OF PROGRESSIVE EFFECTS; AND OF THE CONTINUED ACTION OF CAUSES.
§ 1. In the last four chapters we have traced the general outlines of
the theory of the generation of derivative laws from ultimate ones. In
the present chapter our attention will be directed to a particular case
of the derivation of laws from other laws, but a case so general, and so
important, as not only to repay, but to require, a separate examination.
This is, the case of a complex phenomenon resulting from one simple law,
by the continual addition of an effect to itself.
There are some phenomena, some bodily sensations for example, which are
essentially instantaneous, and whose existence can only be prolonged by
the prolongation of the existence of the cause by which they are
produced. But most phenomena are in their own nature permanent; having
begun to exist, they would exist for ever unless some cause intervened
having a tendency to alter or destroy them. Such, for example, are all
the facts or phenomena which we call bodies. Water, once produced, will
not of itself relapse into a state of hydrogen and oxygen; such a change
requires some agent having the power of decomposing the compound. Such,
again, are the positions in space, and the movements, of bodies. No
object at rest alters its position without the intervention of some
conditions extraneous to itself; and when once in motion, no object
returns to a state of rest, or alters either its direction or its
velocity, unless some new external conditions are superinduced. It,
therefore, perpetually happens that a temporary cause gives rise to a
permanent effect. The contact of iron with moist air for a few hours,
produces a rust which may endure for centuries; or a projectile force
which launches a cannon ball into space, produces a motion which would
continue for ever unless some other force counteracted it.
Between the two examples which we have here given, there is a difference
worth pointing out. In the former (in which the phenomenon produced is a
substance, and not a motion of a substance), since the rust remains for
ever and unaltered unless some new cause supervenes, we may speak of the
contact of air a hundred years ago as even the proximate cause of the
rust which has existed from that time until now. But when the effect is
motion, which is itself a change, we must use a different language. The
permanency of the effect is now only the permanency of a series of
changes. The second foot, or inch, or mile of motion, is not the mere
prolonged duration of the first foot, or inch, or mile, but another fact
which succeeds, and which may in some respects be very unlike the
former, since it carries the body through a different region of space.
Now, the original projectile force which set the body moving is the
remote cause of all its motion, however long continued, but the
proximate cause of no motion except that which took place at the first
instant. The motion at any subsequent instant is proximately caused by
the motion which took place at the instant preceding. It is on that, and
not on the original moving cause, that the motion at any given moment
depends. For, suppose that the body passes through some resisting
medium, which partially counteracts the effect of the original impulse,
and retards the motion: this counteraction (it needs scarcely here be
repeated) is as strict an example of obedience to the law of the
impulse, as if the body had gone on moving with its original velocity;
but the motion which results is different, being now a compound of the
effects of two causes acting in contrary directions, instead of the
single effect of one cause. Now, what cause does the body obey in its
subsequent motion? The original cause of motion, or the actual motion at
the preceding instant? The latter: for when the object issues from the
resisting medium, it continues moving, not with its original, but with
its retarded velocity. The motion having once been diminished, all that
which follows is diminished. The effect changes, because the cause which
it really obeys, the proximate cause, the real cause in fact, has
changed. This principle is recognised by mathematicians when they
enumerate among the causes by which the motion of a body is at any
instant determined, the _force generated_ by the previous motion; an
expression which would be absurd if taken to imply that this "force" was
an intermediate link between the cause and the effect, but which really
means only the previous motion itself, considered as a cause of further
motion. We must, therefore, if we would speak with perfect precision,
consider each link in the succession of motions as the effect of the
link preceding it. But if, for the convenience of discourse, we speak of
the whole series as one effect, it must be as an effect produced by the
original impelling force; a permanent effect produced by an
instantaneous cause, and possessing the property of self-perpetuation.
Let us now suppose that the original agent or cause, instead of being
instantaneous, is permanent. Whatever effect has been produced up to a
given time, would (unless prevented by the intervention of some new
cause) subsist permanently, even if the cause were to perish. Since,
however, the cause does not perish, but continues to exist and to
operate, it must go on producing more and more of the effect; and
instead of an uniform effect, we have a progressive series of effects,
arising from the accumulated influence of a permanent cause. Thus, the
contact of iron with the atmosphere causes a portion of it to rust; and
if the cause ceased, the effect already produced would be permanent, but
no further effect would be added. If, however, the cause, namely,
exposure to moist air, continues, more and more of the iron becomes
rusted, until all which is exposed is converted into a red powder, when
one of the conditions of the production of rust, namely, the presence of
unoxidized iron, has ceased, and the effect cannot any longer be
produced. Again, the earth causes bodies to fall towards it, that is,
the existence of the earth at a given instant, causes an unsupported
body to move towards it at the succeeding instant: and if the earth were
annihilated, as much of the effect as is already produced would
continue; the object would go on moving in the same direction, with its
acquired velocity, until intercepted by some body or deflected by some
other force. The earth, however, not being annihilated, goes on
producing in the second instant an effect similar and of equal amount
with the first, which two effects being added together, there results an
accelerated velocity; and this operation being repeated at each
successive instant, the mere permanence of the cause, though without
increase, gives rise to a constant progressive increase of the effect,
so long as all the conditions, negative and positive, of the production
of that effect, continue to be realized.
