CHAPTER V

THE LAWS OF MECHANICS

    WE have now seen that there is some reason for believing and no reason for disbelieving that there exist real counterparts to the figures which all people agree in perceiving by touch. We have also seen that, when this assumption is made, it is possible to find laws between tactually and visually perceived figures and the visually perceived distances and positions of the latter relative to the body of the observer, and so to reason from a visually perceived figure of a given shape to the real counterpart of a tactual figure of definite shape events in which do cause the visual perception that we have, and which, if we performed the proper actions, would give us the corresponding tactual perception. We are thus justified in talking about geometrical relations, shapes, and distances as existing even when we do not perceive them, it being understood that we mean the real counterpart to which there would correspond a tactually perceived figure with the geometrical qualities in question if as a matter of fact we did have such a perception. Thus such a statement as: 'I cannot perceive an atom but I believe that atoms are shaped like dumb-bells' means that I believe that there exist in the real counterpart realities qualitatively like those, events in which cause visual perception of dumb-bellshaped figures, but the real quantity that corresponds to the sizes of lines and volumes and surfaces in perceived objects is so small that no perception is actually produced.

    Now the laws of mechanics are laws that are found to enable us to connect the spatial positions of bodies at various times. In particular the so-called Laws of Motion are conditions which, it is believed, all material bodies obey, no matter what other information may be needed for a complete determination of their state. It follows that the laws of mechanics are well worth discussion. In the first place, if what has been said before be correct they are laws about reality or can become so by a simple change of statement like the one I have given above about the shape of an atom. In the second place, they give rise to much the most certain and important physical science that exists, and so they are good examples of causal laws at their best. In the third place, they raise a number of interesting questions, and their obscurity compared with the certainty of the structure that has been built upon them is yet another example of the truth mentioned by Mr Russell that in most sciences it is the middle parts that are simple and evident, and the foundations and the higher applications that are obscure and difficult.

    I think that our best plan will be to state the three laws in a form similar to Newton's if not absolutely identical with it, and then to consider the notions involved in them and any problems that may arise with regard to them and the kind of evidence for them. Our discussions may of course necessitate a different statement at the end from what we had at the beginning.

    The laws of motion as usually stated are

1. Every body persists in a state of rest or of uniform motion in a straight line unless acted upon by some external force.
2. The measure of the force acting upon a body at any moment in a given direction is the rate of change of momentum of the body in that direction at that moment.
3. For any pair of bodies A and B which exert forces on each other the forces are equal in magnitude, have the same line of action, and are opposite in sense.

    The first law has been by some people supposed to be a barren tautology, and by others to be an a priori truth. But the most serious question that must necessarily precede any decision on these points is: What precisely is meant by the motion of a body in a straight line, and by its uniformity? Until you are clear about this you obviously cannot be sure whether that of which you are certain a priori is the first law of motion, because you do not know what is meant by the first law of motion.

    Everyone knows that the paths in which bodies appear to travel depend, as we put it, on whether the observer is at rest or in motion on his own account. And the velocity which we attribute to the body will depend on the body relatively to which the distance travelled is measured and on the measurement of time adopted.

    Hence, if you take all motion to be relative, you cannot talk about the direction or the velocity of a body, because no body has an unique direction or velocity. Each one will have dozens of different and equally good velocities and directions according to the bodies relatively to which its motion is determined. Similarly for the measurement of time. It is always measured by the recurrence of certain events, and, according as you take different processes to be uniform, you will reach different results as to the uniformity or lack of it in a given motion. Newton of course clearly recognised this. He distinguished between absolute space, time, and motion, and the relative spaces, times, and motions; and admitted that it was perfectly possible that no change was uniform and therefore capable of giving a measure of absolute time, and that no body was absolutely at rest or in uniform absolute motion in a straight line. But he thought that he could give a criterion of absolute motion, and, by experiment, prove the reality of at any rate absolute rotation. On the other hand, most philosophers have a strong objection to absolute motion, space, or time; and, putting this aside, there is felt to be a reasonable doubt whether it can be necessary or possible to start -- as mechanics presumably does - - from empirical data, to reach laws which are stated in terms of two entities which certainly cannot be perceived, and finally to return and explain empirically perceived motion by means of these laws.

    Thus most of the recent discussions on mechanics have set the following problem before themselves: To state Newton's Laws of Motion compatibly with the belief that all motion is relative to material bodies.

    Now we have seen that, if all motion be relative, you cannot talk about the motion of a body. But of course it might be possible to find one material system or a number of systems such that the motions of other bodies relatively to these obeyed Newton's laws. This has been attempted by Neumann, Streintz, and Lange.

    Neumann's¹ theory is that there exists somewhere in space a body a, motions relative to which obey the Newtonian Laws. He compares the assumption of a to that of a luminiferous ether, but holds that it still more closely resembles the theory of electric fluids. It, like them, has a certain indeterminateness, since any other body a' that moved relatively to a with an uniform translational velocity and without rotation would do equally well for our fundamental body of reference. I do not agree with Mr Russell's² criticism of Neumann's body a. He admits that Neumann does not incur the fallacy of supposing that a is fixed (no small triumph for a relativist!). But he goes on to add that 'any statement as to its rest or motion . . . on his theory . . . would be wholly unmeaning. Nevertheless it seems evident that the question whether one body is at rest or in motion must have as good a meaning as the same question concerning any other body; and this seems sufficient to condemn Neumann's attempted escape from absolute motion.' This criticism seems to me to rest on a misunderstanding of those people who wish to keep both Newton's laws and relative motion. Relative motion without any further information is indeterminate. But it is perfectly possible that the relative motions of all bodies except one with respect to that one should be found to obey certain laws. It is not true to say that it is meaningless to make any statement about a's rest or motion. This assumes that it is only those motions that obey Newton's laws that are real, which is tantamount to the assumption of absolute motion in a more rigid sense even than Newton's. For it is quite certain that a has motions relatively to other bodies, since other bodies have motions relatively to a; though ex hypothesi it will not, like other bodies, have one definite relative rest or motion to which Newton's laws will apply. a in fact, like all other bodies, will have a large number of relative motions at the same time; but, unlike other bodies, there will not be one of its relative motions that can be picked out as obeying Newton's laws.

    The trouble about Neumann's body a is that it is as incapable of being perceived directly as the points of absolute space. Hence it makes it no easier to see how the laws of Mechanics could have been discovered and justified by empirical observations. If a be essential to the laws of mechanics, then those laws must tacitly assert relations to a. But the laws of mechanics must have been induced from empirical observations of motions relative to bodies other than a. Now this must mean that regularities are discovered when we refer motions to some bodies A, B, C . . . etc. Hence the observed regularities are regularities in motions relative to A or B or C . . . . Thus the argument for a will have to be as follows: If you replace the motions relative to A or B or C by motions relative to a body a and suppose that these relative motions obey the Newtonian laws you will be able to explain the observed regularities in motions relative to A and B and C. . . . But I fail to see that this furnishes any evidence for the existence of a; all that we know is that, relative to it, we can ascribe such velocities in such directions to observable bodies that, when subjected to the Newtonian laws, they will account for the observable relative motions of the bodies. But we must note that, when we have assumed the Newtonian laws and the body a, we have still to fix the Newtonian masses and forces and the particular velocities and directions of the assumed motions relative to a, so that the results shall agree with the observed relative motions. Can we suppose that agreement with observed regularities will suffice to determine all these variables, and to prove the truth of the laws, and to prove the existence of a? The fact is that if the Newtonian laws be true of some relative motion there must be regularities of some kind amongst motions relative to any body in the universe that you choose to take, and, since all relative motions are equally real, the various laws thus reached will be merely mathematical transformations of each other. The sole advantage of the Newtonian system with a body a will be that the laws will be much simpler. But this is not the slightest ground for believing in the existence of the body a. To do so on such grounds would be strictly comparable to holding that, because the equation of a curve becomes simpler by suitable changes of axes and shifting of origin, there is reason to believe that the particular position of the origin and the particular inclinations of the axes that give the simplest equation are real in a sense in which other axes and origins are not. Thus I conclude that Neumann is quite wrong in comparing the evidence for the reality of a to that for the reality of the ether. He is no doubt nearer the mark when be says that it is more comparable to the electric fluids; for the electric fluids are merely mathematical fictions, which however have a slightly better claim to be considered as possible realities than a.

    In fact we can come to a general conclusion about all efforts to state the Newtonian laws in terms of relative motion. Either they mean that, if we take all motions relatively to axes defined by certain perceptible bodies like 'fixed' stars, Newton's laws are approximately obeyed. Or they mean that we can so replace the perceived relative motions by motions relative to an imaginary body that, if we assume that these motions obey the Newtonian laws, we can account for all the observed relative motions. This, however, will not give the smallest reason for supposing that the imagined body is real or that the Newtonian laws are truer than others that would result if the motions were taken relatively to some other body. It will merely tell us that, in so far as the results of this hypothesis agree with the observed relative motions, there will be some laws connecting the motions of bodies relative to any body that we choose to take, which will be compatible with the Newtonian laws and will be merely mathematical transformations of them, but will be much more complex. The Newtonian laws and the imagined body will be comparable, not to a fruitful scientific hypothesis which is constantly verified by experience; but to a happy choice of axes and origin which renders otherwise intractable problems soluble. That would not of course be any disparagement to the Newtonian laws nor to the genius of their discoverer. It would still have been one of the most important discoveries ever made that such a transformation can be found. But it would be more than this. For, in connecting all the observed regularities in the motions of bodies relative to other bodies, it would have shown that there are general laws connecting all motions, no matter to what body they are taken as relative. Otherwise it would be perfectly possible that there should be motions that obey no laws; and this would also involve the proposition that the motions of all bodies relatively to some bodies were without regularity.

    But, it may be said, if a be purely imaginary, what is the use of calling it a body? Have not the axes in a become indistinguishable from the supposed axes in absolute space? Moreover, as Mr Russell asks, if the axes be material how can you deny that they may be subject to forces? And, if they be subject to forces, they will be accelerated and therefore will not be axes relative to which Newton's laws will hold. Mr Russell remarks that the question whether all bodies be subject to forces must be at least significant, and that this is fatal to the relative view. Now I do not think that this is a fair argument against the theory in question, because it seems to me to rest on the assumption of the opposite view. If Newton's laws be really laid down for motions relative to absolute space then it is reasonable to hold that they must be true or false whatever bodies we may be considering, and that it must be always significant to ask this question. But suppose we were to put the laws in the form that they hold as far as we can gather for the motions of all observable bodies relative to axes defined by the fixed stars. It would follow from this that if the laws hold rigidly of the motions of other bodies the fixed stars cannot be acted on by forces. What would the statement mean then, which we should be much more likely to admit, that the fixed stars too are very probably acted on by forces? If we are going to use 'forces' in the same sense as before and in the only one that Mr Russell could admit, it would mean: Very probably the fixed stars do move with accelerated motions relative to axis for motions relative to which the Newtonian laws hold. Now the only axes for motions relative to which we have any reason to suppose the Newtonian laws hold are defined by the fixed stars themselves. And it is quite certain that the stars cannot be accelerated relative to axes defined by themselves. Hence the meaning of the phrase would be: Very likely there are bodies such that, if axes be defined by them, the fixed stars have accelerations relatively to those axes which obey Newton's laws. This would of course assure us that our original belief that motions of observable bodies relative to axes defined by the fixed stars obey the Newtonian laws cannot be strictly accurate. Hence we can always give a meaning to Mr Russell's question on a purely relative theory. His question is : 'Are your bodies of reference beta₁ subject to Newtonian forces or not?' We answer (1) that this question, on the relative theory, must mean: Are there any bodies of reference beta₂ relative to which your bodies of reference beta₁ have accelerations that obey Newton's laws? And (2) the answer to this question is (a) that, if you hold that the observable bodies X which you said obeyed Newton's laws for their motions relative to beta₁ really obey them accurately, then there cannot be a set of bodies beta₂ relative to which the bodies beta₁ have accelerations that obey Newton's laws. In that case the bodies beta₁ are not subject to Newtonian forces. But (b) if, on the other hand, you admit - - as of course you must do -- that the motions of the other bodies X relative to axes defined by beta₁ can only be known to obey Newton's laws within the limits of experimental error, then it is perfectly possible that there may be a set of bodies beta₂ relative to which both the bodies beta₁ and the bodies X are accelerated according to Newtonian laws, but that the nature of their respective motions is such that the motions of X relative to beta₁ also approximately obey Newton's laws. In this case the bodies beta₁ may be subject to Newtonian forces. Of course the same sort of question may be asked of beta₂ and the same sort of answer can be given and so on ad indefinitum. But I do not see that there is anything objectionable in such a regress; it certainly makes no difference to mechanics as on empirical science, and Mr Russell's question can be made significant on the relative theory of any set of bodies of which he chooses to ask it.

