Chapter VI: completes the preliminary course in the fundamentals of (2)

I do not wish to speak too definitely of the Einstein theories in these introductory remarks, and so shall refrain from mentioning explicitly in this place the situation which they bring up and upon which what I have just said has direct bearing. It will be recognized when it arises. What must be pointed out here, however, is that we are putting the thing which the scientist calls the "observed value" on a footing of vastly greater consequence than we should have been willing offhand to concede to it. So far as any single observer is concerned, his own best observed values are themselves the external world; he cannot properly go behind the conditions surrounding his observations and speak of a real external world beyond these observations. Any world which he may think of as so existing is purely a conceptual world, one which for some reason he infers to exist behind the deceptive observations. Provided he makes this reservation he is quite privileged to speculate about this concealed world, to bestow upon it any characteristics that he pleases; but it can have no real existence for him until he becomes able to observe it. The only reality he knows is the one he can directly observe.

LAWS OF NATURE

The observations which we have been discussing, and which we have been trying to endow with characteristics of "reality" which they are frequently not realized to possess, are the raw material of physical science. The finished product is the result of bringing together a large number of these observations.]* [The whole underlying thought behind the making of observations, in fact, is to correlate as many as possible of them, to obtain some generalization, and finally to express this in some simple mathematical form. This formulation is then called a "law of nature."]35

[Much confusion exists because of a misunderstanding in the lay mind of what is meant by a "law of nature." It is perhaps not a well chosen term. One is accustomed to associate the word law with the idea of necessity or compulsion. In the realm of nature the term carries no such meaning. The laws of nature are man's imperfect attempts to explain natural phenomena; they are not inherent in matter and the universe, not an iron bar of necessity running through worlds, systems and suns. Laws of nature are little more than working hypotheses, subject to change or alteration or enlargement or even abandonment, as man's vision widens and deepens. No sanctity attaches to them, and if any one, or all, of them fail to account for any part, or all, of the phenomena of the universe, then it or they must be supplemented or abandoned.]102

[The test of one of these laws is that it can be shown to include all the related phenomena hitherto known and that it enables us to predict new phenomena which can then be verified. If new facts are discovered that are not in agreement with one of these generalized statements, the assumptions on which the latter is based are examined, those which are not in accordance with the new facts are given up, and the statement is modified so as to include the new facts.]10 [And if one remembers that the laws of physics were formerly based on a range of observations much narrower than at present available, it seems natural that in the light of this widening knowledge one law or another may be seen to be narrow and insufficient. New theories and laws do not necessarily disprove old ones, but explain certain discrepancies in them and penetrate more deeply into their underlying principles, thereby broadening our ideas of the universe. To follow the new reasoning we must rid ourselves of the prejudice behind the old, not because it is wrong but because it is insufficient. The universe will not be distorted to fit our rules, but will teach us the rules of existence.]125

[Always, however, we must guard against the too easy error of attributing to these rules anything like absolute truth.]* [The modern scientist has attained a very business-like point of view toward his "laws of nature." To him a law is fundamentally nothing but a short-hand way of expressing the results of a large number of experiments in a single statement. And it is important to remember that this mere shortening of the description of a lot of diverse occurrences is by no means any real explanation of how and why they happened. In other words, the aim of science is not ultimately to explain but only to discover the relations that hold good among physical quantities and to embody all these relations in as few and as simple physical laws as possible.]221 [This is inherently the method of relativism.]* [Under it a set of phenomena is observed. There are two or many observers, and they write down their several findings. These are reviewed by a final observer or judge, who strains out the bias due to the different viewpoints of the original observers. He then writes down, not any absolute law of nature governing the observed phenomena, but a law as general as possible expressing their interrelations.]121 [And through this procedure modern science and philosophy reveal with increasing emphasis that we superimpose our human qualities on external nature to such an extent that]106 [we have at once the strongest practical justification, in addition to the arguments of reason, for our insistence that the contact between objective and subjective represented by the observation is the only thing which we shall ever be able to recognize as real. We may indulge in abstract metaphysical speculation to our heart's content, if we be metaphysically inclined; we may not attempt to impose the dicta of metaphysics upon the physical scientist.]*

CONCEPTS AND REALITIES

[From the inquiry and criticism which have gone on for centuries has emerged the following present-day attitude of mind toward the sum total of our knowledge. The conceptual universe in our minds in some mysterious way parallels the real universe, but is totally unlike it. Our conceptions (ideas) of matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space, and time stand in the same or similar relation to reality as the x's and y's of the mathematician do to the entities of his problem. Matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space, and time do not exist actually and really as we conceive them, nor do they have actually and really the qualities and characteristics with which we endow them. The concepts are simply representations of things outside ourselves; things which, while real, have an essential nature not known to us. Matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space and time are merely devices, symbols, which enable us to reason about reality. They are parts of a conceptual mechanism in our minds which operates, or enables our minds to operate, in the same sequence of events as the sequence of phenomena in the external universe, so that when we perceive by our senses a group of phenomena in the external universe, we can reason out what result will flow from the interaction of the realities involved, and thus predict what the situation will be at a given stage in the sequence.

