravity asserts that every body tends

to approach towards every other body, with a certain determinate force;

but, even supposing the law to hold true, it does not assert that the

body *will* approach. No single law of nature can warrant us in making

an absolute prediction. We must know all the laws of nature and all

the existing agents acting according to those laws before we can say

what will happen. To assume, then, that scientific method can take

everything within its cold embrace of uniformity, is to imply that the

Creator cannot outstrip the intelligence of his creatures, and that

the existing Universe is not infinite in extent and complexity, an

assumption for which I see no logical basis whatever.

*The Indeterminate Problem of Creation.*

A second and very serious misapprehension concerning the import of a

law of nature may now be pointed out. It is not uncommonly supposed

that a law determines the character of the results which shall take

place, as, for instance, that the law of gravity determines what force

of gravity shall act upon a given particle. Surely a little reflection

must render it plain that a law by itself determines nothing. It is

*law plus agents obeying law which has results*, and it is no function

of law to govern or define the number and place of its own agents.

Whether a particle of matter shall gravitate, depends not only upon the

law of Newton, but also upon the distribution of surrounding particles.

The theory of gravitation may perhaps be true throughout all time

and in all parts of space, and the Creator may never find occasion

to create those possible exceptions to it which I have asserted to

be conceivable. Let this be as it may; our science cannot certainly

determine the question. Certain it is, that the law of gravity does not

alone determine the forces which may be brought to bear at any point of

space. The force of gravitation acting upon any particle depends upon

the mass, distance, and relative position of all the other particles

of matter within the bounds of space at the instant in question.

Even assuming that all matter when once distributed through space at

the Creation was thenceforth to act in an invariable manner without

subsequent interference, yet the actual configuration of matter at any

moment, and the consequent results of the law of gravitation, must have

been entirely a matter of free choice.

Chalmers has most distinctly pointed out that the existing

*collocations* of the material world are as important as the laws

which the objects obey. He remarks that a certain class of writers

entirely overlook the distinction, and forget that mere laws without

collocations would have afforded no security against a turbid and

disorderly chaos.[604] Mill has recognised[605] the truth of Chalmers’

statement, without drawing the proper inferences from it. He says[606]

of the distribution of matter through space, “We can discover nothing

regular in the distribution itself; we can reduce it to no uniformity,

to no law.” More lately the Duke of Argyll in his well-known work on

the *Reign of Law* has drawn attention to the profound distinction

between laws and collocations of causes.

[604] *First Bridgewater Treatise* (1834), pp. 16–24.

[605] *System of Logic*, 5th edit. bk. III. chap. V. § 7; chap. XVI.

§ 3.

[606] *System of Logic*, vol. i. p. 384.

The original conformation of the material universe, as far as we can

tell, was free from all restriction. There was unlimited space in

which to frame it, and an unlimited number of material particles,

each of which could be placed in any one of an infinite number of

different positions. It should be added, that each particle might be

endowed with any one of an infinite number of quantities of *vis viva*

acting in any one of an infinite number of different directions. The

problem of Creation was, then, what a mathematician would call *an

indeterminate problem*, and it was indeterminate in a great number of

ways. Infinitely numerous and various universes might then have been

fashioned by the various distribution of the original nebulous matter,

although all the particles of matter should obey the law of gravity.

Lucretius tells us how in the original rain of atoms some of these

little bodies diverged from the rectilinear direction, and coming into

contact with other atoms gave rise to the various combinations of

substances which exist. He omitted to tell us whence the atoms came,

or by what force some of them were caused to diverge; but surely these

omissions involve the whole question. I accept the Lucretian conception

of creation when properly supplemented. Every atom which existed in

any point of space must have existed there previously, or must have

been created there by a previously existing Power. When placed there

it must have had a definite mass and a definite energy. Now, as before

remarked, an unlimited number of atoms can be placed in unlimited space

in an unlimited number of modes of distribution. Out of infinitely

infinite choices which were open to the Creator, that one choice must

have been made which has yielded the Universe as it now exists.

It would be a mistake, indeed, to suppose that the law of gravity,

when it holds true, is no restriction on the distribution of

force. That law is a geometrical law, and it would in many cases

be mathematically impossible, as far as we can see, that the force

of gravity acting on one particle should be small while that on a

neighbouring particle is great. We cannot conceive that even Omnipotent

Power should make the angles of a triangle greater than two right

angles. The primary laws of thought and the fundamental notions of the

mathematical sciences do not seem to admit of error or alteration. Into

the metaphysical origin and meaning of the apparent necessity attaching

to such laws I have not attempted to inquire in this work, and it is

not requisite for my present purpose. If the law of gravity were the

only law of nature and the Creator had chosen to render all matter

obedient to that law, there would doubtless be restrictions upon the

effects derivable from any one distribution of matter.

*Hierarchy of Natural Laws.*

A further consideration presents itself. A natural law like that

of gravity expresses a certain uniformity in the action of agents

submitted to it, and this produces, as we have seen, certain

geometrical restrictions upon the effects which those agents may

produce. But there are other forces and laws besides gravity. One

force may override another, and two laws may each be obeyed and may

each disguise the action of the other. In the intimate constitution of

matter there may be hidden springs which, while acting in accordance

with their own fixed laws, may lead to sudden and unexpected changes.

So at least it has been found from time to time in the past, and so

there is every reason to believe it will be found in the future. To

the ancients it seemed incredible that one lifeless stone could make

another leap towards it. A piece of iron while it obeys the magnetic

force of the loadstone does not the less obey the law of gravity.

A plant gravitates downwards as regards every constituent cell or

fibre, and yet it persists in growing upwards. Life is altogether an

exception to the simpler phenomena of mineral substances, not in the

sense of disproving those laws, but in superadding forces of new and

inexplicable character. Doubtless no law of chemistry is broken by the

action of the nervous cells, and no law of physics by the pulses of the

nervous fibres, but something requires to be added to our sciences in

order that we may explain these subtle phenomena.

Now there is absolutely nothing in science or in scientific method to

warrant us in assigning a limit to this hierarchy of laws. When in

many undoubted cases we find law overriding law, and at certain points

in our experience producing unexpected results, we cannot venture to

affirm that we have exhausted the strange phenomena which may have been

provided for in the original constitution of matter. The Universe might

have been so designed that it should go for long intervals through the

same round of unvaried existence, and yet that events of exceptional

character should be produced from time to time. Babbage showed in that

most profound and eloquent work, *The Ninth Bridgewater Treatise*, that

it was theoretically possible for human artists to design a machine,

consisting of metallic wheels and levers, which should work invariably

according to a simple law of action during any finite number of steps,

and yet at a fixed moment, however distant, should manifest a single

breach of law. Such an engine might go on counting, for instance, the

natural numbers until they would reach a number requiring for its

expression a hundred million digits. “If every letter in the volume

now before the reader’s eyes,” says Babbage,[607] “were changed into

a figure, and if all the figures contained in a thousand such volumes

were arranged in order, the whole together would yet fall far short

of the vast induction the observer would have had in favour of the

truth of the law of natural numbers.... Yet shall the engine, true to

the prediction of its director, after the lapse of myriads of ages,

fulfil its task, and give that one, the first and only exception to

that time-sanctioned law. What would have been the chances against the

appearance of the excepted case, immediately prior to its occurrence?”

[607] *Ninth Bridgewater Treatise*, p. 140.

As Babbage further showed,[608] a calculating engine, after proceeding

through any required number of motions according to a first law, may

be made suddenly to suffer a change, so that it shall then commence

to calculate according to a wholly new law. After giving the natural

numbers for a finite time, it might suddenly begin to give triangular,

or square, or cube numbers, and these changes might be conceived

theoretically as occurring time after time. Now if such occurrences can

be designed and foreseen by a human artist, it is surely within the

capacity of the Divine Artist to provide for analogous changes of law

in the mechanism of the atom, or the construction of the heavens.

[608] *Ibid.* pp. 34–43.

Physical science, so far as its highest speculations can be trusted,

gives some indication of a change of law in the past history of the

Universe. According to Sir W. Thomson’s deductions from Fourier’s

*Theory of Heat*, we can trace down the dissipation of heat by

conduction and radiation to an infinitely distant time when all

things will be uniformly cold. But we cannot similarly trace the

heat-history of the Universe to an infinite distance in the past.

