OF THE APPLICATION OF MATHEMATICS TO THE INDUCTIVE SCIENCES.
1. ALL objects in the world which can be made the subjects
of our contemplation are subordinate to the conditions of
Space, Time, and Number; and on this account, the doctrines
of pure mathematics have most numerous and extensive
applications in every department of our investigations of
nature. And there is a peculiarity in these Ideas, which has
caused the mathematical sciences to be, in all cases, the
first successful efforts of the awakening speculative powers
of nations at the commencement of their intellectual
progress. Conceptions derived from these Ideas are, from the
very first, perfectly precise and clear, so as to be fit
elements of scientific truths. This is not the case with the
other conceptions which form the subjects of scientific
inquiries. The conception of _statical force_, for instance,
was never presented in a distinct form till the works of
Archimedes appeared: the conception of _accelerating force_
was confused, in the mind of Kepler and his contemporaries,
and only became clear enough for purposes of sound
scientific reasoning in the succeeding century: the just
conception of chemical _composition_ of elements gradually,
in modern times, emerged from the erroneous and vague
notions of the ancients. If we take works published on such
subjects before the epoch when the foundations of the true
science were laid, we find the knowledge not only small, but
worthless. The writers did not see any evidence in what we
now consider as the axioms of the science; nor any
inconsistency where we now see self-contradiction. But this
was never the case with speculations concerning {160} space
and number. From their first rise, these were true as far as
they went. The Geometry and Arithmetic of the Greeks and
Indians, even in their first and most scanty form, contained
none but true propositions. Men's intuitions upon these
subjects never allowed them to slide into error and
confusion; and the truths to which they were led by the
first efforts of their faculties, so employed, form part of
the present stock of our mathematical knowledge.
2. But we are here not so much concerned with mathematics in
their pure form, as with their application to the phenomena
and laws of nature. And here also the very earliest history
of civilization presents to us some of the most remarkable
examples of man's success in his attempts to attain to
science. Space and time, position and motion, govern all
visible objects; but by far the most conspicuous examples of
the relations which arise out of such elements, are
displayed by the ever-moving luminaries of the sky, which
measure days, and months, and years, by their motions, and
man's place on the earth by their position. Hence the
sciences of space and number were from the first cultivated
with peculiar reference to Astronomy. I have elsewhere[18\2]
quoted Plato's remark,--that it is absurd to call the
science of the relations of space _geometry_, the measure of
the earth, since its most important office is to be found in
its application to the heavens. And on other occasions also
it appears how strongly he, who may be considered as the
representative of the scientific and speculative tendencies
of his time and country, had been impressed with the
conviction, that the formation of a science of the celestial
motions must depend entirely upon the progress of
mathematics. In the Epilogue to the Dialogue on the
Laws[19\2], he declares mathematical knowledge to be the
first and main requisite for the astronomer, and describes
the portions of it which he holds necessary for astronomical
speculators to cultivate. These seem to be, Plane Geometry,
Theoretical Arithmetic, the Application of Arithmetic {161}
to planes and to solids, and finally the doctrine of
Harmonics. Indeed the bias of Plato appears to be rather to
consider mathematics as the essence of the science of
astronomy, than as its instrument; and he seems disposed, in
this as in other things, to disparage observation, and to
aspire after a science founded upon demonstration alone. 'An
astronomer,' he says in the same place, 'must not be like
Hesiod and persons of that kind, whose astronomy consists in
noting the settings and risings of the stars; but he must be
one who understands the revolutions of the celestial
spheres, each performing its proper cycle.'
[Note 18\2: _Hist. Ind. Sc._ b. iii. c. ii.]
[Note 19\2: _Epinomis_, p. 990.]
A large portion of the mathematics of the Greeks, so long as
their scientific activity continued, was directed towards
Astronomy. Besides many curious propositions of plane and
solid Geometry, to which their astronomers were led, their
Arithmetic, though very inconvenient in its fundamental
assumptions (as being sexagesimal not decimal), was
cultivated to a great extent; and the science of
Trigonometry, in which problems concerning the relations of
space were resolved by means of tables of numerical results
previously obtained, was created. Menelaus of Alexandria
wrote six Books on Chords, probably containing methods of
calculating Tables of these quantities; such Tables were
familiarly used by the later Greek astronomers. The same
author also wrote three Books on Spherical Trigonometry,
which are still extant.
3. The Greeks, however, in the first vigour of their pursuit
of mathematical truth, at the time of Plato and soon after,
had by no means confined themselves to those propositions
which had a visible bearing on the phenomena of nature; but
had followed out many beautiful trains of research,
concerning various kinds of figures, for the sake of their
beauty alone; as for instance in their doctrine of Conic
Sections, of which curves they had discovered all the
principal properties. But it is curious to remark, that
these investigations, thus pursued at first as mere matters
of curiosity and intellectual gratification, were destined,
two thousand years later, to play a very important part in
{162} establishing that system of the celestial motions
which succeeded the Platonic scheme of cycles and epicycles.
If the properties of the conic sections had not been
demonstrated by the Greeks, and thus rendered familiar to
the mathematicians of succeeding ages, Kepler would probably
not have been able to discover those laws respecting the
orbits and motions of the planets which were the occasion of
the greatest revolution that ever happened in the history of
science.
4. The Arabians, who, as I have elsewhere said, added little
of their own to the stores of science which they received
from the Greeks, did however make some very important
contributions in those portions of pure mathematics which
are subservient to astronomy. Their adoption of the Indian
mode of computation by means of the Ten Digits, 1, 2, 3, 4,
5, 6, 7, 8, 9, 0, and by the method of Local Values, instead
of the cumbrous sexagesimal arithmetic of the Greeks, was an
improvement by which the convenience and facility of
numerical calculations were immeasurably augmented. The
Arabians also rendered several of the processes of
trigonometry much more commodious, by using the Sine of an
arc instead of the Chord; an improvement which Albategnius
appears to claim for himself[20\2]; and by employing also
the Tangents of arcs, or, as they called them[21\2],
_upright shadows_.
[Note 20\2: Delambre, _Ast., M. A._, p. 12.]
[Note 21\2: _Ibid._ p. 17.]
5. The constant application of mathematical knowledge to the
researches of Astronomy, and the mutual influence of each
science on the progress of the other, has been still more
conspicuous in modern times. Newton's Method of Prime and
Ultimate Ratios, which we have already noticed as the first
correct exposition of the doctrine of a Limit, is stated in
a series of Lemmas, or preparatory theorems, prefixed to his
_Treatise on the System of the World_. Both the properties
of curve lines and the doctrines concerning force and
motion, which he had to establish, required that the common
mathematical processes should be methodized and extended. If
Newton had not been a most {163} expert and inventive
mathematician, as well as a profound and philosophical
thinker, he could never have made any one of those vast
strides in discovery of which the rapid succession in his
work strikes us with wonder[22\2]. And if we see that the
great task begun by him, goes on more slowly in the hands of
his immediate successors, and lingers a little before its
full completion, we perceive that this arises, in a great
measure, from the defect of the mathematical methods then
used. Newton's synthetical modes of investigation, as we
have elsewhere observed, were an instrument[23\2], powerful
indeed in his mighty hand, but too ponderous for other
persons to employ with effect. The countrymen of Newton
clung to it the longest, out of veneration for their master;
and English cultivators of physical astronomy were, on that
very account, left behind the progress of mathematical
science in France and Germany, by a wide interval, which
they have only recently recovered. On the Continent, the
advantages offered by a familiar use of symbols, and by
attention to their symmetry and other relations, were
accepted without reserve. In this manner the Differential
Calculus of Leibnitz, which was in its origin and
signification identical with the Method of Fluxions of
Newton, soon surpassed its rival in the extent and
generality of its application to problems. This Calculus was
applied to the science of mechanics, to which it, along with
the symmetrical use of co-ordinates, gave a new form; for it
was soon seen that the most difficult problems might in
general be reduced to finding integrals, which is the
reciprocal process of that by which differentials are found;
so that all difficulties of physical astronomy were reduced
to difficulties of symbolical calculation, these, indeed,
being often sufficiently stubborn. Clairaut, Euler, and
D'Alembert employed the increased resources of mathematical
science upon the Theory of the Moon, and other questions
relative to the system of the world; and thus began to
pursue such inquiries in the course in which mathematicians
{164} are still labouring up to the present day. This course
was not without its checks and perplexities. We have
elsewhere quoted[24\2] Clairaut's expression when he had
obtained the very complex differential equations which
contain the solution of the problem of the moon's motion:
'Now integrate them who can!' But in no very long time they
were integrated, at least approximately; and the methods of
approximation have since then been improved; so that now,
with a due expenditure of labour, they may be carried to any
extent which is thought desirable. If the methods of
astronomical observation should hereafter reach a higher
degree of exactness than they now profess, so that
irregularities in the motions of the sun, moon, and planets,
shall be detected which at present escape us, the
mathematical part of the theory of universal gravitation is
in such a condition that it can soon be brought into
comparison with the newly-observed facts. Indeed at present
the mathematical theory is in advance of such observations.
It can venture to suggest what may afterwards be detected,
as well as to explain what has already been observed. This
has happened recently; for Professor Airy has calculated the
law and amount of an inequality depending upon the mutual
attraction of the Earth and Venus; of which inequality (so
small is it,) it remains to be determined whether its effect
can be traced in the series of astronomical observations.
[Note 22\2: _Hist. Ind. Sc._ b. vii. c. ii.]
[Note 23\2: _Ibid._ p. 175.]
[Note 24\2: _Hist. Ind. Sc._ b. vi. c. vi. sect. 7.]
6. As the influence of mathematics upon the progress of
astronomy is thus seen in the cases in which theory and
observation confirm each other, so this influence appears in
another way, in the very few cases in which the facts have
not been fully reduced to an agreement with theory. The most
conspicuous case of this kind is the state of our knowledge
of the Tides. This is a portion of astronomy: for the
Newtonian theory asserts these curious phenomena to be the
result of the attraction of the sun and moon. Nor can there
be any doubt that this is true, as a general statement; yet
the subject is up to the present time a blot {165} on the
perfection of the theory of universal gravitation; for we
are very far from being able in this, as in the other parts
of astronomy, to show that theory will exactly account for
the time, and magnitude, and all other circumstances of the
phenomenon at every place on the earth's surface. And what
is the portion of our mathematics which is connected with
this solitary signal defect in astronomy? It is the
mathematics of the Motion of Fluids; a portion in which
extremely little progress has been made, and in which all
the more general problems of the subject have hitherto
remained entirely insoluble. The attempts of the greatest
mathematicians, Newton, Maclaurin, Bernoulli, Clairaut,
Laplace, to master such questions, all involve some
gratuitous assumption, which is introduced because the
problem cannot otherwise be mathematically dealt with: these
assumptions confessedly render the result defective, and how
defective, it is hard to say. And it was probably precisely
the absence of a theory which could be reasonably expected
to agree with the observations, which made Observations of
this very curious phenomenon, the Tides, to be so much
neglected as till very recently they were. Of late years
such observations have been pursued, and their results have
been resolved into empirical laws, so that the rules of the
phenomena have been ascertained, although the dependence of
these rules upon the lunar and solar forces has not been
shown. Here then we have a portion of our knowledge relating
to facts undoubtedly dependent upon universal gravitation,
in which Observation has outstripped Theory in her progress,
and is compelled to wait till her usual companion overtakes
her. This is a position of which Mathematical Theory has
usually been very impatient, and we may expect that she will
be no less so in the present instance.
