Chapter I: Part 1

DR. ALBERT EINSTEIN IN HIS STUDY AT BERLIN.]

EASY LESSONS IN
EINSTEIN

A DISCUSSION OF THE MORE
INTELLIGIBLE FEATURES OF
THE THEORY OF RELATIVITY

BY

EDWIN E. SLOSSON, M.S., Ph.D.

_Literary Editor of The Independent, Associate in the Columbia School
of Journalism. Author of “Great American Universities,”
“Major Prophets of To-day,” “Six Major Prophets,”
“Creative Chemistry,” etc._

_With an Article by Albert Einstein
and a Bibliography_

ILLUSTRATED

NEW YORK
HARCOURT, BRACE AND HOWE
1920

COPYRIGHT, 1920, BY
HARCOURT, BRACE AND HOWE, INC.

THE QUINN & BODEN COMPANY
RAHWAY, N. J.

_Deepest of all illusory Appearances, for hiding Wonder, as for many other ends, are your two grand fundamental world-enveloping Appearances, Space and Time._--CARLYLE.

_Henceforth Space in itself and Time in itself sink into mere shadows and only a kind of union of the two can be maintained as self-existent._--MINKOWSKI.

A PREFATORIAL DIALOGUE

(The Purpose of which is to Prevent the Prospective Reader from buying the Book under False Pretenses)

SCENE: A street car in uniform movement of translation in any direction.

TIME: The present.

_The Reader_: (looking over the top of a morning paper): Here’s something queer--a whole page taken with a new discovery in physics--“Eclipse Observations Confirm Einstein’s Theory of Relativity.” Anything about it in your paper?

_The Author_: Yes. Here’s a cartoon on it by McCutcheon.

_The Reader_: Must be something to it then. McCutcheon always knows what’s news. (Reads on with audible fragments) “Most sensational discovery in the history of science”--“Greatest achievement of the human intellect”--“Upsets Galileo, Newton, and Euclid”--“Revolution in philosophy and theology.” It looks as though I ought to know something about this, doesn’t it?

_The Author_: I think you will have to sometime. And you might as well do it now and get it over with.

_The Reader_: (running down the column and hitting the high spots): “Parallel lines meet”--“a man moving with the speed of light never grows old”--“gravitation due to a warp in space”--“length of a measuring stick depends upon direction of its motion”--“mass is latent energy”--“time as a fourth dimension”--why, the man is crazy, isn’t he?

_The Author_: Well, definitions of insanity are so uncertain that it is not safe to say who is crazy. But it seems there’s method in his madness--otherwise how could he have hit upon the exact extent of the sun’s attraction on light?

_The Reader_: (Picks up his paper and reads aloud with concentrated attention) “Postulate I. Every law of nature which holds good with respect to a coördinate system K must also hold good for any other system K′, provided that K and K′ are in uniform movement of translation.” Say, do you know anything about this business?

_The Author_: Well, yes, a little. I have followed the controversy--at a safe distance--for a number of years.

_The Reader_: Can you tell me in plain language what it is all about?

_The Author_: Yes. Just that. I can tell you what it is _about_, though I can’t tell you what it _is_. Einstein says that there are only twelve men in the world capable of understanding his latest paper.

_The Reader_: Are you one of the twelve?

_The Author_: No, nor the thirteenth. But without plunging into the mathematics of it, we might talk over some of the interesting aspects of the theory of relativity and in the end I could put you on track of the twelve so you could read up on the subject if you liked.

_The Reader_: All right. That’s fair. This is a slow car anyhow. Go ahead.