But while our conceptual universe has thus a mechanical aspect, we do not regard the real universe as mechanical in its nature.]283 [This may be illustrated by a little story. Entering his friend's house, a gentleman is seized unawares from behind. He turns his head but sees nothing. His hat and coat are removed and deposited in their proper places by some invisible agent, seats and tables and refreshments appear in due time where they are required, all without any apparent cause. The visitor shivers with horror and asks his host for an explanation. He is then told that the ideas "order" and "regularity" are at work, and that it is they who acquit themselves so well of their tasks. These ideas cannot be seen nor felt nor seized nor weighed; they reveal their existence only by their thoughtful care for the welfare of mankind. I think the guest, coming home, will relate that his friend's house is haunted. The ghosts may be kind, benevolent, even useful; yet ghosts they are. Now in Newtonian mechanics, absolute space and absolute time and force and inertia and all the other apparatus, altogether imperceptible, appearing only at the proper time to make possible a proper building up of the theory, play the same mysterious part as the ideas "order" and "regularity" in my story. Classical mechanics is haunted.]116

[As a matter of fact, we realize this and do not allow ourselves to be imposed upon with regard to the true nature of these agencies.]* [We use a mechanistic terminology and a mechanistic mode of reasoning only because we have found by experience that they facilitate our reasoning. They are the tools which we find produce results. They are adapted to our minds, but perhaps it would be better to say that our minds are so constructed as to render our conceptual universe necessarily mechanical in its aspect in order that our minds may reason at all. Two things antithetic are involved--subject (our perceiving mind which builds up concepts) and object (the external reality); and having neither complete nor absolute knowledge of either, we cannot affirm which is more truly to be said to be mechanistic in its nature, though we may suspect that really neither is. We no longer think of cause and effect as dictated by inherent necessity, we simply regard them as sequences in the routine of our sense-impressions of phenomena. In a word, we have at length grasped the idea that our notions of reality, at present at least, whatever they may become ultimately, are not absolute, but simply relative. We see, too, that we do not explain the universe, but only describe our perceptions of its contents.

The so-called laws of nature are simply statements of formulæ which resume or sum up the relationships and sequences of phenomena. Our effort is constantly to find formulæ which will describe the widest possible range of phenomena. As our knowledge increases, that is, as we perceive new phenomena, our laws or formulæ break down, that is, they fail to afford a description in brief terms of all of our perceptions. It is not that the old laws are untrue, but simply that they are not comprehensive enough to include all of our perceptions. The old laws are often particular or limiting instances of the new laws.]283

[From what we have said of the reality of observations it follows that we must support that school of psychology, and the parallel school of philosophy, which hold that concepts originate in perceptions. But this does not impose so strong a restriction upon conceptions as might appear. The elements of all our concepts do come to us from outside; we manufacture nothing out of whole cloth. But when perception has supplied a sufficient volume of raw material, we may group its elements in ways foreign to actual occurrence in the perceptual world, and in so doing get conceptual results so entirely different from what we have consciously perceived that we are strongly tempted to look upon them as having certainly been manufactured in our minds without reference to the externals. Of even more significance is our ability to abstract from concrete objects and concrete incidents the essential features which make them alike and different. But unlike the Greeks, we see that our concept of coldness is not something with which we were endowed from the beginning, but merely an abstraction from concrete experiences with concrete objects that have been cold.

THE CONCEPTS OF SPACE AND TIME

When we have formed the abstract ideas of coldness and warmth, and have had experience indicating that the occurrence of these properties varies in degree, we are in a position to form the secondary abstract notion covered by the word "temperature." When we have formed the abstract ideas of size and position and separation, we are similarly in a position to form a secondary abstraction to which we give the name "space." Not quite so easy to trace to its definite source but none the less clearly an abstraction based on experience, is our idea of what we call "time." None of us are deceived as to the reality of these abstractions.]* [We do not regard space as real in the sense that we regard a chair as real; it is merely an abstract idea convenient for the location of material objects like the chair.]198 [Nor do we regard time as real in this sense. Things occupy space, events occupy time; space and time themselves we realize are immaterial and unreal; space does not exist and time does not happen in the same sense that material objects exist and events occur. But we find it absolutely necessary to have, among the mental machinery mentioned above as the apparatus by aid of which we keep track of the external world, these vessels for that world to exist in and move in.