7. It would be easy to show from the history of other
sciences, for example, Mechanics and Optics, how essential
the cultivation of pure mathematics has been to their
progress. The parabola was already familiar among
mathematicians when Galileo discovered that it was the
theoretical path of a Projectile; and the {166} extension
and generalization of the Laws of Motion could never have
been effected, unless the Differential and Integral Calculus
had been at hand, ready to trace the results of every
hypothesis which could be made. D'Alembert's mode of
expressing the Third Law of Motion in its most general
form[25\2], if it did not prove the law, at least reduced
the application of it to analytical processes which could be
performed in most of those cases in which they were needed.
In many instances the demands of mechanical science
suggested the extension of the methods of pure analysis. The
problem of Vibrating Strings gave rise to the Calculus of
Partial Differences, which was still further stimulated by
its application to the motions of fluids and other
mechanical problems. And we have in the writings of Lagrange
and Laplace other instances equally remarkable of new
analytical methods, to which mechanical problems, and
especially cosmical problems, have given occasion.
[Note 25\2: _Hist. Ind. Sc._ b. vi. c. vi. sect. 7.]
8. The progress of Optics as a science has, in like manner,
been throughout dependent upon the progress of pure
mathematics. The first rise of Geometry was followed by some
advances, slight ones no doubt, in the doctrine of
Reflection and in Perspective. The law of Refraction was
traced to its consequences by means of Trigonometry, which
indeed was requisite to express the law in a simple form.
The steps made in Optical science by Descartes, Newton,
Euler, and Huyghens, required the geometrical skill which
those philosophers possessed. And if Young and Fresnel had
not been, each in his peculiar way, persons of eminent
mathematical endowments, they would not have been able to
bring the Theory of Undulations and Interferences into a
condition in which it could be tested by experiments. We may
see how unexpectedly recondite parts of pure mathematics may
bear upon physical science, by calling to mind a
circumstance already noticed in the History of
Science[26\2];--that Fresnel obtained one of the {167} most
curious confirmations of the theory (the laws of Circular
Polarization by reflection) through an interpretation of an
algebraical expression, which, according to the original
conventional meaning of the symbols, involved an impossible
quantity. We have already remarked, that in virtue of the
principle of the generality of symbolical language, such an
interpretation may often point out some real and important
analogy.
[Note 26\2: _Hist. Ind. Sc._ b. ix. c. xiii. sect. 2.]
9. From this rapid sketch it may be seen how important an
office in promoting the progress of the physical sciences
belongs to mathematics. Indeed in the progress of many
sciences, every step has been so intimately connected with
some advance in mathematics, that we can hardly be surprised
if some persons have considered mathematical reasoning to be
the most essential part of such sciences; and have
overlooked the other elements which enter into their
formation. How erroneous this view is we shall best see by
turning our attention to the other Ideas besides those of
space, number, and motion, which enter into some of the most
conspicuous and admired portions of what is termed exact
science; and by showing that the clear and distinct
development of such Ideas is quite as necessary to the
progress of exact and real knowledge as an acquaintance with
arithmetic and geometry.
{{169}}
BOOK III.
THE
PHILOSOPHY
OF THE
MECHANICAL SCIENCES.
IT is only because we subject trains of phenomena, that is,
all change whatever, to the law of causality--to the
relation of cause and effect--that experience or empirical
knowledge becomes possible.
KANT, _Kr. d. R. V._ 11 Th. 1 Abth. 11 Buch. 2 Haupt.
Quicquid premit vel trahit alterum, tantundem ab eo premitur
vel trahitur ... Si corpus aliquod in corpus aliud impingens
motum ejus vi suâ quomodocunque mutaverit, idem quoque
vicissim in motu proprio eandem mutationem in partem
contrariam vi alterius (ob æqualitatem pressionis, mutuæ)
subibit ... Obtinet etiam hæc Lex in attractionibus.
NEWTON, _Princip._ ad init.
{{171}}
BOOK III.
THE PHILOSOPHY OF THE MECHANICAL SCIENCES.
CHAPTER I.
OF THE MECHANICAL SCIENCES.
IN the History of the Sciences, that class of which we here
speak occupies a conspicuous and important place; coming
into notice immediately after those parts of Astronomy which
require for their cultivation merely the ideas of space,
time, motion, and number. It appears from our History, that
certain truths concerning the _equilibrium_ of bodies were
established by Archimedes;--that, after a long interval of
inactivity, his principles were extended and pursued further
in modern times:--and that to these doctrines concerning
equilibrium and the forces which produce it, (which
constitute the science _Statics_,) were added many other
doctrines concerning the _motions_ of bodies, considered
also as produced by forces, and thus the science of
_Dynamics_ was produced. The assemblage of these sciences
composes the province of _Mechanics_. Moreover, philosophers
have laboured to make out the laws of the equilibrium of
_fluid_ as well as solid bodies; and hence has arisen the
science of _Hydrostatics_. And the doctrines of Mechanics
have been found to have a most remarkable bearing upon the
motions of the heavenly bodies; with reference to which,
indeed, they were at first principally studied. The
explanation of those cosmical facts by means of mechanical
{172} principles and their consequences, forms the science
of _Physical Astronomy_. These are the principal examples of
mechanical science; although some other portions of Physics,
as Magnetism and Electrodynamics, introduce mechanical
doctrines very largely into their speculations.
Now in all these sciences we have to consider _Forces_. In
all mechanical reasonings forces enter, either as producing
motion, or as prevented from doing so by other forces. Thus
force, in its most general sense, is the _cause_ of motion,
or of tendency to motion; and in order to discover the
principles on which the mechanical sciences truly rest, we
must examine the nature and origin of our knowledge of Causes.
In these sciences, however, we have not to deal with Cause
in its more general acceptation, in which it applies to all
kinds of agency, material or immaterial;--to the influence
of thought and will, as well as of bodily pressure and
attractive force. Our business at present is only with such
causes as immediately operate upon matter. We shall
nevertheless, in the first place, consider the nature of
Cause in its most general form; and afterwards narrow our
speculations so as to direct them specially to the
mechanical sciences.
{{173}}
CHAPTER II.
OF THE IDEA OF CAUSE.
1. WE see in the world around us a constant succession of
causes and effects connected with each other. The laws of
this connexion we learn in a great measure from experience,
by observation of the occurrences which present themselves
to our notice, succeeding one another. But in doing this,
and in attending to this succession of appearances, of which
we are aware by means of our senses, we supply from our own
minds the Idea of Cause. This Idea, as we have already shown
with respect to other Ideas, is not derived from experience,
but has its origin in the mind itself;--is introduced into
our experience by the active, and not by the passive part of
our nature.
By Cause we mean some quality, power, or efficacy, by which
a state of things produces a succeeding state. Thus the
motion of bodies from rest is produced by a cause which we
call _Force_: and in the particular case in which bodies
fall to the earth, this force is termed _Gravity_. In these
cases, the Conceptions of Force and Gravity receive their
meaning from the Idea of Cause which they involve: for Force
is conceived as the Cause of Motion. That this Idea of Cause
is not derived from experience, we prove (as in former
cases) by this consideration: that we can make assertions,
involving this idea, which are rigorously necessary and
universal; whereas knowledge derived from experience can
only be true as far as experience goes, and can never
contain in itself any evidence whatever of its necessity. We
assert that 'Every event must have a cause:' and this
proposition we know to be true, not only probably, and
generally, and as far as we can see: {174} but we cannot
suppose it to be false in any single instance. We are as
certain of it as of the truths of arithmetic or geometry. We
cannot doubt that it must apply to all events past and
future, in every part of the universe, just as truly as to
those occurrences which we have ourselves observed. _What_
causes produce what effects;--what is the cause of any
particular event;--what will be the effect of any peculiar
process;--these are points on which experience may enlighten
us. Observation and experience may be requisite, to enable
us to judge respecting such matters. But that every event
has _some_ cause, Experience cannot prove any more than she
can disprove. She can add nothing to the evidence of the
truth, however often she may exemplify it. This doctrine,
then, cannot have been acquired by her teaching; and the
Idea of Cause, which the doctrine involves, and on which it
depends, cannot have come into our minds from the region of
observation.
2. That we do, in fact, apply the Idea of Cause in a more
extensive manner than could be justified, if it were derived
from experience only, is easily shown. For from the
principle that everything must have a cause, we not only
reason concerning the succession of the events which occur
in the progress of the world, and which form the course of
experience; but we infer that the world itself must have a
cause;--that the chain of events connected by common
causation, must have a First Cause of a nature different
from the events themselves. This we are entitled to do, if
our Idea of Cause be independent of, and superior to,
experience: but if we have no Idea of Cause except such as
we gather from experience, this reasoning is altogether
baseless and unmeaning.
3. Again; by the use of our powers of observation, we are
aware of a succession of appearances and events. But none of
our senses or powers of external observation can detect in
these appearances the power or quality which we call Cause.
Cause is that which connects one event with another; but no
sense or perception discloses to us, or can disclose, any
connexion {175} among the events which we observe. We see
that one occurrence follows another, but we can never see
anything which shows that one occurrence _must_ follow
another. We have already noticed[1\3], that this truth has
been urged by metaphysicians in modern times, and generally
assented to by those who examine carefully the connexion of
their own thoughts. The arguments are, indeed, obvious
enough. One ball strikes another and causes it to move
forwards. But by what compulsion? Where is the necessity? If
the mind can see any circumstance in this case which makes
the result inevitable, let this circumstance be pointed out.
But, in fact, there is no such discoverable necessity; for
we can conceive this event not to take place at all. The
struck ball may stand still, for aught we can see. 'But the
laws of motion will not allow it to do so.' Doubtless they
will not. But the laws of motion are learnt from experience,
and therefore can prove no necessity. Why should not the
laws of motion be other than they are? Are they necessarily
true? That they are necessarily such as do actually regulate
the impact of bodies, is at least no obvious truth; and
therefore this necessity cannot be, in common minds, the
ground of connecting the impact of one ball with the motion
of another. And assuredly, if this fail, no other ground of
such necessary connexion can be shown. In this case, then,
the events are not seen to be necessarily connected. But if
this case, where one ball moves another by impulse, be not
an instance of events exhibiting a necessary connexion, we
shall look in vain for any example of such a connexion.
There is, then, no case in which events can be observed to
be necessarily connected: our idea of causation, which
implies that the event is necessarily connected with the
cause, cannot be derived from observation.
[Note 1\3: Book 3. chap. ii.]