    There is one further point to notice before leaving this subject. The relativist is not obliged in fact, and least of all on Mr Russell's theories of causation in general and in mechanics, to make his axes real bodies. What he wants is to find such axes as will give the laws of the actual relative motions of bodies in the world in the simplest possible form. If he can simplify his laws by taking to define his axes bodies which he merely imagines in places where no bodies actually are and by ascribing such relative motions to them with respect to the real bodies in the universe as are compatible with the relative motions of the latter, he is at perfect liberty to do so. For instance, it would be perfectly legitimate, after we have found that the first law of motion is approximately obeyed for motions relative to the fixed stars and have gone on to discover that the fixed stars are actually in relative motion to each other, to ask ourselves whether we cannot, by imagining bodies in places where they cannot be observed and ascribing to them certain motions relative to the real bodies in the universe which are compatible with the latter's actual motions relative to each other, get a set of axes such that the motions which we must ascribe to real bodies relative to them accurately obey the First Law. There are only two plausible objections that could be brought against this. (1) It might be said that the accelerations relative to imaginary bodies are imaginary accelerations and therefore cannot be the effects of real causes nor the causes of real effects. But, as Mr Russell well shows, on any view of mechanics this difficulty, if it be a difficulty, will arise over the resolution and composition of forces; so it is no special objection to the relativist theory. But actually when we have banished the view that causation involves activity and have grasped the fact that it means nothing but laws, the difficulty vanishes. If the laws can be more simply stated in terms of motions that do not really take place than in terms of any that do there is no objection to using the simpler form when you know that you can always get back to the actual motions when you wish. The imaginary bodies and their imaginary motions are to be regarded as parameters, and the fact that they do not exist in nature is no more reason for refusing to state laws of nature in terms of them than is the fact that the cycloidal arches of Westminster Bridge were not actually made by rolling circles on straight lines a reason why an engineer should not simplify his calculations about them by imagining them so produced and introducing parameters accordingly. (2) It might be objected that if you introduce imaginary bodies and motions you might as well introduce absolute space. To this the answer is simple. If you only mean that as a mathematical device the statement of laws in terms of absolute space is as good as the statement in terms of imaginary bodies and motions we may agree in the main. But you must not suppose that this proves the reality of absolute space any more than it proves that of the imagined bodies and motions. And the relativist might reasonably add that his plan involves a less violent effort of imagination, since we are acquainted directly with correlates to real bodies and real relative motions, whilst we have no direct experience of anything corresponding to real points and real absolute motions.

    But of course with these objections removed the theory of the relativity of motion is still by no means out of the wood. There are first of all the supposed experimental arguments for absolute rotation which have to be considered; and then there is an argument for the necessity of absolute translation as being what we really mean by motion to be discussed; and finally we must ask by what positive arguments the theory of the relativity of motion proposes to support its position.

    We will begin with Newton's bucket, and the paradoxes that have been held to spring from a denial of absolute translation. One very startling paradox is mentioned by Neumann³. He points out that if we take the case of a material point revolving round another under their mutual attraction and imagine all the other bodies in the universe annihilated it would follow that the two points must run into each other along the straight line determined by them at the moment at which this cosmic disaster happened. For, on the relative theory, all motions must be changes of distance between points or of angles determined by at least three points. Now the last possibility vanishes because there are no longer three points in the universe. Hence the result follows that the two particles will now run into each other according to the ordinary law of attraction. I confess I do not see why the relative theory should make difficulties for itself by talking of 'material points.' This particular trouble would not arise in so acute a form if you considered two finite bodies like the sun and the earth; for you could define a plane in the sun by taking three marks on it and then observe the change of angle between it and the line joining one of the spots to the centre of the earth. But of course this is a part of the general difficulty about any relative theory. If it makes its points finite and perceptible you cannot talk of the distance and the plane that they define. And, if you make them material points, then, though you have avoided this difficulty, your points have ceased to be perceptible and it is difficult to see what advantage you now have over the absolute theory of space. But this is a question for geometry rather than for mechanics. Still we shall see that it is impossible to state the third law or to apply the laws of motion to rotating bodies without introducing particles, so that Neumann's paradox is a well-founded one.

    But the more celebrated argument for absolute rotation is Newton's bucket experiment. Everyone knows what this is. A bucket with water in it is rotated about its axis.

1. The bucket rotates relatively to the water and the surface of the latter remains level.
2. The water finally takes up the rotation of the bucket. There is then no relative motion between the bucket and the water, but the surface of the latter becomes depressed in the middle and assumes a paraboloidal form.
3. The bucket is now stopped and the water goes on rotating. It is now rotating relatively to the bucket and it still retains the paraboloidal form.
4. Finally, the water ceases to rotate relatively to the bucket, and the surface is once more flat.

This experiment, Newton held, distinguished between absolute and relative rotation and gave a measure of the former. We will discuss these two arguments in turn.

    To begin with Neumann's. If Neumann's assumption that all the bodies in the universe be annihilated be taken literally it is clear that it must include the annihilation of the body a. For the two particles that are left cannot possibly be the body a, since that must consist of at least four particles. But, since Neumann holds that the laws of mechanics can only be stated in terms of motions relative to a he goes too far in saying that the particles would run together along the straight line joining them at the moment according to the law of gravitation. For to know how they would run together involves an integration based on the assumption that the first law of motion still holds, and Neumann has been insisting that that law is meaningless until we introduce a. But we may agree that the particles must keep to this straight line however they may move in it. What one does not quite see is what relevance the whole example has to Neumann's particular theory. For Neumann's theory is a particular account of the statement of Newton's laws in terms of relative motion. This particular paradox might be valid as a ground for rejecting relative motion altogether, but it cannot be any ground for accepting Neumann's particular theory of the body a; all that could result from it would be that the most terrible consequences would follow the annihilation of the body a when we are given the relativity of all motion and a's peculiar position among bodies.

    But the question is: Does this paradox, which certainly proves nothing in favour of Neumann's theory, prove something in favour of the theory of absolute rotation which contradicts Neumann's? Mach⁴ holds that it does not. He accuses Neumann of 'making a meaningless assumption for the purpose of eliminating a contradiction,' and of 'making too free a use of intellectual experiment.' I suppose that the 'contradiction' is to be found in the two propositions:

1. The existence of other bodies not exerting forces cannot be relevant to the motion of a body, and
2. that, if the relative theory of motion be true, the annihilation of all other bodies does change the motion of the remaining two.

The 'meaningless assumption' must be the denial of the relative theory. It is impossible, however, to agree that there is any contradiction to be removed, on the relative theory; and I do not see that either Neumann or Mach (who accept the relative theory) can possibly hold that there is. It is merely forgotten as usual that, on the relative theory, every body has at the same time a number of different motions in different directions, according to the body to which the motion is taken as relative. When any bodies are taken to cease to exist the motions relative to these naturally cease to exist also. But the other proposition which asserts that the disappearance of bodies not exerting forces is irrelevant means that it is irrelevant to motions relative to a Newtonian frame of reference. If the bodies that constitute this disappear too then naturally the Newtonian laws cease to apply, since the motions to which alone they apply have ceased to exist. But there is no contradiction in this to the other statement. There is merely a paradox because unsophisticated persons do not believe that motion is purely relative to other bodies. Thus the whole question is whether this belief which makes Neumann's result a paradox does or does not rest on a 'meaningless assumption.' To assert that it does is to assert without proof that what nearly everyone believes is not only false but meaningless.

    The other criticism of Mach's is worthy of more serious consideration. Apparently Mach's objection is that this result rests on too violent an abstraction. He says: 'When experimenting in thought it is permissible to modify unimportant circumstances in order to bring out new features in a given case; but it is not to be antecedently assumed that the universe is without influence on the phenomenon in question⁵. This is clearly very unsatisfactory. 'Unimportant' is purely subjective in this reference; apparently things become important when the abstraction of them lands Mach's theories in paradoxes. But then their importance is relative to Mach rather than to the universe at large. In fact the whole argument is irrelevant. Either motion is relative, as Mach believes, or it is not. If so then all bodies in the universe are equally relevant and Neumann's result follows strictly. If not⁶, we may certainly agree with Mach that the laws of mechanics which have been induced from motions that took place in presence of the rest of the material universe would very likely not hold when there were only two bodies in it. But that is not the least ground for supposing that the laws that hold under these circumstances would be such that the bodies would move in absolute space in the same way as they would on the relative theory when there were only two bodies left. If absolute motion be a 'meaningless assumption' this would be a meaningless suggestion; and if not, there is no reason why the laws of absolute motion when there are only two bodies in the universe should agree with those of relative motion under like circumstances, when it is notorious that, for many relative motions, they do not when there are more bodies. Mach's arguments in fact are both logically fallacious, one is a petitio principii and the other an ignoratio elenchi. The whole paradox arises out of the fact that we instinctively believe that motion is not purely relative, and the whole question is whether this belief in any form can be justified or refuted.

    Before we discuss this question, however, we must consider Newton's bucket. This celebrated experiment, together with gyroscopes and Foucault's pendulum, has been the great argument used by believers in absolute rotation, among whom we may mention Newton, Kant (in a peculiar and characteristic modification), Maxwell, Streintz, and Mr Russell. What does the experiment immediately prove? It proves that the relative rotation of the water and the bucket is irrelevant to the shape of the surface of the water. The water is flat when both bucket and water are at rest relative to the earth, and paraboloidal when both are moving with the same velocity relatively to it. Hence both states are compatible with relative rest of the bucket and the water. On the other hand both are also compatible with relative motion of the two. The point that is relevant to the shape of the surface is the motion of the water relative to something other than the bucket. I must confess that this seems to me to be perfectly compatible with the relative theory. The water might well become curved when it moved relatively to the fixed stars, but have its shape quite indifferent to its motion relative to the bucket. So far I agree with Mach. But he then proceeds to add the statement that: 'No one is competent to say how the experiment would turn out if the sides of the vessel were increased in thickness and mass till they were ultimately several leagues thick⁷.' The precise relevance of this remark I have never, after the longest consideration, been able to fathom. So far as I can see Mach's position would be that we can account for the motions that are perceived by Newton's laws if, wherever Newton uses absolute space, velocity, or acceleration, we substitute position, velocity, or acceleration relative to axes defined by the fixed stars. This is at least a plausible position; but, if it is to be of any use, it must enable us to give at least a probable account of what would happen if the walls of the bucket became several leagues thick. If Mach merely wishes to tell us that we cannot be perfectly certain what will happen under circumstances that we cannot observe this is no doubt quite true, but it is surely rather platitudinous. But how the proposition that we cannot be certain of what will happen under a given set of circumstances can increase the probability of any proposition except that we live in an uncertain world is more than I can understand.

    I conclude then that, whilst Newton's bucket might possibly be a test between absolute and relative rotation for those people who already believe in absolute rotation, there is nothing in it that is incompatible with a purely relative theory of motion, and therefore it cannot decide between the two rival theories.

    We pass then to the Foucault pendulum which is supposed to prove to us that the earth rotates absolutely. Let us once more begin by considering what it is precisely that we observe in the Foucault experiment. We swing a large pendulum and we note the plane in which it begins to swing⁸. We will suppose that we make a chalk-mark on the floor in the line where this plane cuts it. As time goes on, if we draw lines on the same principle we shall find that they make angles with each other. From a consideration of these angles and the lapse of time we can calculate the velocity of the earth about its axis; and we find that this value agrees with that reached from observations made on the fixed stars. It is surely perfectly plain that what we observe is not absolute rotation, but the change of position of a plane with respect to its former position as marked out by material lines. Hence the conclusion that the earth rotates about its axis with a certain absolute angular velocity must be deduced from these observed premises by reasoning. The results of that reasoning must contain a reference to absolute time and space which was clearly, not contained in the premises which were observed. But new entities cannot get into a conclusion unless they were introduced somewhere in the course of the reasoning as supplementary assumptions. As a matter of fact they were of course introduced in the laws of motion and in particular in the first law. For it was on the first law and on that alone that it was decided that the plane in which the pendulum swings is fixed in absolute space. Thus the Foucault pendulum only proves absolute rotation if you have already assumed laws of motion in terms of absolute space and time. By itself it merely exhibits a particular case of relative motion.

