University of Virginia Library

Chapter 5
The Continuity of Motion

§55. Like the Indestructibility of Matter, the Continuity of Motion, or, more strictly, of that something which has Motion for one of its sensible forms, is a truth on which depends the possibility of exact Science, and therefore of a Philosophy which unifies the results of exact Science. Motions, visible and invisible, of masses and of molecules, form the larger half of the phenomena to be interpreted; and if such motions might either proceed from nothing or lapse into nothing, there could be no scientific interpretation of them.

This second fundamental truth, like the first, is not self-evident to primitive men nor to the uncultured among ourselves. Contrariwise, to uninstructed minds the opposite seems self-evident. The facts that a stone thrown up soon loses its ascending motion, and that after the blow its fall gives to the Earth, it remains quiescent, apparently prove that the principle of activity* which the stone manifested may disappear absolutely. Accepting the dicta of unaided perception, all men once believed, and most believe still, that motion can pass into nothing, and ordinarily does so pass. But the establishment of certain facts having opposite implications, led to inquiries which have proved these appearances to be illusive. The discovery that the celestial motions do not diminish, raised the suspicion that a moving body, when not interfered with, will go on for ever without change of velocity; and suggested the question whether bodies which lose their motion, do not at the same time communicate as much motion to other bodies. It was a familiar fact that a stone would glide further over a smooth surface, as that of ice, presenting no small objects to which it could part with its motion by collision, than over a surface strewn with such small objects; and that a stick hurled into the air would travel a far greater distance than if hurled into a dense medium like water. Thus the primitive notion that moving bodies have an inherent tendency to stop — a notion which the Greeks did not get rid of, and which lasted till the time of Galileo — began to give way. It was further shaken by such experiments as those of Hooke, which proved that a top spins the longer in proportion as it is prevented from communicating motion to surrounding matter.

To explain here all disappearances of visible motions is out of the question. It must suffice to state, generally, that the molar motion which disappears when a bell is struck by its clapper, re-appears in the bell's vibrations and in the waves of air they produce; that when a moving mass is stopped by coming against a mass that is immovable, the motion which does not show itself in sound shows itself in molecular motion; and that when bodies rub against one another, the motion lost by friction is gained in the motion of molecules. But one aspect of this general truth, as it is displayed in the motions of masses, we must carefully contemplate; for, otherwise, the doctrine of the Continuity of Motion will be misapprehended.

§56. As expressed by Newton, the first law of motion is that "every body must persevere in its state of rest, Or of uniform motion in a straight line, unless it be compelled to change that state by forces impressed upon it."

With this truth may be associated the truth that a body describing a circular orbit round a centre which detains it by a tractive force, moves in that orbit with undiminished velocity.

The first of these abstract truths is never realized in the concrete, and the second of them is but approximately realized. Uniform motion in a straight line implies the absence of a resisting medium; and it further implies the absence of forces, gravitative or other exercised by neighbouring masses: conditions never fulfilled. So, too, the maintenance of a circular orbit by any celestial body, implies that there are no perturbing bodies, and that there is an exact adjustment between its velocity and the tractive force of its primary: neither requirement ever being conformed to. In actual orbits, sensibly elliptical as they are, the velocity is sensibly variable. And along with great eccentricity there goes great variation.

With the case of these celestial bodies which, moving in eccentric orbits, display at one time little motion and at another much motion, may be associated as partially analogous the case of the pendulum. With speed now increasing and now decreasing, the pendulum alternates between extremes at which motion ceases.

How shall we so conceive these allied phenomena as to express rightly the truth common to them? The first law of motion, nowhere literally fulfilled, is yet, in a sense, implied by these facts which seem at variance with it. Though in a circular orbit the direction of the motion is continually being changed, yet the velocity remains unchanged. Though in an elliptical orbit there is now acceleration and now retardation, yet the average speed is constant through successive revolutions. Though the pendulum comes to a momentary rest at the end of each swing, and then begins a reverse motion, yet the oscillation, considered as a whole, is continuous: friction and atmospheric resistance being absent, this alternation of states would go on for ever.

What, then, do these cases show us in common? That which vision familiarizes us with in Motion, and that which has thus been made the dominant element in our conception of Motion, is not the element of which we can allege continuity. If we regard Motion simply as change of place, then the pendulum shows us both that the rate of this change may vary from instant to instant, and that, ceasing at intervals, it may be afresh initiated.