It is obvious that this state of things is merely a case of the
Composition of Causes. A cause which continues in action, must on a
strict analysis be considered as a number of causes exactly similar,
successively introduced, and producing by their combination the sum of
the effects which they would severally produce if they acted singly. The
progressive rusting of the iron is in strictness the sum of the effects
of many particles of air acting in succession upon corresponding
particles of iron. The continued action of the earth upon a falling body
is equivalent to a series of forces, applied in successive instants,
each tending to produce a certain constant quantity of motion; and the
motion at each instant is the sum of the effects of the new force
applied at the preceding instant, and the motion already acquired. In
each instant, a fresh effect, of which gravity is the proximate cause,
is added to the effect of which it was the remote cause: or (to express
the same thing in another manner) the effect produced by the earth's
influence at the instant last elapsed, is added to the sum of the
effects of which the remote causes were the influences exerted by the
earth at all the previous instants since the motion began. The case,
therefore, comes under the principle of a concurrence of causes
producing an effect equal to the sum of their separate effects. But as
the causes come into play not all at once, but successively, and as the
effect at each instant is the sum of the effects of those causes only
which have come into action up to that instant, the result assumes the
form of an ascending series; a succession of sums, each greater than
that which preceded it; and we have thus a progressive effect from the
continued action of a cause.
Since the continuance of the cause influences the effect only by adding
to its quantity, and since the addition takes place according to a fixed
law (equal quantities in equal times), the result is capable of being
computed on mathematical principles. In fact, this case, being that of
infinitesimal increments, is precisely the case which the differential
calculus was invented to meet. The questions, what effect will result
from the continual addition of a given cause to itself, and what amount
of the cause, being continually added to itself, will produce a given
amount of the effect, are evidently mathematical questions, and to be
treated, therefore, deductively. If, as we have seen, cases of the
Composition of Causes are seldom adapted for any other than deductive
investigation, this is especially true in the case now examined, the
continual composition of a cause with its own previous effects; since
such a case is peculiarly amenable to the deductive method, while the
undistinguishable manner in which the effects are blended with one
another and with the causes, must make the treatment of such an instance
experimentally, still more chimerical than in any other case.
§ 2. We shall next advert to a rather more intricate operation of the
same principle, namely, when the cause does not merely continue in
action, but undergoes, during the same time, a progressive change in
those of its circumstances which contribute to determine the effect. In
this case, as in the former, the total effect goes on accumulating by
the continual addition of a fresh effect to that already produced, but
it is no longer by the addition of equal quantities in equal times; the
quantities added are unequal, and even the quality may now be different.
If the change in the state of the permanent cause be progressive, the
effect will go through a double series of changes, arising partly from
the accumulated action of the cause, and partly from the changes in its
action. The effect is still a progressive effect, produced however, not
by the mere continuance of a cause, but by its continuance and its
progressiveness combined.
A familiar example is afforded by the increase of the temperature as
summer advances, that is, as the sun draws nearer to a vertical
position, and remains a greater number of hours above the horizon. This
instance exemplifies in a very interesting manner the twofold operation
on the effect, arising from the continuance of the cause, and from its
progressive change. When once the sun has come near enough to the
zenith, and remains above the horizon long enough, to give more warmth
during one diurnal rotation than the counteracting cause, the earth's
radiation, can carry off, the mere continuance of the cause would
progressively increase the effect, even if the sun came no nearer and
the days grew no longer; but in addition to this, a change takes place
in the accidents of the cause (its series of diurnal positions), tending
to increase the quantity of the effect. When the summer solstice has
passed, the progressive change in the cause begins to take place the
reverse way; but, for some time, the accumulating effect of the mere
continuance of the cause exceeds the effect of the changes in it, and
the temperature continues to increase.