    But, it will be asked, does not the fact that the angular velocity deduced from this experiment agrees with that reached from the observations of fixed stars prove anything? Certainly it proves something, but not, so far as I can see, anything about absolute space and motion. What it proves is that the plane in which the pendulum swings is, within the limits of experimental error, at rest with respect to axes determined by the fixed stars. This tells us that, if the sky had always been covered with clouds, we might have reached Newton's laws if we had referred all motions to systems of coordinates defined by the planes of Foucault's pendulums and the perpendiculars to them. This seems to be the best that can be said for Streintz's⁹ position that we must define what he calls Fundamental Bodies and Fundamental Systems by their non-rotation as proved by gyroscopes and pendulums and then enunciate the Newtonian laws with respect to them. But Streintz ruins his position by coming to a statesmanlike compromise (worthy of the best traditions of English public life) and accepting absolute rotation, whilst he holds that absolute translation is meaningless. This is a double error. In the first place we have seen that such experiments do not prove absolute rest or motion apart from the assumption of Newton's laws in terms of absolute space and motion. In the second place these laws are stated in terms of absolute translation, which Streintz says is meaningless. And, finally, if you accept absolute rotation you have accepted absolute direction at any rate, and, although this does not positively necessitate absolute space, it does not seem to be an easier assumption or a more likely one.

    I hope that I have now shown conclusively that it is circular to pretend that such experiments prove absolute rotation; they merely give a ground for deciding between absolute and relative motions to those people who have already accepted absolute motion and formulated the laws of mechanics in terms of it.

    Mach uses his well-known argument against the Foucault pendulum experiment that it cannot be said to prove that the earth would rotate even if there were no fixed stars. He says that: 'The universe is not twice given with an earth at rest and an earth in motion; but only once with its relative motions alone determinable. It is accordingly not permitted to us to say how things would be if the earth did not rotate¹⁰.' Mr Russell¹¹ severely criticises this argument, but I cannot accept his criticism as it stands. After repeating the first of the two sentences that I have quoted from Mach, Mr Russell adds: 'Hence any argument that the rotation of the earth could be inferred if there were no heavenly bodies is futile.' I do not know if Mach drew this conclusion in his first edition from which Mr Russell quotes. As will be seen in the quotation which I give his conclusion is different in the English translation. His argument seems to me to be the perfectly sound one that the earth is never given (i.e. in experience) as absolutely at rest or in motion, and that he does not see how we can pass from the observed relative rest or motion to a supposed absolute rest or motion divided in certain proportions between the two terms. The conclusion that Mr Russell gives and that I am unable to find in my edition of Mach is obviously foolish on Mach's own principles. It is in fact ambiguous. It might mean either that if there were no heavenly bodies there might be nothing relatively to which the earth could rotate, or else that, if the heavenly bodies ceased to exist, the earth might cease to rotate relatively to something that remains observable whether there be heavenly bodies or not, and relatively to which it did rotate when there were heavenly bodies. The first alternative is disproved by the Foucault experiment, since that does give an observable relative rotation whether the fixed stars can be observed or not. But if Mach, as interpreted by Mr Russell, merely means that, if there were no heavenly bodies, the Foucault experiment might cease to give the same result, I really do not see how we could disprove this, or how it is relevant to the question of absolute or relative motion. It is in fact a repetition of Mach's argument against Neumann which we have already seen to be intrinsically valid but a platitude, and, in the use to which it is put, an ignoratio elenchi.

    I must confess that I do not understand Mr Russell's condemnation of the argument in itself. For he condemns it, not because he thinks that its present use is an ignoratio elenchi, but because he holds that it rests on an entirely false view of the nature of mechanics as a science. He says: 'The logical basis of the argument is that all propositions are essentially concerned with actual existents and not with entities which may or may not exist. For if . . . the whole dynamical world with its laws can be considered without reference to existence then it can be no part of the meaning of those laws to assert that the matter to which they apply exists . . .¹².' This seems to me a very curious argument. I can only meet it by asking whether it be or be not true that mechanics is a science the laws of which are based on empirical observations. Mr Russell at least is hardly likely to deny this. Now those empirical observations have all been made in the presence of the heavenly bodies. Also all laws based on observation are only probable, and it is generally and reasonably held that their probability diminishes as the state of affairs to which it is proposed to apply them departs more and more from that under which they were originally observed to be obeyed. Hence it seems to me that Mach is perfectly correct in holding that it is quite possible that the Foucault experiment might not succeed if all the heavenly bodies were annihilated. Such a conclusion, as we have seen, is not relevant to what he is trying to prove, but it seems to me a valid conclusion as against Mr Russell's criticisms. It does not rest at all on the logical basis that 'all propositions are essentially concerned with existents' but on the proposition that all general laws that rest on empirical evidence must depend for their truth on observations of the existent, that they can never be more than probable, and that their probability diminishes the further the state of affairs under consideration differs from that from observations of which they were originally induced. Nor does this conclusion conflict in the slightest degree with the fact that rational dynamics makes no mention of the fixed stars, or that the problems with which works on rational dynamics are full have no reference to real distributions of matter. For what is Rational Dynamics? It is the mathematical results that follow from the assumption of Newton's laws. True, Newton's laws, on the relative theory, are those that are found to hold of the existent world for those motions that take place relatively to axes defined by the fixed stars. But this does not prevent rational dynamics from dropping the fixed stars, replacing them by axes in an assumed absolute space, and supposing that all motions relative to such axes obey the same laws as those that are observed to hold in the real world for motions relative to the fixed stars, and working out the results of these assumptions for any assigned imaginary distribution of masses. You might perfectly well make up a non- Newtonian rational dynamics, and the only difference would be that its results would not be applicable to the motions of real bodies relative to the fixed stars. In fact we may end this discussion by proposing the following dilemma to Mr Russell. Either Newtonian dynamics is purely rational or not. If so I fail to see how it can throw any light on the existent and how Mr Russell can be justified in such statements as that Foucault's pendulum and Newton's bucket prove the existence of absolute apace and motion. But if not, then, though no appeal to the existent is needed in proving its conclusions from its premises, you cannot be sure that the conclusions would apply to an existent so different from that from which the premises were originally induced as a universe that consisted of nothing but the earth with all the heavenly bodies annihilated would be from the total universe that we know and observe.

    I agree then with Mach that none of the arguments so far produced are incompatible with a purely relative theory of motion, so long as the theory understands itself.

    Experimental arguments from the motions of matter then will not help us, and we must pass to other ones. Let us begin by asking on what grounds precisely relativists deny absolute space and motion. I think there are three. The first is that they cannot be perceived and that we can do without them. The second is that they are meaningless. And the third is that space involves contradictions. Arguments like the second are always cheap and easy and rather contemptible. Taken literally the argument asks us to believe that the words 'absolute space' and 'absolute motion' are mere noises or marks on paper which convey nothing to anybody. The beat way to refute this opinion is actually to state what the difference between the two views means, and this I do lower down. In the meanwhile it is only necessary to note (what many people forget) that to say that absolute motion is meaningless and to produce further arguments against it is inconsistent. You cannot refute a noise or overthrow a mark by a syllogism; you must confine yourself to saying that you do not understand what Newton and Euler and other supporters of absolute motion were talking about, and that you strongly suspect that they themselves did not know what they were talking about. And you will run some risk of having the first statement believed and the second doubted. The third point need not detain us either. The arguments to prove that space cannot be real because, if it were, contradictory propositions would be true about it, all rest on sheer errors about infinity and continuity. For their refutation we have merely to refer to the relevant chapters in Mr Russell's Principles of Mathematics.

    The sole argument then that is left to the relativists against the belief of common-sense in absolute motion is that it is not perceptible. Is this any argument at all? It will be remembered that we have tried to show in the earlier parts of the present essay that in all probability nothing that is perceptible is real. Hence we should be more disposed to believe that absolute motion is an appearance if we could perceive it than to believe that it cannot be real because it is imperceptible. We must therefore try to show what is meant by the distinction between absolute and relative motion on our view. We have expressed a belief in the existence of a real counterpart to those spatial relations which we agree in perceiving by the sense of touch and to those objects of visual perception which science, based on the belief of a real counterpart to tactual space, is able to justify. But what is perceived by sight or touch always has other qualities beside spatial ones. The spatial qualities that we perceive involve relations, and those relations must have terms. Those terms we took to be real but unknown qualities in real counterpart relations, and we supposed that the changes in these caused our perceptions of colour and temperature and also of perceived spatial relations. Now the question of absolute space for us simply is this: Do the real correlates to perceived spatial relations hold directly between these unknown terms or is there a continuum of other terms between which real relations eternally hold, so that the correlates of perceived and changeable spatial relations are really complex and arise from the terms between which they hold being 'at' now one and now another of this continuum of terms which stand in eternal relations to each other? The latter supposition is what would correspond to absolute space on our theory. We may at once refer to these supposed terms as 'geometrical points,' since it is admitted that such points could not be perceived. We can at least see that the question at issue is not meaningless. We cannot perceive either points, if there be such things, or those other terms between which we believe the relations correlated to the felt spatial relations hold. But, whilst points cause no perception in us at all we believe that the other terms or events in them are the causes of perceptions. We will call the latter set of terms 'material qualities.' The question then can be put as follows: Granted that there are material qualities in certain real relations closely correlated with those spatial relation that we perceive by touch between sensible things, and granted that events in these material qualities cause perceptions of pieces of matter of various shapes and sizes and in various motions, are we to say that these real correlated relations are simple and variable relations between such material qualities? The affirmative answer to this question is the relative theory. The other alternative is that there is a real continuum of geometrical points which eternally stand to each other in certain relations and do not produce any perception in us. The real correlated relations to the spatial relations that we perceive will then not be simple but will arise out of a certain relation of material qualities to variable geometrical points + the eternal relations of these points to each other. This is the absolute theory in terms of our view of perception and of the probable nature of the real. Since on our theory, we cannot perceive either geometrical points or material qualities, the fact that geometrical points are imperceptible is very much less important than it might be on some other theories.

    It will be said, however, that the admission that unlike material qualities, geometrical points and their eternal relations produce no perceptions in us, takes away all reason for believing in them. But this is not strictly true. For common-sense holds that not all motion is relative, and, if it be correct in that opinion, something like geometrical points will be needed to make our theory correspond to the views which common-sense holds and can justify.

    The question then is: Can common-sense justify this belief? Is it ultimate, or does it depend on bad reasoning or on psychological causes that are competent to produce the belief but are equally or only compatible with the truth of the relative theory? If the belief were ultimate it would, I think, be very unwise not to give due weight to it, since we have now seen that there is absolutely no argument against its possibility. For the question turns on what is meant by motion; and that, after all, must ultimately rest on what people find, after careful reflexion on all the possibilities, that they actually do mean by it. If, after looking at all the alternatives, people still think that it is essential to what they mean by motion to be able to say of two bodies that move relatively to each other that the real motion of one is so much and that of the other so much, or that one really moves and the other really stands still, it would be unwise to accept the relative theory which makes such statements falsehoods.

    Now it is not to be denied that, if bodies have absolute motions at all, people constantly make mistakes as to what they are. We feel instinctively that the earth stands still and that motions relative to it are absolute motions. But the fact that people often or always make mistakes in assigning the magnitude of a quality q in given cases, does not prove that there is no such quality as q, or that it is not an essential part of the notion of that process in a particular case of which it was wrongly assigned. The very fact that people always assign it somewhere and in some degree in the process shows that they at least believe that the quality in question is in general essentially bound up with the meaning of process. It might very well be possible to explain their mistakes by psychological causes without being able to explain away that characteristic about which their judgments were mistaken.

    In the first place, it is clear that in many cases of perceived motion something is perceived in one body which cannot be observed in the others. Thus, if I see someone walking up the Hall there is a positive perceptible difference from what I should see if the end of the Hall moved toward him, although, on the relative theory there can be no difference as far as motion is concerned, since that merely consists in the shortening of the distance between the man and the wall. Thus, it seems necessary to hold that, even if we cannot perceive absolute motion, we can perceive distinctions in the case of two bodies that move relatively to each other which are interpreted in terms of absolute motion. This, however, need not trouble the relativist. For, in the case under discussion there are differences of relative motion that account for the difference perceived. The man who is said to be 'really' moving really is moving relatively to my body, whilst the wall at the end of the Hall is not. Hence, while it might remain true that all motion is relative, still the man who is said by common-sense to move will indeed have a relative motion which the end of the Hall, which common sense calls stationary, will not have. Again, we are quite sure, with respect to our own bodies, whether we move or something else moves relatively to us but then the actual perceptible difference is that whilst in both cases we perceive the same relative motion between our own body and the other one, in the one case we also have certain muscular sensations and perceive that we are also moving relatively to other bodies, and in the other we have no muscular sensations and no perception of relative motion between our own body and other bodies than the given one, while the given one is now perceived to be in relative motion to others beside our own.

    I do not think that there can be any doubt that there is a peculiar perception both tactual and visual which we can call perception of motion. And motion so perceived is undoubtedly absolute, in the sense that the direct object of the perception is not the mere change of a relation between two bodies but is ascribed as a quality to one or both of them. But all this is in the realm of appearances. When you come to ask what it is in reality that causes the perception of such appearances you can always, it seems to me, explain them just as well in terms of real relative motion as in terms of real absolute motion. The belief that motion is absolute, which common-sense holds, seems to spring from a confusion between these two different senses of motion; viz. motion as the object of a perception and therefore an appearance, and real motion. Apparent, i.e. perceptible motion is always absolute in the sense described above. It is always ascribed as a quality to the moving body or bodies. But our previous investigations have led us to believe that all such perceptible qualities are appearances. And, at any rate, the motion that is thus absolute is not that which mechanics takes to be absolute. The perceptible absolute motion of the earth is zero, whilst the perceptible motion of telegraph-posts from a train window though quite as absolute as any other perceptible motion is not that which mechanics has to ascribe to them.