4. But it may be said, we have not any such Idea of Cause,
implying necessary connexion with effect, and a quality by
which this connexion is produced. {176} We see nothing but
the succession of events; and by _cause_ we mean nothing but
a certain succession of events;--namely, a constant,
unvarying succession. Cause and effect are only two events
of which the second invariably follows the first. We delude
ourselves when we imagine that our idea of causation
involves anything more.
To this I reply by asking, what then is the meaning of the
maxim above quoted, and allowed by all to be universally and
necessarily true, that every event must have a cause? Let us
put this maxim into the language of the explanation just
noticed; and it becomes this:--'Every event must have a
certain other event invariably preceding it.' But why must
it? Where is the necessity? Why must like events always be
preceded by like, except so far as other events interfere?
That there is such a necessity, no one can doubt. All will
allow that if a stone ascend because it is thrown upwards in
one case, a stone which ascends in another case has also
been thrown upwards, or has undergone some equivalent
operation. All will allow that in this sense, every kind of
event must have some other specific kind of event preceding
it. But this turn of men's thoughts shows that they see in
events a connexion which is not mere succession. They see in
cause and effect, not merely what does, often or always,
precede and follow, but what _must_ precede and follow. The
events are not only conjoined, they are connected. The cause
is more than the prelude, the effect is more than the
sequel, of the fact. The cause is conceived not as a mere
occasion; it is a power, an efficacy, which has a real
operation.
5. Thus we have drawn from the maxim, that Every Effect must
have a Cause, arguments to show that we have an Idea of
Cause which is not borrowed from experience, and which
involves more than mere succession. Similar arguments might
be derived from any other maxims of universal and necessary
validity, which we can obtain concerning Cause: as, for
example, the maxims that Causes are measured by their
Effects, and that Reaction is equal and opposite to {177}
Action. These maxims we shall soon have to examine; but we
may observe here, that the necessary truth which belongs to
them, shows that they, and the Ideas which they involve, are
not the mere fruits of observation; while their meaning,
including, as it does, something quite different from the
mere conception of succession of events, proves that such a
conception is far from containing the whole import and
signification of our Idea of Cause.
The progress of the opinions of philosophers on the points
discussed in this chapter, has been one of the most
remarkable parts of the history of Metaphysics in modern
times: and I shall therefore briefly notice some of its
features.
{{178}}
CHAPTER III.
MODERN OPINIONS RESPECTING THE IDEA OF CAUSE.
1. TOWARDS the end of the seventeenth century there existed
in the minds of many of the most vigorous and active
speculators of the European literary world, a strong
tendency to ascribe the whole of our Knowledge to the
teaching of Experience. This tendency, with its
consequences, including among them the reaction which was
produced when the tenet had been pushed to a length
manifestly absurd, has exercised a very powerful influence
upon the progress of metaphysical doctrines up to the
present time. I proceed to notice some of the most prominent
of the opinions which have thus obtained prevalence among
philosophers, so far as the Idea of Cause is concerned.
Locke was one of the metaphysicians who produced the
greatest effect in diffusing this opinion, of the exclusive
dependence of our knowledge upon experience. Agreeably to
this general system, he taught[2\3] that our ideas of Cause
and Effect are got from observation of the things about us.
Yet notwithstanding this tenet of his, he endeavoured still
to employ these ideas in reasoning on subjects which are far
beyond all limits of experience: for he professed to prove,
from our idea of Causation, the existence of the Deity[3\3].
[Note 2\3: _Essay on the Human Understanding_, b. ii. c. xxvi.]
[Note 3\3: B. iv. c. x.]
Hume noticed this obvious inconsistency; but declared
himself unable to discover any remedy for a defect so fatal
to the most important parts of our knowledge. He could see,
in our belief of the succession of cause and effect, nothing
but the habit of associating in our minds what had often
been {179} associated in our experience. He therefore
maintained that we could not, with logical propriety, extend
our belief of such a succession to cases entirely distinct
from all those of which our experience consisted. We see, he
said, an actual _conjunction_ of two events; but we can in
no way detect a necessary _connexion_; and therefore we have
no means of inferring cause from effect, or effect from
cause[4\3]. The only way in which we recognize Cause and
Effect in the field of our experience, is as an unfailing
Sequence: we look in vain for anything which can assure us
of an infallible Consequence. And since experience is the
only source of our knowledge, we cannot with any justice
assert that the world in which we live must necessarily have
had a cause.
[Note 4\3: Hume's _Phil. of the Human Mind_, vol. i. p.
94.]
2. This doctrine, taken in conjunction with the known
skepticism of its author on religious points, produced a
considerable fermentation in the speculative world. The
solution of the difficulty thus thrown before philosophers,
was by no means obvious. It was vain to endeavour to find in
experience any other property of a Cause, than a constant
sequence of the effect. Yet it was equally vain to try to
persuade men that they had no idea of Cause; or even to
shake their belief in the cogency of the familiar arguments
concerning the necessity of an original cause of all that is
and happens. Accordingly these hostile and apparently
irreconcilable doctrines,--the indispensable necessity of a
cause of every event, and the impossibility of our knowing
such a necessity,--were at last allowed to encamp side by
side. Reid, Beattie, and others, formed one party, who
showed how widely and constantly the idea of a cause
pervades all the processes of the human mind: while another
sect, including Brown, and apparently Stewart, maintained
that this idea is always capable of being resolved into a
constant sequence; and these latter reasoners tried to
obviate the dangerous and shocking inferences which some
persons might try to draw from their opinion, by declaring
the {180} maxim that "Every event must have a cause," to be
an instinctive law of belief, or a fundamental principle of
the human mind[5\3].
[Note 5\3: Stewart's _Active Powers_, vol. i. p. 347.
Browne's _Lectures_, vol. i. p. 115.]
3. While this series of discussions was going on in Britain,
a great metaphysical genius in Germany was unravelling the
perplexity in another way. Kant's speculations originated,
as he informs us, in the trains of thought to which Hume's
writings gave rise; and the _Kritik der Reinen Vernunft_, or
_Examination of the Pure Reason_, was published in 1787,
with the view of showing the true nature of our knowledge.
Kant's solution of the difficulties just mentioned differs
materially from that above stated. According to Brown[6\3],
succession observed and cause inferred,--the memory of past
conjunctions of events and the belief of similar future
conjunctions,--are facts, independent, so far as we can
discover, but inseparably combined by a law of our mental
nature. According to Kant, causality is an inseparable
condition of our experience: a connexion in events is
requisite to our apprehending them _as_ events. Future
occurrences must be connected by causation as the past have
been, because we cannot think of past, present, and future,
without such connexion. We cannot fix the mind upon
occurrences, without including these occurrences in a series
of causes and effects. The relation of Causation is a
condition under which we think of events, as the relations
of space are a condition under which we see objects.
[Note 6\3: _Lectures_, vol. i. p. 114.]
4. On a subject so abstruse, it is not easy to make our
distinctions very clear. Some of Brown's illustrations
appear to approach very near to the doctrine of Kant. Thus
he says[7\3], 'The _form_ of bodies is the relation of their
elements to each other in space,--the _power_ of bodies is
their relation to each other in time.' Yet notwithstanding
such approximations in expression, the Kantian doctrine
appears to be different from {181} the views of Stewart and
Brown, as commonly understood. According to the Scotch
philosophers, the cause and the effect are two things,
connected in our minds by a law of our nature. But this view
requires us to suppose that we can conceive the law to be
absent, and the course of events to be unconnected. If we
can understand what is the special force of this law, we
must be able to imagine what the case would be if the law
were non-existing. We must be able to conceive a mind which
does not connect effects with causes. The Kantian doctrine,
on the other hand, teaches that we cannot imagine events
liberated from the connexion of cause and effect: this
connexion is a condition of our conceiving any real
occurrences: we cannot think of a real sequence of things,
except as involving the operation of causes. In the Scotch
system, the past and the future are in their nature
independent, but bound together by a rule; in the German
system, they share in a common nature and mutual relation,
by the act of thought which makes them past and future. In
the former doctrine cause is a tie which binds; in the
latter it is a character which pervades and shapes events.
The Scotch metaphysicians only assert the _universality_ of
the relation; the German attempts further to explain its
_necessity_.
[Note 7\3: _Lectures_, vol. i. p. 127.]
This being the state of the case, such illustrations as that
of Dr. Brown quoted above, in which he represents _cause_ as
a relation of the same kind with _form_, do not appear
exactly to fit his opinions. Can the relations of figure be
properly said to be connected with each other by a law of
our nature, or a tendency of our mental constitution? Can we
ascribe it to a law of our thoughts, that we believe the
three angles of a triangle to be equal to two right angles?
If so, we must give the same reason for our belief that two
straight lines cannot inclose a space; or that three and two
are five. But will any one refer us to an ultimate law of
our constitution for the belief that three and two are five?
Do we not see that they are so, as plainly as we see that
they are three and two? Can we imagine laws of our
constitution abolished, so that three and two shall {182}
make something different from five;--so that an inclosed
space shall lie between two straight lines;--so that the
three angles of a plane triangle shall be greater than two
right angles? We cannot conceive this. If the numbers _are_
three and two; if the lines _are_ straight; if the triangle
_is_ a rectilinear triangle, the consequences are
inevitable. We cannot even imagine the contrary. We do not
want a law to direct that things should be what they are.
The relation, then, of cause and effect, being of the same
kind as the necessary relations of figure and number, is not
properly spoken of as established in our minds by a special
law of our constitution: for we reject that loose and
inappropriate phraseology which speaks of the relations of
figure and number as 'determined by laws of belief.'
5. In the present work, we accept and adopt, as the basis of
our inquiry concerning our knowledge, the existence of
necessary truths concerning causes, as there exist necessary
truths concerning figure and number. We find such truths
universally established and assented to among the
cultivators of science, and among speculative men in
general. All mechanicians agree that reaction is equal and
opposite to action, both when one body presses another, and
when one body communicates motion to another. All reasoners
join in the assertion, not only that every observed change
of motion has had a cause, but that every change of motion
must have a cause. Here we have certain portions of
substantial and undoubted knowledge. Now the essential point
in the view which we must take of the idea of cause is
this,--that our view must be such as to form a solid basis
for our knowledge. We have, in the Mechanical Sciences,
certain universal and necessary truths on the subject of
causes. Now any view which refers our belief in causation to
mere experience or habit, cannot explain the possibility of
such necessary truths, since experience and habit can never
lead to a perception of necessary connexion. But a view
which teaches us to acknowledge axioms concerning cause, as
we acknowledge axioms {183} concerning space, will lead us
to look upon the science of mechanics as equally certain and
universal with the science of geometry; and will thus
materially affect our judgment concerning the nature and
claims of our scientific knowledge.
Axioms concerning Cause, or concerning Force, which as we
shall see, is a modification of Cause, will flow from an
Idea of Cause, just as axioms concerning space and number
flow from the ideas of space and number or time. And thus
the propositions which constitute the science of Mechanics
prove that we possess an idea of cause, in the same sense in
which the propositions of geometry and arithmetic prove our
possession of the ideas of space and of time or number.
