OF THE FOUR METHODS OF EXPERIMENTAL INQUIRY.
§ 1. The simplest and most obvious modes of singling out from among the
circumstances which precede or follow a phenomenon, those with which it
is really connected by an invariable law, are two in number. One is, by
comparing together different instances in which the phenomenon occurs.
The other is, by comparing instances in which the phenomenon does occur,
with instances in other respects similar in which it does not. These two
methods may be respectively denominated, the Method of Agreement, and
the Method of Difference.
In illustrating these methods, it will be necessary to bear in mind the
twofold character of inquiries into the laws of phenomena; which may be
either inquiries into the cause of a given effect, or into the effects
or properties of a given cause. We shall consider the methods in their
application to either order of investigation, and shall draw our
examples equally from both.
We shall denote antecedents by the large letters of the alphabet, and
the consequents corresponding to them by the small. Let A, then, be an
agent or cause, and let the object of our inquiry be to ascertain what
are the effects of this cause. If we can either find, or produce, the
agent A in such varieties of circumstances, that the different cases
have no circumstance in common except A; then whatever effect we find to
be produced in all our trials, is indicated as the effect of A. Suppose,
for example, that A is tried along with B and C, and that the effect is
_a b c_; and suppose that A is next tried with D and E, but without B
and C, and that the effect is _a d e_. Then we may reason thus: _b_ and
_c_ are not effects of A, for they were not produced by it in the second
experiment; nor are _d_ and _e_, for they were not produced in the
first. Whatever is really the effect of A must have been produced in
both instances; now this condition is fulfilled by no circumstance
except _a_. The phenomenon _a_ cannot have been the effect of B or C,
since it was produced where they were not; nor of D or E, since it was
produced where they were not. Therefore it is the effect of A.
For example, let the antecedent A be the contact of an alkaline
substance and an oil. This combination being tried under several
varieties of circumstances, resembling each other in nothing else, the
results agree in the production of a greasy and detersive or saponaceous
substance: it is therefore concluded that the combination of an oil and
an alkali causes the production of a soap. It is thus we inquire, by the
Method of Agreement, into the effect of a given cause.
In a similar manner we may inquire into the cause of a given effect. Let
_a_ be the effect. Here, as shown in the last chapter, we have only the
resource of observation without experiment: we cannot take a phenomenon
of which we know not the origin, and try to find its mode of production
by producing it: if we succeeded in such a random trial it could only be
by accident. But if we can observe _a_ in two different combinations, _a
b c_, and _a d e_; and if we know, or can discover, that the antecedent
circumstances in these cases respectively were A B C and A D E; we may
conclude by a reasoning similar to that in the preceding example, that A
is the antecedent connected with the consequent _a_ by a law of
causation. B and C, we may say, cannot be causes of _a_, since on its
second occurrence they were not present; nor are D and E, for they were
not present on its first occurrence. A, alone of the five circumstances,
was found among the antecedents of _a_ in both instances.
For example, let the effect _a_ be crystallization. We compare instances
in which bodies are known to assume crystalline structure, but which
have no other point of agreement; and we find them to have one, and as
far as we can observe, only one, antecedent in common: the deposition of
a solid matter from a liquid state, either a state of fusion or of
solution. We conclude, therefore, that the solidification of a
substance from a liquid state is an invariable antecedent of its
crystallization.
In this example we may go farther, and say, it is not only the
invariable antecedent but the cause; or at least the proximate event
which completes the cause. For in this case we are able, after detecting
the antecedent A, to produce it artificially, and by finding that _a_
follows it, verify the result of our induction. The importance of thus
reversing the proof was strikingly manifested when by keeping a phial of
water charged with siliceous particles undisturbed for years, a chemist
(I believe Dr. Wollaston) succeeded in obtaining crystals of quartz: and
in the equally interesting experiment in which Sir James Hall produced
artificial marble, by the cooling of its materials from fusion under
immense pressure: two admirable examples of the light which may be
thrown upon the most secret processes of nature by well-contrived
interrogation of her.
But if we cannot artificially produce the phenomenon A, the conclusion
that it is the cause of _a_ remains subject to very considerable doubt.
Though an invariable, it may not be the unconditional antecedent of _a_,
but may precede it as day precedes night or night day. This uncertainty
arises from the impossibility of assuring ourselves that A is the _only_
immediate antecedent common to both the instances. If we could be
certain of having ascertained all the invariable antecedents, we might
be sure that the unconditional invariable antecedent, or cause, must be
found somewhere among them. Unfortunately it is hardly ever possible to
ascertain all the antecedents, unless the phenomenon is one which we can
produce artificially. Even then, the difficulty is merely lightened, not
removed: men knew how to raise water in pumps long before they adverted
to what was really the operating circumstance in the means they
employed, namely, the pressure of the atmosphere on the open surface of
the water. It is, however, much easier to analyse completely a set of
arrangements made by ourselves, than the whole complex mass of the
agencies which nature happens to be exerting at the moment of the
production of a given phenomenon. We may overlook some of the material
circumstances in an experiment with an electrical machine; but we shall,
at the worst, be better acquainted with them than with those of a
thunder-storm.