    But it is not unnatural that the absoluteness which attaches to the apparent motions which common-sense perceives should be carried over into the account that science gives of the real causes of the perceptions of these appearances and so it is supposed that motion must be absolute in the widely different meaning that that term has in mechanics. Now the theory of absolute motion is indeed a possible explanation of these appearances, though, in as far as it is admitted that the appearances often cause us to make mistakes about the real absolute motion, it can hardly be called a completely satisfactory one. But it is also possible, as we have seen, to explain these distinctions in the appearances in terms of purely relative motion + muscular sensation. We must remember that, even if A and B could only move relatively to each other, so that there were no sense in saying that it was really A that moves whilst B stood still, yet there might still be reason to believe that the cause of the change of their mutual distance was in one of the terms only, and this would make a distinction between the two which, owing to the prejudice carried over from apparent absolute motion, would come to be held as a mark of real absolute motion. And I do not think that the phenomena of motion offer us any means of deciding between the two possible alternatives.

    Thus, my conclusion is that common-sense believes in real absolute motion because it really does perceive different qualities in moving objects which depend on motion but are not accounted for by the merely symmetrical relation of change of distance between them. But perceptible motion, like all that is perceived, is an appearance; it is not the motion that is measurable and to which it seems likely that there directly corresponds a real change in the spatial counterpart. The real motions that cause the perceptions of these appearances, being called by the same name, and being so intimately related to the former, are supposed by a kind of subreption to be absolute in the sense that each body has its own motion. When this view is worked out it leads to the theory of absolute space and motion that I suggested above. And that theory will explain the facts, and is not meaningless. On the other hand, a theory of purely relative real motion will equally explain the apparent absolute motions that people perceive. Hence the fact that common-sense quite rightly believes that the motions that it perceives are absolute is no reason why we should think that the motions with which we deal in science are so. The question remains a perfectly open one. And we have already come to the same conclusion from the experimental attempts at a proof of absolute motion. At the same time it ought to be added that the absolute theory lays claim to a much greater knowledge about the real than does the relative one, and therefore I think that, on philosophic grounds, the relative theory is to be preferred. For mathematical purposes the absolute view is undoubtedly better, and we have seen that problems treated in terms of absolute motion can be applied to the existing world when for absolute motion is substituted motion relative to axes defined by the fixed stars.

    We have thus decided in the same sense as Mach, though by somewhat different arguments, that 'a straight line' for the purposes of the Laws of Motion means a straight line relative to axes defined by the fixed star, and that 'uniform velocity' means velocity that is uniform relative to such axes¹³. But now the question arises: What do we mean by uniformity? We need a discussion about time and its measurement to answer this question. An uniform motion is presumably one in which equal spaces are covered in equal times. But what are equal times, and how do you know when they are equal?

    We generally measure equal times by the rotation of the earth. But how do we know that the successive rotations of the earth take place in equal times? As a matter of fact we do not even believe it to be strictly true, for we talk of the slow change of the length of the second so measured owing to the tidal drag, and attempts have been made to calculate this change. It must be noted that this is not a question of whether time be absolute or relative; the question is one of measurement which applies equally to either theory, and rests on the fact that quantities of duration cannot be moved about in time as metre-scales can be in space and superposed. We thus seem to be in the same position as we should be in the measurement of space if we had to measure a length in England by considering how long it looks as compared with what we can remember as to the length of the standard metre in Paris from the time when we last saw it. Whether temporal relations be directly between events, or directly between absolute moments and only indirectly between events, the fact remains that it is only the relations between events that can be observed and measured, and so this difficulty arises on either view.

    Neumann¹⁴, discusses this question. His method is (i) to state the First Law of Motion in terms of the same time instead of equal times, and then (ii) to construct from it a definition of equal times. On this view the actual statement of the first law becomes: 'Any two bodies A and B under the action of no forces describe straight lines relative to the body a, and successive equal lengths of A's path correspond to successive equal lengths of B's path.' You take any arbitrary distance on A's path and note where B is when A is at the beginning of this distance. You then note where B is on its path when A is at the end of this distance. Call these two distances l_a and l_b. You now let A run on any other position, let us say l_a' from the point from which l_a was measured. When it reaches this point you note that B is at a position l_b' from the point from which l_b was measured. And you find that, if no forces act upon either, l_b'/l_b = l_a'/l_a. This then is Neumann's account of the Law of Inertia.

    From it he now proceeds to give a definition of equal times. Equal times are those in which a body under the action of no forces describes equal lengths. Thus, the point of Neumann's definition is to replace equal times for one body by the same time for several bodies in the statement of the Law of Inertia. That law then states the relation between successive distances passed over in the same time by different bodies all unacted upon by forces. He then defines equality of time by means of the law of inertia so stated. This of course does not mean that he defines what is meant by the equality of two durations, but that he tells us what is to be taken as the measure of it in mechanics.

    Streintz¹⁵ criticises Neumann's position as follows. He says that Neumann's form of the law of inertia would still be obeyed if, under the action of no forces, any two bodies obeyed laws of motion of the form x = kf(t) and x' = k'f(t) where f is any function. This of course is obviously true. Streintz goes on to state that this is incompatible with Neumann's statement that equal times are those in which a body under no forces covers equal spaces. For, he says, all that we can tell from the fact that a body obeying Neumann's law of inertia has covered two spaces x₂ - x₁and x₄ - x₃ which are equal is that

f(t₂)-f(t₁) = f(t₄)-f(t₃).

    I think this criticism rests on a complete misunderstanding of Neumann's position. He is not trying to prove from his law of inertia that a body that obeys it will describe equal spaces in equal times. He is defining the times in which such a body passes over equal spaces as equal. It is clearly impossible to refute a definition; the best you can do is to show that it covers some cases where the times are admitted not to equal, or leaves out others in which we believe that the times are equal. But, with Neumann's definition, Streintz's hypothesis that x = kf(t) for a body moving under the action of no forces reduces to x = kt. The fact is that Neumann's statement of the law of inertia is not simply convertible, and Streintz is quite justified in pointing this out; but, on the other, Neumann is also justified in taking a class of motions which the law says fulfil the conditions (viz. those performed under the action of no force) and defining the times which bodies animated with these motions take to cover equal spaces as equal: provided always that he can pick out this class of motions that are performed under the action of no force by some criterion other than their fulfilling the conditions laid down in the law.

    The real criticism on Neumann's definition is that he does not tell us how to distinguish bodies under the action of no forces from others. His law of inertia, as we have now seen, cannot be a criterion of this until we have found at least one motion which is known, independently of that law, to be performed under the action of no forces. For his statement of the law merely tells us about the relation between the corresponding spaces passed over by two bodies both under the action of no forces.

    Streintz¹⁶ himself rests the measurement of time on an axiom, which is that, if any action takes place twice under exactly the same circumstances, it will always take exactly the same time to accomplish itself. This position needs a little further explanation before it can be criticised. The example that Streintz gives shows that he does not mean that the conditions are to remain constant throughout the whole of' each example of the process. They may change during it in any way you like. But at corresponding points of each process the conditions must be the same. Under these conditions Streintz considers it axiomatic that the duration of the two examples of the process will be identical. The axiom is plausible, and we must examine it to see whether it may not contain an implicit circle. What precisely do we mean by 'corresponding points of each process'? Clearly not points at equal times after the beginning of each process; for this assumes the measurement of equal times. It must therefore be held to mean at the same observable stage of the processes. Thus, if we drop the same body from the same point above the earth's surface at different times equal durations will be supposed to have elapsed each time it reaches the earth, because it will be supposed to have been subject to the same conditions at corresponding points in its fall.

    The axiom is not therefore logically vicious. On the other hand, are the circumstances ever the same in two different experiments? And, if so, how do we know it? It is perfectly clear that they never are precisely the same, and therefore the axiom, even if true, is useless as it stands as a means of telling when equal times have elapsed. It would need modification to the form: If any process can be repeated and, if at corresponding stages the relevant circumstances are the same, it will take place in equal times. But what are 'relevant circumstances'? Since ex hypothesi the two events as such do not differ in any observable characteristic the only way in which a circumstance can be relevant is to alter the time between any two corresponding stages in each of its successive occurrences. Hence to say that in corresponding stages no relevant circumstances are different, just means that in corresponding stages no circumstances are different that would make the time between a pair in one process differ from that between the corresponding pair in the repeated process. So now the definition of equal times based on the only form in which the axiom can be applied to their actual determination is sheerly circular.

    Lange¹⁷ gives a peculiar theory of the nature of the First Law of Motion both with regard to space and time. He asserts that his theory of time- measurement is identical with Neumann's in principle. This, as we have seen, would be a most serious objection to it if he happens to be correct in this statement. But his discussion is much more general than Neumann's and deserves a short consideration. It is best put in the second Appendix to the work with which we are dealing. Lange holds that the full statement of the First Law involves two conventions and two verifiable propositions. He say that you must not take the motions with which mechanics deal as being either (a) relative to absolute space and absolute time, or (b) relative to any axes defined by existing bodies or to actual events. He gives no new arguments for either position, and the only point that interests us at present is his discussion as to what these motions do involve. To answer this question be introduces an axiom called by him the 'Principle of Particular Determination.' I had better quote Lange's own words about this principle. 'Wir haben . . . ein Theorem, welches in Bezug auf eine beliebige Anzahl gleichartiger Elemente der wissenschaftlichen Betrachtung . . . die gleiche Behauptung . . . aufstellt. Der zum Verstandniss dieser Behauptung vorauszusetzende Massstab der quantitativen Bestimmung . . . wird so normirt dass jenes Theorem für eines der Elemente gilt und zwar eben per conventionem gilt. Der Wahrheitsgehalt, d.h. das mehr als bloss Conventionelle des Theorems liegt dann nur darin, dass nach jener auf Grund eines Elementes erfolgten conventionellen Normirung des Massstabes das Theorem sofort fur alle ubrigen Elemente gilt.'

    Now in the First Law we are told that all material particles left to themselves describe straight lines with uniform velocity. Lange takes any one particle left to itself and defines its motion as being one that covers equal spaces in equal times relatively to a certain set of axes. This is conventional. The times are defined as equal in which this particular particle covers equal spaces. The experimental part of the law is supposed to be that every other particle when left to itself describes relatively to the same axes equal spaces in times defined as equal by the motion of the first particle. It follows that the experimental law can be put in the form: 'If out of all the particles moving under the action of no force any one be chosen, then every one of the rest will describe equal spaces whilst the first describes equal spaces, and the times in which the first does this are defined as being equal.' This is practically the same as Neumann's account of the law of inertia. The definition of equal times as those in which the assigned particle describes equal spaces only differs from Neumann's definition by referring to an assigned particle instead of to any particle left to itself. I fail to see that this is any improvement. In fact the whole method of treatment suffers from the same defect as Neumann's. It has to assume that you can know when a particle is under the action of no forces, and it offers no test by which this can be determined without assuming that you have been able to determine it independently in at least one case. Lange's attempt to define a straight line for the purposes of the first law of motion seems to me to labour under the same defect and some others besides. If I understand him aright his position is as follows. If three material points be shot out together from a single point of space with their directions not in one plane and left to themselves they will in general describe curves. You can prove as a mere matter of geometry that there is always at least one set of axes relative to which all these curves are straight lines. The physical part of the law is the further assertion that any fourth material point left to itself will describe a straight line relative to axes so defined. Clearly we have the old trouble about how you are to judge when material points are left to themselves apart from the laws of motion. Moreover the statement that the three points are left to themselves is ambiguous. Does it mean left to their mutual actions but free from those of any other bodies, or that each is free from the action of all other bodies? Lange might reply that it is of no importance whether we can tell in particular cases that bodies are left to themselves, because his object is not to point out an actual inertial system (this can only be done by experience) but to define what is meant by such a system. But this does not meet the objection that the notion of being 'left to itself' enters into the definition of the system and therefore must presumably be relevant, and yet it must mean 'left to itself' not in some vague general way but in a strictly mechanical sense, and we can only give a meaning to this after we have stated and accepted the laws of motion.