_The Author_: (See following pages)--

EASY LESSONS IN EINSTEIN

“_A warp in nature has been found,_
_No line is straight, no circle round;_
_For Isaac Newton had unsound_
_Ideas of gravitation._”

Why is it that our newspapers are sending out their reporters to interview astronomers as well as actresses and devoting pages to speculations on the nature of space and time as well as on the state of the market? It is--to get at the bottom of it--merely because a few photographs taken during the eclipse of the sun on May 29, 1919, by two telescopes, one at Sobral in northern Brazil and the other on the island of Principe off the west coast of Africa, showed an abnormal shift of less than one-324,000th of a right angle in the position of the stars. When these photograph films were laid over films taken before the eclipse it was found that the star-images about the darkened disk of the sun did not exactly coincide with the images when the sun was not in their midst. Measured with a micrometer the displacement of the stars from their ordinary positions was found to be 1.60 seconds of arc on the African plates and 1.98 seconds on the Brazilian plates. Average these two observations and you get 1.79. This is extremely close to the 1.73 predicted by Professor Einstein of Berlin and twice as large as the deflection calculated according to Newton’s law of gravitation which would be .87 of a second.

When the announcement of this result was made at the meeting of the Royal Society of London on November 6 all eyes were turned toward Sir Oliver Lodge, for last February he had been rash enough to express the hope, if not the prediction, that the results of the eclipse expedition would support Newton rather than Einstein. But instead of taking part in the discussion Sir Oliver got up and walked out. It was suspected that he had “gone off mad,” as we Americans would put it, because the starlight would not follow his preferred path. But he put a stop to any such rumors by a letter to _The Times_ in which he explains that his departure was not due to any dissatisfaction with the universe but to the necessity of catching the 6 o’clock train. He frankly acknowledges that “the eclipse result is a great victory for Einstein; the quantitative agreement is too close to allow much room for doubt” but he adds “a caution against a strengthening of great and complicated generalizations concerning space and time on the strength of this splendid result: I trust that it may be accounted for, with reasonable simplicity in terms of the ether of space.”

This caution is wise, but we cannot hold our breath till 1922, when the next eclipse comes, to see if these observations are verified and we may in the meantime consider some of the implications of Einstein’s theory of relativity.

Sir Joseph Thomson, President of the Royal Society, in making the momentous announcement in the session of the Society, said:

If his theory is right, it makes us take an entirely new view of gravitation. If it is sustained that Einstein’s reasoning holds good--and it has sustained two very severe tests in connection with the perihelion of Mercury and the present eclipse--then it is the result of one of the highest achievements of human thought. The weak point in the theory is the great difficulty in expressing it. It would seem that no one can understand the new law of gravitation without a thorough knowledge of the theory of invariants and of the calculus of variations.

What is this theory of relativity and why is it so important? The mathematics of it are too much for most of us, but we can get some notion of it by a familiar illustration.

Suppose you wake up some morning in a Pullman berth and look out of the window to see where you are. You find your view blocked by a passing train on the next track. Now if you do not feel any jar of your car and cannot catch sight of the landscape beyond the other train you cannot tell whether (1) your train is moving forward and the other train is standing still, or (2) your train is standing still and the other train is moving backward, or (3) whether both trains are moving in opposite directions, or (4) whether both trains are moving in the same direction, but your train faster. It is obvious that the trains are getting past one another. You can measure their speed of parting as accurately as you please. But all you can perceive is the relative motion of the two trains. You begin to wonder whether there is any such thing as absolute motion; whether there is any real difference between rest and motion. Is there any possible way of telling whether your train is in motion or not if all you can see out of the window is some object that itself be moving? Suppose the windows were all curtained, how could you find out whether you were moving forward or backward or standing still?

You discuss this curious question with your fellow passengers at the breakfast table and one of them makes the brilliant suggestion that it might be possible to determine the absolute motion of the car by reference to the air. If the car is moving forward the air would stream from front to rear and the reverse if it were moving backward. “Suppose,” says the ingenious experimentalist, “that you stand at one end of the car and I at the other. We will shout at each other alternately and time the passage of the sound with our stop watches. Since sound is carried by air waves it will take longer for the shout to go against the air current than with it, and from that measurement it might be possible for us not only to determine which way the car is moving but how to calculate how fast it travels, assuming, of course, that there is no wind blowing.” That strikes you as a crucial experiment, but you point out one possible difficulty, that the doors at the ends of the car may be closed and the air inside is being carried along with the car, so no difference would be observable in the speed of the sound even though the car were moving. “All right,” replies your scientific friend, “we will make a preliminary test to see if the enclosed air is carried along with the car, and if we find that it is not then we will try the second experiment with the sound signals to see which way the air current is moving. These two experiments must settle it, for either the air is moving with the car or it is moving through the car. Can you conceive of any other possibility than these two?” No, you cannot, so you proceed to try the two experiments. First you visit both ends of the car and find both doors open; the air then is not being carried along with the car. You turn then with confidence to the second experiment and you find, of course, that there is a difference in the speed of sound whether it moves with the air drift or against it.