For a certain negative value of the time the formulæ give impossible

values, indicating that there was some initial distribution of heat

which could not have resulted, according to known laws of nature,[609]

from any previous distribution.[610] There are other cases in which a

consideration of the dissipation of energy leads to the conception of

a limit to the antiquity of the present order of things.[611] Human

science, of course, is fallible, and some oversight or erroneous

simplification in these theoretical calculations may afterwards be

discovered; but as the present state of scientific knowledge is the

only ground on which erroneous inferences from the uniformity of nature

and the supposed reign of law are founded, I am right in appealing to

the present state of science in opposition to these inferences. Now

the theory of heat places us in the dilemma either of believing in

Creation at an assignable date in the past, or else of supposing that

some inexplicable change in the working of natural laws then took

place. Physical science gives no countenance to the notion of infinite

duration of matter in one continuous course of existence. And if in

time past there has been a discontinuity of law, why may there not be

a similar event awaiting the world in the future? Infinite ingenuity

could have implanted some agency in matter so that it might never yet

have made its tremendous powers manifest. We have a very good theory

of the conservation of energy, but the foremost physicists do not

deny that there may possibly be forms of energy, neither kinetic nor

potential, and therefore of unknown nature.[612]

[609] Professor Clifford, in his most interesting lecture on “The

First and Last Catastrophe” (*Fortnightly Review*, April 1875,

p. 480, reprint by the Sunday Lecture Society, p. 24), objects that I

have erroneously substituted “known laws of nature” for “known laws

of conduction of heat.” I quite admit the error, without admitting

all the conclusions which Professor Clifford proceeds to draw; but I

maintain the paragraph unchanged, in order that it may be discussed

in the Preface.

[610] Tait’s *Thermodynamics*, p. 38. *Cambridge Mathematical

Journal*, vol. iii. p. 174.

[611] Clerk Maxwell’s *Theory of Heat*, p. 245.

[612] Maxwell’s *Theory of Heat*, p. 92.

We can imagine reasoning creatures dwelling in a world where the

atmosphere was a mixture of oxygen and inflammable gas like the

fire-damp of coal-mines. If devoid of fire, they might have lived

through long ages unconscious of the tremendous forces which a single

spark would call into play. In the twinkling of an eye new laws might

come into action, and the poor reasoning creatures, so confident about

their knowledge of the reign of law in their world, would have no time

to speculate upon the overthrow of all their theories. Can we with our

finite knowledge be sure that such an overthrow of our theories is

impossible?

*The Ambiguous Expression, “Uniformity of Nature.”*

I have asserted that serious misconception arises from an erroneous

interpretation of the expression Uniformity of Nature. Every law of

nature is the statement of a certain uniformity observed to exist among

phenomena, and since the laws of nature are invariably obeyed, it seems

to follow that the course of nature itself is uniform, so that we can

safely judge of the future by the present. This inference is supported

by some of the results of physical astronomy. Laplace proved that

the planetary system is stable, so that no perturbation which planet

produces upon planet can become so great as to cause disruption and

permanent alteration of the planetary orbits. A full comprehension of

the law of gravity shows that all such disturbances are essentially

periodic, so that after the lapse of millions of years the planets will

return to the same relative positions, and a new cycle of disturbances

will then commence.

As other branches of science progress, we seem to gain assurance

that no great alteration of the world’s condition is to be expected.

Conflict with a comet has long been the cause of fear, but now it is

credibly asserted that we have passed through a comet’s tail without

the fact being known at the time, or manifested by any more serious a

phenomenon than a slight luminosity of the sky. More recently still

the earth is said to have touched the comet Biela, and the only result

was a beautiful and perfectly harmless display of meteors. A decrease

in the heating power of the sun seems to be the next most probable

circumstance from which we might fear the extinction of life on the

earth. But calculations founded on reasonable physical data show

that no appreciable change can be going on, and experimental data to

indicate a change are wholly wanting. Geological investigations show

indeed that there have been extensive variations of climate in past

times; vast glaciers and icebergs have swept over the temperate regions

at one time, and tropical vegetation has flourished near the poles

at another time. But here again the vicissitudes of climate assume a

periodic character, so that the stability of the earth’s condition does

not seem to be threatened.

All these statements may be reasonable, but they do not establish the

Uniformity of Nature in the sense that extensive alterations or sudden

catastrophes are impossible. In the first place, Laplace’s theory of

the stability of the planetary system is of an abstract character, as

paying regard to nothing but the mutual gravitation of the planetary

bodies and the sun. It overlooks several physical causes of change

and decay in the system which were not so well known in his day as at

present, and it also presupposes the absence of any interruption of the

course of things by conflict with foreign astronomical bodies.

It is now acknowledged by astronomers that there are at least two ways

in which the *vis viva* of the planets and satellites may suffer loss.

The friction of the tides upon the earth produces a small quantity

of heat which is radiated into space, and this loss of energy must

result in a decrease of the rotational velocity, so that ultimately

the terrestrial day will become identical with the year, just as the

periods of revolution of the moon upon its axis and around the earth

have already become equal. Secondly, there can be little doubt that

certain manifestations of electricity upon the earth’s surface depend

upon the relative motions of the planets and the sun, which give

rise to periods of increased intensity. Such electrical phenomena

must result in the production and dissipation of heat, the energy

of which must be drawn, partially at least, from the moving bodies.

This effect is probably identical (p. 570) with the loss of energy of

comets attributed to the so-called resisting medium. But whatever be

the theoretical explanation of these phenomena, it is almost certain

that there exists a tendency to the dissipation of the energy of the

planetary system, which will, in the indefinite course of time, result

in the fall of the planets into the sun.

It is hardly probable, however, that the planetary system will be left

undisturbed throughout the enormous interval of time required for the

dissipation of its energy in this way. Conflict with other bodies is

so far from being improbable, that it becomes approximately certain

when we take very long intervals of time into account. As regards

cometary conflicts, I am by no means satisfied with the negative

conclusions drawn from the remarkable display on the evening of the

27th of November, 1872. We may often have passed through the tail of a

comet, the light of which is probably an electrical manifestation no

more substantial than the aurora borealis. Every remarkable shower of

shooting stars may also be considered as proceeding from a cometary

body, so that we may be said to have passed through the thinner parts

of innumerable comets. But the earth has probably never passed, in

times of which we have any record, through the nucleus of a comet,

which consists perhaps of a dense swarm of small meteorites. We can

only speculate upon the effects which might be produced by such a

conflict, but it would probably be a much more serious event than any

yet registered in history. The probability of its occurrence, too,

cannot be assigned; for though the probability of conflict with any one

cometary nucleus is almost infinitesimal, yet the number of comets is

immensely great (p. 408).

It is far from impossible, again, that the planetary system may be

invaded by bodies of greater mass than comets. The sun seems to be

placed in so extensive a portion of empty space that its own proper

motion would not bring it to the nearest known star (α Centauri) in

less than 139,200 years. But in order to be sure that this interval of

undisturbed life is granted to our globe, we must prove that there are

no stars moving so as to meet us, and no dark bodies of considerable

size flying through intervening space unknown to us. The intrusion

of comets into our system, and the fact that many of them have

hyperbolic paths, is sufficient to show that the surrounding parts of

space are occupied by multitudes of dark bodies of some size. It is

quite probable that small suns may have cooled sufficiently to become

non-luminous; for even if we discredit the theory that the variation of

brightness of periodic stars is due to the revolution of dark companion

stars, yet there is in our own globe an unquestionable example of a

smaller body which has cooled below the luminous point.

Altogether, then, it is a mere assumption that the uniformity of nature

involves the unaltered existence of our own globe. There is no kind

of catastrophe which is too great or too sudden to be theoretically

consistent with the reign of law. For all that our science can tell,

human history may be closed in the next instant of time. The world may

be dashed to pieces against a wandering star; it may be involved in a

nebulous atmosphere of hydrogen to be exploded a second afterwards; it

may be scorched up or dissipated into vapour by some great explosion in

the sun; there might even be within the globe itself some secret cause

of disruption, which only needs time for its manifestation.

There are some indications, as already noticed (p. 660), that violent

disturbances have actually occurred in the history of the solar system.

Olbers sought for the minor planets on the supposition that they were

fragments of an exploded planet, and he was rewarded with the discovery

of some of them. The retrograde motion of the satellites of the more

distant planets, the abnormal position of the poles of Uranus and the

excessive distance of Neptune, are other indications of some violent

event, of which we have no other evidence. I adduce all these facts

and arguments, not to show that there is any considerable probability,

as far as we can judge, of interruption within the scope of human

history, but to prove that the Uniformity of Nature is theoretically

consistent with the most unexpected events of which we can form a

conception.