    The question then arises: Have we not some measure of the equality of the duration of two events which do not completely overlap, independent of the laws of motion? It is perfectly clear that people did measure times and did judge them to be equal or unequal before the First Law of Motion was thought of; and there must therefore be some supposed criterion independent of any assumption as to the presence or absence of forces. Streintz's axiom, we saw, would not work as stated, but it seems to be much more promising than any that we have yet noticed. I think, however, that the best account of the criterion of the equality of non-coincident durations is to be found in an article in Mind¹⁸ by Rhodes. It will not do as it stands, but by introducing certain modifications I think it can be made fairly satisfactory.

    Recurrent events are either isochronous or anisochronous. We want a test for isochronism. Rhodes says that there is no intrinsic difference between a series of isochronous events and one of anisochronous events. He goes so far as to say that there is nothing in the motion of a pendulum taken alone to make us believe that it is any more likely to be isochronous than are the motions of a weather-cock. This, I think, is exaggerated. It does not seem to me that we lack all power of making judgments correctly about the relative magnitudes of durations that do not overlap completely. The trouble with us is that these judgments are very likely to be affected by subjective factors, and that to a much greater degree than immediate judgments about spatial equality or inequality. Further, whilst we can make our immediate judgments about space almost indefinitely more accurate by means of measuring-rods -- though the assumption of their constancy of length raises rather difficult problems -- we have not as yet found anything that corresponds to a measuring-rod for time. It is for this purpose that we want isochronous recurrent processes. Thus, whilst I differ from Rhodes in believing that anybody in his senses would recognise that a pendulum is very much more nearly isochronous than a weather-cock by merely looking at the movements of each, I agree that we want something more than the intrinsic qualities of the pendulum's motion to assure us that it really is isochronous. It might be much more nearly so than the movements of the weather-cock and yet be far from perfect isochronism.

    Rhodes's method of determining isochronism is as follows. Suppose there is a process P whose isochronism we want to test. Let P' be any other process that goes on contemporarily with P. Then, if P' be recurrent, it is either isochronous or anisochronous. Thus the a priori possibilities are

1. P and P' both anisochronous;
2. P and P' both isochronous;
3. P isochronous and P' anisochronous; and
4. P anisochronous and P' isochronous.

Now in case (ii) it is certain that to every complete recurrent event in the process P there will correspond the same number of events in the process P'. You cannot of course ensure that it shall be a whole number, but it will always be the same whole number + the same fraction of an event. By a fraction of an event we mean an example of one of the recurrent events that reaches a certain determinate stage other than the end one of it. This is a necessary characteristic of two isochronous recurrent events, whether there be any causal connexion between them or not. Does it completely distinguish the case (ii) from all the other possible cases! It is clear that it cuts out the cases (iii) and (iv) entirely. We are left then with (i). Rhodes of course admits that two anisochronous series of events might have this quality which two isochronous series must have. Hence the condition for isochronism that he has given, though necessary, is not sufficient. He therefore adds that the two processes P and P' must be 'independent of each other.' Unfortunately he does not tell us how this is to be discovered, and so his theory becomes useless.

    I think, however, that we can complete the criterion in terms of the theory of probability, on the assumption that we can make rough immediate judgments about the equality or inequality of durations that do not entirely overlap. For the present purpose the statement that P and P' are both isochronous but are not independent means that each is accelerated or retarded according to the same law. Now it is fair to assume that the more different the two processes are from each other the less likely is this to be the case. It is not a priori wildly improbably that all pendulums are actually retarded according to the same law even when swinging in vacuo; but it is very unlikely that the rotation of the earth, the swings of a pendulum and the vibrations of an electron are all retarded according to the same law. Let us call the proposition 'P is isochronous' p_i. Let us denote the contradictory of a proposition q by -q. Let us call the proposition that to an event in P there always corresponds the same number of events in P', r. Finally I shall adopt the notation p/q = 'the probability of p given that q is true.' p/f = 'the probability of p given the knowledge that everyone is supposed to have before the experiment is performed.

    We want to compare p_i/r with - p_i/r. We know that

r/p_i-p'_i=0, r/- p_i p'_i=0, and r/p_i p'_i=1

Now p_i <--> p_i p'_i .v. p_i -p'_i,

Therefore p_i/r = p_i p'_i/ r+p_i -p'_i/ r
= [(p_i p'_i/f x r/p_i p'_i)/(r/f)] + [(p_i -p'_i/f x r/p_i-p'_i)/(r/f)]
= (p_i p'_i/f)/(r/f)
Again -p_i = - p_ip'_i .v. - p_i-p'_i
Therefore

- p_i/r = - p_ip'_i/r + - p_i-p'_i/r
= [(- p_ip'_i/f x r/- p_ip'_i)/(r/f)] + [(-p_i-p'_i/f x r/- p_i-p'_i)/(r/f)]
=(-p_ip'_i/f x r/- p_ip'_i)/(r/f)

Hence we have that
[(p_i/r)/(- p_i/r)] =(p_ip'_i/f)/(- p_i-p'_i/f x r/- p_i-p'_i)

Let us now consider this fraction critically. The first point to notice is that all the terms in it, being values of probabilities, must be proper fractions. Hence there is a general expectation that the denominator which is the product of two fractions will be pretty small. Further, from what we have already said about the possibility of judging roughly of isochronism by immediate inspection, and in view of the fact that the experiment is being used on two processes that appear to be isochronous to see whether they really are so, we see that p_ip'_i/f is very much greater than -p_i-p'_i/f. It follows that p_i/r must be very much greater than -p_i/r unless r/- p_i-p'_i is practically equal to one. But it is perfectly obvious that is a very small fraction, since it is only under very special circumstances that two non- isochronous series of events should be so connected that to every complete event in one there shall correspond the same number of events in the other. It is clear from observing series of events which immediate inspection shows to be anisochronous that in general this connexion does not hold. And from what we have said above it is clear that the less alike the two processes P and P' are to each other the smaller is r/- p_i-p'_i, for the less likely it is that both are accelerated or retarded according to the same law. On the other hand the fact that they resemble each other little in other respects is no reason for thinking them unlikely to be really isochronous if they seem to be so, i.e. for p_ip'_i/f to be small. Hence we conclude that if two series of recurrent events appear to be isochronous by themselves, and if to each event in one there always corresponds a fixed number of events in the other, it is enormously more probable that each is isochronous than that neither is so, whilst it is impossible that one should be isochronous and the other not. I think therefore that Rhodes's method of recognising isochronism is really satisfactory and non-circular with the modifications that we have made in it. There are two further considerations not mentioned by Rhodes which may be added touching the effect of the duration of the experiments. (i) If there be the smallest difference between the laws of two anisochronous processes it is certain that the longer they are compared the more it will show itself. So that in the case of two processes which give the test for isochronism however long we try them together we are forced to decide either for real isochronism or anisochronism + exact identity of law. And the latter is much less likely than the former, as we have said, if the two processes be of very different kinds. (ii) If we consider a single recurrent process and find that (a) it recurs pretty quickly, and (b) that it continues as long as we care to observe, or at any rate for a very long time, and (c) that successive occurrences seem to us to take the same time, we can get a fair test of equality of time. For (a) and (b) tell us together either that the process is not retarded at all or that the retardation is very small at each recurrence. And (c) tells us that it must be pretty regular. But if two successive occurrences are judged to take the same time and it can be proved that the retardation in each must he very small we must be committing a very small error in counting them as equal. Thus even a single process which is actually not perfectly isochronous may when properly criticised give us a very accurate measure of equality of two times, provided we admit that we are capable of comparing pretty accurately the times taken by two successive events of the same kind when the time occupied by each is short. And there seems no reason to doubt this.

    We can now define equal times as those in which a recurrent event takes place in an isocronous recurrent series; isochronism being measured according to the method indicated above. Since this method makes no mention of forces we can state the Laws of Motion without any further trouble as regards space and time. The first law will run as follows: A body under the action of no forces describes a straight line relative to axes determined by the fixed stars so that, whenever it covers equal spaces, the same number of events takes place in some recurrent series which appears to be isochronous and is found to be so related to some other series that appears to be isochronous that to every event in the former there corresponds the same number of events in the latter. It must be added that, whilst it is perfectly possible that a series which seems isochronous and fulfils the conditions with respect to another apparently isochronous series might not obey them when tested with respect to another apparently isochronous series, yet, as a matter of fact, this does not generally happen. Most perceptibly isochronous series obey the test indifferently with each other within the limits of observation; and this is what we should expect from real isochronism, whilst we should not expect this agreement if we were mistaking for isochronism connected anisochronous series. But, when once the science of mechanics has been based on the laws of motion, and these upon this method of measuring equality of time, it is perfectly possible and logical to show that the world will be more easily and fully explained if we assume certain small and imperceptible departures from isochronism in processes on whose assumed absolute isochronism the laws of motion were originally built¹⁹.

    The remaining problem of the laws of motion is that of Force. What is meant by force, and how do we know whether it is present or absent? It would seem that, in order to arrive at the law that when no forces act on a body it moves in a straight line with uniform velocity relative to a certain set of axes, we must have been able to recognise, in some cases at least, the presence or absence of force by other characteristics than those of motion. To discuss this point it will be best to turn to the Second Law of Motion. This law offers us a measure of force and not a definition of it. It says that the force acting on a body in a given direction at a given moment is measured by the rate of change of its momentum in that direction at that moment with respect to time. Momentum is measured by the product of mass and velocity, and, if mass be constant, this will mean that the measure of the force acting on a body at a given moment is the mass x the acceleration at that moment in the given direction. This is supposed to be relative to the axes and the measure of time that we have already discussed. It follows at once from this that, if the velocity be constant in a given direction, the force will be zero in that direction. Hence the converse of the First Law follows from the Second. The law itself does not follow however. The fact that when a body is at rest or moves with uniform velocity in a given direction the measure of the force in that direction is zero does not prove that, when no force acts on a body, it will move in a straight line with uniform or zero velocity. We must therefore discuss the Second Law in order to come to any conclusion about the nature of force. On the subject of the position of force in mechanics a long discussion has raged; some holding that it is necessary to retain it, others that it ought to be rejected as a mere relic of animism. We cannot perhaps expect to say much that is new on such a hackneyed subject, but we cannot pass it by without discussion.

    The whole question of force is closely bound up with the view that is taken of causation. Kirchhoff and Mach who banished forces from mechanics, supposed, it would seem, that the notion of cause is obsolete, and that the case of force in mechanics is a peculiarly disgraceful example of the survival of obsolete prejudices. We should prefer to say that force is bound up with the activity view of causation, although it will be obvious enough that we do not hold that, with the departure of activity, science must or can drop causation. I think that, as far as Mach is concerned, any lengthy discussion on this point would be merely verbal. It is quite clear that, rightly or wrongly, he retains what we have called causal laws, and what we have held to be practically the whole of causation. In this sense then Mach does not drop causation from Mechanics. On the other hand be does drop the activity view of causation, and he seems to hold that the name 'cause' is no longer applicable after that view has been dropped. Any argument on such a question as this would obviously be merely verbal; and it is not worth while to embark on it. We have merely to note that, in our sense of causation, there undoubtedly is causation in mechanics, and that we mean to go on using the word.

    The classical definition of force is that it is the immediate cause of motion. But this is far too vague. For motions with a uniform velocity can and do continue without the action of force. We should have to say that it is the cause of acceleration, instead of merely saying that the acceleration at a given moment is the measure of the force at that moment. But it is clear that people did not mean that everything that caused an acceleration was a force in the sense in which mechanics uses the term. When I will to move my arm it might be said that my volition is the cause of the motion. But it would not be said that a volition was a mechanical force. Yet again when one billiard ball hits another it would be said that the first causes the acceleration of the second. But no one would call a billiard ball a force, We saw that the activity view of causation makes the cause a substance in a certain state, and the effect a state of a substance. Hence I think we can fairly state the common view of force in mechanics as follows: A body in a state of exercising force is the cause of a state of acceleration in another body.

    Now we have already rejected the activity view of causation which makes substances in a certain state the causes of states in other substances. We cannot therefore make any exception in the present case. Force seems here, as activity is everywhere, to be a mere qualitas occulta, the mere turning of a causal law connecting observable or inferable states of two substances at different times into a quality of one of them. But the case for force can be argued a little further by its upholders. Some people think that some of their perceptions are either perceptions of force, or, at worst, of something that proves the existence of peculiar qualities to which the name forces may appropriately be given. If this were true forces would not be occult qualities and we could state laws, in our own sense of causation, about them. We must therefore enquire for a moment into these alleged perceptions of force.

    We may agree with Höfler²⁰ that the term force is ambiguous in ordinary life. It covers tensions and pressures and also supposed permanent real qualities of bodies, such as the force of gravitation. Höfler adds a third meaning, viz. mass-accelerations. This is not even I think what is meant by force except by people who merely wish to keep the name force after they have given up the common belief in it as something special; it is certainly not the idea that the word force calls up in the mind of any unsophisticated person. The people who wish to identify force with mass-accelerations seem to me to have forgotten that there is such a thing as the science of Statics where forces are believed to operate as much as in Dynamics but where there are no accelerations. Hence, I think, we shall do best to say that, if there are forces, then they are the causes of mass-accelerations in dynamics and of the straightening of strings and the extension of spring-balances in statics. The question is whether people are right in thinking that the qualities called forces can be perceived, and whether they have any special perceptions which tell them independently that there must be these special qualities even though they cannot perceive them.