The mode of discovering and proving laws of nature, which we have now
examined, proceeds on the following axiom: Whatever circumstances can be
excluded, without prejudice to the phenomenon, or can be absent
notwithstanding its presence, is not connected with it in the way of
causation. The casual circumstances being thus eliminated, if only one
remains, that one is the cause which we are in search of: if more than
one, they either are, or contain among them, the cause; and so, _mutatis
mutandis_, of the effect. As this method proceeds by comparing different
instances to ascertain in what they agree, I have termed it the Method
of Agreement: and we may adopt as its regulating principle the following
canon:--
FIRST CANON.
_If two or more instances of the phenomenon under investigation have
only one circumstance in common, the circumstance in which alone all the
instances agree, is the cause (or effect) of the given phenomenon._
Quitting for the present the Method of Agreement, to which we shall
almost immediately return, we proceed to a still more potent instrument
of the investigation of nature, the Method of Difference.
§ 2. In the Method of Agreement, we endeavoured to obtain instances
which agreed in the given circumstance but differed in every other: in
the present method we require, on the contrary, two instances resembling
one another in every other respect, but differing in the presence or
absence of the phenomenon we wish to study. If our object be to discover
the effects of an agent A, we must procure A in some set of ascertained
circumstances, as A B C, and having noted the effects produced, compare
them with the effect of the remaining circumstances B C, when A is
absent. If the effect of A B C is _a b c_, and the effect of B C, _b c_,
it is evident that the effect of A is _a_. So again, if we begin at the
other end, and desire to investigate the cause of an effect _a_, we must
select an instance, as _a b c_, in which the effect occurs, and in which
the antecedents were A B C, and we must look out for another instance in
which the remaining circumstances, _b c_, occur without _a_. If the
antecedents, in that instance, are B C, we know that the cause of _a_
must be A: either A alone, or A in conjunction with some of the other
circumstances present.
It is scarcely necessary to give examples of a logical process to which
we owe almost all the inductive conclusions we draw in daily life. When
a man is shot through the heart, it is by this method we know that it
was the gun-shot which killed him: for he was in the fulness of life
immediately before, all circumstances being the same, except the wound.
The axioms implied in this method are evidently the following. Whatever
antecedent cannot be excluded without preventing the phenomenon, is the
cause, or a condition, of that phenomenon: Whatever consequent can be
excluded, with no other difference in the antecedents than the absence
of a particular one, is the effect of that one. Instead of comparing
different instances of a phenomenon, to discover in what they agree,
this method compares an instance of its occurrence with an instance of
its non-occurrence, to discover in what they differ. The canon which is
the regulating principle of the Method of Difference may be expressed as
follows:
SECOND CANON.
_If an instance in which the phenomenon under investigation occurs, and
an instance in which it does not occur, have every circumstance in
common save one, that one occurring only in the former; the circumstance
in which alone the two instances differ, is the effect, or the cause, or
an indispensable part of the cause, of the phenomenon._
§ 3. The two methods which we have now stated have many features of
resemblance, but there are also many distinctions between them. Both
are methods of _elimination_. This term (employed in the theory of
equations to denote the process by which one after another of the
elements of a question is excluded, and the solution made to depend on
the relation between the remaining elements only) is well suited to
express the operation, analogous to this, which has been understood
since the time of Bacon to be the foundation of experimental inquiry:
namely, the successive exclusion of the various circumstances which are
found to accompany a phenomenon in a given instance, in order to
ascertain what are those among them which can be absent consistently
with the existence of the phenomenon. The Method of Agreement stands on
the ground that whatever can be eliminated, is not connected with the
phenomenon by any law. The Method of Difference has for its foundation,
that whatever cannot be eliminated, is connected with the phenomenon by
a law.
Of these methods, that of Difference is more particularly a method of
artificial experiment; while that of Agreement is more especially the
resource employed where experimentation is impossible. A few reflections
will prove the fact, and point out the reason of it.
It is inherent in the peculiar character of the Method of Difference,
that the nature of the combinations which it requires is much more
strictly defined than in the Method of Agreement. The two instances
which are to be compared with one another must be exactly similar, in
all circumstances except the one which we are attempting to investigate:
they must be in the relation of A B C and B C, or of _a b c_ and _b c_.