6. The idea of cause, like the ideas of space and time, is a
part of the _active_ powers of the mind. The relation of
cause and effect is a relation or condition under which
events are apprehended, which relation is not given by
observation, but supplied by the mind itself. According to
the views which explain our apprehension of cause by
reference to habit, or to a supposed law of our mental
nature, causal connexion is a consequence of agencies which
the mind passively obeys; but according to the view to which
we are led, this connexion is a result of faculties which
the mind actively exercises. And thus the relation of cause
and effect is a condition of our apprehending successive
events, a part of the mind's constant and universal
activity, a source of necessary truths; or, to sum all this
in one phrase, a Fundamental Idea.
{{184}}
CHAPTER IV.
OF THE AXIOMS WHICH RELATE TO THE IDEA OF CAUSE.
1. _Causes are abstract Conceptions._--WE have now to
express, as well as we can, the fundamental character of
that Idea of Cause of which we have just proved the
existence. This may be done, at least for purposes of
reasoning, in this as in former instances, by means of
axioms. I shall state the principal axioms which belong to
this subject, referring the reader to his own thoughts for
the axiomatic evidence which belongs to them.
But I must first observe, that in order to express general
and abstract truths concerning cause and effect, these
terms, _cause_ and _effect_, must be understood in a general
and abstract manner. When one event gives rise to another,
the first _event_ is, in common language, often called the
cause, and the second the effect. Thus the meeting of two
billiard-balls may be said to be the cause of one of them
turning aside out of the path in which it was moving. For
our present purposes, however, we must not apply the term
cause to such occurrences as this meeting and turning, but
to a certain conception, _force_, abstracted from all such
special events, and considered as a quality or property by
which one body affects the motion of the other. And in like
manner in other cases, cause is to be conceived as some
abstract quality, power, or efficacy, by which change is
produced; a quality not identical with the events, but
disclosed by means of them. Not only is this abstract mode
of conceiving force and cause useful in expressing the
fundamental principles of science; but it supplies us with
the only mode by which such principles can be {185} stated
in a general manner, and made to lead to substantial truth
and real knowledge.
Understanding _cause_, therefore, in this sense, we proceed
to our Axioms.
2. First Axiom. _Nothing can take place without a Cause._
Every event, of whatever kind, must have a cause in the
sense of the term which we have just indicated; and that it
must, is a universal and necessary proposition to which we
irresistibly assent as soon as it is understood. We believe
each appearance to come into existence,--we conceive every
change to take place,--not only with something preceding it,
but something by which it is made to be what it is. An
effect without a cause;--an event without a preceding
condition involving the efficacy by which the event is
produced;--are suppositions which we cannot for a moment
admit. That the connexion of effect with cause is universal
and necessary, is a universal and constant conviction of
mankind. It persists in the minds of all men, undisturbed by
all the assaults of sophistry and skepticism; and, as we
have seen in the last chapter, remains unshaken, even when
its foundations seem to be ruined. This axiom expresses, to
a certain extent, our Idea of Cause; and when that idea is
clearly apprehended, the axiom requires no proof, and indeed
admits of none which makes it more evident. That
notwithstanding its simplicity, it is of use in our
speculations, we shall hereafter see; but in the first
place, we must consider the other axioms belonging to this
subject.
3. Second Axiom. _Effects are proportional to their Causes,
and Causes are measured by their Effects._
We have already said that _cause_ is that quality or power,
in the circumstances of each case, by which the effect is
produced; and this power, an abstract property of the
condition of things to which it belongs, can in no way fall
directly under the cognizance of the senses. Cause, of
whatever kind, is not apprehended as including objects and
events which share its nature by being co-extensive with
certain portions of it, as space and time are. It cannot
therefore, like them, be {186} measured by repetition of its
own parts, as space is measured by repetition of inches, and
time by repetition of minutes. Causes may be greater or
less; as, for instance, the force of a man is greater than
the force of a child. But how much is the one greater than
the other? How are we to compare the abstract conception,
force, in such cases as these?
To this, the obvious and only answer is, that we must
compare causes by means of their effects;--that we must
compare force by something which force can do. The child can
lift one fagot; the man can lift ten such fagots: we have
here a means of comparison. And whether or not the rule is
to be applied in this manner, that is, by the number of
things operated on, (a question which we shall have to
consider hereafter,) it is clear that this form of rule,
namely, a reference to some effect or other as our measure,
is the right, because the only possible form. The cause
determines the effect. The cause being the same, the effect
must be the same. The connexion of the two is governed by a
fixed and inviolable rule. It admits of no ambiguity. Every
degree of intensity in the cause has some peculiar
modification of the effect corresponding to it. Hence the
effect is an unfailing index of the amount of the cause; and
if it be a measurable effect, gives a measure of the cause.
We can have no other measure; but we need no other, for this
is exact, sufficient and complete.
It may be said, that various effects are produced by the
same cause. The sun's heat melts wax and expands
quicksilver. The force of gravity causes bodies to move
downwards if they are free, and to press down upon their
supports if they are supported. Which of the effects is to
be taken as the measure of heat, or of gravity, in these
cases? To this we reply, that if we had merely different
states of the same cause to compare, any of the effects
might be taken. The sun's heat on different days might be
measured by the expansion of quicksilver, or by the quantity
of wax melted. The force of gravity, if it were different at
different places, might be measured by the spaces through
which a given weight would bend an elastic {187} support, or
by the spaces through which a body would fall in a given
time. All these measures are consistent with the general
character of our idea of cause.
4. _Limitation of the Second Axiom._--But there may be
circumstances in the nature of the case which may further
determine the kind of effect which we must take for the
measure of the cause. For example, if causes are conceived
to be of such a nature as to be capable of addition, the
effects taken as their measure must conform to this
condition. This is the case with mechanical causes. The
weights of two bodies are the causes of the pressure which
they exert downwards; and these weights are capable of
addition. The weight of the two is the sum of the weight of
each. We are therefore not at liberty to say that weights
shall be measured by the spaces through which they bend a
certain elastic support, except we have first ascertained
that the whole weight bends it through a space equal to the
sum of the inflections produced by the separate weights.
Without this precaution, we might obtain inconsistent
results. Two weights, each of the magnitude 3 as measured by
their effects, might, if we took the inflections of a spring
for the effects, be together equal to 5 or to 7 by the same
kind of measurement. For the inflection produced by two
weights of 3 might, for aught we can see beforehand, be more
or less than twice as great as the inflection produced by
one weight of 3. That forces are capable of addition, is a
condition which limits, and, as we shall see, in some cases
rigorously fixes, the kind of effects which are to be taken
as their measures.
Causes which are thus capable of addition are to be measured
by the repeated addition of equal quantities. Two such
causes are _equal_ to each other when they produce exactly
the same effect. So far our axiom is applied directly. But
these two causes can be _added_ together; and being thus
added, they are _double_ of one of them; and the cause
composed by addition of _three_ such, is _three_ times as
great as the first; and so on for any measure whatever. By
this means, and by this {188} means only, we have a complete
and consistent measure of those causes which are so
conceived as to be subject to this condition of being added
and multiplied.
Causes are, in the present chapter, to be understood in the
widest sense of the term; and the axiom now under our
consideration applies to them, whenever they are of such a
nature as to admit of any measure at all. But the cases
which we have more particularly in view are _mechanical_
causes, the causes of the motion and of the equilibrium of
bodies. In these cases, forces are conceived as capable of
addition; and what has been said of the measure of causes in
such cases, applies peculiarly to mechanical forces. Two
weights, placed together, may be considered as a single
weight, equal to the _sum_ of the two. Two pressures,
pushing a body in the same direction at the same point, are
identical in all respects with some single pressure, their
_sum_, pushing in like manner; and this is true whether or
not they put the body in motion. In the cases of mechanical
forces, therefore, we take some certain effect, velocity
generated or weight supported, which may fix the _unit_ of
force; and we then measure all other forces by the
successive repetition of this unit, as we measure all spaces
by the successive repetition of our unit of lineal measure.
But these steps in the formation of the science of Mechanics
will be further explained, when we come to follow our axioms
concerning cause into their application in that science. At
present we have, perhaps, sufficiently explained the axiom
that causes are measured by their effects, and we now
proceed to a third axiom, also of great importance.
5. Third Axiom. _Reaction is equal and opposite to Action._
In the case of mechanical forces, the action of a cause
often takes place by an operation of one body upon another;
and in this case, the action is always and inevitably
accompanied by an _opposite_ action. If I press a stone with
my hand, the stone presses my hand in return. If one ball
strike another and put it in motion, the second ball
diminishes the motion of {189} the first. In these cases the
operation is mutual; the Action is accompanied by a
Reaction. And in all such cases the Reaction is a force of
exactly the same nature as the Action, exerted in an
opposite direction. A pressure exerted upon a body at rest
is resisted and balanced by another pressure; when the
pressure of one body puts another in motion, the body,
though it yields to the force, nevertheless exerts upon the
pressing body a force like that which it suffers.
Now the axiom asserts further, that this Reaction is
_equal_, as well as opposite, to the Action. For the
Reaction is an effect of the Action, and is determined by
it. And since the two, Action and Reaction, are forces of
the same nature, each may be considered as cause and as
effect; and they must, therefore, determine each other by a
common rule. But this consideration leads necessarily to
their equality: for since the rule is mutual, if we could
for an instant suppose the Reaction to be less than the
Action, we must, by the same rule, suppose the Action to be
less than the Reaction. And thus Action and Reaction, in
every such case, are rigorously equal to each other.
It is easily seen that this axiom is not a proposition which
is, or can be, proved by experience; but that its truth is
anterior to special observation, and depends on our
conception of Action and Reaction. Like our other axioms,
this has its source in an Idea; namely, the Idea of Cause,
under that particular condition in which cause and effect
are mutual. The necessary and universal truth which we
cannot help ascribing to the axiom, shows that it is not
derived from the stores of experience, which can never
contain truths of this character. Accordingly, it was
asserted with equal confidence and generality by those who
did not refer to experience for their principles, and by
those who did. Leonicus Tomæus, a commentator of Aristotle,
whose work was published in 1552, and therefore at a period
when no right opinions concerning mechanical reaction were
current, at least in his school, says, in his remarks on the
Author's Questions concerning the communication of motion,
that 'Reaction is equal and {190} contrary to Action.' The
same principle was taken for granted by all parties, in all
the controversies concerning the proper measure of force, of
which we shall have to speak: and would be rigorously true,
as a law of motion, whichever of the rival interpretations
of the measure of the term 'Action' we were to take.
6. _Extent of the Third Axiom._--It may naturally be asked
whether this third Axiom respecting causation extends to any
other cases than those of mechanical action, since the
notion of Cause in general has certainly a much wider
extent. For instance, when a hot body heats a cold one, is
there necessarily an equal reaction of the second body upon
the first? Does the snowball cool the boy's hand exactly as
much as the hand heats the snow? To this we reply, that, in
every case in which one body acts upon another by its
physical qualities, there must be some reaction. No body can
affect another without being itself also affected. But in
any physical change the _action_ exerted is an abstract term
which may be variously understood. The hot hand may _melt_ a
cool body, or may _warm_ it: which kind of effect is to be
taken as action? This remains to be determined by other
considerations.