Space and time, then, are concepts.]* [It is not strange, however, that when confronted with the vast and bewildering complexity of the universe and the difficulty of keeping separate and distinct in our minds our perceptions and conceptions, we should at times and as respects certain things project our conceptions illegitimately into the perpetual universe and mistake them for perceptions. The most notable example perhaps of this projection has occurred in the very case of space and time, most fundamental of all of our concepts. We got to think of these as absolute, as independent of each other and of all other things, and as always existing and continuing to exist whether or not we or anything else existed--space as a three-dimensional, uniform continuum, having the same properties in all directions; time as a one-dimensional, irreversible continuum, flowing in one direction. It is difficult to get back to the idea that space and time so described and defined are concepts merely, for the idea of their absolute existence is ingrained in us as the result probably of long ancestral experience.]283

[Newton's definitions of course represent the classical idea of time and space. He tells us that "absolute, true and mathematical time flows in virtue of its own nature, uniformly and without reference to any external object;" and that "absolute space, by virtue of its own nature and without reference to any external object, always remains the same and is immovable." Of course from modern standpoints it is absurd to call either of these pronouncements a definition; but they represent about as well as any words can the ideas which Newton had about time and space, and they make it clear enough that he regarded both as having real existence in the external world.

If space and time are to be the vessels of our universe, and if the only thing that really matters is measured results, it is plain enough that we must have, from the very beginning, means of measuring space and time. Whether we believe space and time to have real existence or not, it is obvious that we can measure neither directly. We shall have to measure space by measuring from one material object to another; we shall have to measure time by some similar convention based on events. We shall later have something further to say about the measurement of time; for the present we need only point out that]* [Newtonian time is measured independently of space; and the existence is presupposed of a suitable timekeeper.]10

[The space of Galileo and Newton was conceived of as empty, except in so far as certain parts of it were occupied by matter. Positions of bodies in this space were in general determined by reference to]283 [a "coordinate system" of some kind. This is again something that demands a certain amount of discussion.

THE REFERENCE FRAME FOR SPACE

The mathematician, following the lead of the great French all-around genius, Descartes, shows us very clearly how to set up, for the measurement of space, the framework known as the Cartesian coordinate system. The person of most ordinary mathematical attainments will realize that to locate a point in a plane we must have two measurements; and we could probably show this person, without too serious difficulty, that we can locate a point in any surface by two measurements. An example of this is the location of points on the earth's surface by means of their latitude and longitude. It is equally clear that if we add a third dimension and attempt to locate points in space, we must add a third measurement. In the case of points on the earth's surface, this might be the elevation above sea level, which would define the point not as part of the spherical surface of the earth but as part of the solid sphere. Or we may fall back on Dr. Slosson's suggestion that in order to define completely the position of his laboratory, we must make a statement about Broadway, and one about 116th Street, and one telling how many flights of stairs there are to climb. In any event, it should be clear enough that the complete definition of a point in space calls for three measurements.

The mathematician formulates all this with the utmost precision. He asks us to]* [pick out any point whatever in space and call it $O$. We then draw or conceive to be drawn through this point three mutually perpendicular lines called coordinate axes, which we may designate $OX$, $OY$ and $OZ$, respectively. Finally, we consider the three planes also mutually perpendicular like the two walls and the floor of a room that meet in one common corner, which are formed by the lines $OX$ and $OY$, $OY$ and $OZ$, and $OZ$ and $OX$, respectively. These three planes are called coordinate planes. And then any other point $P$ in space can be represented with respect to $O$ by its perpendicular distances from each of the three coordinate planes--the distances $x$, $y$, $z$ in the figure. These quantities are called the coordinates of the point.]272

[To the layman there seems something altogether naive in this notion of the scientist's setting up the three sides of a box in space and using them as the basis of all his work. The layman somehow feels that while it is perfectly all right for him to tell us that he lives at 1065 (one coordinate) 156th Street (two coordinates) on the third floor (three coordinates), it is rather trivial business for the serious-minded scientist to consider the up-and-down, the forward-and-back, the right-and-left of every point with which he has occasion to deal. There seems to the layman something particularly inane and foolish and altogether puerile about a set of coordinate axes, and you simply can't make him believe that the serious-minded scientist has to monkey with any such funny business. He can't be induced to take this coordinate-axis business seriously. Nevertheless, the fact is that the scientist takes it with the utmost seriousness. It is necessary for him to define the positions of points; and he does do it by means of a set of coordinate axes.

The scientist, however, is not interested in points of empty space. The point is to him merely part again of the conceptual machinery which he uses in his effort to run along with the external world. He knows there are no real points, but it suits his convenience to keep track of certain things that are real by representing them as points. But these things are in practically every instance material bodies; and in practically every instance, instead of staying put in one spot, they insist upon moving about through space. The scientist has to use his coordinate system, not merely to define a single position of such a "point," but to keep track of the path over which it moves and to define its position in that path at given moments.