It is true that this similarity of circumstances needs not extend to
such as are already known to be immaterial to the result. And in the
case of most phenomena we learn at once, from the commonest experience,
that most of the coexistent phenomena of the universe may be either
present or absent without affecting the given phenomenon; or, if
present, are present indifferently when the phenomenon does not happen
and when it does. Still, even limiting the identity which is required
between the two instances, A B C and B C, to such circumstances as are
not already known to be indifferent; it is very seldom that nature
affords two instances, of which we can be assured that they stand in
this precise relation to one another. In the spontaneous operations of
nature there is generally such complication and such obscurity, they are
mostly either on so overwhelmingly large or on so inaccessibly minute a
scale, we are so ignorant of a great part of the facts which really take
place, and even those of which we are not ignorant are so multitudinous,
and therefore so seldom exactly alike in any two cases, that a
spontaneous experiment, of the kind required by the Method of
Difference, is commonly not to be found. When, on the contrary, we
obtain a phenomenon by an artificial experiment, a pair of instances
such as the method requires is obtained almost as a matter of course,
provided the process does not last a long time. A certain state of
surrounding circumstances existed before we commenced the experiment;
this is B C. We then introduce A; say, for instance, by merely bringing
an object from another part of the room, before there has been time for
any change in the other elements. It is, in short (as M. Comte
observes), the very nature of an experiment, to introduce into the
pre-existing state of circumstances a change perfectly definite. We
choose a previous state of things with which we are well acquainted, so
that no unforeseen alteration in that state is likely to pass
unobserved; and into this we introduce, as rapidly as possible, the
phenomenon which we wish to study; so that in general we are entitled to
feel complete assurance that the pre-existing state, and the state which
we have produced, differ in nothing except the presence or absence of
that phenomenon. If a bird is taken from a cage, and instantly plunged
into carbonic acid gas, the experimentalist may be fully assured (at all
events after one or two repetitions) that no circumstance capable of
causing suffocation had supervened in the interim, except the change
from immersion in the atmosphere to immersion in carbonic acid gas.
There is one doubt, indeed, which may remain in some cases of this
description; the effect may have been produced not by the change, but by
the means employed to produce the change. The possibility, however, of
this last supposition generally admits of being conclusively tested by
other experiments. It thus appears that in the study of the various
kinds of phenomena which we can, by our voluntary agency, modify or
control, we can in general satisfy the requisitions of the Method of
Difference; but that by the spontaneous operations of nature those
requisitions are seldom fulfilled.
The reverse of this is the case with the Method of Agreement. We do not
here require instances of so special and determinate a kind. Any
instances whatever, in which nature presents us with a phenomenon, may
be examined for the purposes of this method; and if all such instances
agree in anything, a conclusion of considerable value is already
attained. We can seldom, indeed, be sure that the one point of agreement
is the only one; but this ignorance does not, as in the Method of
Difference, vitiate the conclusion; the certainty of the result, as far
as it goes, is not affected. We have ascertained one invariable
antecedent or consequent, however many other invariable antecedents or
consequents may still remain unascertained. If A B C, A D E, A F G, are
all equally followed by _a_, then _a_ is an invariable consequent of A.
If _a b c_, _a d e_, _a f g_, all number A among their antecedents, then
A is connected as an antecedent, by some invariable law, with _a_. But
to determine whether this invariable antecedent is a cause, or this
invariable consequent an effect, we must be able, in addition, to
produce the one by means of the other; or, at least, to obtain that
which alone constitutes our assurance of having produced anything,
namely, an instance in which the effect, _a_, has come into existence,
with no other change in the pre-existing circumstances than the addition
of A. And this, if we can do it, is an application of the Method of
Difference, not of the Method of Agreement.
It thus appears to be by the Method of Difference alone that we can
ever, in the way of direct experience, arrive with certainty at causes.
The Method of Agreement leads only to laws of phenomena (as some writers
call them, but improperly, since laws of causation are also laws of
phenomena): that is, to uniformities, which either are not laws of
causation, or in which the question of causation must for the present
remain undecided. The Method of Agreement is chiefly to be resorted to,
as a means of suggesting applications of the Method of Difference (as in
the last example the comparison of A B C, A D E, A F G, suggested that A
was the antecedent on which to try the experiment whether it could
produce _a_); or as an inferior resource, in case the Method of
Difference is impracticable; which, as we before showed, generally
arises from the impossibility of artificially producing the phenomena.
And hence it is that the Method of Agreement, though applicable in
principle to either case, is more emphatically the method of
investigation on those subjects where artificial experimentation is
impossible: because on those it is, generally, our only resource of a
directly inductive nature; while, in the phenomena which we can produce
at pleasure, the Method of Difference generally affords a more
efficacious process, which will ascertain causes as well as mere laws.