In all cases of physical change produced by one body in
another, it is generally possible to assume such a meaning
of action, that the reaction shall be of the same nature as
the action; and when this is done, the third axiom of
causation, that reaction is equal to action, is universally
true. Thus if a hot body heat a cold one, the change may be
conceived as the transfer of a certain substance, _heat_ or
_caloric_, from the first body to the second. On this
supposition, the first body _loses_ just as much heat as the
other _gains_; action and reaction are equal. But if the
reaction be of a different kind to the action we can no
longer apply the axiom. If a hot body _melt_ a cold one, the
latter _cools_ the former: here, then, is reaction; but so
long as the action and reaction are stated in this form, we
cannot assert any equality between them.
In treating of the secondary mechanical sciences, we {191}
shall see further in what way we may conceive the physical
action of one body upon another, so that the same axioms
which are the basis of the science of Mechanics shall apply
to changes not at first sight manifestly mechanical.
The three axioms of causation which we have now stated are
the fundamental maxims of all reasoning concerning causes as
to their quantities; and it will be shown in the sequel that
these axioms form the basis of the science of Mechanics,
determining its form, extent, and certainty. We must,
however, in the first place, consider how we acquire those
conceptions upon which the axioms now established are to be
employed.
[2d Ed.] [The Axiom that _Reaction is equal and opposite to
Action_, may appear to be at variance with a maxim
concerning Cause which is commonly current; namely, that the
'Cause precedes Effect, and Effect follows Cause.' For it
may be said, if _A_, the Action, and _R_, the Reaction, can
be considered as mutually the cause of each other, _A_ must
precede _R_, and yet must follow it, which is impossible.
But to this I reply, that in those cases of direct Causation
to which the maxim applies, the Cause and Effect are not
successive, but simultaneous. If I press against some
obstacle, the obstacle resists and returns the pressure at
the instant it is exerted, not after any interval of time,
however small. The common maxim, that the effect follows the
cause, has arisen from the practice of considering, as
examples of cause and effect, not instantaneous forces or
causes, and the instantaneous changes which they produce;
but taking, instead of this latter, the _cumulative_ effects
produced in the course of time, and compared with like
results occurring without the action of the cause. Thus, if
we alter the length of a clock-pendulum, this change
produces, as its effect, a subsequent change of rate in the
clock: because the rate is measured by the accumulated
effects of the pendulum's gravity, before and after the
change. But the pendulum produces its mechanical effect upon
the escapement, at the moment of its contact, and each wheel
upon the next, at the moment of _its_ contact. As has {192}
been said in a Review of this work, 'The time lost in cases
of indirect physical causation is consumed in the movements
which take place among the parts of the mechanism in action,
by which the active forces so transformed into momentum are
transported over intervals of space to new points of action,
the motion of matter in such cases being regarded as a mere
carrier of force.' (_Quarterly Rev._ No. cxxxv. p. 212.)
This subject I have further treated in the _Memoirs of the
Cambridge Philosophical Society_, vol. vii. part iii.] [In
this Third Edition I add this discussion.]
_Discussion of the Question:--Are Cause and Effect
successive or simultaneous?_
I HAVE at various times laid before this Society
dissertations on the metaphysical grounds and elements of
our knowledge, and especially on the foundations of the
science of mechanics. As these speculations have not failed
to excite some attention, both here and elsewhere, I am
tempted to bring forward in the same manner some additional
disquisitions of the same kind. Indeed, the immediate
occasion of the present memoir is of itself an evidence that
such subjects are not supposed to be without their interest
for the general reader; for I am led to the views and
reasonings which I am now about to lay before the Society,
by some remarks in one of our most popular Reviews, (_The
Quarterly Review_, Article on the _History_ and _Philosophy
of the Inductive Sciences_, June 1841). A writer of singular
acuteness and comprehensiveness of view has there made
remarks upon the doctrines which I had delivered in the
_Philosophy of the Inductive Sciences_, which remarks appear
to me in the highest degree instructive and philosophical. I
am not, however, going here to discuss fully the doctrines
contained in this critique. With respect to its general
tendency, I will only observe, that the author does not
accept, in the form in which I had given it, the account of
the origin and ground of necessary and universal truths. I
had stated that our knowledge is derived from Sensations and
Ideas; and that Ideas, which are the conditions of
perception, such as _space_, _time_, _likeness_, _cause_,
make universal and necessary knowledge possible; whereas, if
knowledge were derived from Sensation alone, it could not
have those characters. I have moreover {193} enumerated a
long series of Fundamental Ideas as the bases of a
corresponding series of sciences, of which sciences I have
shown also, by an historical survey, that they claim to
possess universal truths, and have their claims allowed. I
have gone further: for I have stated the Axioms which flow
from these Fundamental Ideas, and which are the logical
grounds of necessity and universality in the truths of each
science, when the science is presented in the form of a
demonstrated system. The Reviewer does not assent to this
doctrine, nor to the argument by which it is supported;
namely, that Experience cannot lead to universal truths,
except by means of a universal Idea supplied by the mind,
and infused into the particular facts which observation
ministers. He considers that the existence of universal
truths in our knowledge may be explained otherwise. He holds
that it is a sufficient account of the matter to say that we
pass from special experience to universal truth in virtue of
'the inductive propensity--the irresistible impulse of the
mind to generalize _ad infinitum_.' I shall not here dwell
upon very strong reasons which may be assigned, as I
conceive, for not accepting this as a full and satisfactory
explanation of the difficulty. Instead of doing so, I shall
here content myself with remarking, that even if we adopt
the Reviewer's expressions, we must still contend that there
are _different forms_ of the _impulse of the mind to
generalize_, corresponding to each of the Fundamental Ideas
of our system. These Fundamental Ideas, if they be nothing
else, must at least be accepted as a classification of the
modes of action of the Inductive Propensity,--as so many
different paths and tendencies of the Generalizing Impulse:
and the Axioms which I have stated as the express results of
the Fundamental Ideas, and as the steps by which those Ideas
make universal truths possible, are still no less worthy of
notice, if they are stated as the results of our
Generalizing Impulse; and as the steps by which that
Impulse, in its many various forms, makes universal truths
possible. The Generalizing Impulse in that operation by
which it leads us to the Axioms of Geometry, and to those of
Mechanics, takes very different courses; and these courses
may well deserve to be separately studied. And perhaps, even
if we accept this view of the philosophy of our knowledge,
no simpler or clearer way can be found of describing and
distinguishing these fundamentally different operations of
the Inductive Propensity, than by saying, {194} that in the
one case it proceeds according to the Idea of Space, in
another according to the Idea of Mechanical Cause; and the
like phraseology may be employed for all the other cases.
This then being understood, my present object is to consider
some very remarkable, and, as appears to me, novel views of
the Idea of Cause which the Reviewer propounds. And these
may be best brought under our discussion by considering them
as an attempt to solve the question, Whether, according to
our fundamental apprehensions of the relation of Cause and
Effect, effect follows cause in the order of time, or is
simultaneous with it.
At first sight, this question may seem to be completely
decided by our fundamental convictions respecting cause and
effect, and by the axioms which have been propounded by
almost all writers, and have obtained universal currency
among reasoners on this subject. That the cause must precede
the effect,--that the effect must follow the cause,--are, it
might seem, self-evident truths, assumed and assented to by
all persons in all reasonings in which those notions occur.
Such a doctrine is commonly asserted in general terms, and
seems to be verified in all the applications of the idea of
cause. A heavy body produces motion by its weight; the
motion produced is subsequent in time to the pressure which
the weight exerts. In a machine, bodies push or strike each
other, and so produce a series of motions; each motion, in
this case, is the result of the motions and configurations
which have preceded it. The whole series of such motions
employs time; and this time is filled up and measured by the
series of causes and effects, the effects being, in their
turn, causes of other effects. This is the common mode of
apprehending the universal course of events, in which the
chain of causation, and the progress of time, are
contemplated as each the necessary condition and
accompaniment of the other.
But this, the Critic remarks, is not true in _direct_
causation. 'If the antecedence and consequence in question
be understood as the interposition of an interval of time,
however small, between the action of the cause and the
production of the effect, we regard it as inadmissible. In
the production of motion by force, for instance, though the
effect be cumulative with continued exertion of the cause,
yet each elementary or individual action is, to our
apprehension, _instanter_ accompanied with its corresponding
increment of momentum in the body moved. In all dynamical
{195} reasonings no one has ever thought of interposing an
instant of time between the action and its resulting
momentum; nor does it appear necessary.' This is so evident,
that it appears strange it should have the air of novelty;
yet, so far as I am aware, the matter has never before been
put in the same point of view. But this being the case, the
question occurs, how it is that time _seems_ to be employed
in the progress from cause to effect? How is it that the
opinion of the effect being subsequent to the cause has
generally obtained? And to this the Critic's answer is
obvious:--it is so in cases of indirect or of _cumulative_
effect. If a ball _A_ strikes another, _B_, and puts it in
motion, and _B_ strikes _C_, and puts it in motion, _A_'s
impact may be considered as the cause, though not the direct
cause, of _C_'s motion. Now time, namely the time of _B_'s
motion after it is struck by _A_, and before it strikes _C_,
intervenes between _A_'s impact and the beginning of _C_'s
motion: that is, between the cause and its effect. In this
sense, the effect is subsequent to the cause. Again, if a
body be put in motion by a series of impulses acting at
finite intervals of time, all in the same direction, the
motion at the end of all these intervals is the effect of
all the impulses, and exists after they have all acted. It
is the accumulated effect, and subsequent to each separate
action of the cause. But in this case, each impulse produces
its effect instantaneously, and the time is employed, not in
the transition from any cause to its effect, but in the
intervals between the action of the several causes, during
which intervals the body goes on with the velocity already
communicated to it. In each impulse, force produces motion:
and the motion goes on till a new change takes place, by the
same kind of action. The force may be said, in the language
employed by the Critic, to be transformed into momentum; and
in the successive impulses, successive portions of force are
thus transformed; while in the intervening intervals, the
force thus transformed into momentum is carried by the body
from one place to another, where a new change awaits it.
'The cause is absorbed and transformed into effect, and
therein treasured up.' Hence, as the Writer says, 'The time
lost in cases of indirect physical causation is that
consumed in the movements which take place among the parts
of the mechanism set in action, by which the active forces
so transformed into mechanism are transported over intervals
of space to new points of action, the motion of matter in
such cases being {196} regarded as a mere carrier of
force':--and when force is directly counteracted by force,
their mutual destruction must be conceived, as the Reviewer
says, to be instantaneous. We can therefore hardly resist
his conclusion, that men have been misled in assuming
sequence as a feature in the relation of cause and effect;
and we may readily assent to his suggestion, that sequence,
when observed, is to be held as a sure indication of
indirect action, accompanied with a movement of parts.