TIME AND THE COORDINATE SYSTEM

This introduces the concept of time into intimate relationship with the spatial coordinate system. And at once we feel the lack of a concrete, visualized fourth dimension.]* [If we want to fix objects in the floor alone, the edge of the room running toward the ceiling would become unnecessary and could be dropped from our coordinate system. That is, we need only two coordinates to fix the position of a point in a plane. Suppose instead of discarding the third coordinate, we use it to represent units of time. It then enables us to record the time it took a moving point in the floor to pass from position to position. Certain points in the room would be vertically above the corresponding points occupied by the moving point in its path across the floor; and the vertical height above the floor of such points corresponds to a value of the time-coordinate which indicates the time it took the point to move from position to position.]152 [Just as the path of the point across the floor is a continuous curve (for the mathematician, it should be understood, this term "curve" includes the straight line, as a special case in which the curvature happens to be zero); so the series of points above these in the room forms a continuous curve which records for us, not merely the path of the point across the floor, but in addition the time of its arrival at each of its successive positions. In the algebraic work connected with such a problem, the third coordinate behaves exactly the same, regardless of whether we consider it to represent time or a third spatial dimension; we cannot even tell from the algebra what it does represent.

When we come to the more general case of a point moving freely through space, we have but three coordinates at our disposal; there is not a fourth one by aid of which we can actually diagram its time-space record. Nevertheless, we can write down the numerical and algebraic relations between its three space-coordinates and the time which it takes to pass from one position to another; and by this means we can make all necessary calculations. Its motion is completely defined with regard both to space and to time. We are very apt to call attention to the fact that if we did have at our disposal a fourth, space-coordinate, we could use it to represent the time graphically, as before, and actually construct a geometric picture of the path of our moving point with regard to space and time. And on this account we are very apt to speak as though the time measurements constituted a fourth coordinate, regardless of any question of our ability to construct a picture of this coordinate. The arrival of a point in a given position constitutes an event; and this event is completely defined by means of four coordinates--three in space, which we can picture on our coordinate axes, and one in time which we cannot.

The set of coordinate axes in space, together with the zero point from which we measure time, constitute what we call a frame of reference. If we are not going to pay any attention to time, we can think of the space coordinate system alone as constituting our reference frame. This expression appears freely throughout the subsequent text, and always with one or the other of these interpretations.

We see, then, how we can keep track of a moving point by keeping track of the successive positions which it occupies in our reference frame.]* [Now we have implied that these coordinate axes are fixed in space; but there is nothing to prevent us from supposing that they move.]272 [If they do, they carry with them all their points; and any motion of these points which we may speak about will be merely motion with reference to the coordinate system. If we find something outside our coordinate system that is not moving, the motion of points in our system with regard to those outside it will be a combination of their motion with regard to our coordinate axes and that of these axes with regard to the external points. This will be a great nuisance; and it represents a state of affairs which we shall try to avoid. We shall avoid it, if at all, by selecting a coordinate system with reference to which we, ourselves, are not moving; one which partakes of any motion which we may have. Or perhaps we shall sometimes wish to reverse the process, in studying the behavior of some group of bodies, and seek a set of axes which is at rest with respect to these bodies; one which partakes of any motion they may have.

THE CHOICE OF A COORDINATE FRAME

All this emphasizes the fact that our coordinate axes are not picked out for us in advance by nature, and set down in some one particular spot. We select them for ourselves, and we select them in the most convenient way. But different observers, or perhaps the same observer studying different problems, will find it advantageous to utilize different coordinate systems.]* [The astronomer has found it possible, and highly convenient, to select a coordinate frame such that the great majority of the stars have, on the whole, no motion with respect to it.]283 [Such a system would be most unsuited for investigations confined to the earth; for these we naturally select a framework attached to the earth, with its origin O at the earth's center if our investigation covers the entire globe and at some more convenient point if it does not, and in either event accompanying the earth in its rotation and revolution. But such a framework, as well as the one attached to the fixed stars, would be highly inconvenient for an investigator of the motions of the planets; he would doubtless attach his reference frame to the sun.]101

[In this connection a vital question suggests itself. Is the expression of natural law independent of or dependent upon the choice of a system of coordinates? And to what extent shall we be able to reconcile the results of one observer using one reference frame, and a second observer using a different one? The answer to the second question is obvious.]* [True, if any series of events is described using two different sets of axes, the descriptions will be different, depending upon the time system adopted and the relative motion of the axes. But if the connection between the reference systems is known, it is possible by mathematical processes to deduce the quantities observed in one system if those observed in the other are known.]35 [This process of translating the results of one observer into those of another is known as a transformation; and the mathematical statement of the rule governing the transformation is called the equation or the equations (there are usually several of them) of the transformation.]* [Transformations of this character constitute a well-developed branch of mathematics.]35