    It is obvious that perceptions of straightened strings, or extended spring-balances, or accelerated bodies are not themselves perceptions of forces. And it is by no means clear that the causal laws which explain such perceptible effects ever need among their data any special qualities like forces, or indeed anything but positions, motions, and certain characteristic coefficients. If there be any perceptions whose objects can reasonably be held to be forces or to necessitate a belief in special qualities which we can call forces its their cause they must be our sensations of tension and pressure. The names of these obviously beg the question of whether their objects really are forces. But it is tolerably obvious that, when we consider, we do not hold that they are any more perceptions of forces than are perceptions of stretched strings or accelerated masses. When we feel our skin pulled or pressed we localise the felt pull or pressure in our skin as we should a felt temperature. But, if we suppose that forces are qualities which are localised somewhere in the body that is causing the pull or pressure, since that is not in general our own body, it is clear that the immediate object of our perception is not what is believed to be the quality of force.

    Indeed when we say that we feel a pull we mean that we have a perception whose object is localised at a certain part of our body and which is supposed to be caused by a force in another body which is pulling it. We must say then that common-sense holds, not that it directly perceives forces in its experiences of pulls and pressures, but that it perceives a quality localised in the body the existence of which is supposed to be a mark of the existence of force in the body that is said to be pulling or pressing.

    But, after all, is there any better reason to assume a new kind of quality as the cause of these perceptions than for the cause of the straightening and stretching of strings? On the contrary it seems quite clear that these perceptions are the result of the shortening and lengthening of parts of our skins. They thus depend on the general effects in bodies with regard to which we saw no good reason to suppose that it was necessary to assume forces, as special qualities of bodies, as their causes. Hence these feelings cannot furnish the smallest additional evidence for the necessity of assuming real forces.

    The only other experiences that we have which might seem to offer any evidence for the reality of forces are sensations of effort that we have in pushing, pulling, and raising bodies. We must not confuse these with the sensations that we have already discussed. The two always occur together, and this might be expected, if there really were forces and these sensations wire closely connected with them, from the Third Law of Motion. It is, however, perfectly easy to distinguish the two. Can we say that the perceptions of effort are the perception of the quality of force in our bodies? I do not think that Poincare's rhetorical question: 'Do you suppose that the sun feels muscular effort in keeping the planets in their elliptical paths?²¹' has any particular weight, although he thinks that it settles the matter. Naturally if the sun has no mind it does not perceive anything at all. But that is quite irrelevant. The true question is: Does the sun in keeping the earth from moving off at a tangent have that sort of quality which we who have minds perceive when we forcibly drag a heavy body? Is it a real quality, and, if so, may it not be a quality that only belongs to organic bodies? Even if we conclude that it is a real quality in our own bodies it is a sheer inference to suppose that it exists in other bodies as different from our own as the sun or a billiard-ball. And, as in the case of temperature, such an inference is very precarious.

    I conclude then that we have no reason to suppose that inorganic bodies, with which, after all, we are mainly concerned in mechanics, have any qualities in them comparable to what we perceive in our own bodies when we say that we exercise force or have force exerted upon us. Such qualities cannot be directly perceived in other bodies, the inference from our bodies to non- organic ones is precarious in the extreme, and there seems every reason to believe that the causes of these peculiar perceptions in us are not special qualities called forces in foreign bodies, but the same qualities and states as cause the tensions of strings and the changes of velocity in bodies. It only remains then to ask whether the states and qualities that cause these various effects are something special to which the name of forces can be given, or are just the positions, velocities, etc., of bodies. What we have now seen is that certain peculiar experiences of ours give no reason for adopting the former alternative.

    Mr Russell has an argument which, if true, would demolish forces altogether as real qualities. We have seen that, if forces be real, they must be the causes of accelerations among other effects; though it does not of course follow that the causes of accelerations must be forces; for that is the question that we are now discussing. Now Mr Russell's argument is that an acceleration is not an event and therefore cannot be an effect. Hence it cannot have a cause. Hence, if there are no causes of accelerations and forces are the causes of accelerations or nothing, there can be no real forces. Let us then consider Mr Russell's arguments to prove that accelerations are not real events. There are two of them, one narrower and one more sweeping. The first argument²² is that, whatever may be the case with a resultant acceleration, it is certain that a component acceleration is not an event. The components of a vector are merely the products of analysis. The vector from London to Cambridge can indeed be resolved into a vector from Euston to Bletchley and another from Bletchley to Cambridge; but a sane person in coming to Cambridge from London does not pass through Bletchley, and so the journeys represented by these two vectors are not in any sense implied in the journey represented by the vector from London to Cambridge. This is the only one that is actually made. The others are not parts of it which must exist if it does, but operations whose final result, if they were made, -- which they are not -- would be the same. Hence component forces at any rate cannot be real. The other argument²³ is that no acceleration, component or resultant, can be a real event. This argument is based on Weierstrass's account of the nature of a differential coefficient. The momentary acceleration is the differential coefficient of space by time for that particular value of the time. As such it is a limit and a mere number, and so cannot be an effect of anything. Hence that which, if it were anything, would be the cause of an acceleration is a mere nothing and cannot be real.

    These arguments are interesting, and, if a belief that accelerations are real events be truly incompatible with the theory of the nature of a differential coefficient, it must be rejected, since that theory is undoubtedly true. But I do not think that there is this incompatibility. No doubt an acceleration as a limit is a mere number, but then no one ever supposed that, by saying that the acceleration of a body at a given moment was, x feet per second per second, he meant that it was identical with x feet per second per second. I can see no reason whatever why velocity and acceleration should not be real states of bodies with magnitudes measured by the numbers which are reached as the limits of certain mathematical expressions. Whether an acceleration be a real event or not is of course another question; but I do not see that the fact that, if it were, its magnitude would be measured by the value of a limit could prove that it could not be real. It will be well to make it quite clear that this possibility interferes in no way with Weierstrass's solution of the Zenonian paradoxes about motion. The essence of that solution is the insistence on the fact that every value of a variable is a constant, so that at every moment the arrow in Zeno's paradox is at one definite point equally, whether in a finite time it moves or not. The difference between the two cases is simply that, in the one, it is at different points at different moments, whilst, in the other, it is at the same point at different moments. Now this would meet all the difficulties just as well if at every moment the moving arrow had a quality with an intensive magnitude measured by the differential coefficient of space with respect to time for the body at that moment. The moving arrow would still, at any moment, be at a single point of space, and, at different moments, at different points. It would merely have a quality at each point which the arrow that was at the same point at different moments would not have. Hence Weierstrass's unquestionably correct account of motion and his denial of infinitesimals does not touch the possibility of the existence of a physical quality of velocity or acceleration; and I cannot agree with Mr Russell that Weierstrass's theory necessitates a more integrated form of the laws of motion.

    We pass then to the other argument that, at any rate, only resultant accelerations could be real. It is quite clear that Mr Russell's argument is valid as regards the ordinary analysis into components in three directions at right angles to each other. These three directions are chosen merely for convenience in handling the problem under discussion, and correspond to nothing that actually takes place. But is it certain that nothing corresponds when the components are apparently actually given? For instance, in the case of a system of gravitating particles, are there no accelerations along the lines joining the given particle to each of the others? It is quite certain that there are no motions along these lines in general, but only along that of the resultant force on the particle at each moment. It is also true that in a finite time tau the position reached by the particle in question is not what it would be if it were successively dragged along the lines joining it to the particles at the beginning of the motion for a time tau along each. If we grant, however, that accelerations may be states of particles, whose magnitude is measured by the differential coefficient of space by time at the moment for the particle, then it would be possible that at each moment the various particles did produce these states with their appropriate magnitudes, even though no states corresponding to an arbitrary selection of components existed. But, although I do not think it possible to disprove the existence of such states, I see very little evidence for their reality. And it is certain that the laws of mechanics can be stated without them. Hence it seems much more philosophical to state any causal laws that are involved in a form that avoids velocities and accelerations. At the same time you cannot work out any problems in mechanics in this integrated form. You must also remember that, although a statement of the relations between accelerations is not directly about what exists, yet the fact that such statements bring you back in the end to what does exist and can be observed makes them laws about the existent. It was no doubt important for Mr Russell to show that, if you insist that causal laws shall only connect actually existent states or events, then the laws of motion will involve relations between configurations at three moments. But, when you have passed so far away from the ordinary view of causation as Mr Russell has done and as we have also, it seems rather needless purism to deny the name of causal laws to uniformities that connect, if not existent states, yet terms which only are connected as they are in virtue of the nature of the existent.

    But, even when we state laws in terms of accelerations, we do not need to make any mention of forces. Hence we can conclude that, if accelerations do not exist, forces, as causes of them, cannot exist; whilst, if they do exist, there is not the slightest reason to suppose that their causes are special qualities called forces. Wherever forces nominally occur in dynamics it is always in the form of special laws over and above the laws of motion, connecting the configuration or states of a system with the accelerations of its parts. Thus, when we talk of the electrical attraction between two charged particles, we seem to be talking of force as a cause of their accelerations. But, when we work out a problem, the only way in which the alleged force enters is in the fact that coefficients e, e' are to be attached to each particle and that the acceleration in one due to the presence of the other is ee'/mr², where r is their distance apart. Instead of the force determining the acceleration it is determined by two coefficients and the relative positions of the particles according to a certain law. The only use that force has in dynamics is that it is a convenient phrase when we do not want explicitly to take into consideration the other bodies in the system. We can then talk of a field of force; but we merely mean by this that bodies placed in a certain position all have a certain definite acceleration imparted to them that only depends on their own masses and other coefficients.

    Our best plan will now be to try and state the laws of motion without force, although not necessarily in terms of configurations, for the reason that I have given above. For a reason into which we shall enter later the laws must be stated in terms of particles and not of definite bodies.

    I should state the First Law in the following terms: 'Changes in the magnitude or the direction of the velocity of any particle relative to axes determined by the fixed stars, and measured by processes shown to be isochronous by the method discussed above, cannot be determined by purely immanent causal laws.' The first law then tells us that we must look for the other data in any causal law that shall enable us to predict such changes relative to the fixed stars in events or qualities of something outside the system consisting of the particle and the stars. Is the evidence for this law empirical or a priori? The only possible way to show that it was a priori would be to connect it with our axiom about causality, and it is easy to show that it cannot be deduced from that. Our axiom, it will be remembered, said that, if any system had been quiescent for a finite time and then began to change, the change could only be explained by a causal law transeunt to the system. It is clear that this is both too wide and too narrow for the first law of motion. It is too wide in as far as it tells us that in any system of particles that had been relatively at rest for a finite time the cause of any change from rest must be looked for outside the system. For our law refers to a particular system, viz. that of the particle and the fixed stars. And to that system it does not apply because we know that the system of the fixed stars and any particle is not quiescent, since the fact of stellar motion in the line of sight renders it almost certain that the stars are in relative motion to each other. But again it is too narrow. At best it could only tell us that a particle at rest relatively to the fixed stars would remain at rest; it could not tell us that a particle in motion with reference to them might not move in circles, or stop, or perpetually alter its velocity without our having to go outside the system for explanation.

    If there be any evidence then for the law it must be empirical. Everyone knows the type of direct empirical evidence that is offered for the law. We are told that if we throw a stone along a smooth flat surface it will travel a long way in a straight line, and that the smoother the surface is made the further it will travel. This always appears to me to be a bad example of the law, though I think it is of importance in view of the historical prejudices which the law had to meet. With regard to the first point, it is a bad example because we need to commit a circle in order to be sure that it is an example at all. The reason is as follows. What do you mean by smooth? If you mean a geometrical quality then it is clear that the necessity of mentioning it in the example shows that you have not reached an immanent causal law. The future position of the body is foretold, not from its present position and velocity alone, but from these + the geometrical qualities of the surface on which it moves. Hence, as it stands, we have no example here of the supposed law. But if, on the other hand, you mean by smooth that the surface is one that has no effect on the motion of the body, then the experiment and its completion in thought merely come to this, that continued movement in a straight line is a proof that the surface is not having any causal influence. But it is only a proof by assuming the law, and so the experiment can give no support to the law.