But yet if we turn for a moment to other kinds of causation,
we seem to be compelled at every step to recognize the truth
of the usual maxim upon this subject, that effects are
subsequent to causes. Is not poison, taken at a certain
moment, the cause of disorder and death which follow at a
_subsequent_ period? Is not a man's early prudence often the
cause of his prosperity in _later_ life, and his folly,
though for a moment it may produce gratification, _finally_
the cause of his ruin? And even in the case of mechanism,
if, in a clock which goes rightly, we alter the length of
the pendulum, is not this alteration the cause of an
alteration which _afterwards_ takes place in the rate of the
clock's going? Are not all these, and innumerable other
cases, instances in which the usual notion of the effect
following the cause is verified? and are they not
irreconcileable with the new doctrine of cause and effect
being simultaneous?
In order to disentangle this apparent confusion, let us
first consider the case last mentioned, of a clock, in which
some alteration is made which affects the rate of going.
So long as the parts of the clock remain unaltered, its rate
will remain unaltered; and any part which is considered as
capable of alteration, may be considered as, if we please,
the cause of the unaltered rate, by being itself unaltered.
But we do not usually introduce the positive idea of cause,
to correspond with this negation of change. If we speak of
the rate as unaltered, we may also say that it is so because
there is _no cause_ of alteration. The steady rate is the
indication of the absence of any cause of alteration; and
the rate of going measures the progress of time, in a state
of things in which causes of change are thus excluded. If an
alteration takes place in any part of the clock, once for
all, the rate is altered; but the new rate is steady as the
old rate was, and, like it, measures the uniform progress of
time. But the difference between the new rate and the old is
occasioned by {197} the difference of the parts of the
clock; and the new rate may very properly be said to be
caused by the change of the parts, and to be subsequent to
it: for it does prevail after the change, and does not
prevail before.
But how is this view to be reconciled with the one just
quoted from the Reviewer, and, as it appeared,
satisfactorily proved by him; according to which all
mechanical effects are simultaneous with their causes, and
not subsequent to them? We have here the two views in close
contact, and in seeming opposition.
In the going of a clock, the parts are in motion; and these
motions are determined by forces arising from the form and
connexion of the parts of the mechanism. Each of the forces
thus exerted at any instant produces its effect at the same
instant; and thus, so far as the term _cause_ refers to such
instantaneous forces, the cause and the effect are
simultaneous. But if such instantaneous forces act at
successive intervals of time, the motion during each
interval is unaltered, and by its uniform progress measures
the progress of time. And thus the motion of the machine
consists of a series of intervals, during each of which the
motion is uniform, and measures the time; separated from
each other by a series of changes, at each of which the
change measures the instantaneous force, and is simultaneous
with it. And if, in this case, we suppose, at any point of
time, the instantaneous forces to cease, the succession of
them being terminated, from that point of time the motion
would be uniform. And since the rate of the motion in each
interval of time is determined by the instantaneous force
which last acted and by the preceding motion, the rate of
the motion in each interval of time is determined by all the
preceding instantaneous forces. Hence, when the series of
instantaneous forces stops, the rate at which the motion
goes on permanently, from that point of time, is determined
by the antecedent series of such forces, which series may be
considered as an aggregate cause; and hence it appears, that
the _permanent_ effect is determined by the _aggregate_
cause; and in this sense the effect is subsequent to the
cause.
Thus we obtain, in this case, a solution of the difficulty
which is placed before us. The instantaneous effect or
change is simultaneous with the instantaneous force or cause
by which it is {198} produced. But if we consider a series
of such instantaneous forces as a single aggregate cause,
and the final condition as a permanent effect of this cause,
the effect is subsequent to the cause. In this case, the
cause is immediately succeeded by the effect. The cause acts
in time: the effect goes on in time. The times occupied by
the cause and by the effect succeed each other, the one
ending at the point of time at which the other begins. But
the time which the cause occupies is really composed of a
series of instants of uniform motion interposed between
instantaneous forces; and during the time that this series
of causes is going on, to make up the aggregate cause, a
series of effects is going on to make up the final effect.
There is a progressive cause and a progressive effect which
go on together, and occupy the same finite time; and this
simultaneous progression is composed of all the simultaneous
instantaneous steps of cause and effect. The aggregate cause
is the sum of the progression of causes; the final effect is
the last term of the progression of effects. At each step,
as the Reviewer says, cause is transformed into effect; and
it is treasured up in the results during the intermediate
intervals; and the time occupied is not the time which
intervenes between cause and effect at each step, but the
time which intervenes between these transformations.
I have supposed forces to act at distinct instants, and to
cease to act in the intervals between; and then, the
aggregate of such intervals to make up a finite time, during
which an aggregate force acts. But if the action of the
force be rigorously continuous, it will easily be seen that
all the consequences as to cause and effect will be the
same; the discontinuous action being merely the usual
artifice by which, in mathematical reasonings, we obtain
results respecting continuous changes. It will still be
true, that the uniform motion which takes place after a
continuous force has acted, is the effect subsequent to the
cause; while the change which takes place at any instant by
the action of the force, is the instantaneous effect
simultaneous with the cause.
It may be objected, that this solution does not appear
immediately to apply: for the motion of a clock is not
uniform during any portion of the time. The parts move by
intervals of varied motion and of rest; or by oscillations
backwards and forwards; and the succession of forces which
acts during any {199} oscillation, or any cycle of motion,
is repeated during the succeeding oscillation or cycle, and
so on indefinitely; and if an alteration be made in the
parts, it is not a change once for all, but recurs in its
operation in every cycle of the motion.
But it will be found that this circumstance does not prevent
the same explanation from being still applicable with a
slight modification. Instead of uniform motion in the
intervals of causation, we shall have to speak of _steady
going_: and instead of considering all the forces which
affect the motion as causes of change of uniform motion, we
shall have to speak of changes in the parts of the mechanism
as causes of _change of rate of going_. With this
modification, it will still be true, that any instantaneous
cause produces its instantaneous effect simultaneously,
while the permanent effect is subsequent to the change which
is its cause. The steady going of the clock is assumed as a
normal condition, in which it measures the progress of time;
and in this assumption, the notion of cause and effect is
not brought into view. But a steady rate thus denoting the
mean passage of time, a change in the rate indicates a cause
of change. The _change of rate_, as an instantaneous
_transition_ from one rate to another, is _simultaneous_
with the change in the parts. But then the _changed rate_ as
a continued _condition_ in which, no new change supervening,
the rate again measures the progress of time, is
_subsequent_ to the change of parts, for it begins when that
ends, and continues when the progress of that has ceased.
If, however, this be a satisfactory solution of the
difficulty in the case of mechanism, how shall we apply the
same views to the other cases? Growth, the effect of food,
is subsequent to the act of taking food; disorder, the
effect of poison, is subsequent to the introduction of
poison into the system. Can we say that the animal would
continue unchanged if it were not to take food; and that
food is the cause of a change, namely, of growth? This is
manifestly false; for if the animal were not to take food,
it would soon perish. But the analogy of the former case, of
the clock, will enable us to avoid this perplexity. As we
assumed a steady rate of going in the clock to be the
measure of time when we considered the effect of mechanism,
so we assume a steady rate of action in the animal functions
to be the measure of the progress of time when we consider
the causes which act upon the {200} development and health
of animals. Digestion, and of course nutrition, are a part
of this normal condition; they are involved in the steady
going of the animal mechanism, and we must suppose these
functions to go regularly on, in order that the animal may
preserve its character of animal. Food and digestion may be
considered as causes of the continued existence of the
animal, in the same way in which the form of the parts of a
clock is the cause of the steady going of a clock. And when
we come to consider causes of change, this kind of
causation, which produces a normal condition of things,
merely measuring the flow of time, is left out of our
account. We can conceive an uniform condition of animal
existence, the animal neither growing nor wasting. This
being taken as the normal condition, any deviation from this
condition indicates a cause, and is taken as the evidence
and measure of the cause of change. And thus, in a growing
animal, the food partly keeps the animal in continued animal
existence, and partly, and in addition to this, causes its
growth. Food, in the former view, is always circulating in
the system, and is supposed to be uniformly administered;
the cycles of nutrition being merged in the notion of
uniform existence, as the oscillations of the pendulum in a
clock are merged in the notion of uniform going; and the
elementary steps of nutrition which are, in this view,
supposed to take place at each instant, produce their
instantaneous effect, for they are requisite in the cycle of
animal processes which goes on from instant to instant. But
on the other hand, in considering growth, we compare the
state of an animal with a preceding state, and consider the
nutriment taken in the intervening time as the cause of the
change: hence this nutriment, as an aggregate, is considered
as the cause of growth of the animal; and in this view the
effect is subsequent to the cause. But yet here, as in the
case of mechanism, the progressive effect is simultaneous,
step by step, with the progressive cause. There is a series
of operations; as for instance, intussusception, digestion,
assimilation, growth: each of these is a progressive
operation; and in the progress of each operation, the steps
of the effect and the instantaneous forces are simultaneous.
But the end of one operation is the beginning of the next,
or at least in part, and hence we have time occupied by the
succession. The end of intussusception is the beginning of
digestion, the end of digestion the beginning of
assimilation, {201} and so on. These aggregate effects
succeed each other; and hence growth is subsequent to the
taking of food; though each instantaneous force of animal
life, no less than of mechanism, produces an effect
simultaneous with its action. Each of these separate
operations is an aggregate operation, and occupies time; and
each aggregate effect is a condition of the action of the
cause in the next operation.
Again; if an animal in a permanent condition, neither waxing
nor wasting, may be taken as the normal state in which the
functions of life measure time, in order that we may
consider growth as an effect, to be referred to food as
cause; we may, for other purposes, consider, as the normal
condition, an animal waxing and then wasting, according to
the usual law of animal life: and we must take this, the
healthy progress of an animal, as our normal condition, if
we have to consider causes which produce disease. If we have
to refer the morbid condition of an animal to the influence
of poison, for example, we must consider how far the
condition deviates from what it would have been if the
poison had not been taken into the frame. The usual progress
of the animal functions including its growth, is the measure
of time; the deviation from this usual progress is the
indication of cause; and the effect of the poison is
subsequent to the cause, because the poison acts through the
cycle of the animal functions just mentioned, which occupies
time; and because the taking the poison into the system, not
any subsequent action of the animal forces in the system, is
considered as the event which we must contemplate as a
cause. To resume the analogy of the clock: the rate of the
clock is altered by altering the parts; but this alteration
itself may occupy time; as if we alter the rate of a clock
by applying a drop of acid, which gradually eats off a part
of the pendulum, the corrosion, as an aggregate effect,
occupies time; and the rates before and after the change are
separated by this time. But the application of the drop is
the cause; and thus, in this case the final effect is
subsequent to the cause, though here, as in the case of
mechanism, the instantaneous forces always produce a
simultaneous effect.