§ 4. There are, however, many cases in which, though our power of
producing the phenomenon is complete, the Method of Difference either
cannot be made available at all, or not without a previous employment of
the Method of Agreement. This occurs when the agency by which we can
produce the phenomenon is not that of one single antecedent, but a
combination of antecedents, which we have no power of separating from
each other, and exhibiting apart. For instance, suppose the subject of
inquiry to be the cause of the double refraction of light. We can
produce this phenomenon at pleasure, by employing any one of the many
substances which are known to refract light in that peculiar manner. But
if, taking one of those substances, as Iceland spar for example, we wish
to determine on which of the properties of Iceland spar this remarkable
phenomenon depends, we can make no use, for that purpose, of the Method
of Difference; for we cannot find another substance precisely resembling
Iceland spar except in some one property. The only mode, therefore, of
prosecuting this inquiry is that afforded by the Method of Agreement; by
which, in fact, through a comparison of all the known substances which
have the property of doubly refracting light, it was ascertained that
they agree in the circumstance of being crystalline substances; and
though the converse does not hold, though all crystalline substances
have not the property of double refraction, it was concluded, with
reason, that there is a real connexion between these two properties;
that either crystalline structure, or the cause which gives rise to that
structure, is one of the conditions of double refraction.
Out of this employment of the Method of Agreement arises a peculiar
modification of that method, which is sometimes of great avail in the
investigation of nature. In cases similar to the above, in which it is
not possible to obtain the precise pair of instances which our second
canon requires--instances agreeing in every antecedent except A, or in
every consequent except _a_; we may yet be able, by a double employment
of the Method of Agreement, to discover in what the instances which
contain A or _a_, differ from those which do not.
If we compare various instances in which _a_ occurs, and find that they
all have in common the circumstance A, and (as far as can be observed)
no other circumstance, the Method of Agreement, so far, bears testimony
to a connexion between A and _a_. In order to convert this evidence of
connexion into proof of causation by the direct Method of Difference, we
ought to be able, in some one of these instances, as for example A B C,
to leave out A, and observe whether by doing so, _a_ is prevented. Now
supposing (what is often the case) that we are not able to try this
decisive experiment; yet, provided we can by any means discover what
would be its result if we could try it, the advantage will be the same.
Suppose, then, that as we previously examined a variety of instances in
which _a_ occurred, and found them to agree in containing A, so we now
observe a variety of instances in which _a_ does not occur, and find
them agree in not containing A; which establishes, by the Method of
Agreement, the same connexion between the absence of A and the absence
of _a_, which was before established between their presence. As, then,
it had been shown that whenever A is present _a_ is present, so it being
now shown that when A is taken away _a_ is removed along with it, we
have by the one proposition A B C, _a b c_, by the other B C, _b c_,
the positive and negative instances which the Method of Difference
requires.
This method may be called the Indirect Method of Difference, or the
Joint Method of Agreement and Difference; and consists in a double
employment of the Method of Agreement, each proof being independent of
the other, and corroborating it. But it is not equivalent to a proof by
the direct Method of Difference. For the requisitions of the Method of
Difference are not satisfied, unless we can be quite sure either that
the instances affirmative of _a_ agree in no antecedent whatever but A,
or that the instances negative of _a_ agree in nothing but the negation
of A. Now if it were possible, which it never is, to have this
assurance, we should not need the joint method; for either of the two
sets of instances separately would then be sufficient to prove
causation. This indirect method, therefore, can only be regarded as a
great extension and improvement of the Method of Agreement, but not as
participating in the more cogent nature of the Method of Difference. The
following may be stated as its canon:--
THIRD CANON.
_If two or more instances in which the phenomenon occurs have only one
circumstance in common, while two or more instances in which it does not
occur have nothing in common save the absence of that circumstance; the
circumstance in which alone the two sets of instances differ, is the
effect, or the cause, or an indispensable part of the cause, of the
phenomenon._
We shall presently see that the Joint Method of Agreement and Difference
constitutes, in another respect not yet adverted to, an improvement upon
the common Method of Agreement, namely, in being unaffected by a
characteristic imperfection of that method, the nature of which still
remains to be pointed out. But as we cannot enter into this exposition
without introducing a new element of complexity into this long and
intricate discussion, I shall postpone it to a subsequent chapter, and
shall at once proceed to a statement of two other methods, which will
complete the enumeration of the means which mankind possess for
exploring the laws of nature by specific observation and experience.
§ 5. The first of these has been aptly denominated the Method of
Residues. Its principle is very simple. Subducting from any given
phenomenon all the portions which, by virtue of preceding inductions,
can be assigned to known causes, the remainder will be the effect of the
antecedents which had been overlooked, or of which the effect was as yet
an unknown quantity.