But if what we may call the translation-element in Motion is not continuous, what is continuous? If, like Galileo, we watch a swinging chandelier, and observe, not the isochronism of its oscillations but the recurring reversal of direction, we are impressed with the fact that though, at the end of each swing, the translation through space ceases, yet there is something which does not cease; for the translation recommences in the opposite direction. And on remembering that when a violent push was given to the chandelier it described a larger arc, and was a longer time before the resistance of the air brought it to rest, we are shown that what continues to exist during its alternating movements is some correlative of the muscular effort which put it in motion. The truth forced on our attention is that translation through space is not itself an existence; and that hence the cessation of Motion, considered simply as translation, is not the cessation of an existence, but is the cessation of a certain sign of an existence.

Still there remains a difficulty If that element in the chandelier's motion of which alone we can allege continuity is the correlative of the muscular effort which moved, the chandelier, what becomes of this element at either extreme of the oscillation? Arrest the chandelier in the middle of its swing, and it gives a blow to the hand-exhibits some principle of activity such as muscular effort can give. But touch it at either turning point and it displays no such principle of activity. This has disappeared just as much as the translation through space has disappeared. How then, can it be alleged that though the Motion through space is not continuous, the principle of activity implied by the Motion is continuous?

Unquestionably the facts show that the principle of activity continues to exist under some form. When not perceptible it must be latent. How is it latent? A clue to the answer is gained on observing that though the chandelier when seized at the turning point of its swing, gives no impact in the direction of its late movement, it forthwith begins to pull in the opposite direction; and on observing, further, that its pull is great when the swing has been made extensive by a violent push. Hence the loss of visible activity at the highest point of the upward motion, is accompanied by the production of an invisible activity which generates the subsequent motion downwards. To conceive this latent activity gained, as an existence equal to the perceptible activity lost, is not easy; but we may help ourselves so to conceive it by considering cases of another class.

§57. When one who pushes against a door that has stuck fast, produces by great effort no motion, but eventually by a little greater effort bursts the door open, swinging it back and tumbling headlong into the room, he has evidence that the first muscular strain which did not produce transfer of matter through space, was yet equivalent to a certain amount of such transfer. Again, when a railway-porter gradually stops a detached carriage by pulling at the buffer, he shows us that (supposing friction, etc., absent) the slowly-diminished motion of the carriage over a certain space, is the equivalent of the constant backward strain put upon the carriage while it is traveling through that space. Carrying with us the conception thus reached, we will now consider a case which makes it more definite.

When used as a plaything, a ball fastened to the end of an india-rubber string yields a clear idea of the correlation between perceptible activity and latent activity. If, retaining one end of the string, a boy throws the ball from him horizontally, its motion is resisted by the increasing strain on the string; and the string, stretched more and more as the ball recedes, presently brings it to rest. Where now exists the principle of activity which the moving ball displayed? It exists in the strained thread of india-rubber. Under what form of changed molecular state it exists we need not ask. It suffices that the string is the seat of a tension generated by the motion of the ball, and equivalent to it. When the ball has been arrested the stretched string begins to generate in it an opposite motion, and continues to accelerate that motion until the ball comes back to the point at which the stretching of the string commenced — a point at which, but for loss by atmospheric resistance and molecular redistribution, its velocity would be equal to the original velocity. Here the truth that the principle of velocity, alternating between visible and invisible modes, does not cease to exist when the translation through space ceases to exist, is readily comprehensible; and it becomes easy to understand the corollary that at each point in the path of the ball, the quantity of its perceptible activity, plus the quantity which is latent in the stretched string, yields a constant sum.