[When we inquire about the invariance of natural law it is necessary to be rather sure of just what we mean by this expression. The statement that a given body is moving with a velocity of 75 miles per hour is of course not a natural law; it is a mere numerical observation. But aside from such numerical results, we have a large number of mathematical relations which give us a more or less general statement of the relations that exist between velocities, accelerations, masses, forces, times, lengths, temperatures, pressures, etc., etc. There are some of these which we would be prepared to state at once as universally valid--distance travelled equals velocity multiplied by time, for instance. We do not believe that any conceivable change of reference systems could bring about a condition in which the product of velocity and time, as measured from a certain framework, would fail to equal distance as measured from this same framework. There are other relations more or less of the same sort which we probably believe to be in the same invariant category; there are others, perhaps, of which we might be doubtful; and presumably there are still others which we should suspect of restricted validity, holding in certain reference systems only and not in others.

The question of invariance of natural law, then, may turn out to be one which may be answered in the large by a single statement; it may equally turn out to be one that has to be answered in the small, by considering particular laws in connection with particular transformations between particular reference systems. Or, perhaps, we may find ourselves justified in taking the stand that an alleged "law of nature" is truly such a law only in the event that it is independent of the change from one reference system to another. In any event, the question may be formulated as follows:

Observer A, using the reference system R, measures certain quantities $t$, $w$, $x$, $y$, $z$. Observer B, using the reference system S, measures the same items and gets the values $t'$, $w'$, $x'$, $y'$, $z'$. The appropriate transformation equations for calculating the one set of values from the other is found. If a mathematical relation of any sort is found to exist between the values $t$, $w$, $x$, $y$, $z$, will the same relation exist between the values $t'$, $w'$, $x'$, $y'$, $z'$? If it does not, are we justified in still calling it a law of nature? And if it does not, and we refrain from calling it such a law, may we expect in every case to find some relation that will be invariant under the transformation, and that may therefore be recognized as the natural law connecting $t$, $w$, $x$, $y$ and $z$?

I have found it advisable to discuss this point in such detail because here more than in any other single place the competing essayists betray uncertainty of thought and sloppiness of expression. It doesn't amount to much to talk about the invariance of natural laws and their persistence as we pass from one coordinate system to another, unless we are fairly well fortified with respect to just what we mean by invariance and by natural law. We don't expect the velocity of a train to be 60 miles per hour alike when we measure it with respect to a signal tower along the line and with respect to a moving train on the other track. We don't expect the angular displacement of Mars to change as rapidly when he is on the other side of the sun as when he is on our side. But we do, I think, rather expect that in any phenomenon which we may observe, we shall find a natural law of some sort which is dependent for its validity neither upon the units we employ, nor the place from which we make our measurements, nor anything else external to the phenomenon itself. We shall see, later, whether this expectation is justified, or whether it will have to be discarded in the final unravelling of the absolutist from the relativistic philosophy which, with Einstein, we are to undertake.]*

III

THE RELATIVITY OF UNIFORM MOTION

Classical Ideas on the Subject; the Ether and the Apparent Possibility of Absolute Motion; the Michelson-Morley Experiment and the Final Negation of This Possibility

BY VARIOUS CONTRIBUTORS AND THE EDITOR

When we speak of a body as being "in motion," we mean that this body is changing its position "in space." Now it is clear that the position of an object can only be determined with reference to other objects: in order to describe the place of a material thing we must, for example, state its distances from other things. If there were no such bodies of reference, the words "position in space" would have no definite meaning for us.]24 [The number of such external bodies of reference which it is necessary to cite in order to define completely the position of a given body in space depends upon the character of the space dealt with. We have seen that when we visualize the space of our experience as a surface of any character, two citations are sufficient; and that when we conceive of it as surrounding us in three dimensions we require three. It will be realized that the mathematician is merely meeting this requirement when he sets up his system of coordinate axes to serve as a reference frame.]*

[What is true of "place" must be true also of "motion," since the latter is nothing but change of place. In fact, it would be impossible to ascribe a state of motion or of rest to a body poised all alone in empty space. Whether a body is to be regarded as resting or as moving, and if the latter at what speed, depends entirely upon the objects to which we refer its positions in space.]24 [As Einstein sits at his desk he appears to us to be at rest; but we know that he is moving with the rotation of the earth on its axis, with the earth in its orbit about the sun, and with the solar system in its path through space--a complex motion of which the parts or the whole can be detected only by reference to appropriately chosen ones of the heavenly bodies. No mechanical test has ever been devised which will detect this motion,]182 [if we reserve for discussion in its proper place the Foucault pendulum experiment which will reveal the axial rotation of our globe.]* [No savage, if he were to "stand still," could be convinced that he was moving with a very high velocity or in fact that he was moving at all.]30 [You drop a coin straight down a ship's side: from the land its path appears parabolic; to a polar onlooker it whirls circle-wise; to dwellers on Mars it darts spirally about the sun; to a stellar observer it gyrates through the sky]263 [in a path of many complications. To you it drops in a straight line from the deck to the sea.]* [Yet its various tracks in ship-space, sea-space, earth-space, sun-space, star-space, are all equally real,]263 [and the one which will be singled out for attention depends entirely upon the observer, and the objects to which he refers the motion.]* [The earth moves in the solar system, which is itself approaching a distant star-cluster. But we cannot say whether we are moving toward the cluster, or the cluster toward us,]18 [or both, or whether we are conducting a successful stern chase of it, or it of us,]* [unless we have in mind some third body with reference to which the motions of earth and star-cluster are measured.]18 [And if we have this, the measurements made with reference to it are of significance with regard to it, rather than with regard to the earth and the star-cluster alone.]*

