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 forever unless some cause intervened having a
tendency to alter or destroy them. Such, for example, are all the facts of
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 forever 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
forever 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 need 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 recognized 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 a 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 can not any longer be produced. Again, the earth causes bodies to
fall toward it; that is, the existence of the earth at a given instant
causes an unsupported body to move toward 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 farther 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 continued 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 a 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 a 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 forever 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, and which if
that something else had preceded, would not have been required, is not the
cause, however invaluable 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 a 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, etc.; 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.