Suppose, as before, that we have the antecedents A B C, followed by the
consequents _a b c_, and that by previous inductions (founded, we will
suppose, on the Method of Difference) we have ascertained the causes of
some of these effects, or the effects of some of these causes; and are
thence apprised that the effect of A is _a_, and that the effect of B is
_b_. Subtracting the sum of these effects from the total phenomenon,
there remains _c_, which now, without any fresh experiments, we may know
to be the effect of C. This Method of Residues is in truth a peculiar
modification of the Method of Difference. If the instance A B C, _a b
c_, could have been compared with a single instance A B, _a b_, we
should have proved C to be the cause of _c_, by the common process of
the Method of Difference. In the present case, however, instead of a
single instance A B, we have had to study separately the causes A and B,
and to infer from the effects which they produce separately, what effect
they must produce in the case A B C where they act together. Of the two
instances, therefore, which the Method of Difference requires,--the one
positive, the other negative,--the negative one, or that in which the
given phenomenon is absent, is not the direct result of observation and
experiment, but has been arrived at by deduction. As one of the forms of
the Method of Difference, the Method of Residues partakes of its
rigorous certainty, provided the previous inductions, those which gave
the effects of A and B, were obtained by the same infallible method, and
provided we are certain that C is the _only_ antecedent to which the
residual phenomenon _c_ can be referred; the only agent of which we had
not already calculated and subducted the effect. But as we can never be
quite certain of this, the evidence derived from the Method of Residues
is not complete unless we can obtain C artificially and try it
separately, or unless its agency, when once suggested, can be accounted
for, and proved deductively from known laws.
Even with these reservations, the Method of Residues is one of the most
important among our instruments of discovery. Of all the methods of
investigating laws of nature, this is the most fertile in unexpected
results; often informing us of sequences in which neither the cause nor
the effect were sufficiently conspicuous to attract of themselves the
attention of observers. The agent C may be an obscure circumstance, not
likely to have been perceived unless sought for, nor likely to have been
sought for until attention had been awakened by the insufficiency of the
obvious causes to account for the whole of the effect. And _c_ may be so
disguised by its intermixture with _a_ and _b_, that it would scarcely
have presented itself spontaneously as a subject of separate study. Of
these uses of the method, we shall presently cite some remarkable
examples. The canon of the Method of Residues is as follows:--
FOURTH CANON.
_Subduct from any phenomenon such part as is known by previous
inductions to be the effect of certain antecedents, and the residue of
the phenomenon is the effect of the remaining antecedents._
§ 6. There remains a class of laws which it is impracticable to
ascertain by any of the three methods which I have attempted to
characterize; namely, the laws of those Permanent Causes, or
indestructible natural agents, which it is impossible either to exclude
or to isolate; which we can neither hinder from being present, nor
contrive that they shall be present alone. It would appear at first
sight that we could by no means separate the effects of these agents
from the effects of those other phenomena with which they cannot be
prevented from coexisting. In respect, indeed, to most of the permanent
causes, no such difficulty exists; since though we cannot eliminate
them as coexisting facts, we can eliminate them as influencing agents,
by simply trying our experiment in a local situation beyond the limits
of their influence. The pendulum, for example, has its oscillations
disturbed by the vicinity of a mountain: we remove the pendulum to a
sufficient distance from the mountain, and the disturbance ceases: from
these data we can determine by the Method of Difference, the amount of
effect due to the mountain; and beyond a certain distance everything
goes on precisely as it would do if the mountain exercised no influence
whatever, which, accordingly, we, with sufficient reason, conclude to be
the fact.
The difficulty, therefore, in applying the methods already treated of to
determine the effects of Permanent Causes, is confined to the cases in
which it is impossible for us to get out of the local limits of their
influence. The pendulum can be removed from the influence of the
mountain, but it cannot be removed from the influence of the earth: we
cannot take away the earth from the pendulum, nor the pendulum from the
earth, to ascertain whether it would continue to vibrate if the action
which the earth exerts upon it were withdrawn. On what evidence, then,
do we ascribe its vibrations to the earth's influence? Not on any
sanctioned by the Method of Difference; for one of the two instances,
the negative instance, is wanting. Nor by the Method of Agreement; for
though all pendulums agree in this, that during their oscillations the
earth is always present, why may we not as well ascribe the phenomenon
to the sun, which is equally a coexistent fact in all the experiments?
It is evident that to establish even so simple a fact of causation as
this, there was required some method over and above those which we have
yet examined.
As another example, let us take the phenomenon Heat. Independently of
all hypothesis as to the real nature of the agency so called, this fact
is certain, that we are unable to exhaust any body of the whole of its
heat. It is equally certain, that no one ever perceived heat not
emanating from a body. Being unable, then, to separate Body and Heat, we
cannot effect such a variation of circumstances as the foregoing three
methods require; we cannot ascertain, by those methods, what portion of
the phenomena exhibited by any body is due to the heat contained in it.
If we could observe a body with its heat, and the same body entirely
divested of heat, the Method of Difference would show the effect due to
the heat, apart from that due to the body. If we could observe heat
under circumstances agreeing in nothing but heat, and therefore not
characterized also by the presence of a body, we could ascertain the
effects of heat, from an instance of heat with a body and an instance of
heat without a body, by the Method of Agreement; or we could determine
by the Method of Difference what effect was due to the body, when the
remainder which was due to the heat would be given by the Method of
Residues. But we can do none of these things; and without them the
application of any of the three methods to the solution of this problem
would be illusory. It would be idle, for instance, to attempt to
ascertain the effect of heat by subtracting from the phenomena exhibited
by a body, all that is due to its other properties; for as we have never
been able to observe any bodies without a portion of heat in them,
effects due to that heat might form a part of the very results, which we
were affecting to subtract in order that the effect of heat might be
shown by the residue.