*Possible States of the Universe.*

When we give the rein to scientific imagination, it becomes apparent

that conflict of body with body must not be regarded as the rare

exception, but as the general rule and the inevitable fate of each

star system. So far as we can trace out the results of the law of

gravitation, and of the dissipation of energy, the universe must be

regarded as undergoing gradual condensation into a single cold solid

body of gigantic dimensions. Those who so frequently use the expression

Uniformity of Nature seem to forget that the Universe might exist

consistently with the laws of nature in the most diverse conditions. It

might consist, on the one hand, of a glowing nebulous mass of gaseous

substances. The heat might be so intense that all elements, even carbon

and silicon, would be in the state of gas, and all atoms, of whatever

nature, would be flying about in chemical independence, diffusing

themselves almost uniformly in the neighbouring parts of space. There

would then be no life, unless we can apply that name to the passage

through each part of space of similar average trains of atoms, the

particular succession of atoms being governed only by the theory of

probability, and the law of divergence from a mean exhibited in the

Arithmetical Triangle. Such a universe would correspond partially to

the Lucretian rain of atoms, and to that nebular hypothesis out of

which Laplace proposed philosophically to explain the evolution of the

planetary system.

According to another extreme supposition, the intense heat-energy of

this nebulous mass might be radiated away into the unknown regions

of outer space. The attraction of gravity would exert itself between

each two particles, and the energy of motion thence arising would,

by incessant conflicts, be resolved into heat and dissipated.

Inconceivable ages might be required for the completion of this

process, but the dissipation of energy thus proceeding could end only

in the production of a cold and motionless universe. The relation of

cause and effect, as we see it manifested in life and growth, would

degenerate into the constant existence of every particle in a fixed

position relative to every other particle. Logical and geometrical

resemblances would still exist between atoms, and between groups of

atoms crystallised in their appropriate forms for evermore. But time,

the great variable, would bring no variation, and as to human hopes and

troubles, they would have gone to eternal rest.

Science is not really adequate to proving that such is the inevitable

fate of the universe, for we can seldom trust our best-established

theories far from their data. Nevertheless, the most probable

speculations which we can form as to the history, especially of our own

planetary system, is that it originated in a heated revolving nebulous

mass of gas, and is in a state of excessively slow progress towards

the cold and stony condition. Other speculative hypotheses might

doubtless be entertained. Every hypothesis is pressed by difficulties.

If the whole universe be cooling, whither does the heat go? If we are

to get rid of it entirely, outer space must be infinite in extent, so

that it shall never be stopped and reflected back. But not to speak

of metaphysical difficulties, if the medium of heat undulations be

infinite in extent, why should not the material bodies placed in

it be infinite also in number and aggregate mass? It is apparent

that we are venturing into speculations which surpass our powers of

scientific inference. But then I am arguing negatively; I wish to show

that those who speak of the uniformity of nature, and the reign of

law, misinterpret the meaning involved in those expressions. Law is

not inconsistent with extreme diversity, and, so far as we can read

the history of this planetary system, it did probably originate in

heated nebulous matter, and man’s history forms but a brief span in

its progress towards the cold and stony condition. It is by doubtful

and speculative hypotheses alone that we can avoid such a conclusion,

and I depart least from undoubted facts and well-established laws

when I assert that, whatever uniformities may underlie the phenomena

of nature, constant variety and ever-progressing change is the real

outcome.

*Speculations on the Reconcentration of Energy.*

There are unequivocal indications, as I have said, that the material

universe, as we at present see it, is progressing from some act of

creation, or some discontinuity of existence of which the date may be

approximately fixed by scientific inference. It is progressing towards

a state in which the available energy of matter will be dissipated

through infinite surrounding space, and all matter will become cold

and lifeless. This constitutes, as it were, the historical period of

physical science, that over which our scientific foresight may more

or less extend. But in this, as in other cases, we have no right to

interpret our experience negatively, so as to infer that because the

present state of things began at a particular time, there was no

previous existence. It may be that the present period of material

existence is but one of an indefinite series of like periods. All that

we can see, and feel, and infer, and reason about may be, as it were,

but a part of one single pulsation in the existence of the universe.

After Sir W. Thomson had pointed out the preponderating tendency

which now seems to exist towards the conversion of all energy into

heat-energy, and its equal diffusion by radiation throughout space,

the late Professor Rankine put forth a remarkable speculation.[613] He

suggested that the ethereal, or, as I have called it, the *adamantine*

medium in which all the stars exist, and all radiation takes place, may

have bounds, beyond which only empty space exists. All heat undulations

reaching this boundary will be totally reflected, according to the

theory of undulations, and will be reconcentrated into foci situated in

various parts of the medium. Whenever a cold and extinct star happens

to pass through one of these foci, it will be instantly ignited and

resolved by intense heat into its constituent elements. Discontinuity

will occur in the history of that portion of matter, and the star will

begin its history afresh with a renewed store of energy.

[613] *Report of the British Association* (1852), Report of Sections,

p. 12.

This is doubtless a mere speculation, practically incapable of

verification by observation, and almost free from restrictions afforded

by present knowledge. We might attribute various shapes to the

adamantine medium, and the consequences would be various. But there

is this value in such speculations, that they draw attention to the

finiteness of our knowledge. We cannot deny the possible truth of such

an hypothesis, nor can we place a limit to the scientific imagination

in the framing of other like hypotheses. It is impossible, indeed, to

follow out our scientific inferences without falling into speculation.

If heat be radiated into outward space, it must either proceed *ad

infinitum*, or it must be stopped somewhere. In the latter case we fall

upon Rankine’s hypothesis. But if the material universe consist of a

finite collection of heated matter situated in a finite portion of an

infinite adamantine medium, then either this universe must have existed

for a finite time, or else it must have cooled down during the infinity

of past time indefinitely near to the absolute zero of temperature. I

objected to Lucretius’ argument against the destructibility of matter,

that we have no knowledge whatever of the laws according to which

it would undergo destruction. But we do know the laws according to

which the dissipation of heat appears to proceed, and the conclusion

inevitably is that a finite heated material body placed in a perfectly

cold infinitely extended medium would in an infinite time sink to

zero of temperature. Now our own world is not yet cooled down near to

zero, so that physical science seems to place us in the dilemma of

admitting either the finiteness of past duration of the world, or else

the finiteness of the portion of medium in which we exist. In either

case we become involved in metaphysical and mechanical difficulties

surpassing our mental powers.

*The Divergent Scope for New Discovery.*

In the writings of some recent philosophers, especially of Auguste

Comte, and in some degree John Stuart Mill, there is an erroneous

and hurtful tendency to represent our knowledge as assuming an

approximately complete character. At least these and many other

writers fail to impress upon their readers a truth which cannot be

too constantly borne in mind, namely, that the utmost successes which

our scientific method can accomplish will not enable us to comprehend

more than an infinitesimal fraction of what there doubtless is to

comprehend.[614] Professor Tyndall seems to me open to the same charge

in a less degree. He remarks[615] that we can probably never bring

natural phenomena completely under mathematical laws, because the

approach of our sciences towards completeness may be asymptotic, so

that however far we may go, there may still remain some facts not

subject to scientific explanation. He thus likens the supply of novel

phenomena to a convergent series, the earlier and larger terms of which

have been successfully disposed of, so that comparatively minor groups

of phenomena alone remain for future investigators to occupy themselves

upon.

[614] Mr. C. J. Monroe objects that in this statement I do injustice

to Comte, who, he thinks, did impress upon his readers the inadequacy

of our mental powers compared with the vastness of the subject matter

of science. The error of Comte, he holds, was in maintaining that

science had been carried about as far as it is worth while to carry

it, which is a different matter. In either case, Comte’s position is

so untenable that I am content to leave the question undecided.

[615] *Fragments of Science*, p. 362.

On the contrary, as it appears to me, the supply of new and unexplained

facts is divergent in extent, so that the more we have explained, the

more there is to explain. The further we advance in any generalisation,

the more numerous and intricate are the exceptional cases still

demanding further treatment. The experiments of Boyle, Mariotte,

Dalton, Gay-Lussac, and others, upon the physical properties of gases,

might seem to have exhausted that subject by showing that all gases

obey the same laws as regards temperature, pressure, and volume. But

in reality these laws are only approximately true, and the divergences

afford a wide and quite unexhausted field for further generalisation.

The recent discoveries of Professor Andrews have summed up some of

these exceptional facts under a wider generalisation, but in reality

they have opened to us vast new regions of interesting inquiry,

and they leave wholly untouched the question why one gas behaves

differently from another.