There might, I admit, be practical difficulties in the way of carrying out such a delicate experiment on a moving train, but we need not bother with them, for probably the current of air through the car would be so strong as to blow your hat out of the back door and that would settle the question to your satisfaction--or at least it would settle the question in the affirmative.

But imagine your amazement if this second experiment should give negative results like the first one; if you could detect no difference in time whether the sound was sent forward or back or across the car. You would then have proved by experiment (1) that the air did not move with the car and (2) that the air did not move through the car. You might suppose from this that your car is at rest, but suppose the people on the other train passing yours tried the same experiments and got the same result, namely, that they, too, were at rest as regards the air. You would then be in a quandary, for your two indisputable experiments had apparently given contradictory results. You might get out of it by saying that there was no air, but if not what carried the sound waves--and the hat?

CONTRADICTORY EXPERIMENTS

Now this is the quandary in which physicists have been in for the last thirty-three years. Is there any way of _discovering_ absolute motion among the heavenly bodies? We can observe and measure with great accuracy their relative motion. The sun is seen to pass across the sky from east to west and man at first assumed that the earth was still and the sun went around it. This is the natural and instinctive assumption, for when you first glance out of your Pullman window you get the impression that the other train is the moving one. But for the last three hundred years it has been the fashion to assume the earth was moving and not the sun. That assumption has the advantage of simplifying the calculations of the astronomers, though I never could see why we should have to give up our simple notions of sunrise and sunset to save them a little trouble figuring.

The earth moves--if it does move--so quietly and silently that we feel no jar or engine-beat to tell us of its motion. If the earth were perpetually shrouded by clouds could we find out its motion through space or even its rotation? And do we actually get any proof on this point from observation of the heavenly bodies? We see them moving about relatively to each other and we can represent their movements most easily by supposing that the moon goes around the earth and that the earth and the rest of the planets go around the sun. But is this whole solar system in motion? So it seems when we compare it with the stars. But who knows if the solar system and all the visible stars are not altogether moving off through space at the rate of a mile or a thousand miles a second? How can we tell unless we have something that is still and fixed to measure the motion by?

It seemed until recently that we had such a fixture, the ether. We know of the sun and stars only from the light that comes from them to us. Light, as we can prove by simple experiments, consists of wave motion. Now, can you have wave motion without something to wave? Sound waves are conveyed by air but there is no air between the earth and the sun. So as nothing could be found to fill this empty space scientists had to invent something to satisfy their sense of the fitness of things. The ether was the product of their excogitations. It was a British invention, devised in the Royal Institution, whence have come so many useful theories and discoveries.

The ether, as Salisbury said, is simply the nominative of the verb “to undulate.” It was conceived of as a sort of transparent jelly filling all space, more rigid than any solid, more frictionless than any fluid, more easily penetrated than any gas. It must be more elastic than steel and yet so rarefied that ordinary matter passes through it without the slightest effort. The ether is supposed to slip between the particles of the rushing earth as the wind blows through the branches of a tree.

For many years after its invention the ether had nothing to do except to carry light about from one place to another. But when the electro-magnetic waves of the wireless telegraph were produced something was needed also to carry them and this new task was laid upon the shoulders of the uncomplaining ether. When Röntgen discovered the X-rays, whose waves are 10,000 times shorter than the shortest light waves, these were turned over to the ether to run. In fact, it got so that whenever a physicist found any action that he could not explain by ordinary matter he said: “Let the ether do it,” and that hypothetical substance apparently answered every purpose until it came to this question of relative motion.