If, therefore, there were no other methods of experimental investigation
than these three, we should be unable to determine the effects due to
heat as a cause. But we have still a resource. Though we cannot exclude
an antecedent altogether, we may be able to produce, or nature may
produce for us, some modification in it. By a modification is here
meant, a change in it, not amounting to its total removal. If some
modification in the antecedent A is always followed by a change in the
consequent _a_, the other consequents _b_ and _c_ remaining the same; or
_vice versâ_, if every change in _a_ is found to have been preceded by
some modification in A, none being observable in any of the other
antecedents; we may safely conclude that _a_ is, wholly or in part, an
effect traceable to A, or at least in some way connected with it through
causation. For example, in the case of heat, though we cannot expel it
altogether from any body, we can modify it in quantity, we can increase
or diminish it; and doing so, we find by the various methods of
experimentation or observation already treated of, that such increase or
diminution of heat is followed by expansion or contraction of the body.
In this manner we arrive at the conclusion, otherwise unattainable by
us, that one of the effects of heat is to enlarge the dimensions of
bodies; or what is the same thing in other words, to widen the distances
between their particles.
A change in a thing, not amounting to its total removal, that is, a
change which leaves it still the same thing it was, must be a change
either in its quantity, or in some of its variable relations to other
things, of which variable relations the principal is its position in
space. In the previous example, the modification which was produced in
the antecedent was an alteration in its quantity. Let us now suppose the
question to be, what influence the moon exerts on the surface of the
earth. We cannot try an experiment in the absence of the moon, so as to
observe what terrestrial phenomena her annihilation would put an end to;
but when we find that all the variations in the _position_ of the moon
are followed by corresponding variations in the time and place of high
water, the place being always either the part of the earth which is
nearest to, or that which is most remote from, the moon, we have ample
evidence that the moon is, wholly or partially, the cause which
determines the tides. It very commonly happens, as it does in this
instance, that the variations of an effect are correspondent, or
analogous, to those of its cause; as the moon moves farther towards the
east, the high water point does the same: but this is not an
indispensable condition; as may be seen in the same example, for along
with that high water point there is at the same instant another high
water point diametrically opposite to it, and which, therefore, of
necessity, moves towards the west, as the moon, followed by the nearer
of the tide waves, advances towards the east: and yet both these motions
are equally effects of the moon's motion.
That the oscillations of the pendulum are caused by the earth, is proved
by similar evidence. Those oscillations take place between equidistant
points on the two sides of a line, which, being perpendicular to the
earth, varies with every variation in the earth's position, either in
space or relatively to the object. Speaking accurately, we only know by
the method now characterized, that all terrestrial bodies tend to the
earth, and not to some unknown fixed point lying in the same direction.
In every twenty-four hours, by the earth's rotation, the line drawn from
the body at right angles to the earth coincides successively with all
the radii of a circle, and in the course of six months the place of that
circle varies by nearly two hundred millions of miles; yet in all these
changes of the earth's position, the line in which bodies tend to fall
continues to be directed towards it: which proves that terrestrial
gravity is directed to the earth, and not, as was once fancied by some,
to a fixed point of space.
The method by which these results were obtained, may be termed the
Method of Concomitant Variations: it is regulated by the following
canon:--
FIFTH CANON.
_Whatever phenomenon varies in any manner whenever another phenomenon
varies in some particular manner, is either a cause or an effect of that
phenomenon, or is connected with it through some fact of causation._
The last clause is subjoined, because it by no means follows when two
phenomena accompany each other in their variations, that the one is
cause and the other effect. The same thing may, and indeed must happen,
supposing them to be two different effects of a common cause: and by
this method alone it would never be possible to ascertain which of the
suppositions is the true one. The only way to solve the doubt would be
that which we have so often adverted to, viz. by endeavouring to
ascertain whether we can produce the one set of variations by means of
the other. In the case of heat, for example, by increasing the
temperature of a body we increase its bulk, but by increasing its bulk
we do not increase its temperature; on the contrary, (as in the
rarefaction of air under the receiver of an air-pump,) we generally
diminish it: therefore heat is not an effect, but a cause, of increase
of bulk. If we cannot ourselves produce the variations, we must
endeavour, though it is an attempt which is seldom successful, to find
them produced by nature in some case in which the pre-existing
circumstances are perfectly known to us.
It is scarcely necessary to say, that in order to ascertain the uniform
concomitance of variations in the effect with variations in the cause,
the same precautions must be used as in any other case of the
determination of an invariable sequence. We must endeavour to retain all
the other antecedents unchanged, while that particular one is subjected
to the requisite series of variations; or in other words, that we may be
warranted in inferring causation from concomitance of variations, the
concomitance itself must be proved by the Method of Difference.