The science of crystallography is that perhaps in which the most

precise and general laws have been detected, but it would be untrue

to assert that it has lessened the area of future discovery. We can

show that each one of the seven or eight hundred forms of calcite is

derivable by geometrical modifications from an hexagonal prism; but

who has attempted to explain the molecular forces producing these

modifications, or the chemical conditions in which they arise? The law

of isomorphism is an important generalisation, for it establishes a

general resemblance between the forms of crystallisation of natural

classes of elements. But if we examine a little more closely we find

that these forms are only approximately alike, and the divergence

peculiar to each substance is an unexplained exception.

By many similar illustrations it might readily be shown that in

whatever direction we extend our investigations and successfully

harmonise a few facts, the result is only to raise up a host of other

unexplained facts. Can any scientific man venture to state that there

is less opening now for new discoveries than there was three centuries

ago? Is it not rather true that we have but to open a scientific book

and read a page or two, and we shall come to some recorded phenomenon

of which no explanation can yet be given? In every such fact there is a

possible opening for new discoveries, and it can only be the fault of

the investigator’s mind if he can look around him and find no scope for

the exercise of his faculties.

*Infinite Incompleteness of the Mathematical Sciences.*

There is one privilege which a certain amount of knowledge should

confer; it is that of becoming aware of the weakness of our powers

compared with the tasks which they might undertake if stronger. To the

poor savage who cannot count twenty the arithmetical accomplishments of

the schoolboy are miraculously great. The schoolboy cannot comprehend

the vastly greater powers of the student, who has acquired facility in

algebraic processes. The student can but look with feelings of surprise

and reverence at the powers of a Newton or a Laplace. But the question

at once suggests itself, Do the powers of the highest human intellect

bear a finite ratio to the things which are to be understood and

calculated? How many further steps must we take in the rise of mental

ability and the extension of mathematical methods before we begin to

exhaust the knowable?

I am inclined to find fault with mathematical writers because they

often exult in what they can accomplish, and omit to point out that

what they do is but an infinitely small part of what might be done.

They exhibit a general inclination, with few exceptions, not to do

so much as mention the existence of problems of an impracticable

character. This may be excusable as far as the immediate practical

result of their researches is in question, but the custom has the

effect of misleading the general public into the fallacious notion

that mathematics is a *perfect* science, which accomplishes what it

undertakes in a complete manner. On the contrary, it may be said that

if a mathematical problem were selected by chance out of the whole

number which might be proposed, the probability is infinitely slight

that a human mathematician could solve it. Just as the numbers we can

count are nothing compared with the numbers which might exist, so the

accomplishments of a Laplace or a Lagrange are, as it were, the little

corner of the multiplication-table, which has really an infinite extent.

I have pointed out that the rude character of our observations prevents

us from being aware of the greater number of effects and actions

in nature. It must be added that, if we perceive them, we should

usually be incapable of including them in our theories from want of

mathematical power. Some persons may be surprised that though nearly

two centuries have elapsed since the time of Newton’s discoveries, we

have yet no general theory of molecular action. Some approximations

have been made towards such a theory. Joule and Clausius have measured

the velocity of gaseous atoms, or even determined the average distance

between the collisions of atom and atom. Thomson has approximated to

the number of atoms in a given bulk of substance. Rankine has formed

some reasonable hypotheses as to the actual constitution of atoms. It

would be a mistake to suppose that these ingenious results of theory

and experiment form any appreciable approach to a complete solution

of molecular motions. There is every reason to believe, judging from

the spectra of the elements, their atomic weights and other data, that

chemical atoms are very complicated structures. An atom of pure iron

is probably a far more complicated system than that of the planets

and their satellites. A compound atom may perhaps be compared with

a stellar system, each star a minor system in itself. The smallest

particle of solid substance will consist of a great number of such

stellar systems united in regular order, each bounded by the other,

communicating with it in some manner yet wholly incomprehensible. What

are our mathematical powers in comparison with this problem?

After two centuries of continuous labour, the most gifted men have

succeeded in calculating the mutual effects of three bodies each

upon the other, under the simple hypothesis of the law of gravity.

Concerning these calculations we must further remember that they are

purely approximate, and that the methods would not apply where four or

more bodies are acting, and all produce considerable effects upon each

other. There is reason to believe that each constituent of a chemical

atom goes through an orbit in the millionth part of the twinkling of

an eye. In each revolution it is successively or simultaneously under

the influence of many other constituents, or possibly comes into

collision with them. It is no exaggeration to say that mathematicians

have the least notion of the way in which they could successfully

attack so difficult a problem of forces and motions. As Herschel has

remarked,[616] each of these particles is for ever solving differential

equations, which, if written out in full, might belt the earth.

[616] *Familiar Lectures on Scientific Subjects*, p. 458.

Some of the most extensive calculations ever made were those required

for the reduction of the measurements executed in the course of the

Trigonometrical Survey of Great Britain. The calculations arising out

of the principal triangulation occupied twenty calculators during

three or four years, in the course of which the computers had to solve

simultaneous equations involving seventy-seven unknown quantities.

The reduction of the levellings required the solution of a system of

ninety-one equations. But these vast calculations present no approach

whatever to what would be requisite for the complete treatment of

any one physical problem. The motion of glaciers is supposed to

be moderately well understood in the present day. A glacier is a

viscid, slowly yielding mass, neither absolutely solid nor absolutely

rigid, but it is expressly remarked by Forbes,[617] that not even an

approximate solution of the mathematical conditions of such a moving

mass can yet be possible. “Every one knows,” he says, “that such

problems are beyond the compass of exact mathematics;” but though

mathematicians may know this, they do not often enough impress that

knowledge on other people.

[617] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 406.

The problems which are solved in our mathematical books consist of

a small selection of those which happen from peculiar conditions to

be solvable. But the very simplest problem in appearance will often

give rise to impracticable calculations. Mr. Todhunter[618] seems to

blame Condorcet, because in one of his memoirs he mentions a problem

to solve which would require a great and impracticable number of

successive integrations. Now, if our mathematical sciences are to cope

with the problems which await solution, we must be prepared to effect

an unlimited number of successive integrations; yet at present, and

almost beyond doubt for ever, the probability that an integration taken

haphazard will come within our powers is exceedingly small.

[618] *History of the Theory of Probability*, p. 398.

In some passages of that remarkable work, the *Ninth Bridgewater

Treatise* (pp. 113–115), Babbage has pointed out that if we had power

to follow and detect the minutest effects of any disturbance, each

particle of existing matter would furnish a register of all that has

happened. “The track of every canoe--of every vessel that has yet

disturbed the surface of the ocean, whether impelled by manual force or

elemental power, remains for ever registered in the future movement of

all succeeding particles which may occupy its place. The furrow which

it left is, indeed, instantly filled up by the closing waters; but they

draw after them other and larger portions of the surrounding element,

and these again, once moved, communicate motion to others in endless

succession.” We may even say that “The air itself is one vast library,

on whose pages are for ever written all that man has ever said or

even whispered. There, in their mutable but unerring characters, mixed

with the earliest as well as the latest sighs of mortality, stand for

ever recorded, vows unredeemed, promises unfulfilled, perpetuating in

the united movements of each particle the testimony of man’s changeful

will.”

When we read reflections such as these, we may congratulate ourselves

that we have been endowed with minds which, rightly employed, can

form some estimate of their incapacity to trace out and account for

all that proceeds in the simpler actions of material nature. It ought

to be added that, wonderful as is the extent of physical phenomena

open to our investigation, intellectual phenomena are yet vastly more

extensive. Of this I might present one satisfactory proof were space

available by pointing out that the mathematical functions employed

in the calculations of physical science form an infinitely small

fraction of the functions which might be invented. Common trigonometry

consists of a great series of useful formulæ, all of which arise out

of the relation of the sine and cosine expressed in one equation,

sin ^{2}*x* + cos ^{2}*x* = 1. But this is not the only trigonometry

which may exist; mathematicians also recognise hyperbolic trigonometry,

of which the fundamental equation is cos ^{2}*x* - sin ^{2}*x* = 1. De

Morgan has pointed out that the symbols of ordinary algebra form but

three of an interminable series of conceivable systems.[619] As the

logarithmic operation is to addition or addition to multiplication, so

is the latter to a higher operation, and so on without limit.

[619] *Trigonometry and Double Algebra*, chap. ix.

We may rely upon it that immense, and to us inconceivable, advances

will be made by the human intellect, in the absence of any catastrophe

to the species or the globe. Within historical periods we can trace

the rise of mathematical science from its simplest germs. We can

prove our descent from ancestors who counted only on their fingers.

How infinitely is a Newton or a Laplace above those simple savages.