Again, the motion of a planet is a progressive effect, produced by
causes at once permanent and progressive. The orbit of a planet is
determined (omitting perturbations) by two causes: first, the action of
the central body, a permanent cause, which alternately increases and
diminishes as the planet draws nearer to or goes further from its
perihelion, and which acts at every point in a different direction; and,
secondly, the tendency of the planet to continue moving in the direction
and with the velocity which it has already acquired. This force also
grows greater as the planet draws nearer to its perihelion, because as
it does so its velocity increases; and less, as it recedes from its
perihelion: and this force as well as the other acts at each point in a
different direction, because at every point the action of the central
force, by deflecting the planet from its previous direction, alters the
line in which it tends to continue moving. The motion at each instant is
determined by the amount and direction of the motion, and the amount
and direction of the sun's action, at the previous instant: and if we
speak of the entire revolution of the planet as one phenomenon (which,
as it is periodical and similar to itself, we often find it convenient
to do,) that phenomenon is the progressive effect of two permanent and
progressive causes, the central force and the acquired motion. Those
causes happening to be progressive in the particular way which is called
periodical, the effect necessarily is so too; because the quantities to
be added together returning in a regular order, the same sums must also
regularly return.
This example is worthy of consideration also in another respect. Though
the causes themselves are permanent, and independent of all conditions
known to us, the changes which take place in the quantities and
relations of the causes are actually caused by the periodical changes in
the effects. The causes, as they exist at any moment, having produced a
certain motion, that motion, becoming itself a cause, reacts upon the
causes, and produces a change in them. By altering the distance and
direction of the central body relatively to the planet, and the
direction and quantity of the force in the direction of the tangent, it
alters the elements which determine the motion at the next succeeding
instant. This change renders the next motion somewhat different; and
this difference, by a fresh reaction upon the causes, renders the next
motion again different, and so on. The original state of the causes
might have been such, that this series of actions modified by reactions
would not have been periodical. The sun's action, and the original
impelling force, might have been in such a ratio to one another, that
the reaction of the effect would have been such as to alter the causes
more and more, without ever bringing them back to what they were at any
former time. The planet would then have moved in a parabola, or an
hyperbola, curves not returning into themselves. The quantities of the
two forces were, however, originally such, that the successive reactions
of the effect bring back the causes, after a certain time, to what they
were before; and from that time all the variations continue to recur
again and again in the same periodical order, and must so continue
while the causes subsist and are not counteracted.
§ 3. In all cases of progressive effects, whether arising from the
accumulation of unchanging or of changing elements, there is an
uniformity of succession not merely between the cause and the effect,
but between the first stages of the effect and its subsequent stages.
That a body _in vacuo_ falls sixteen feet in the first second,
forty-eight in the second, and so on in the ratio of the odd numbers, is
as much an uniform sequence as that when the supports are removed the
body falls. The sequence of spring and summer is as regular and
invariable as that of the approach of the sun and spring: but we do not
consider spring to be the cause of summer; it is evident that both are
successive effects of the heat received from the sun, and that,
considered merely in itself, spring might continue for ever, without
having the slightest tendency to produce summer. As we have so often
remarked, not the conditional, but the unconditional invariable
antecedent is termed the cause. That which would not be followed by the
effect unless something else had preceded, is not the cause, however
invariable the sequence may in fact be.
It is in this way that most of those uniformities of succession are
generated, which are not cases of causation. When a phenomenon goes on
increasing, or periodically increases and diminishes, or goes through
any continued and unceasing process of variation reducible to an uniform
rule or law of succession, we do not on this account presume that any
two successive terms of the series are cause and effect. We presume the
contrary; we expect to find that the whole series originates either from
the continued action of fixed causes, or from causes which go through a
corresponding process of continuous change. A tree grows from half an
inch high to a hundred feet; and some trees will generally grow to that
height, unless prevented by some counteracting cause. But we do not call
the seedling the cause of the full-grown tree; the invariable antecedent
it certainly is, and we know very imperfectly on what other antecedents
the sequence is contingent, but we are convinced that it is contingent
on something; because the homogeneousness of the antecedent with the
consequent, the close resemblance of the seedling to the tree in all
respects except magnitude, and the graduality of the growth, so exactly
resembling the progressively accumulating effect produced by the long
action of some one cause, leave no possibility of doubting that the
seedling and the tree are two terms in a series of that description, the
first term of which is yet to seek. The conclusion is further confirmed
by this, that we are able to prove by strict induction the dependence of
the growth of the tree, and even of the continuance of its existence,
upon the continued repetition of certain processes of nutrition, the
rise of the sap, the absorptions and exhalations by the leaves, &c.; and
the same experiments would probably prove to us that the growth of the
tree is the accumulated sum of the effects of these continued processes,
were we not, for want of sufficiently microscopic eyes, unable to
observe correctly and in detail what those effects are.
This supposition by no means requires that the effect should not, during
its progress, undergo many modifications besides those of quantity, or
that it should not sometimes appear to undergo a very marked change of
character. This may be either because the unknown cause consists of
several component elements or agents, whose effects, accumulating
according to different laws, are compounded in different proportions at
different periods in the existence of the organized being; or because,
at certain points in its progress, fresh causes or agencies come in, or
are evolved, which intermix their laws with those of the prime agent.