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_ can not 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 can not 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, can not 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 further, 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 can not 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 analyze 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 principal 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 endeavored 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
gunshot which killed him: for he was in the fullness of life immediately
before, all circumstances being the same, except the wound.
The axioms implied in this method are evidently the following. Whatever
antecedent can not 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 can not 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 co-existent
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. Comté 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 any
thing, 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
any thing, 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 can
not 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 can not
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 connection
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
connection between A and _a_. In order to convert this evidence of
connection 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
connection 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 can not 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 can not be prevented from co-existing. In
respect, indeed, to most of the permanent causes, no such difficulty
exists; since, though we can not eliminate them as co-existing 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 every thing 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 can not be removed from the influence of the earth: we can not 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
co-existent 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 can not effect
such a variation of circumstances as the foregoing three methods require;
we can not 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 can not 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 _vicè versa_, 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 can not 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 can not 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 toward 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 toward the west, as the moon, followed by
the nearer of the tide waves, advances toward 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 toward 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 endeavoring 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 can not ourselves
produce the variations, we must endeavor, though it is an attempt which is
seldom successful, to find them produced by nature in some case in which
the preexisting 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 endeavor 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 _toward_ 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 connection 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 any thing 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 relations to the variations of A. In that case,
when A diminishes, _a_ will be seen to approach not toward zero, but
toward 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 can not, 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 can not 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, etc.,
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 toward 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 a 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,(136) “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
connection 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, can not 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
connection 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 _a 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.