It might at first appear that the Method of Concomitant Variations
assumes a new axiom, or law of causation in general, namely, that every
modification of the cause is followed by a change in the effect. And it
does usually happen that when a phenomenon A causes a phenomenon _a_,
any variation in the quantity or in the various relations of A, is
uniformly followed by a variation in the quantity or relations of _a_.
To take a familiar instance, that of gravitation. The sun causes a
certain tendency to motion in the earth; here we have cause and effect;
but that tendency is _towards_ the sun, and therefore varies in
direction as the sun varies in the relation of position; and moreover
the tendency varies in intensity, in a certain numerical correspondence
to the sun's distance from the earth, that is, according to another
relation of the sun. Thus we see that there is not only an invariable
connexion between the sun and the earth's gravitation, but that two of
the relations of the sun, its position with respect to the earth and its
distance from the earth, are invariably connected as antecedents with
the quantity and direction of the earth's gravitation. The cause of the
earth's gravitating at all, is simply the sun; but the cause of its
gravitating with a given intensity and in a given direction, is the
existence of the sun in a given direction and at a given distance. It is
not strange that a modified cause, which is in truth a different cause,
should produce a different effect.
Although it is for the most part true that a modification of the cause
is followed by a modification of the effect, the Method of Concomitant
Variations does not, however, presuppose this as an axiom. It only
requires the converse proposition; that anything on whose modifications,
modifications of an effect are invariably consequent, must be the cause
(or connected with the cause) of that effect; a proposition, the truth
of which is evident; for if the thing itself had no influence on the
effect, neither could the modifications of the thing have any influence.
If the stars have no power over the fortunes of mankind, it is implied
in the very terms, that the conjunctions or oppositions of different
stars can have no such power.
Although the most striking applications of the Method of Concomitant
Variations take place in the cases in which the Method of Difference,
strictly so called, is impossible, its use is not confined to those
cases; it may often usefully follow after the Method of Difference, to
give additional precision to a solution which that has found. When by
the Method of Difference it has first been ascertained that a certain
object produces a certain effect, the Method of Concomitant Variations
may be usefully called in, to determine according to what law the
quantity or the different relations of the effect follow those of the
cause.
§ 7. The case in which this method admits of the most extensive
employment, is that in which the variations of the cause are variations
of quantity. Of such variations we may in general affirm with safety,
that they will be attended not only with variations, but with similar
variations, of the effect: the proposition, that more of the cause is
followed by more of the effect, being a corollary from the principle of
the Composition of Causes, which, as we have seen, is the general rule
of causation; cases of the opposite description, in which causes change
their properties on being conjoined with one another, being, on the
contrary, special and exceptional. Suppose, then, that when A changes
in quantity, _a_ also changes in quantity, and in such a manner that we
can trace the numerical relation which the changes of the one bear to
such changes of the other as take place within our limits of
observation. We may then, with certain precautions, safely conclude that
the same numerical relation will hold beyond those limits. If, for
instance, we find that when A is double, _a_ is double; that when A is
treble or quadruple, _a_ is treble or quadruple; we may conclude that if
A were a half or a third, _a_ would be a half or a third, and finally,
that if A were annihilated, _a_ would be annihilated, and that _a_ is
wholly the effect of A, or wholly the effect of the same cause with A.
And so with any other numerical relation according to which A and _a_
would vanish simultaneously; as for instance, if _a_ were proportional
to the square of A. If, on the other hand, _a_ is not wholly the effect
of A, but yet varies when A varies, it is probably a mathematical
function not of A alone, but of A and something else: its changes, for
example, may be such as would occur if part of it remained constant, or
varied on some other principle, and the remainder varied in some
numerical relation to the variations of A. In that case, when A
diminishes, _a_ will be seen to approach not towards zero, but towards
some other limit: and when the series of variations is such as to
indicate what that limit is, if constant, or the law of its variation if
variable, the limit will exactly measure how much of _a_ is the effect
of some other and independent cause, and the remainder will be the
effect of A (or of the cause of A).
These conclusions, however, must not be drawn without certain
precautions. In the first place, the possibility of drawing them at all,
manifestly supposes that we are acquainted not only with the variations,
but with the absolute quantities both of A and _a_. If we do not know
the total quantities, we cannot, of course, determine the real numerical
relation according to which those quantities vary. It is therefore an
error to conclude, as some have concluded, that because increase of heat
expands bodies, that is, increases the distance between their particles,
therefore the distance is wholly the effect of heat, and that if we
could entirely exhaust the body of its heat, the particles would be in
complete contact. This is no more than a guess, and of the most
hazardous sort, not a legitimate induction: for since we neither know
how much heat there is in any body, nor what is the real distance
between any two of its particles, we cannot judge whether the
contraction of the distance does or does not follow the diminution of
the quantity of heat according to such a numerical relation that the two
quantities would vanish simultaneously.