    The fact is that there is no non-circular direct experimental evidence of the first law. What an experiment of the kind just discussed did do was to upset the scholastic theory of motion based on the maxim; Cessante causâ cessat effectus. The application of this maxim to motion was that a motion only continued as long as that which originally caused it continued to act, or at any rate, that, when it ceased, the motion died away of itself. Now these views the experiment does tend to undermine, and so it makes way for the true view. When it is seen that the stone goes on for some distance and the distance depends on the nature of the surface it suggests that motion does not have to be kept up by the same causes as produce it, since a flat surface is not a cause of motion in the first instance. And, when was further seen that the more you polished the surface the further the stone went, it followed that it was much more reasonable to ascribe the gradual cessation of motion to the nature of the surface than to some immanent causal law in the nature of all moving bodies. Thus, the value of the experiments that are alleged to prove the first law is not that they do prove it but that they render improbable certain plausible alternative views that are incompatible with it.

    The true proof of the law is to be found in the explanation that it offers of projectiles' paths and of planetary motion. It was found that, under certain circumstances, by holding that motions were compounded out of unchanging velocities in certain directions and changeable ones that obeyed definite laws in which the presence and qualities of other bodies were data, these motions could be completely accounted for. And wherever the velocities did remain constant in magnitude and direction no mention of the states of other bodies was needed.

    With regard to the second law I agree with Mr Russell's statement²⁴. This is that, in an isolated material system of n particles, there are certain constant coefficients called masses (m₁, m₂ . . . m_n) to be associated with the particles respectively. Then, for any particle in the system, the acceleration multiplied by the mass is a function of the masses, relative positions, and velocities of the particles at the moment. It will be the same function for the system at all times. And in general it will involve other coefficients beside masses.

    There are two points to notice about this law. In the first place the other coefficients that enter into the expression are the marks of what are called 'forces of nature' such as gravitation, and electrical charges, and the elasticity constants of bodies. We have already seen that they are not anything that need lead us to believe in forces in the old sense. What is called a 'mechanical explanation' consists in replacing them by masses, positions, velocities, and geometrical conditions. We shall have to discuss the meaning of such attempts toward the end of this chapter.

    The other point to notice is that the law is only asserted for an isolated system. When we ask what this means we find that it means a system whose states at all moments can be determined by a purely immanent causal law from a sufficient number of observations made within it. How wide a system must be in order to be isolated is a question that cannot be answered a priori, but we learn by experience what kind of terms are relevant and what are not. All systems, if our discussion on space and motion be correct, must be assumed to include the fixed stars, or, at any rate, axes defined by bodies that are known to be at rest or in uniform translational motions in straight lines with respect to them.

    The nature of the evidence for the Second Law is once more entirely indirect. It is in fact precisely the same evidence as proves the first law, viz. that all mechanical processes can be analysed in this particular way. As Poincaré points out, if they appear to fail, you can always avoid the difficulty by assuming enough hidden particles in proper positions and motions to which these two laws do apply.

    To discuss the notion of mass we need the third law. Now it is with the third law that the introduction of particles instead of finite bodies becomes essential, so that this will be the best place for the discussion of this question foreshadowed on p. 333. The third law states that, in an isolated system of particles, you can take the particles by pairs, and, considering their accelerations along the lines joining them, these will always be opposite in direction, and the product of them by the respective masses of the pair will be equal. So far we have only spoken of masses as coefficients introduced in the second law and apparently comparable to such other coefficients as e, the electric charges, or gamma, the gravitational constant. But here we learn something fresh about the coefficients. Granted that for any two particles in the system f_rs and f_sr are opposite in sign, which is a law of nature, it is of course a mere convention that coefficients m_r and m_s can be found such that m_rf_rs+m_sf_sr = 0. But where we pass out of the world of convention and back to laws of nature is in the fact that, if we take any third particle, and, with the same notation as before, have m_rf_rt + m_tf_tr = 0 where mt is merely chosen to make this equation hold, we shall find that, with the value of m_t thus reached, the equation m_sf_st + m_tf_ts = 0 holds. And you will find that, when the relative masses of the particles in a given isolated system have thus been determined, they are, in general²⁵, fixed and independent of any other variations in the states of the particles.

    Now it is quite clear that such a proposition as the Third Law has no definite meaning except for particles, because there is no line that can be called the line joining two finite bodies. At the same time all that can ever be observed is finite bodies and their states. How then do we reach the notion of particles and the laws stated in terms of them? And why do we state the other laws also in terms of particles and not of finite bodies?

    The analyses of motion which, I have argued, led to the first and second laws were analyses of the motions of bodies of finite size, rigid, and moving translationally. Such were the experiments with projectiles, and falling bodies, and the simple pendulum; such too were the motions of the planets about the sun to which Newton first applied this analysis.

    It is, of course, true that the earth and the other planets also rotate on their own axes; but Newton found that it was not necessary to take this fact into account when he laid down the laws that govern the motions of the planets about the sun, and of the moon about the earth. Hence something very like the first two laws of motion was reached without having recourse to particles. They were not indeed directly observed to hold of finite rigid bodies in their translational motions, but they could be analysed out of those motions. Similarly, owing to the large distances of the planets from each other and from the sun, it was possible for many purposes to treat them as points and talk of the line joining them. This enabled the third law to be applied to them in a rough way, but it was not their motions that were the evidence on which the third law was based. That evidence was of two kinds. It came partly from dynamics and partly from statics. Also, whilst it is much the most paradoxical of the three laws to the untutored mind, it is curious that it is the only one of the three for which there is a kind of direct sensible evidence. The dynamical evidence for the law with regard to finite bodies came from the experiments of Huyghens and Wren on the percussion of spheres.

    They were not, of course, entirely clear as to the distinction between mass and weight, but then weight is an accurate enough measure of mass at a given place on the earth. The result of their experiments was to show that, when the spheres ran into each other along the line joining their centres, the sum of their momenta before and after impact was constant, whatever might be the elasticity of the bodies. Owing to their use of spheres they could get a clear account of what was meant by the line joining the two bodies at the moment of impact. When we combine this result with the second law of motion we see that the accelerations of the bodies which ended in the permanent change of their velocities, and presumably took place while they were in contact, must have been in opposite directions along the common normal, and must have obeyed the third law. Similarly, dynamical experiments can be performed with floating magnets which support the law for finite bodies that can be treated as points.

    I will not interrupt the argument for the moment with the consideration of the statical proofs, and the direct experiences of the truth of the law for finite bodies, but will pass at once to the introduction of particles from what was already known. What was already known then was that the three laws held for the translational motions of bodies which were rigid and approximately spherical like the planets and the balls in the impact experiments. Here you could give a definite meaning to the velocity or acceleration of the body in a given direction, and to the line joining two bodies under certain circumstances. But, with such limitations, it is clear that there is an immense number of motions with which Newton's laws could not deal. They could do nothing with pure rotations, since they are all about translational motion. They could not be directly applied to bodies that are not fairly rigid, for how can you talk of the velocity or acceleration of a body whose different parts have different ones. And, except for bodies so far apart with respect to their size, as to be treated as points, or bodies that only act when in contact where you can define directions by the common normal at the point of contact, how can you give a meaning to the Third Law?

    The passage to particles from finite bodies is best seen in the treatment of rotation which first showed that Newton's laws could deal with motions of that kind as well as with the translational motions in reference to which they were formulated. The first problem involving the rotation of rigid bodies that was solved was that of the time of swing of a compound pendulum. This was first fully discussed by Huyghens. Huyghens' solution, however, was based on principles which, though compatible with Newton's, and indeed, mutually deducible from them, were formally different. It involved, in fact, the conceptions of work and kinetic energy instead of those of force and momentum. The statement of the Newtonian method of dealing with such cases is to be found in D'Alembert's principle.

    But the essence of both solutions is to imagine the body divided up into an indefinite number of masses located at geometrical points. You thus get rid of rotation on the part of these elements, because a point cannot rotate. On the other hand, you can replace the rotation of a finite body about an axis by the translatory motion of its elements in closed curves. Now the laws of motion, as we saw, were obtained from just such motions of finite bodies. It is then obviously worth while to enquire whether, if you assume that their translatory motions obey the laws of motion for finite bodies, you will be able to account for the motions of the whole system. At any rate, it is clear that all the laws can be stated with much greater precision for particles than for finite bodies, and in fact that the third cannot otherwise be stated in a quite definite form. We must notice at once that neither D'Alembert nor Huyghens have any right to assume a priori either (a) that the same laws that applied to the large bodies would apply to the little ones, or (b) that, if they did, it would be possible to work out the results of applying them to an infinite number of particles with no other conditions imposed than that they form elements of a rigid body. Naturally, unless we could actually prove that this assumption led to the laws that had been found to hold for finite bodies, we should have no justification for making it. But it can easily be shown that if the third law of motion applies to any system of bodies the position of the mass- centre of these bodies (defined in the usual way) will either continue to remain unchanged, or will alter with uniform velocity along a straight line relative to axes defined by fixed stars so long as the system is isolated, no matter what special laws there may be determining the mass-acceleration of each particle in terms of functions of the masses, positions, velocities, etc. of the others. But when our system of particles is subject to the condition that the distances between all of them remain constant, we have the case of a rigid body, and so we get back to the first law of motion for finite rigid bodies from the analysis of the motions of which it was first deduced. So that the hypothesis that the mass-points obey the same laws of motion as the finite rigid bodies, is at any rate self-consistent as far as concerns the first law, whilst it allows of much greater rigour of statement.

    But we must not stop here, for, by the same procedure, we can show, what was not at all clear before, that the laws of motion, though stated in terms of the translatory motion of finite bodies, are also capable of dealing with their rotations. For, as we have seen, we can reduce rotation of a finite body to the description of closed curves by its particles. But then we can prove that, if these particles obey the third law, and if we assume in addition that the oppositely directed accelerations that they determine in each other are along the line joining them, the sum of the angular momenta of all the particles in an isolated system about any axis fixed relatively to the fixed stars is constant. Similarly, from the other laws of motion and the special laws for the system, we can determine the rotation of any set of particles in the system about any axis. Hence the mere assumption of particles accurately obeying the same laws as were laid down -- though imperfectly at best -- for the translatory motions of finite bodies make the latter, even in their pure rotations susceptible of mechanical treatment. Hence the object of science is much better attained by stating the laws in terms of particles. The laws that were originally discovered for finite bodies become consequences, and new consequences are discovered for the rotations of such bodies. The only additional assumptions that have had to be made are (a) that the opposite accelerations postulated by the third law shall be in the line joining the particles, and (b) that the mass of a finite body is the sum of the masses of its particles. With regard to the first point, we saw that the weakness of the third law, as stated for finite bodies, was in the matter of direction, and hence any assumption that we make with regard to direction when we come to deal with particles, where direction first becomes definite, can hardly be said to contradict anything that has been observed of finite bodies. But, indeed, nothing in the interactions of finite bodies would point to the oppositely determined accelerations having any other directions than that along the line of junction. If it made an angle with it, our two magnet poles in water would rotate with ever increasing velocity, which is contrary to all experience. The assumption about mass is necessary to make the first law apply to rigid bodies on the assumption that it applies to their constituent particles. There is no evidence against masses being additive, and, with regard to bodies of finite size, conclusive evidence for it.

    I think it should now be clear why we state the laws of motion in terms of particles, although they must have been discovered by considering and analysing the motions of finite bodies. And it is also clear in what sense the statement in terms of particles is justified by those experiences of finite bodies and their motions which alone we can have. Owing to the fact that works on dynamics begin for didactic purposes with the dynamics of the particle, and then pass through D'Alembert's principle to the dynamics of finite bodies, we are often inclined to think that principle merely a mathematical development. But we must remember that the order of knowledge is (1) laws for the translatory motions of finite bodies; (2) the application of these laws to hypothetical particles with the increased precision of statement that this allows; (3) proof that this gives the laws for finite rigid bodies in their translational motions, and that it also enables us to account for the rotations of such bodies without any new assumptions. The last is a real discovery made by the hypothetical method.