Aided by this illustration, we can vaguely conceive what happens between bodies connected, not by a stretched string, but by a traction exercised by an invisible agency. It matters not to our general conception that the intensity of this traction varies in a different manner: decreasing as the square of the distance increases, but being practically constant for terrestrial distances. Notwithstanding these differences there is a truth common to the two cases. The weight of something held in the hand shows that between one body in space and another there exists a strain. This downward pull affects the hand as it might be affected by a stretched elastic string. Hence, when a body projected upwards and gradually retarded by gravity, finally stops, we must regard the principle of activity manifested as having become latent in the strain between it and the Earth — a strain of which the quantity is to be conceived as the product of its intensity and the distance through which it acts. Carrying a step further our illustration of the stretched string, will elucidate this. To simulate the action of gravity at terrestrial distances, let us imagine that when the attached moving body has stretched the elastic string to its limit, say at the distance of ten feet (from which point it is prevented from contracting back), a second like string could instantly be tied to the end of the first and to the body, which continuing its course stretched this second string, and so on with a succession of such strings, till the body was arrested. Then, obviously, the quantity of the principle of activity which the moving body possessed, but which has now become latent in the stretched strings, is measured by the number of such strings over which the strain extends. Now though the tractive force of the Earth is not exercised in a like way — though gravity, utterly unknown in its nature, is probably a resultant of actions pervading the ethereal medium; yet the above analogy suggests the belief that the principle of activity exhibited by a stone thrown up and presently arrested, has not ceased to exist, but has become so much imperceptible or latent activity in the medium occupying space; and that when the stone falls, this is re-transformed into its equivalent of perceptible activity. If we conceive the process at all, we must conceive it thus: otherwise, we have to conceive that a power has been changed into a space-relation, and this is inconceivable.

Here, then, is the solution of the difficulty The space-element of Motion is not in itself a thing. Change of position is not an existence, but the manifestation of an existence. This existence (supposing it not transferred by collision or friction) may cease to display itself as translation; but it can do so only by displaying itself as strain. And this principle of activity now shown by translation, now by strain, and often by the two together, is alone that which in Motion we can call continuous.

§58. What is this principle of activity? Vision gives us no idea of it. If by a mirror we cast the image of an illuminated object on to a dark wall, and then suddenly changing the attitude of the mirror make the reflected image pass from side to side, no thought arises that there is present in the image a principle of activity. Before we can conceive the presence of this, we must regard the visual impression as symbolizing something tangible. Sight of a moving body suggests a principle of activity which would be appreciable by skin and muscles were the body laid hold of. This principle of activity which Motion shows us, is the objective sense of effort. By pushing and pulling we get feelings which, generalized and abstracted, yield our ideas of resistance and tension. Now displayed by changing position and now by unchanging strain, this principle of activity is ultimately conceived by us under the single form of its equivalent muscular effort. So that the continuity of Motion, as well as the indestructibility of Matter, is really known to us in terms of Force. Here, however, the Force is of the kind known as Energy — a word applied to the force, molar or molecular, possessed by matter in action, as distinguished from the passive force by which matter maintains its shape and occupies space: a force which physicists appear to think needs no name.

§59. And now we reach the truth to be here especially noted. All proofs of the Continuity of Motion involve the postulate that the quantity of Energy is constant. Observe what results when we analyze the reasonings by which the Continuity of Motion is shown.

A particular planet is identified by its constant power to affect our eyes in a special way. Further, such planet has not been seen to move by the astronomer; but its motion is inferred from a comparison of its present position with the position it before occupied. This comparison proves to be a comparison between the different impressions produced on him by the different adjustments of his observing instruments. And the validity of the inferences. drawn depends on the truth of the assumption that these masses of matter, celestial and terrestrial, continue to affect his senses in the same ways under the same conditions. On going a step further back, it turns out that difference in the adjustment of his observing instrument, and by implication in the planet's position, is meaningless until shown to correspond with a certain calculated position which the planet must occupy, supposing that no motion has been lost. And if, finally, we examine the implied calculation, we find that it takes into account those accelerations and retardations which ellipticity of the orbit involves, as well as those variations of motion caused by adjacent planets — we find, that is, that the motion is concluded to be indestructible not from the uniform velocity of the planet, but from the constant quantity of motion exhibited after allowances have been made for the motions communicated to, or received from, other celestial bodies. And when we ask how this is estimated, we discover that the estimate assumes certain laws of force or energy; which laws, one and all, embody the postulate that energy cannot be destroyed.

Similarly with the à priori conclusion that Motion is continuous. That which defies suppression in thought (disciplined thought, of course), is the force which the motion indicates. We can imagine retardation to result from the actions of other bodies. But to imagine this we must imagine loss of some of the energy implied by the motion. We are obliged to conceive this energy as impressed in the shape of reaction on the bodies causing the retardation. And the motion communicated to them, we are compelled to regard as a product of the communicated energy. We can mentally diminish the velocity or space-element of motion, by diffusing the momentum or force-element over a larger mass of matter; but the quantity of this force-element is unchangeable in thought.*

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