[We can express all this by saying "All motions are relative; there is no such thing as absolute motion." This line of argument has in fact been followed by many natural philosophers. But is its result in agreement with actual experience? Is it really impossible to distinguish between rest and motion of a body if we do not take into consideration its relations to other objects? In fact it can easily be seen that, at least in many cases, no such distinction is possible.

WHO IS MOVING?

Imagine yourself sitting in a railroad car with veiled windows and running on a perfectly straight track with unchanging velocity: you would find it absolutely impossible to ascertain by any mechanical means whether the car were moving or not. All mechanical instruments behave exactly the same, whether the car be standing still or in motion.]24 [If you drop a ball you will see it fall to the floor in a straight line, just as though you had dropped it while standing on the station platform. Furthermore, if you drop the ball from the same height in the two cases, and measure the velocities with which it strikes the car floor and the station platform, or the times which it requires for the descent, you will find these identical in the two cases.]182

[Any changes of speed or of direction (as when the car speeds up or slows down or rounds a curve) can be detected by observing the behavior of bodies in the car, without apparent reference to any outside objects. This becomes particularly obvious with sudden irregularities of motion, which manifest themselves by shaking everything in the car. But a uniform motion in a straight line does not reveal itself by any phenomenon within the vehicle.]24

[Moreover, if we remove the veil from our window to the extent that we may observe the train on the adjoining track, we shall be able to make no decision as to whether we or it be moving. This is indeed an experience which we have all had.]* [Often when seated in a train about to leave the station, we have thought ourselves under way, only to perceive as the motion becomes no longer uniform that another train has been backing into the station on the adjoining track. Again, as we were hurried on our journey, we have, raising suddenly our eyes, been puzzled to say whether the passing train were moving with us or against us or indeed standing still; or more rarely we have had the impression that both it and we seemed to be at rest, when in truth both were moving rapidly with the same speed.]82 [Even this phrase "in truth" is a relative one, for it arises through using the earth as an absolute reference body. We are indeed naive if we cannot appreciate that there is no reason for doing this beyond convenience, and that to an observer detached from the earth it were just as reasonable to say that the rails are sliding under the train as that the train is advancing along the rails. One of my own most vivid childhood recollections is of the terror with which, riding on a train that passed through a narrow cut, I hid my head in the maternal lap to shut out the horrid sight of the earth rushing past my window. The absence of a background in relatively slow retrograde motion was sufficient to prevent my consciousness from drawing the accustomed conclusion that after all it was really the train that was moving.]*

MECHANICAL RELATIVITY

[So we can enunciate the following principle: When a body is in uniform rectilinear motion relatively to a second body, then all phenomena take place on the first in exactly the same manner as on the second; the physical laws for the happenings on both bodies are identical.]24 [And between a system of bodies, nothing but relative motion may be detected by any mechanical means whatever; any attempt to discuss absolute motion presupposes a super-observer on some body external to the system. Even then, the "absolute" motion is nothing but motion relative to this super-observer. By no mechanical means is uniform straight-line motion of any other than relative character to be detected. This is the Principle of Mechanical Relativity.

There is nothing new in this. It was known to Galileo, it was known to Newton, it has been known ever since. But the curious persistence of the human mind in habits of thought which confuse relativity with absolutism brought about a state of affairs where we attempted to know this and to ignore it at the same time. We shall have to return to the mathematical mode of reasoning to see how this happened. The mathematician has a way all his own of putting the statement of relativity which we have made. He recalls, what we have already seen, that the observer on the earth who is measuring his "absolute" motion with respect to the earth has merely attached his reference framework to the earth; that the passenger in the train who measures all motion naively with respect to his train is merely carrying his coordinate axes along with his baggage, instead of leaving them on the solid ground; that the astronomer who deals with the motion of the earth about the sun, or with that of the "fixed" stars against one another, does so simply by the artifice of hitching his frame of reference to the sun or to one of the fixed stars. So the mathematician points out that dispute as to which of two bodies is in motion comes right down to dispute as to which of two sets of coordinate axes is the better one, the more nearly "natural" or "absolute." He therefore phrases the mechanical principle of relativity as follows:

Among all coordinate systems that are merely in uniform straight-line motion to one another, no one occupies any position of unique natural advantage; all such systems are equivalent for the investigation of natural laws; all systems lead to the same laws and the same results.