In contrast with this, let us consider a case in which the absolute
quantities are known; the case contemplated in the first law of motion;
viz. that all bodies in motion continue to move in a straight line with
uniform velocity until acted upon by some new force. This assertion is
in open opposition to first appearances; all terrestrial objects, when
in motion, gradually abate their velocity and at last stop; which
accordingly the ancients, with their _inductio per enumerationem
simplicem_, imagined to be the law. Every moving body, however,
encounters various obstacles, as friction, the resistance of the
atmosphere, &c., which we know by daily experience to be causes capable
of destroying motion. It was suggested that the whole of the retardation
might be owing to these causes. How was this inquired into? If the
obstacles could have been entirely removed, the case would have been
amenable to the Method of Difference. They could not be removed, they
could only be diminished, and the case, therefore, admitted only of the
Method of Concomitant Variations. This accordingly being employed, it
was found that every diminution of the obstacles diminished the
retardation of the motion: and inasmuch as in this case (unlike the case
of heat) the total quantities both of the antecedent and of the
consequent were known; it was practicable to estimate, with an approach
to accuracy, both the amount of the retardation and the amount of the
retarding causes, or resistances, and to judge how near they both were
to being exhausted; and it appeared that the effect dwindled as rapidly,
and at each step was as far on the road towards annihilation, as the
cause was. The simple oscillation of a weight suspended from a fixed
point, and moved a little out of the perpendicular, which in ordinary
circumstances lasts but a few minutes, was prolonged in Borda's
experiments to more than thirty hours, by diminishing as much as
possible the friction at the point of suspension, and by making the body
oscillate in a space exhausted as nearly as possible of its air. There
could therefore be no hesitation in assigning the whole of the
retardation of motion to the influence of the obstacles; and since,
after subducting this retardation from the total phenomenon, the
remainder was an uniform velocity, the result was the proposition known
as the first law of motion.
There is also another characteristic uncertainty affecting the inference
that the law of variation which the quantities observe within our limits
of observation, will hold beyond those limits. There is of course, in
the first instance, the possibility that beyond the limits, and in
circumstances therefore of which we have no direct experience, some
counteracting cause might develop itself; either a new agent, or a new
property of the agents concerned, which lies dormant in the
circumstances we are able to observe. This is an element of uncertainty
which enters largely into all our predictions of effects; but it is not
peculiarly applicable to the Method of Concomitant Variations. The
uncertainty, however, of which I am about to speak, is characteristic of
that method; especially in the cases in which the extreme limits of our
observation are very narrow, in comparison with the possible variations
in the quantities of the phenomena. Any one who has the slightest
acquaintance with mathematics, is aware that very different laws of
variation may produce numerical results which differ but slightly from
one another within narrow limits; and it is often only when the absolute
amounts of variation are considerable, that the difference between the
results given by one law and by another becomes appreciable. When,
therefore, such variations in the quantity of the antecedents as we have
the means of observing, are small in comparison with the total
quantities, there is much danger lest we should mistake the numerical
law, and be led to miscalculate the variations which would take place
beyond the limits; a miscalculation which would vitiate any conclusion
respecting the dependence of the effect upon the cause, that could be
founded on those variations. Examples are not wanting of such mistakes.
"The formulæ," says Sir John Herschel,[33] "which have been empirically
deduced for the elasticity of steam, (till very recently,) and those for
the resistance of fluids, and other similar subjects," when relied on
beyond the limits of the observations from which they were deduced,
"have almost invariably failed to support the theoretical structures
which have been erected on them."
In this uncertainty, the conclusion we may draw from the concomitant
variations of _a_ and A, to the existence of an invariable and exclusive
connexion between them, or to the permanency of the same numerical
relation between their variations when the quantities are much greater
or smaller than those which we have had the means of observing, cannot
be considered to rest on a complete induction. All that in such a case
can be regarded as proved on the subject of causation is, that there is
some connexion between the two phenomena; that A, or something which can
influence A, must be _one_ of the causes which collectively determine
_a_. We may, however, feel assured that the relation which we have
observed to exist between the variations of A and _a_, will hold true in
all cases which fall between the same extreme limits; that is, wherever
the utmost increase or diminution in which the result has been found by
observation to coincide with the law, is not exceeded.
The four methods which it has now been attempted to describe, are the
only possible modes of experimental inquiry--of direct induction _à
posteriori_, as distinguished from deduction: at least, I know not, nor
am able to imagine, any others. And even of these, the Method of
Residues, as we have seen, is not independent of deduction; though, as
it also requires specific experience, it may, without impropriety, be
included among methods of direct observation and experiment.
These, then, with such assistance as can be obtained from Deduction,
compose the available resources of the human mind for ascertaining the
laws of the succession of phenomena. Before proceeding to point out
certain circumstances, by which the employment of these methods is
subjected to an immense increase of complication and of difficulty, it
is expedient to illustrate the use of the methods, by suitable examples
drawn from actual physical investigations. These, accordingly, will form
the subject of the succeeding chapter.