Thus we have in every case a _uniform_ state, or a state
which is considered as uniform, or at least _normal_; and
which is taken as the indication and measure of _time_; and
we have also _change_, {202} which is contemplated as a
deviation from uniformity, and is taken as the indication
and measure of _cause_. The uniform state may be one which
never exists, being purely imaginary; as the case in which
no forces act; and the case in which animal functions go on
permanently, the animal neither growing nor wasting. The
normal state may also be a state in which change is
constantly taking place, as, in fact, even a state of motion
is a state of change; such states also are, in a further
sense, that of a clock going by starts, and that of an
animal constantly growing: in these cases the changes are
all merged in a wider view of uniformity, so that these are
taken as the normal states. And in all these cases,
successive changes which take place are separated by
intervals of time, measured by the normal progress; and each
change is produced by some _simultaneous_ instantaneous
cause. But taking the cause in a larger sense, we group
these instantaneous causes, and perhaps omit in our
contemplation some of the intervening intervals; and thus
assign the cause to a _preceding_, and the effect to a
_succeeding_ time.
I may observe further, as a corollary from what has been
said, that the measure of time is different, when we
consider different kinds of causation; and in each case, is
_homogeneous_ with the changes which causation effects. In
the consideration of mechanical causes, we measure time by
mechanical changes;--by uniform motion, or uniform
succession of cycles of motion; by the rotation of a wheel,
or the oscillation of a pendulum. But if we have to consider
physiological changes, the progress of time is
physiologically measured;--by the normal progress of vital
operations; by the circulation, digestion or development of
the organized body; by the pulse, or by the growth. These
different measures of time give to time, so far as it is
exhibited by facts and events, a different character in the
different cases. Phenomenal time has a different nature and
essence according to the kind of the changes which we
consider, and which gives us our sole phenomenal indication
of cause.
I fear that I am travelling into matters too abstruse and
metaphysical for the occasion: but before I conclude, I will
present one other aspect of the subject.
In stating the difficulty, I referred to cases of moral as
well as physical causation; as when prudence produces
prosperity, or {203} when folly produces ruin. It may be
asked, whether we are here to apply the same
explanation;--whether we are to assume a normal condition of
human existence, in which neither prudence nor folly are
displayed, neither prosperity nor adversity
produced;--whether we are to conceive the progress of such a
state to measure the progress of time, and deviations from
it to denote causes of the kind mentioned. It may be asked
further, whether, if we do make this supposition, we can
resolve the influence of such causes as prudence or
imprudence into instantaneous acts, which produce their
effects immediately: and which occupy time only by being
separated by intervals of the inactive normal moral
condition. To this I must here reply, that the discussion of
such questions would carry me too far, and would involve
speculations not included within the acknowledged domain of
this Society, from which I therefore abstain. But I may say,
before quitting the subject, that I do not think the
suppositions above suggested are untenable; and that in
order to include moral causation under the maxims of
causation in general, we must necessarily make some such
hypothesis. The peculiarity of that kind of causation which
the will and the character exert, and which is exerted upon
the will and the character, would make this case far more
complex and difficult than those already considered; but, at
the same time, would offer us the means of explaining what
may seem harsh, in the above analogy. For instance, we
should have to assume such a maxim as this: that in moral
causation, time is not to be measured by the flow of
mechanical or physiological events;--not by the clock, or by
the pulse. Moral causation has its own clock, its own pulse,
in the progress of man's moral being; and by this measure of
time is the relation of moral cause and effect to be
defined.
That in estimating moral causation, the progress of time is
necessarily estimated by moral changes, and not by
machinery,--by the progress of events, and not by the going
of the clock,--is a truth familiar as a practical maxim to
all who give their thoughts to dramatic or narrative
fictions. Who feels any thing incongruous or extravagantly
hurried in the progress of events in that great exhibition
of moral causation, the tragedy of Othello? If we were asked
what time those vast and terrible {204} and complex changes
of the being and feelings of the characters occupy, we
should say, that, measured on its own scale, the event is of
great extent;--that the transaction is of considerable
magnitude in all ways. But if, with previous critics, we
look into the progress of time by the day and the hour--what
is the measure of this history? Forty-eight hours.
{{205}}
CHAPTER V.
OF THE ORIGIN OF OUR CONCEPTIONS OF FORCE AND MATTER.
1. _Force._--WHEN the faculties of observation and thought
are developed in man, the idea of causation is applied to
those changes which we see and feel in the state of rest and
motion of bodies around us. And when our abstract
conceptions are thus formed and named, we adopt the term
_Force_, and use it to denote that property which is the
cause of motion produced, changed, or prevented. This
conception is, it would seem, mainly and primarily suggested
by our consciousness of the exertions by which we put bodies
in motion. The Latin and Greek words for _Force_, Vis, Ϝὶς,
were probably, like all abstract terms, derived at first
from some sensible object. The original meaning of the Greek
word was a _muscle_ or _tendon_. Its first application as an
abstract term is accordingly to muscular force:
Δεύτερος αὖτ' Αἴας πολὺ μείζονα λᾶαν ἀείρας
ἦκ' ἐπιδινήσας, ἐπέρεισε δὲ ϜÎ͂Ν' ἀπέλεθρον.
Then Ajax a far heavier stone upheaved,
He whirled it, and impressing Force intense
Upon the mass, dismist it.
The property by which bodies affect each other's motions,
was naturally likened to that energy which we exert upon
them with similar effect: and thus the labouring horse, the
rushing torrent, the descending weight, the elastic bow,
were said to exert force. {206} Homer[8\3] speaks of the
_force_ of the river, Ϝὶς ποταμοῖο; and Hesiod[9\3] of the
_force_ of the north wind, Ϝὶς ἀνέμου βορέαο.
[Note 8\3: _Il._ xxi.]
[Note 9\3: _Op. et D._]
Thus man's general notion of force was probably first
suggested by his muscular exertions, that is, by an act
depending upon that muscular sense, to which, as we have
already seen, the perception of space is mainly due. And
this being the case, it will be easily understood that the
_Direction_ of the force thus exerted is perceived by the
muscular sense, at the same time that the force itself is
perceived; and that the direction of any other force is
understood by comparison with force which man must exert to
produce the same effect, in the same manner as force itself
is so understood.
This abstract notion of Force long remained in a very vague
and obscure condition, as may be seen by referring to the
History for the failures of attempts at a science of force
and motion, made by the ancients and their commentators in
the middle ages. By degrees, in modern times, we see the
scientific faculty revive. The conception of Force becomes
so far distinct and precise that it can be reasoned upon in
a consistent manner, with demonstrated consequences; and a
genuine science of Mechanics comes into existence. The
foundations of this science are to be found in the Axioms
concerning causation which we have already stated; these
axioms being interpreted and fixed in their application by a
constant reference to observed facts, as we shall show. But
we must, in the first place, consider further those primary
processes of observation by which we acquire the first
materials of thought on such subjects.
2. _Matter._--The conception of Force, as we have said,
arises with our consciousness of our own muscular exertions.
But we cannot imagine such exertions without also imagining
some bodily substance against which they are exercised. If
we press, we press something: if we thrust or throw, there
must be something {207} to resist the thrust or to receive
the impulse. Without body, muscular force cannot be exerted,
and force in general is not conceivable.
Thus Force cannot exist without _Body_ on which it acts. The
two conceptions, Force and Matter, are co-existent and
correlative. Force implies resistance; and the force is
effective only when the resistance is called into play. If
we grasp a stone, we have no hold of it till the closing of
the hand is resisted by the solid texture of the stone. If
we push open a gate, we must surmount the opposition which
it exerts while turning on its hinges. However slight the
resistance be, there must be some resistance, or there would
be no force. If we imagine a state of things in which
objects do not resist our touch, they must also cease to be
influenced by our strength. Such a state of things we
sometimes imagine in our dreams; and such are the poetical
pictures of the regions inhabited by disembodied spirits. In
these, the figures which appear are conspicuous to the eye,
but impalpable like shadow or smoke; and as they do not
resist the corporeal impressions, so neither do they obey
them. The spectator tries in vain to strike or to grasp
them.
Et ni docta comes tenues sine corpore vitas
Admoneat volitare cavâ sub imagine formæ,
Irruat ac frustra ferro diverberet umbras.
The Sibyl warns him that there round him fly
Bodiless things, but substance to the eye;
Else had he pierced those shapes with life-like face,
And smitten, fierce, the unresisting space.
Neque illum
Prensantem nequicquam umbras et multa volentem
Dicere, preterea vidit.
He grasps her form, and clutches but the shade.
Such may be the circumstances of the unreal world of dreams,
or of poetical fancies approaching to dreams: for in these
worlds our imaginary perceptions are bound by no rigid
conditions of force and reaction. In {208} such cases, the
mind casts off the empire of the idea of cause, as it casts
off even the still more familiar sway of the ideas of space
and time. But the character of the material world in which
we live when awake is, that we have at every instant and at
every place, force operating on matter and matter resisting
force.
3. _Solidity._--From our consciousness of muscular exertion,
we derive, as we have seen, the conception of force, and
with that also the conception of matter. We have already
shown, in a former chapter, that the same part of our frame,
the muscular system, is the organ by which we perceive
extension and the relations of space. Thus the same organ
gives us the perception of body as resisting force, and as
occupying space; and by combining these conditions we have
the conception of _solid_ extended bodies. In reality, this
resistance is inevitably presented to our notice in the very
facts from which we collect the notion of extension. For the
action of the hand and arm by which we follow the forms of
objects, implies that we apply our fingers to their surface;
and we are stopped there by the resistance which the body
offers. This resistance is precisely that which is requisite
in order to make us conscious of cur muscular effort[10\3].
Neither touch, nor any other mere passive sensation, could
produce the perception of extent, as we have already urged:
nor could the muscular sense lead to such a perception,
except the extension of the muscles were felt to be
resisted. And thus the perception of resistance enters the
mind along with the perception of extended bodies. All the
objects with which we have to do are not only extended but
solid.
[Note 10\3: Brown's _Lectures_, i. 466.]
This sense of the term _solidity_, (the general property of
all matter,) is different to that in which we oppose
_solidity_ to _fluidity_. We may avoid ambiguity by opposing
_rigid_ to _fluid_ bodies. By solid bodies, as we now speak
of them, we mean only such as resist the pressure which we
exert, so long as their parts continue in their places. By
fluid bodies, we mean those {209} whose parts are, by a
slight pressure, removed out of their places. A drop of
water ceases to prevent the contact of our two hands, not by
ceasing to have solidity in this sense, but by being thrust
out of the way. If it could remain in its place, it could
not cease to exercise its resistance to our pressure, except
by ceasing to be matter altogether.