The mathematician has thus removed the statement of relativity from its intimate association with the external observed phenomena, and transferred it to the observer and his reference frame. We must either accept the principle of relativity, or seek a set of coordinate axes that have been singled out by nature as an absolute reference frame. These axes must be in some way unique, so that when we refer phenomena to them, the laws of nature take a form of exceptional simplicity not attained through reference to ordinary axes. Where shall we look for such a preferred coordinate system?]*

THE SEARCH FOR THE ABSOLUTE

[Older theory clung to the belief that there was such a thing as absolute motion in space.]197 [As the body of scientific law developed from the sixteenth century onward, the not unnatural hypothesis crept in, that these laws (that is to say, their mathematical formulations rather than their verbal statements) would reveal themselves in especially simple forms, were it possible for experimenters to make their observations from some absolute standpoint; from an absolutely fixed position in space rather than from the moving earth.]264 [Somewhere a set of coordinate axes incapable of motion was to be found,]197 [a fixed set of axes for measuring absolute motion; and for two hundred years the world of science strove to find it,]147 [in spite of what should have been assurance that it did not exist. But the search failed, and gradually the universal applicability of the principle of relativity, so far as it concerned mechanical phenomena, grew into general acceptance.]* [And after the development, by the great mathematicians of the eighteenth century, of Newton's laws of motion into their most complete mathematical form, it was seen that so far as these laws are concerned the absolutist hypothesis mentioned is quite unsupported. No complication is introduced into Newton's laws if the observer has to make his measurements in a frame of reference moving uniformly through space; and for measurements in a frame like the earth, which moves with changing speed and direction about the sun and rotates on its axis at the same time, the complication is not of so decisive a nature as to give us any clue to the earth's absolute motion in space.

But mechanics, albeit the oldest, is yet only one of the physical sciences. The great advance made in the mathematical formulation of optical and electromagnetic theory during the nineteenth century revived the hope of discovering absolute motion in space by means of the laws derived from this theory.]264 [Newton had supposed light to be a material emanation, and if it were so, its passage across "empty space" from sun and stars to the earth raised no problem. But against Newton's theory Huyghens, the Dutch astronomer, advanced the idea that light was a wave motion of some sort. During the Newtonian period and for many years after, the corpuscular theory prevailed; but eventually the tables were turned.]* [Men made rays of light interfere, producing darkness (see page 61). From this, and from other phenomena like polarization, they had deduced that light was a form of wave motion similar to water ripples; for these interfere, producing level surfaces, or reinforce each other, producing waves of abnormal height. But if light were to be regarded as a form of wave motion--and the phenomena could apparently be explained on no other basis--then there must be some medium capable of undergoing this form of motion.]135 [Transmission of waves across empty space without the aid of an intermediary material medium would be "action at a distance," an idea repugnant to us. Trammeled by our tactual, wire-pulling conceptions of a material universe, we could not accustom ourselves to the idea of something--even so immaterial a something as a wave--being transmitted by nothing. We needed a word--ether--to carry light if not to shed it; just as we need a word--inertia--to carry a projectile in its flight.]231 [It was necessary to invest this medium with properties to account for the observed facts. On the whole it was regarded as the perfect fluid.]235 [The ether was imagined as an all-pervading, imponderable substance filling the vast emptiness through which light reaches us, and as well the intermolecular spaces of all matter. Nothing more was known definitely, yet this much served as a good working hypothesis on the basis of which Maxwell was enabled to predict the possibility of radio communication. By its fruits the ether hypothesis justified itself; but does the ether exist?]231

THE ETHER AND ABSOLUTE MOTION

[If it does exist, it seems quite necessary, on mere philosophical grounds, that it shall be eligible to serve as the long-sought reference frame for absolute motion. Surely it does not make sense to speak of a homogeneous medium filling all space, sufficiently material to serve as a means of communication between remote worlds, and in the next breath to deny that motion with respect to this medium is a concept of significance.]* [Such a system of reference as was offered by the ether, coextensive with the entire known region of the universe, must necessarily serve for all motions within our perceptions.]186 [The conclusion seems inescapable that motion with respect to the ether ought to be of a sufficiently unique character to stand out above all other motion. In particular, we ought to be able to use the ether to define, somewhere, a system of axes fixed with respect to the ether, the use of which would lead to natural laws of a uniquely simply description.