Pythagoras is said to have sacrificed a hecatomb when he discovered the

forty-seventh proposition of Euclid, and the occasion was worthy of

the sacrifice. Archimedes was beside himself when he first perceived

his beautiful mode of determining specific gravities. Yet these great

discoveries are the commonplaces of our school books. Step by step we

can trace upwards the acquirement of new mental powers. What could be

more wonderful than Napier’s discovery of logarithms, a new mode of

calculation which has multiplied perhaps a hundredfold the working

powers of every computer, and has rendered easy calculations which were

before impracticable? Since the time of Newton and Leibnitz worlds of

problems have been solved which before were hardly conceived as matters

of inquiry. In our own day extended methods of mathematical reasoning,

such as the system of quaternions, have been brought into existence.

What intelligent man will doubt that the recondite speculations of a

Cayley, a Sylvester, or a Clifford may lead to some new development of

new mathematical power, at the simplicity of which a future age will

wonder, and yet wonder more that to us they were so dark and difficult.

May we not repeat the words of Seneca: “Veniet tempus, quo ista quæ

nunc latent, in lucem dies extrahat, et longioris ævi diligentia:

ad inquisitionem tantorum ætas una non sufficit. Veniet tempus, quo

posteri nostri tam aperta nos nescisse mirentur.”

*The Reign of Law in Mental and Social Phenomena.*

After we pass from the so-called physical sciences to those which

attempt to investigate mental and social phenomena, the same general

conclusions will hold true. No one will be found to deny that there

are certain uniformities of thinking and acting which can be detected

in reasoning beings, and so far as we detect such laws we successfully

apply scientific method. But those who attempt to establish social or

moral sciences soon become aware that they are dealing with subjects

of enormous perplexity. Take as an instance the science of political

economy. If a science at all, it must be a mathematical science,

because it deals with quantities of commodities. But as soon as we

attempt to draw out the equations expressing the laws of demand and

supply, we discover that they have a complexity entirely surpassing

our powers of mathematical treatment. We may lay down the general form

of the equations, expressing the demand and supply for two or three

commodities among two or three trading bodies, but all the functions

involved are so complicated in character that there is not much fear

of scientific method making rapid progress in this direction. If such

be the prospects of a comparatively formal science, like political

economy, what shall we say of moral science? Any complete theory of

morals must deal with quantities of pleasure and pain, as Bentham

pointed out, and must sum up the general tendency of each kind of

action upon the good of the community. If we are to apply scientific

method to morals, we must have a calculus of moral effects, a kind

of physical astronomy investigating the mutual perturbations of

individuals. But as astronomers have not yet fully solved the problem

of three gravitating bodies, when shall we have a solution of the

problem of three moral bodies?

The sciences of political economy and morality are comparatively

abstract and general, treating mankind from simple points of view,

and attempting to detect general principles of action. They are to

social phenomena what the abstract sciences of chemistry, heat, and

electricity are to the concrete science of meteorology. Before we

can investigate the actions of any aggregate of men, we must have

fairly mastered all the more abstract sciences applying to them,

somewhat in the way that we have acquired a fair comprehension of

the simpler truths of chemistry and physics. But all our physical

sciences do not enable us to predict the weather two days hence with

any great probability, and the general problem of meteorology is almost

unattempted as yet. What shall we say then of the general problem of

social science, which shall enable us to predict the course of events

in a nation?

Several writers have proposed to lay the foundations of the science

of history. Buckle undertook to write the *History of Civilisation

in England*, and to show how the character of a nation could be

explained by the nature of the climate and the fertility of the soil.

He omitted to explain the contrast between the ancient Greek nation

and the present one; there must have been an extraordinary revolution

in the climate or the soil. Auguste Comte detected the simple laws

of the course of development through which nations pass. There are

always three phases of intellectual condition,--the theological, the

metaphysical, and the positive; applying this general law of progress

to concrete cases, Comte was enabled to predict that in the hierarchy

of European nations, Spain would necessarily hold the highest place.

Such are the parodies of science offered to us by the *positive*

philosophers.

A science of history in the true sense of the term is an absurd notion.

A nation is not a mere sum of individuals whom we can treat by the

method of averages; it is an organic whole, held together by ties of

infinite complexity. Each individual acts and re-acts upon his smaller

or greater circle of friends, and those who acquire a public position

exert an influence on much larger sections of the nation. There will

always be a few great leaders of exceptional genius or opportunities,

the unaccountable phases of whose opinions and inclinations sway the

whole body. From time to time arise critical situations, battles,

delicate negotiations, internal disturbances, in which the slightest

incidents may change the course of history. A rainy day may hinder a

forced march, and change the course of a campaign; a few injudicious

words in a despatch may irritate the national pride; the accidental

discharge of a gun may precipitate a collision the effects of which

will last for centuries. It is said that the history of Europe depended

at one moment upon the question whether the look-out man upon Nelson’s

vessel would or would not descry a ship of Napoleon’s expedition

to Egypt which was passing not far off. In human affairs, then,

the smallest causes may produce the greatest effects, and the real

application of scientific method is out of the question.

*The Theory of Evolution.*

Profound philosophers have lately generalised concerning the production

of living forms and the mental and moral phenomena regarded as their

highest development. Herbert Spencer’s theory of evolution purports

to explain the origin of all specific differences, so that not even

the rise of a Homer or a Beethoven would escape from his broad

theories. The homogeneous is unstable and must differentiate itself,

says Spencer, and hence comes the variety of human institutions

and characters. In order that a living form shall continue to exist

and propagate its kind, says Darwin, it must be suitable to its

circumstances, and the most suitable forms will prevail over and

extirpate those which are less suitable. From these fruitful ideas

are developed theories of evolution and natural selection which go

far towards accounting for the existence of immense numbers of living

creatures--plants, and animals. Apparent adaptations of organs to

useful purposes, which Paley regarded as distinct products of creative

intelligence, are now seen to follow as natural effects of a constantly

acting tendency. Even man, according to these theories, is no distinct

creation, but rather an extreme case of brain development. His nearest

cousins are the apes, and his pedigree extends backwards until it joins

that of the lowliest zoophytes.

The theories of Darwin and Spencer are doubtless not demonstrated; they

are to some extent hypothetical, just as all the theories of physical

science are to some extent hypothetical, and open to doubt. Judging

from the immense numbers of diverse facts which they harmonise and

explain, I venture to look upon the theories of evolution and natural

selection in their main features as two of the most probable hypotheses

ever proposed. I question whether any scientific works which have

appeared since the *Principia* of Newton are comparable in importance

with those of Darwin and Spencer, revolutionising as they do all our

views of the origin of bodily, mental, moral, and social phenomena.

Granting all this, I cannot for a moment admit that the theory of

evolution will destroy theology. That theory embraces several laws

or uniformities which are observed to be true in the production of

living forms; but these laws do not determine the size and figure of

living creatures, any more than the law of gravitation determines the

magnitudes and distances of the planets. Suppose that Darwin is correct

in saying that man is descended from the Ascidians: yet the precise

form of the human body must have been influenced by an infinite train

of circumstances affecting the reproduction, growth, and health of the

whole chain of intermediate beings. No doubt, the circumstances being

what they were, man could not be otherwise than he is, and if in any

other part of the universe an exactly similar earth, furnished with

exactly similar germs of life, existed, a race must have grown up there

exactly similar to the human race.

By a different distribution of atoms in the primeval world a different

series of living forms on this earth would have been produced. From the

same causes acting according to the same laws, the same results will

follow; but from different causes acting according to the same laws,

different results will follow. So far as we can see, then, infinitely

diverse living creatures might have been created consistently with the

theory of evolution, and the precise reason why we have a backbone,

two hands with opposable thumbs, an erect stature, a complex brain,

about 223 bones, and many other peculiarities, is only to be found in

the original act of creation. I do not, any less than Paley, believe

that the eye of man manifests design. I believe that the eye was

gradually developed, and we can in fact trace its gradual development

from the first germ of a nerve affected by light-rays in some simple

zoophyte. In proportion as the eye became a more accurate instrument

of vision, it enabled its possessor the better to escape destruction,

but the ultimate result must have been contained in the aggregate of

the causes, and these causes, as far as we can see, were subject to the

arbitrary choice of the Creator.