    The question now arises: Are we to consider the analysis into particles as a real analysis, or is it merely a happy mathematical device? We must remember that, whether we state the laws of motion in terms of velocities and accelerations as we have done, or purely in terms of configurations as Mr Russell does, a statement in terms of particles seems to be equally necessary. Thus we cannot argue off-hand that particles, like accelerations and velocities, are merely mathematical apparatus by which we can indeed start from the existent and get back to the existent, but which themselves have nothing existent corresponding to them. In fact we have to ask the question whether Mr Russell's laws, in so far as they are stated in terms of particles, are any more truly causal, in the sense of dealing with what exists, than the laws stated in terms of velocities and accelerations. Let us then begin by clearly understanding what such a question means on our views. People can perceive bodies, but it is clear that they cannot perceive particles. Hence, whilst bodies can exist as appearances, particles, if they exist at all, must be realities. Now let us consider our views about the nature of the laws of mechanics. As laws taken from appearances and stated in terms of them, we might think that they were laws about appearances, but this is not what we really do think, since we hold that it is quite irrelevant whether anyone is looking on or not to the fact that the laws are obeyed. What we mean by this is that, whilst as stated, they are laws about what we perceive, they are laws about that aspect of it and its changes to which we believe there to correspond a real counterpart. The laws that govern the motions of billiard balls are, as stated, laws about appearances and they were found by observing the motions of perceptible billiard balls. But we believe that there exists something corresponding to the felt round objects and to their visible spatial relations, and that these real relations condition the shapes and distances which, under suitable circumstances, events in the terms that have these real relations, enable us to perceive. Thus our mechanical laws, though stated in terms of appearances, are really about the changes in these real relations to which, when we do have the perceptions, there correspond these perceptible changes from which the laws were reached. We see then that the fact that the particles could not possibly be perceived does not militate against their reality, since, with finite and perceptible bodies, the laws of mechanics only hold of them in so far as they correspond in their spatial relations to a real correlate which exists whether we perceive it or not. But the real reason that makes it unwise to assume that particles are more than a mathematical device is the following. A particle differs entirely from any matter that we ever do perceive in that it has no extension at all. In this respect it is utterly different to assume the reality of particles from what it is to assume the reality of molecules or electrons. A molecule is supposed to be of finite size, though imperceptible. This means for us that it is a real thing with qualities and relations like those which determine our perception of bodies of finite size and definite shape, but that, unlike them, it does not have events in it that can cause us to have a perception of a body of a definite shape and minute size. But, with a particle, all is different. It has no size at all, to speak in terms of appearances, and that means that, if it is real it is not in the least like anything that determines any perception that we have. It has in fact every mark of being a mathematical artifact, and, as such, whilst we cannot deny that it might possibly be real, we have no good reason for supposing that it is so. On this ground I think we may conclude that we were justified in not paying too much attention to Mr Russell's insistence on the statement of the laws of motion in terms of nothing but particles, times and positions; since there is no more reason to suppose that particles really exist, than that velocities and accelerations do so.

    The laws of motion then can and must be stated without reference to forces. These are replaced by coefficients which enter the expressions for the amount of acceleration of one particle in terms of the relative positions, velocities, etc., of other particles in the system. We have now a final question to consider. Force is used in statics. Does it here reduce to the same mere name as in dynamics, or must we at length introduce special qualities called forces? As far as the conditions of equilibrium are concerned of course, statics is only a particular case of dynamics, for which the ordinary laws of motion are ample. The only difference that could possibly lead to difficulty is that, whereas dynamics seems to deal wholly with the motions of particles, it is commonly believed that a system may be in equilibrium, in the sense that none of its particles are in motion, and yet 'forces' may be exerted between its parts. Consider, for instance, the case of a bow with the string drawn so that the bow is bent. We hold that all the particles in the bow are at rest (setting aside molecular and electrical theories), and the conditions for this can be determined by the laws of motion. But it is also held that the string and the wood are in a state of tension, and, in fact, that forces are being exerted between one part of the system and the others. Let us try to see precisely what this means. In the first place it means that the shape of the wood is different from what it would be if it did not form part of the system to which it now belongs. It is clear that particles cannot be supposed to change their shape; all that can alter with them is their relative positions. Thus, we must recognise that the same system of particles can have different configurations when it forms part of different systems. But there is nothing in this that has not already been mentioned in the second law of motion. It was known from that that, to determine the positions of the particles in a system at any moment, certain special laws were needed over and above the laws of motions. These laws tell us that, with certain systems of particles, the particles will come to rest in certain relative positions; so that the second law of motion which relates momentary configurations to momentary accelerations will tell us what will be the configuration when there is no acceleration on the part of any of the particles, when once the particular laws for the system under description are known. What happens then is this: We find that the positions of the particles of certain substances are connected with their accelerations according to definite laws when portions of the substance of definite size and shape are made parts of systems, the acceleration produced by which on a particle placed at the point at which the substance is placed we have already determined. We assume that the same laws will be obeyed in any system in which a piece of such a substance forms a part. And, by saying that the piece of the substance in a given system, though in equilibrium, is exerting forces inside the system, we mean that, in the present configurations it would not be in equilibrium out of that system, but that its particles would be accelerated according to the known laws until a new stable configuration was reached.

    Thus once more the alleged 'forces' are nothing but general laws connecting configuration of particles with acceleration in systems of a particular kind. Because these laws hold as a rule wherever a system of particles of the same kind may find itself, we make this law, which is general, into a particular quality of the body, and call it a force. We are more particularly encouraged to do this in statical examples because, if our bodies form parts of such a statical system, they themselves are deformed according to certain laws just like other bodies from the shapes and sizes that they have when they are not parts of the system. These deformations, as we saw, are accompanied by a peculiar kind of feeling, and, because this feeling is peculiar, we assume -- without justification, as I have tried to show, -- that it must have a peculiar cause, viz. a force.

    But, although there is no need to assume something special called forces, we must consider how these special laws of matter comport with the statement of all mechanical laws in terms of particles. The special laws that determine for us the form of our function in the second law and the coefficients that it is to contain are supplied by the special sciences like electricity, magnetism, and what is commonly lumped together under the title of 'Properties of Matter.' Whilst one of these special laws -- that of gravitation -- seems to apply to all kinds of matter at all times and in all states of aggregation, the others do not. The same matter may have different electrical charges at different times, only certain matter is magnetisable, and the constants like rigidity and the coefficients of dilatation and friction vary from one kind and state of matter to another. Now it is true that all these laws, in so far as mechanics can use them, will have to be stated in terms of the relation between the accelerations of particles and the distances, velocities, etc., of other particles together with certain coefficients. But, at the same time, if a given configuration of iron particles is associated with a different acceleration from the same configuration of particles in wood it would seem that we must accept as ultimate the different qualities of material bodies. And, again, if we take the case of charged bodies and find that we can explain their attraction by a law of the form ee'/r², whilst there is nothing similar in the case of apparently precisely similar bodies which we assert to be uncharged, we seem forced to assume really different states of matter. We can now see what precisely is meant by a mechanical explanation of some physical phenomenon. It does not merely mean a proof that all the motions in this sphere of physics obey the laws of motion; for this is generally believed from the outset. What it means is that, by suitable assumptions, the coefficients in the function required by the second law, which depend on the particular nature of the matter under discussion, can be eliminated. By 'suitable assumptions,' for the present purpose, it is meant that, instead of supposing ultimately different kinds of particles in which acceleration and configuration are connected by the same general form of law but with particular different coefficients, we can explain the phenomena by assuming different invisible configuration and numbers of particles of the same kind connected by laws in which there is no difference of coefficient. When this has been done with the separate 'properties of matter' a mechanical explanation would go on to try and show that all these were special cases of a single law connecting distributions of particles with accelerations. The perceived qualities in which bodies differ must pari passu with such explanations be thrust back into different ultimate psychophysiological laws connecting different states of aggregation of the same kind of particles with the clearly different objects that we perceive.

    This tendency on the part of the physicist has been bitterly reviled by philosophers, notably by Prof Ward in his Naturalism and Agnosticism. Let us see what attitude we ought to take up towards it.

    Since we have seen no reason for supposing that particles as such are real, there is nothing to choose between the statement that a piece of wood and a piece of iron of the same shape and size consist of the same number and arrangement of particles with different laws -- or, at any rate, with laws that require a different coefficient -- and the statement that they consist of different numbers of particles, all alike, obeying the same laws, but giving a different observable effect owing to their differences of number and configuration. The really important question is this: Are we justified in supposing that the causes of the perception of a piece of wood and of a piece of iron of the same size and shape are differently aggregated collections of the same kind of small but finite real bodies differing only by their number and arrangement, and such that when, for purposes of the second law of motion, we make the mathematical analysis into particles, we can assume the same laws connecting configuration with acceleration, and explain the observed differences in terms of the differences of number and configuration? This is the sort of question that is raised by molecular theories, and more especially by the electron theory; and an affirmative answer to it would show that these theories do give information about the real.

    Let us suppose for a moment that this could be done; would it have any philosophical importance, and would there be any philosophical objection to it as an account of the real? Our previous results will enable us to answer this question. In the first place we have seen that there is not the least reason, on the assumption that there is a real world events in which cause our perceptions, why, to every difference of quality in the object of our perceptions, there should be one in the cause of the perception. Hence there is no theoretical objection to reducing the ultimate complexity of the real causes of our perceptions below that of the world that we perceive, so long as you are still able to account for the differences in what we perceive. Again, whilst we have seen that there is no intrinsic reason why any of the ordinary sense- qualities should not he real, the causal theory of perception leaves no reason for thinking that they are so. On the other hand, there is some reason for thinking that there is a real counterpart to our objects of spatial perception and more especially to those of touch. Hence, whilst there is no theoretical objection to the reduction of the ultimate differences of the supposed real causes of our perceptions as far as is compatible with the explanation of the differences in their objects, we are positively making fewer debatable assumptions about the real by reducing real qualities and replacing them by laws connecting those that are left and their geometrical relations, as far as is possible. Such procedure does indeed make more assumptions about those qualities and relations that it leaves in the real, but it assumes the reality of fewer ultimate qualities. Now, as the sole ground for assuming real differences of quality is to explain perceived differences of quality, and, as we find that we can best explain the latter by assuming only a few of the former, together with laws about their relations and changes in terms of that real, counterpart to the objects of spatial perception which we have been compelled to assume, the procedure of physics does offer us the least objectionable account of that branch of reality with which it treats that is possible. It would be madness to hold that the information that it offers us is certain, when taken as information about the real; it is, and it must always remain, problematic in the extreme. But, with all its imperfections, it remains the best account of the real causes of our perceptions that we can have in the present state of knowledge; and it is only by pursuing the same methods that a still better account can be reached.

    It may well be that further experience will show, as it seems to be showing, that the Newtonian laws are only special cases of still more general laws; and that Newtonian mass is not a genuine constant under all conditions. But the more general laws will still be laws about positions and velocities of some extended quality or qualities, and, as such, will be capable of the same sort of defence that I have offered for the traditional mechanical physics; that, although its laws were deduced from experiments made with appearances, they are transcriptions of the laws that hold among the real causes of our perceptions of appearances, and among the relations of these real causes to each other in a real spatial counterpart.

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Notes

    ¹ Die Galilei- Newtonische Theorie

    ² Principles of Mathematics, Chap. LVIII. p. 491.

    ³ Op. cit. p. 28.

    ⁴ Principles of Mechanics, Eng. Trans. p. 572. [He is discussing another argument of Neumann's, but it is quite parallel to the present one.]

    ⁵ Loc. cit. (the italics are Machs).

    ⁶ I take the object of Mach's argument to be to show that Neumann's paradox is not really so very paradoxical, because, even if we assume absolute motion, we have no right to assume that, when the universe is reduced to two bodies, the laws induced from such a widely different universe as ours would apply. It is on this view of his meaning that I discuss the passage.

    ⁷ Principles of Mechanics, p. 232.

    ⁸ I shall assume that the pendulum is swinging at the N. Pole. This makes no difference of principle and simplifies the statement.

    ⁹ Die physikalischen Grundlagen der Mechanik, pp. 24 and 25.

    ¹⁰ Principles of Mechanics, p. 232. (Italics are Mach's.)

    ¹¹ Principles of Mechanics, p. 492.

    ¹² Principles of Mathematics, p. 493.

    ¹³ Subject of course to the possibility of defining axes by means of imaginary bodies such that motions of real bodies relative to them shall still more accurately obey the First Law.

    ¹⁴ Op. cit. p. 18 et seq.

    ¹⁵ Op. cit. p. 87 et seq.

    ¹⁶ Op. cit. p. 80 et seq.

    ¹⁷ Die geschichtliche Entwicklung des Bewegungsbegriffes

    ¹⁸ Mind, O. S., Vol. X, 1885, 'The Measurement of Time.'

    ¹⁹ This may be compared with our desertion, first of the earth, and then perhaps of the fixed stars, as axes of reference relative to which the First Law holds, and the substitution of axes defined by imaginary bodies so determined that motions relative to them shall accurately obey the First Law.

    ²⁰ Studien zur gegenwärtigen Philosophie der Mechanik, p. 31.

    ²¹ La Science et l'Hypothèse, p. 130.

    ²² Principles of Mathematics, pp. 484, 485.

    ²³ Ibid., pp. 473, 474.

    ²⁴ Principles of Mathematics, p. 483. [It might seem that the statement that it was the momentary configuration that is relevant is contradicted by the form of Hooke's Law for stretched strings within the elastic limit, viz. mx = -[q(x- x₀)/x₀] where x is the present and x₀ the unstmtreched length. But, in a system so isolated as Mr Russell takes, we could hardly talk about the unstretched length of a body. We could merely say that x₀, like q, is a constant that has to be introduced in connecting the momentary configuration with the momentary acceleration.]

    ²⁵ This is not strictly true of moving charged particles, when the velocity approaches that of light.

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Contents -- Appendix

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