Maxwell's work added fuel to this hope.]* [During the last century, after the units of electricity had been defined, one set for static electrical calculations and one for electromagnetic calculations, it was found that the ratio of the metric units of capacity for the two systems was numerically equal to what had already been found as the velocity with which light is transmitted through the hypothetical ether. One definition refers to electricity at rest, the other to electricity in motion. Maxwell, with little more working basis than this, undertook to prove that electrical and optical phenomena were merely two aspects of a common cause,]235 [to which the general designation of "electromagnetic waves" was applied. Maxwell treated this topic in great fullness and with complete success. In particular, he derived certain equations giving the relations between the various electrical quantities involved in a given phenomenon. But it was found, extraordinarily enough, that these relations were of such character that, when we subject the quantities involved to a change of coordinate axes, the transformed quantities did not preserve these relations if the new axes happened to be in motion with respect to the original ones. This, of course, was taken to indicate that motion really is absolute when we come to deal with electromagnetic phenomena, and that the ether which carries the electromagnetic waves really may be looked to to display the properties of an absolute reference frame.

Reference to the phenomenon of aberration, which Dr. Pickering has discussed adequately in his essay and which I need therefore mention here only by name, indicated that the ether was not dragged along by material bodies over and through which it might pass. It seemed that it must filter through such bodies, presumably via the molecular interstices, without appreciable opposition. Were this not the case, we should be in some doubt as to the possibility of observing the velocity through the ether of material bodies; if the ether adjacent to such bodies is not dragged along or thrown into eddies, but "stands still" while the bodies pass, there seems no imaginable reason for anything other than the complete success of such observations. And of course these are of the utmost importance, the moment we assign to the ether the rôle of absolute reference frame.

THE EARTH AND THE ETHER

One body in motion with respect to the ether is our earth itself. We do not know in advance in what direction to expect this motion or what magnitude to anticipate that it will have. But one thing is clear.]* [In its motion around the sun, the earth has, at opposite points on its orbit, a difference in velocity with respect to the surrounding medium which is double its orbital velocity with respect to the sun. This difference comes to 37 miles per second. The earth should therefore, at some time in the year, show a velocity equal to or greater than 18 1/2 miles per second, with reference to the universal medium. The famous Michelson-Morley experiment of 1887 was carried out with the expectation of observing this velocity.]267

[The ether, of course, and hence velocities through it, cannot be observed directly. But it acts as the medium for the transmission of light.]* [If the velocity of light through the ether is $C$ and that of the earth through the ether is $v$, then the velocity of light past the earth, so the argument runs, must vary from $C-v$ to $C+v$, according as the light is moving exactly in the same direction as the earth, or in the opposite direction,]182 [or diagonally across the earth's path so as to get the influence only of a part of the earth's motion. This of course assumes that $C$ has always the same value; an assumption that impresses one as inherently probable, and one that is at the same time in accord with ordinary astronomical observation.

It is not possible to measure directly the velocity of light (186,330 miles per second, more or less) with sufficient accuracy to give any meaning to the variation in this velocity which might be effected by adding or subtracting that of the earth in its orbit (a mere 18 1/2 miles per second). It is, however, possible to play a trick on the light by sending it back and forth over several paths, and comparing (not measuring absolutely, but merely comparing) with great minuteness the times consumed in these several round trips.

A JOURNEY UPSTREAM AND BACK

The number of letters the Scientific American has received questioning the Michelson-Morley experiment indicates that many people are not acquainted with the fundamental principle on which it is based. So let us look at a simple analogous case. Suppose a swimmer or a rower make a return trip upstream and down, contending with the current as he goes up and getting its benefit when he comes down. Obviously, says snap judgment, since the two legs of the journey are equal, he derives exactly as much benefit from the current when he goes with it as he suffers handicap from it when he goes against it; so the round trip must take exactly the same time as a journey of the same length in still water, the argument applying equally in the case where the "swimmer" is a wave of light in the ether stream.

But let us look now at a numerical case. A man can row in still water at four miles per hour. He rows twelve miles upstream and back, in a current of two miles per hour. At a net speed of two miles per hour he arrives at his turning point in six hours. At a net speed of six miles per hour he makes the down-stream leg in two hours. The elapsed time for the journey is eight hours; in still water he would row the twenty-four miles in six hours.

If we were to attempt an explanation of this result in words we should say that by virtue of the very fact that it does delay him, the adverse current prolongs the time during which it operates; while by virtue of the very fact that it accelerates his progress, the favoring current shortens its venue. The careless observer realizes that distances are equal between the two legs of the journey, and unconsciously assumes that times are equal.

If the journey be made directly with and directly against the stream of water or ether or what not, retardation is effected to its fullest extent. If the course be a diagonal one, retardation is felt to an extent measurable as a component, and depending for its exact value upon the exact angle of the path. Felt, however, it must always be.