Although Agassiz was clearly wrong in holding that every species of

living creature appeared on earth by the immediate intervention of

the Creator, which would amount to saying that no laws of connection

between forms are discoverable, yet he seems to be right in asserting

that living forms are distinct from those produced by purely physical

causes. “The products of what are commonly called physical agents,”

he says,[620] “are everywhere the same (*i.e.* upon the whole surface

of the earth), and have always been the same (*i.e.* during all

geological periods); while organised beings are everywhere different

and have differed in all ages. Between two such series of phenomena

there can be no causal or genetic connection.” Living forms as we now

regard them are essentially variable, but from constant mechanical

causes constant effects would ensue. If vegetable cells are formed

on geometrical principles being first spherical, and then by mutual

compression dodecahedral, then all cells should have similar forms. In

the Foraminifera and some other lowly organisms, we seem to observe the

production of complex forms on geometrical principles. But from similar

causes acting according to similar laws only similar results could

be produced. If the original life germ of each creature is a simple

particle of protoplasm, unendowed with any distinctive forces, then the

whole of the complex phenomena of animal and vegetable life are effects

without causes. Protoplasm may be chemically the same substance, and

the germ-cell of a man and of a fish may be apparently the same, so far

as the microscope can decide; but if certain cells produce men, and

others as uniformly produce a species of fish, there must be a hidden

constitution determining the extremely different results. If this were

not so, the generation of every living creature from the uniform germ

would have to be regarded as a distinct act of creation.

[620] Agassiz, *Essay on Classification*, p. 75.

Theologians have dreaded the establishment of the theories of Darwin

and Huxley and Spencer, as if they thought that those theories could

explain everything upon the purest mechanical and material principles,

and exclude all notions of design. They do not see that those theories

have opened up more questions than they have closed. The doctrine of

evolution gives a complete explanation of no single living form. While

showing the general principles which prevail in the variation of living

creatures, it only points out the infinite complexity of the causes

and circumstances which have led to the present state of things. Any

one of Mr. Darwin’s books, admirable though they all are, consists

but in the setting forth of a multitude of indeterminate problems. He

proves in the most beautiful manner that each flower of an orchid is

adapted to some insect which frequents and fertilises it, and these

adaptations are but a few cases of those immensely numerous ones which

have occurred in the lives of plants and animals. But why orchids

should have been formed so differently from other plants, why anything,

indeed, should be as it is, rather than in some of the other infinitely

numerous possible modes of existence, he can never show. The origin

of everything that exists is wrapped up in the past history of the

universe. At some one or more points in past time there must have been

arbitrary determinations which led to the production of things as they

are.

*Possibility of Divine Interference.*

I will now draw the reader’s attention to pages 149 to 152. I there

pointed out that all inductive inference involves the assumption that

our knowledge of what exists is complete, and that the conditions of

things remain unaltered between the time of our experience and the

time to which our inferences refer. Recurring to the illustration

of a ballot-box, employed in the chapter on the inverse method of

probabilities, we assume when predicting the probable nature of

the next drawing, firstly, that our previous drawings have been

sufficiently numerous to give us knowledge of the contents of the

box; and, secondly, that no interference with the ballot-box takes

place between the previous and the next drawings. The results yielded

by the theory of probability are quite plain. No finite number of

casual drawings can give us sure knowledge of the contents of the

box, so that, even in the absence of all disturbance, our inferences

are merely the best which can be made, and do not approach to

infallibility. If, however, interference be possible, even the theory

of probability ceases to be applicable, for, the amount and nature

of that interference being arbitrary and unknown, there ceases to

be any connection between premises and conclusion. Many years of

reflection have not enabled me to see the way of avoiding this hiatus

in scientific certainty. The conclusions of scientific inference appear

to be always of a hypothetical and provisional nature. Given certain

experience, the theory of probability yields us the true interpretation

of that experience and is the surest guide open to us. But the best

calculated results which it can give are never absolute probabilities;

they are purely relative to the extent of our information. It seems to

be impossible for us to judge how far our experience gives us adequate

information of the universe as a whole, and of all the forces and

phenomena which can have place therein.

I feel that I cannot in the space remaining at my command in the

present volume, sufficiently follow out the lines of thought suggested,

or define with precision my own conclusions. This chapter contains

merely *Reflections* upon subjects of so weighty a character that I

should myself wish for many years--nay for more than a lifetime of

further reflection. My purpose, as I have repeatedly said, is the

purely negative one of showing that atheism and materialism are no

necessary results of scientific method. From the preceding reviews of

the value of our scientific knowledge, I draw one distinct conclusion,

that we cannot disprove the possibility of Divine interference in

the course of nature. Such interference might arise, so far as our

knowledge extends, in two ways. It might consist in the disclosure

of the existence of some agent or spring of energy previously

unknown, but which effects a given purpose at a given moment. Like

the pre-arranged change of law in Babbage’s imaginary calculating

machine, there may exist pre-arranged surprises in the order of

nature, as it presents itself to us. Secondly, the same Power, which

created material nature, might, so far as I can see, create additions

to it, or annihilate portions which do exist. Such events are in a

certain sense inconceivable to us; yet they are no more inconceivable

than the existence of the world as it is. The indestructibility of

matter, and the conservation of energy, are very probable scientific

hypotheses, which accord satisfactorily with experiments of scientific

men during a few years past, but it would be gross misconception of

scientific inference to suppose that they are certain in the sense

that a proposition in geometry is certain. Philosophers no doubt hold

that *de nihilo nihil fit*, that is to say, their senses give them no

means of imagining to the mind how creation can take place. But we are

on the horns of a trilemma; we must either deny that anything exists,

or we must allow that it was created out of nothing at some moment of

past time, or that it existed from eternity. The first alternative is

absurd; the other two seem to me equally conceivable.

*Conclusion.*

It may seem that there is one point where our speculations must end,

namely where contradiction begins. The laws of Identity and Difference

and Duality were the foundations from which we started, and they are,

so far as I can see, the foundations which we can never quit without

tottering. Scientific Method must begin and end with the laws of

thought, but it does not follow that it will save us from encountering

inexplicable, and at least apparently contradictory results. The nature

of continuous quantity leads us into extreme difficulties. Any finite

space is composed of an infinite number of infinitely small spaces,

each of which, again, is composed of an infinite number of spaces of

a second order of smallness; these spaces of the second order are

composed, again, of infinitely small spaces of the third order. Even

these spaces of the third order are not absolute geometrical points

answering to Euclid’s definition of a point, as position without

magnitude. Go on as far as we will, in the subdivision of continuous

quantity, yet we never get down to the absolute point. Thus scientific

method leads us to the inevitable conception of an infinite series

of successive orders of infinitely small quantities. If so, there is

nothing impossible in the existence of a myriad universes within the

compass of a needle’s point, each with its stellar systems, and its

suns and planets, in number and variety unlimited. Science does nothing

to reduce the number of strange things that we may believe. When fairly

pursued it makes absurd drafts upon our powers of comprehension and

belief.

Some of the most precise and beautiful theorems in mathematical science

seem to me to involve apparent contradiction. Can we imagine that a

point moving along a perfectly straight line towards the west would

ever get round to the east and come back again, having performed, as

it were, a circuit through infinite space, yet without ever diverging

from a perfectly straight direction? Yet this is what happens to the

intersecting point of two straight lines in the same plane, when one

line revolves. The same paradox is exhibited in the hyperbola regarded

as an infinite ellipse, one extremity of which has passed to an

infinite distance and come back in the opposite direction. A varying

quantity may change its sign by passing either through zero or through

infinity. In the latter case there must be one intermediate value of

the variable for which the variant is indifferently negative infinity

and positive infinity. Professor Clifford tells me that he has found

a mathematical function which approaches infinity as the variable

approaches a certain limit; yet at the limit the function is finite!

Mathematicians may shirk difficulties, but they cannot make such

results of mathematical principles appear otherwise than contradictory

to our common notions of space.

The hypothesis that there is a Creator at once all-powerful

and all-benevolent is pressed, as it must seem to every candid

investigator, with difficulties verging closely upon logical

contradiction. The existence of the smallest amount of pain and evil

would seem to show that He is either not perfectly benevolent, or

not all-powerful. No one can have lived long without experiencing

sorrowful events of which the significance is inexplicable. But if we

cannot succeed in avoiding contradiction in our notions of elementary

geometry, can we expect that the ultimate purposes of existence shall

present themselves to us with perfect clearness? I can see nothing to

forbid the notion that in a higher state of intelligence much that is

now obscure may become clear. We perpetually find ourselves in the

position of finite minds attempting infinite problems, and can we be

sure that where we see contradiction, an infinite intelligence might

not discover perfect logical harmony?