The perception of solidity, like the perception of
extension, implies an act of the mind, as well as an
impression of the senses: as the perception of extension
implies the idea of space, so the perception of solidity
implies the idea of action and reaction. That an Idea is
involved in our knowledge on this subject, appears, as in
other instances, from this consideration, that the
convictions of persons, even of those who allow of no ground
of knowledge but experience, do in fact go far beyond the
possible limits of experience. Thus Locke says[11\3], that
'the bodies which we daily handle hinder by an
_insurmountable_ force the approach of the parts of our
hands that press them.' Now it is manifest that our
observation can never go to this length. By our senses we
can only perceive that bodies resist the greatest actual
forces that we exert upon them. But our conception of force
carries us further: and since, so long as the body is there
to receive the action of the force, it must suffer the whole
of that action, and must react as much as it suffers: it is
therefore true, that so long as the body remains there, the
force which is exerted upon it can never surmount the
resistance which the body exercises. And thus this doctrine,
that bodies resist the intrusion of other bodies by an
insurmountable force, is, in fact, a consequence of the
axiom that the reaction is always equal to the action.
[Note 11\3: _Essay_, b. ii. c. 4.]
4. _Inertia._--But this principle of the equality of action
and reaction appears also in another way. Not only when we
exert force upon bodies at rest, but when, by our exertions,
we put them in motion, they react. If we set a large stone
in motion, the stone {210} resists; for the operation
requires an effort. By increasing the effort, we can
increase the effect, that is, the motion produced; but the
resistance still remains. And the greater the stone moved,
the greater is the effort requisite to move it. There is, in
every case, a resistance to motion, which shows itself, not
in preventing the motion, but in a reciprocal force, exerted
backwards upon the agent by which the motion is produced.
And this resistance resides in each portion of matter, for
it is increased as we add one portion of matter to another.
We can push a light boat rapidly through the water; but we
may go on increasing its freight, till we are barely able to
stir it. This property of matter, then, by which it resists
the reception of motion, or rather by which it reacts and
requires an adequate force in order that any motion may
result, is called its inertness, or _inertia_. That matter
has such a property, is a conviction flowing from that idea
of a reaction equal and opposite to the action, which the
conception of all force involves. By what laws this inertia
depends on the magnitude, form, and material of the body,
must be the subject of our consideration hereafter. But that
matter has this inertia, in virtue of which, as the matter
is greater, the velocity which the same effort can
communicate to it is less, is a principle inseparable from
the notion of matter itself.
Hermann says that Kepler first introduced this 'most
significant' _inertia_. Whether it is to be found in earlier
writers I know not; Kepler certainly does use it familiarly
in those attempts to assign physical reasons for the motions
of the planets which were among the main occasions of the
discovery of the true laws of mechanics. He assumes the
slowness of the motions of the planets to increase, (other
causes remaining the same,) as the inertia increases; and
though, even in this assumption, there is an errour
involved, (if we adopt that interpretation of the term
_inertia_ to which subsequent researches led,) the
introduction of such a word was one step in determining and
expressing those laws of motion which depend on the
fundamental principle of the equality of action and
reaction. {211}
5. We have thus seen, I trust in a satisfactory manner, the
origin of our conceptions of Force, Matter, Solidity, and
Inertness. It has appeared that the organ by which we obtain
such conceptions is that very muscular frame, which is the
main instrument of our perceptions of space; but that,
besides bodily sensations, these ideal conceptions, like all
the others which we have hitherto considered, involve also
an habitual activity of the mind, giving to our sensations a
meaning which they could not otherwise possess. And among
the ideas thus brought into play, is an idea of action with
an equal and opposite reaction, which forms a foundation for
universal truths to be hereafter established respecting the
conceptions thus obtained.
We must now endeavour to trace in what manner these
fundamental principles and conceptions are unfolded by means
of observation and reasoning, till they become an extensive
yet indisputable science.
{{212}}
CHAPTER VI.
OF THE ESTABLISHMENT OF THE PRINCIPLES OF STATICS.
1. _Object of the Chapter._--IN the present and the
succeeding chapters we have to show how the general axioms
of Causation enable us to construct the science of
Mechanics. We have to consider these axioms as moulding
themselves, in the first place, into certain fundamental
mechanical principles, which are of evident and necessary
truth in virtue of their dependence upon the general axioms
of Causation; and thus as forming a foundation for the whole
structure of the science;--a system of truths no less
necessary than the fundamental principles, because derived
from these by rigorous demonstration.
This account of the construction of the science of
Mechanics, however generally treated, cannot be otherwise
than technical in its details, and will probably be
imperfectly understood by any one not acquainted with
Mechanics as a mathematical science.
I cannot omit this portion of my survey without rendering my
work incomplete; but I may remark that the main purpose of
it is to prove, in a more particular manner, what I have
already declared in general, that there are, in Mechanics no
less than in Geometry, fundamental principles of axiomatic
evidence and necessity;--that these principles derive their
axiomatic character from the Idea which they involve,
namely, the Idea of Cause;--and that through the combination
of principles of this kind, the whole science of Mechanics,
including its most complex and remote results, exists as a
body of solid and universal truths. {213}
2. _Statics and Dynamics._--We must first turn our attention
to a technical distinction of Mechanics into two portions,
according as the forces about which we reason produce rest,
or motion; the former portion is termed _Statics_, the
latter _Dynamics_. If a stone fall, or a weight put a
machine in motion, the problem belongs to Dynamics; but if
the stone rest upon the ground, or a weight be merely
supported by a machine, without being raised higher, the
question is one of Statics.
3. _Equilibrium._--In Statics, forces _balance_ each other,
or keep each other _in equilibrium_. And forces which
directly balance each other, or keep each other in
equilibrium, are necessarily and manifestly equal. If we see
two boys pull at two ends of a rope so that neither of them
in the smallest degree prevails over the other, we have a
case in which two forces are in equilibrium. The two forces
are evidently equal, and are a statical exemplification of
action and reaction, such as are spoken of in the third
axiom concerning causes. Now the same exemplification occurs
in every case of equilibrium. No point or body can be kept
at rest except in virtue of opposing forces acting upon it;
and these forces must always be equal in their opposite
effect. When a stone lies on the floor, the weight of the
stone downwards is opposed and balanced by an equal pressure
of the floor upwards. If the stone rests on a slope, its
tendency to slide is counteracted by some equal and opposite
force, arising, it may be, from the resistance which the
sloping ground opposes to any motion along its surface.
Every case of rest is a case of equilibrium: every case of
equilibrium is a case of equal and opposite forces.
The most complex frame-work on which weights are supported,
as the roof of a building, or the cordage of a machine, are
still examples of equilibrium. In such cases we may have
many forces all combining to balance each other; and the
equilibrium will depend on various conditions of direction
and magnitude among the forces. And in order to understand
what are these conditions, we must ask, in the first place,
what {214} we understand by the magnitude of such
forces;--what is the measure of statical forces.
4. _Measure of Statical Forces._--At first we might expect,
perhaps, that since statical forces come under the general
notion of Cause, the mode of measuring them would be derived
from the second axiom of Causation, that causes are measured
by their effects. But we find that the application of this
axiom is controlled by the limitation which we noticed,
after stating that axiom; namely, the condition that the
causes shall be capable of addition. Further, as we have
seen, a statical force produces no other effect than this,
that it balances some other statical force; and hence the
measure of statical forces is necessarily dependent upon
their balancing, that is, upon the equality of action and
reaction.
That _statical forces are capable of addition_ is involved
in our conception of such forces. When two men pull at a
rope in the same direction, the forces which they exert are
added together. When two heavy bodies are put into a basket
suspended by a string, their weights are added, and the sum
is supported by the string.
Combining these considerations, it will appear that the
measure of statical forces is necessarily given at once by
the fundamental principle of the equality of action and
reaction. Since two opposite forces which balance each other
are equal, each force is measured by that which it balances;
and since forces are capable of addition, a force of any
magnitude is measured by adding together a proper number of
such equal forces. Thus a heavy body which, appended to some
certain elastic branch of a tree, would bend it down through
one inch, may be taken as a unit of weight. Then if we
remove this first body, and find a second heavy body which
will also bend the branch through the same space, this is
also a unit of weight; and in like manner we might go on to
a third and a fourth equal body; and adding together the
two, or the three, or the four heavy bodies, we have a force
twice, or three times, or four times the unit of weight. And
with {215} such a collection of heavy bodies, or _weights_,
we can readily measure all other forces; for the same
principle of the equality of action and reaction leads at
once to this maxim, that any statical force is measured by
the weight which it would support.
As has been said, it might at first have been supposed that
we should have to apply, in this case, the axiom that causes
are measured by their effects in another manner; that thus,
if that body were a unit of weight which bent the bough of a
tree through one inch, _that_ body would be _two_ units
which bent it through _two_ inches, and so on. But, as we
have already stated, the measures of weight must be subject
to this condition, that they are susceptible of being added:
and therefore we cannot take the deflexion of the bough for
our measure, till we have ascertained, that which experience
alone can teach us, that under the burden of two equal
weights, the deflexion will be twice as great as it is with
one weight, which is not true, or at least is neither
obviously nor necessarily true. In this, as in all other
cases, although causes must be measured by their effects, we
learn from experience only how the effects are to be
interpreted, so as to give a true and consistent measure.
With regard, however, to the measure of statical force, and
of weight, no difficulty really occurred to philosophers
from the time when they first began to speculate on such
subjects; for it was easily seen that if we take any uniform
material, as wood, or stone, or iron, portions of this which
are geometrically equal, must also be equal in statical
effect; since this was implied in the very hypothesis of a
uniform material And a body ten times as large as another of
the same substance, will be of ten times the weight. But
before men could establish by reasoning the conditions under
which weights would be in equilibrium, some other principles
were needed in addition to the mere measure of forces. The
principles introduced for this purpose still resulted from
the conception of equal action and reaction; but it required
no small clearness of thought to select them rightly, and to
employ them {216} successfully. This, however, was done, to
a certain extent, by the Greeks; and the treatise of
Archimedes _On the Center of Gravity_, is founded on
principles which may still be considered as the genuine
basis of statical reasoning. I shall make a few remarks on
the most important principle among those which Archimedes
thus employs.
5. _The Center of Gravity._--The most important of the
principles which enter into the demonstration of Archimedes
is this: that "Every body has a center of gravity;" meaning
by the center of gravity, a point at which the whole matter
of the body may be supposed to be collected, to all intents
and purposes of statical reasoning. This principle has been
put in various forms by succeeding writers: for instance, it
has been thought sufficient to assume a case much simpler
than the general one; and to assert that two _equal_ bodies
have their center of gravity in the point midway between
them. It is to be observed, that this assertion not only
implies that the two bodies will _balance_ upon a support
placed at that midway point, but also, that they will
exercise, upon such a support, a _pressure equal to their
sum_; for this point being the center of gravity, the whole
matter of the two bodies may be conceived to be collected
there, and therefore the whole weight will press there. And
thus the principle in question amounts to this, that _when
two equal heavy bodies are supported on the middle point
between them, the pressure upon the support is equal to the
sum of the weights of the bodies_.
A clear understanding of the nature and grounds of this
principle is of great consequence: for in it we have the
foundation of a large portion of the science of Mechanics.
And if this principle can be shown to be necessarily true,
in virtue of our Fundamental Ideas, we can hardly doubt that
there exist many other truths of the same kind, and that no
sound view of the evidence and extent of human knowledge can
be obtained, so long as we mistake the nature of these, its
first principles. {217}