From science, modestly pursued, with a due consciousness of the extreme

finitude of our intellectual powers, there can arise only nobler and

wider notions of the purpose of Creation. Our philosophy will be an

affirmative one, not the false and negative dogmas of Auguste Comte,

which have usurped the name, and misrepresented the tendencies of a

true *positive philosophy*. True science will not deny the existence of

things because they cannot be weighed and measured. It will rather lead

us to believe that the wonders and subtleties of possible existence

surpass all that our mental powers allow us clearly to perceive. The

study of logical and mathematical forms has convinced me that even

space itself is no requisite condition of conceivable existence.

Everything, we are told by materialists, must be here or there,

nearer or further, before or after. I deny this, and point to logical

relations as my proof.

There formerly seemed to me to be something mysterious in the

denominators of the binomial expansion (p. 190), which are reproduced

in the natural constant ε, or

1 + 1/1 + 1/(1 . 2) + 1/(1 . 2 . 3) + ...

and in many results of mathematical analysis. I now perceive, as

already explained (pp. 33, 160, 383), that they arise out of the fact

that the relations of space do not apply to the logical conditions

governing the numbers of combinations as contrasted to those of

permutations. So far am I from accepting Kant’s doctrine that space is

a necessary form of thought, that I regard it as an accident, and an

impediment to pure logical reasoning. Material existences must exist in

space, no doubt, but intellectual existences may be neither in space

nor out of space; they may have no relation to space at all, just as

space itself has no relation to time. For all that I can see, then,

there may be intellectual existences to which both time and space are

nullities.

Now among the most unquestionable rules of scientific method is

that first law that *whatever phenomenon is, is*. We must ignore no

existence whatever; we may variously interpret or explain its meaning

and origin, but, if a phenomenon does exist, it demands some kind

of explanation. If then there is to be competition for scientific

recognition, the world without us must yield to the undoubted existence

of the spirit within. Our own hopes and wishes and determinations are

the most undoubted phenomena within the sphere of consciousness. If men

do act, feel, and live as if they were not merely the brief products of

a casual conjunction of atoms, but the instruments of a far-reaching

purpose, are we to record all other phenomena and pass over these? We

investigate the instincts of the ant and the bee and the beaver, and

discover that they are led by an inscrutable agency to work towards

a distant purpose. Let us be faithful to our scientific method, and

investigate also those instincts of the human mind by which man is led

to work as if the approval of a Higher Being were the aim of life.

INDEX.

Abacus, logical, 104;

arithmetical, 107;

Panchrestus, 182.

Aberration of light, 561;

systematic, 547.

Abscissio infiniti, 79, 713.

Abstract terms, 27;

number, 159.

Abstraction, 704;

logical, 25;

numerical, 158;

of indifferent circumstances, 97.

Accademia del Cimento, 427, 432, 436, 527.

Accident, logical, 700.

Accidental discovery, 529.

Achromatic lenses, 432.

Actinometer, 337.

Adamantine medium, 605, 751.

Adjectives, 14, 30, 31, 35;

indeterminate, 41.

Adrain, of New Brunswick, 375.

Affirmation, 44.

Agassiz, on genera, 726;

on creation of species, 763.

Agreement, 44.

Airy, Sir George Biddell, on perpetual motion, 223;

new property of sphere, 232;

pendulum experiments, 291, 304, 348, 567;

standard clock, 353;

book on *Errors of Observation*, 395;

tides, 488;

extra-polation, 495;

Thales’ eclipse, 537;

interference of light, 539;

density of earth, 291.

Alchemists, 505;

how misled, 428.

Algebra, 123, 155, 164;

Diophantine, 631.

Algebraic, equations, 123;

geometry, 633.

Allotropic state, 663, 670.

Alloys, possible number, 191;

properties, 528.

Alphabet, the Logical, 93, 104, 125;

Morse, 193.

Alphabet, permutations of letters of the, 174, 179.

Alphabetic indexes, 714.

Alternative relations, 67;

exclusive and unexclusive, 205.

Ampère, electricity, 547;

classification, 679.

Anagrams, 128.

Analogy, 627;

of logical and numerical terms, 160;

and generalisation, 596;

in mathematical sciences, 631;

in theory of undulations, 635;

in astronomy, 638;

failure of, 641.

Analysis, logical, 122.

Andrews, Prof. Thomas, experiments on gaseous state, 71, 613, 665, 753.

Angström, on spectrum, 424.

Angular magnitude, 305, 306, 326.

Antecedent defined, 225.

Anticipation of Nature, 509.

Anticipations, of Principle of Substitution, 21;

of electric telegraph, 671.

Apparent, equality, 275;

sequence of events, 409.

Approximation, theory of, 456;

to exact laws, 462;

mathematical principles of, 471;

arithmetic of, 481.

Aqueous vapour, 500.

Aquinas, on disjunctive propositions, 69.

Arago, photometer, 288;

rotating disc, 535;

his philosophic character, 592.

Archimedes, *De Arenæ Numero*, 195;

centre of gravity, 363.

Arcual unit, 306, 330.

Argyll, Duke of, 741.

Aristarchus on sun’s and moon’s distances, 294.

Aristotelian doctrines, 666.

Aristotle, dictum, 21;

singular terms, 39;

overlooked simple identities, 40;

order of premises, 114;

logical error, 117;

definition of time, 307;

on science, 595;

on white swans, 666.

Arithmetic, reasoning in, 167;

of approximate quantities, 481.

Arithmetical triangle, 93, 143, 182, 202, 378, 383;

diagram of, 184;

connection with Logical Alphabet, 189;

in probability, 208.

Asteroids, discovery of, 412, 748.

Astronomy, physical, 459.

Atmospheric tides, 553.

Atomic theory, 662.

Atomic weights, 563.

Atoms, size of, 195;

impossibility of observing, 406.

Augustin on time, 307.

Average, 359, 360;

divergence from, 188;

etymology of, 363.

Axes of crystals, 686.

Axioms of algebra, 164.

Babbage, Charles, calculating machine, 107, 231, 743;

lighthouse signals, 194;

natural constants, 329;

Mosaic history, 412;

universal and general truths, 646;

change of law, 230;

persistence of effects, 757.

Bacon, Francis Lord, *Novum Organum*, 107;

on induction, 121;

biliteral cipher, 193;

First Aphorism, 219;

on causes, 221;

Copernican system, 249, 638;

deficient powers of senses, 278;

observation, 402;

Natural History, 403;

use of hypothesis, 506;

his method, 507;

*experimentum crucis*, 519;

error of his method, 576;

ostensive, clandestine instances, &c., 608, 610;

*latens precessus*, 619.

Bacon, Roger, on the rainbow, 526, 598.

Baily, Francis, 272;

density of earth, 342, 566;

experiments with torsion balance, 370, 397, 432, 567–8;

motions of stars, 572.

Bain, Alexander, on powers of mind, 4;

Mill’s reform of logic, 227.

Baker’s poem, *The Universe*, 621.

Balance, use of the chemical, 292, 351, 354, 369;

delicacy of, 304;

vibrations of, 369.

Ballot, Buys, experiment on sound, 541.

Ballot-box, simile of, 150, 251–6, 765.

Barbara, 55, 57, 88, 105, 141.

Baroko, 85.

Barometer, 659;

Gay Lussac’s standard, 346;

variations, 337, 346, 349.

Bartholinus on double refraction, 585.

Base-line, measurement of, 304.

Bauhusius, verses of, 175.

Baxendell, Joseph, 552.

Beneke, on substitution, 21.

Bennet, momentum of light, 435.

Bentham, George, 15;

bifurcate classification, 695;

infima species, 702;

works on classification, 703;

analytical key to flora, 712.

Bentham, Jeremy, on analogy, 629;

bifurcate classification, 703.

Benzenberg’s experiment, 388.

Bernoulli, Daniel, planetary orbits, 250;

resisting media and projectiles, 467;

vibrations, 476.

Bernoulli, James, 154;

numbers of, 124;

Protean verses, 175;

*De Arte Conjectandi* quoted, 176, 183;

on figurate numbers, 183;

theorem of, 209;

false solution in probability, 213;

solution of inverse problem, 261.

Bessel, F. W., 375;

law of error, 384;

formula for periodic variations, 488;

use of hypothesis, 506;

solar parallax, 560–2;

ellipticity of earth, 565;

pendulum experiments, 604.

Bias, 393, 402.

Biela’s comet, 746.

Bifurcate classification, 694.

Binomial theorem, 190;

discovery of, 231.

Biot, on tension of vapour, 500.

Blind experiments, 433.

Bode’s law, 147, 257, 660.

Boethius, quoted, 33;

on kinds of mean, 360.

Boiling point, 442, 659.

Bonnet’s theory of reproduction, 621.

Boole, George, on sign of equality, 15;

his calculus of logic, 23, 113, 634;

on logical terms, 33;

law of commutativeness, 35;

use of *some*, 41–2;

disjunctive propositions, 70;

Venn on his method, 90;

*Laws of Thought*, 155;

statistical conditions, 168;

propositions numerically definite, 172;

on probability, 199;

general method in probabilities, 206;

Laplace’s solution of inverse problem, 256;

law of error, 377.

Borda, his repeating circle, 290.

Boscovich’s hypothesis, 512.

Botany, 666, 678, 681;

modes of classification, 678;

systematic, 722;

nomenclature of, 727.

Bowen, Prof. Francis, on inference, 118;

classification, 674.

Boyle’s, Robert, law of gaseous pressure, 468, 470, 619;

on hypothesis, 510;

barometer, 659.

Bradley, his observations, 384;

accuracy of, 271;

aberration of light, 535.

Bravais, on law of error, 375.

Brewer, W. H., 142.

Brewster, Sir David, iridescent colours, 419;

spectrum, 429;

Newton’s theory of colours, 518;

refractive indices, 10, 527;

optic axes, 446.

British Museum, catalogue of, 717.

Brodie, Sir B. C., on errors of experiment, 388, 464;

ozone, 663.

Brown, Thomas, on cause, 224.

Buckle, Thomas, on constancy of average, 656;

science of history, 760.

Buffon, on probability, 215;

definition of genius, 576.

Bunsen, Robert, spectrum, 244;

photometrical researches, 273, 324, 441;

calorimeter, 343.

Butler, Bishop, on probability, 197.

Calorescence, 664.

Camestres, 84.

Canton, on compressibility of water, 338.

Carbon, 640, 728;

conductibility of, 442.

Cardan, on inclined plane, 501.

Cards, combinations of, 190.

Carlini, pendulum experiments, 567.

Carnot’s law, 606.

Carpenter, Dr. W. B., 412.

Catalogues, art of making, 714.

Cauchy, undulatory theory, 468.

Cause, 220;

definition of, 224.

Cavendish’s experiment, 272, 566.

Cayley, Professor, 145;

on mathematical tables, 331;

numbers of chemical compounds, 544.

Celarent, 55.

Centre of gravity, 363, 524;

of oscillation, gyration, &c., 364.

Centrobaric bodies, 364.

Certainty, 235, 266.

Cesare, 85.

Chalmers, on collocations, 740.

Chance, 198.

Character, human, 733.

Characteristics, 708.

Chauvenet, Professor W., on treatment of observations, 391.

Chemical affinity, 614;

analysis, 713.

Chladni, 446.

Chloroform, discovery of, 531.

Chronoscope, 616.

Cipher, 32;

Bacon’s, 193.

Circle, circumference of, 389.

Circumstances, indifferent, 419.

Circumstantial evidence, 264.

Clairaut, 650, 651;

on gravity, 463.

Classes, 25;

problem of common part of three, 170.

Classification, 673;

involving induction, 675;

multiplicity of modes, 677;

natural and artificial systems, 679;

in crystallography, 685;

symbolic statement of, 692;

bifurcate, 694;

an inverse and tentative operation, 689;

diagnostic, 710;

by indexes, 714;

of books, 715;

in biological sciences, 718;

genealogical, 719;

by types, 722;

limits of, 730.

Clifford, Professor, on types of compound statements, 143, 529;

first and last catastrophe, 744;

mathematical function, 768.

Clocks, astronomical, 340, 353.

Clouds, 447;

cirrous, 411.

Coincidences, 128;

fortuitous, 261;

measurement by, 292;

method of, 291.

Collective terms, 29, 39.

Collocations of matter, 740.

Colours, iridescent, 419;

natural, 518;

perception of, 437;

of spectrum, 584.

Combinations, 135, 142;

doctrine of, 173;

of letters of alphabet, 174;

calculations of, 180;

higher orders of, 194.

Combinatorial analysis, 176.

Comets, 449;

number of, 408;

hyperbolic, 407;

classification of, 684;

conflict with, 746–7;

Halley’s comet, 537;

Lexell’s comet, 651.

Commutativeness, law of, 35, 72, 177.

Comparative use of instruments, 299.

Compass, variations of, 281.

Complementary statements, 144.

Compossible alternatives, 69.

Compound statements, 144;

events, 204.

Compounds, chemical, 192.

Comte, Auguste, on probability, 200, 214;

on prevision, 536;

his positive philosophy, 752, 760, 768.

Concrete number, 159.

Conditions, of logical symbols, 32;

removal of usual, 426;

interference of unsuspected, 428;

maintenance of similar, 443;

approximation to natural, 465.

Condorcet, 2;

his problem, 253.

Confusion of elements, 237.

Conical refraction, 653.

Conjunction of planets, 293, 657.

Consequent, definition of, 225.

Conservation of energy, 738.

Constant numbers of nature, 328;

mathematical, 330;

physical, 331;

astronomical, 332;

terrestrial, 333;

organic, 333;

social, 334.

Continuity, law of, 615, 729;

sense of, 493;

detection of, 610;

failure of, 619.

Continuous quantity, 274, 485.

Contradiction, law of, 31, 74.

Contrapositive, proposition, 84, 136;

conversion, 83.

Conversion of propositions, 46, 118.

Copernican theory, 522, 625, 638, 647.

Copula, 16.

Cornu, velocity of light, 561.

Corpuscular theory, 520, 538, 667.

Correction, method of, 346.

Correlation, 678, 681.

Cotes, Roger, use of mean, 359;

method of least squares, 377.

Coulomb, 272.

Couple, mechanical, 653.

Creation, problem of, 740.

Crookes’ radiometer, 435.

Cross divisions, 144.

Crystallography, 648, 654, 658, 678, 754;

systems of, 133;

classification in, 685.

Crystals, 602;

Dana’s classification of, 711;

pseudomorphic, 658.

Curves, use of, 392, 491, 496;

of various degrees, 473.

Cuvier, on experiment, 423;

on inferences, 682.

Cyanite, 609.

Cycloid, 633.

Cycloidal pendulum, 461.

Cypher, 124.

D’Alembert, blunders in probability, 213, 214;

on gravity, 463.

Dalton, laws of, 464, 471;

atomic theory, 662.

Darapti, 59.

Darii, 56.

Darwin, Charles, his works, 131;

negative results of observation, 413;

arguments against his theory, 437;

cultivated plants, 531;

his influence, 575;

classification, 718;

constancy of character in classification, 720–1;

on definition, 726;

restoration of limbs, 730;

tendency of his theory, 762, 764.

Davy, Sir H., on new instruments, 270;

nature of heat, 343, 417;

detection of salt in electrolysis, 428.

Day, sidereal, 310;

length of, 289.

Decandolle, on classification, 696.

Decyphering, 124.

Deduction, 11, 49.

Deductive reasoning, 534;

miscellaneous forms of, 60;

probable, 209.

Definition, 39, 62, 711, 723;

purpose of, 54;

of cause and power, 224.

De Morgan, Augustus, negative terms, 14;

Aristotle’s logic, 18;

relatives, 23;

logical universe, 43;

complex propositions, 75;

contraposition, 83;

formal logic quoted, 101;

error of his system, 117;

anagram of his name, 128;

numerically definite reasoning, 168–172;

probability, 198;

belief, 199;

experiments in probability, 207;

probable deductive arguments, 209–210;

trisection of angle, 233;

probability of inference, 259;

arcual unit, 306;

mathematical tables, 331;

personal error, 348;

average, 363;

his works on probability, 394–395;

apparent sequence, 409;

sub-equality, 480;

rule of approximation, 481;

negative areas, 529;

generalisation, 600;

double algebra, 634;

bibliography, 716;

catalogues, 716;

extensions of algebra, 758.

Density, unit of, 316;

of earth, 387;

negative, 642.

Descartes, vortices, 517;

geometry, 632.

Description, 62.

Design, 762–763.

Determinants, inference by, 50.

Development, logical, 89, 97.

Diagnosis, 708.

Dichotomy, 703.

Difference, 44;

law of, 5;

sign of, 17;

representation of, 45;

inference with, 52, 166;

form of, 158.

Differences of numbers, 185.

Differential calculus, 477.

Differential thermometer, 345.

Diffraction of light, 420.

Dimensions, theory of, 325.

Dip-needle, observation of, 355.

Direct deduction, 49.

Direction of motion, 47.

Discontinuity, 620.

Discordance, of theory and experiment, 558;

of theories, 587.