Now whatever we may think about the ether it would seem that if there is any such thing filling all “empty” space we might use it for measuring the motion of the earth through it as we did the air current in the car. If the earth is really revolving around the sun the ether must be whizzing through its pores at the rate of about nineteen miles a second.

But wait--there is the possibility that the earth carries along with it in its flight through space a sort of atmosphere of ether as it does of air. We must first get rid of this possibility by a preliminary experiment to see if a swiftly moving mass of matter does catch up and carry along with it a little of the ether. This would cause a sort of an eddy or disturbance in the ether in the neighborhood of the moving mass as a boat disturbs the water. For instance, a ray of light passing close to a rapidly revolving wheel would be a little deflected and show a distorted image. Sir Oliver Lodge tried this experiment and got negative results. That is, moving matter does not disturb or carry with it the ether. Consequently, it would seem, we are left to the only other logical alternative, that the ether drifts through matter and we should expect to detect this drift by measuring the speed of light in the direction of the earth’s motion. It ought to take longer for light to travel from one point to another if the earth meantime is moving away from the first point and it ought to take less time if the earth is moving toward it. Well, Michelson and Morley tried this experiment--and also got negative results! It did not make any difference whether the ray of light was sent in the direction of the earth’s movement or the reverse or across the line, it traveled invariably at the same speed, 186,000 miles a second. Here then were two unquestionable experiments apparently contradicting each other. One proved that the ether did not travel with the earth. The other proved that the ether did not stand still while the earth traveled through it.

Now when we get contradictory answers to the questions we put to Nature we must assume--unless Nature is nonsensical--that we are asking nonsensical questions. If in the trial of a pickpocket one witness swears that the thief did not run up the street and another witness that he did not run down the street the lawyer does not necessarily say that one of them must be a liar. He meditates a moment and then it occurs to him that possibly the pickpocket did not move or that perhaps he disappeared into the third dimension by climbing up a fire-escape or dropping into a coalhole.

So with our ether quandary. If the ether does not move and does not stand still perhaps there isn’t any ether or perhaps there is a fourth dimension. These are two conceivable ways out of the dilemma though they are not easy to accept, either of them. If there is no ether what carries the light waves? If there is a fourth dimension in what direction does it lie? But it is no harder to believe in or conceive of a fourth dimension than it is the ether, and if the physicist finds that he needs it in his business he will have to have it. Einstein says that he needs a fourth dimension for his formulas.

THE CONUNDRUM OF THE AGES

For twenty-four hundred years philosophic thought has been concerned with the problem of the relation of space and time. Drop into any of the scientific societies of today and you will find them discussing whether space is finite or infinite, whether there is any difference between rest and motion, whether length is absolute or relative, whether time and space have real existence, which are the very questions discussed by Pythagoras and Zeno in the Greek cities of Asia Minor. Now the time spent in these speculations has not been wasted, although it has led to no definite conclusion, for out of it have grown our mathematics and physics. The Wandering Jew, who is the only mortal having the privilege of attending the schools of the Eleatics and those of the present day, would observe one difference, that modern scientists try to put their theories to the test of experiment wherever possible, while the ancients were content with thinking them out.

Of all the guesses that have been given to this riddle of the universe none has been more bold and revolutionary than that contained in a paper of four or five pages contributed in 1905 to the _Annalen der Physik_ by Albert Einstein. The controversy it precipitated has not altogether been confined to the realm of pure reason, for scientists are but human and as such are not entirely uninfluenced by patriotic prejudice.

In this brief paper he proposed a new theory of the universe based upon two postulates. The first was the principle of relativity; that all _motion is relative_. This means, for instance, that we would never know the motion of a smoothly moving train if the windows were darkened and that we could never discover the forward movement of the earth if we could not see the heavenly bodies.

Einstein’s second postulate was that _the velocity of light is independent of the motion of the source_. This is a hard one for our reason to swallow, for it means that nothing can travel faster than light, 186,000 miles a second, and that you cannot make light travel faster than that by giving it a swift send-off. It is the same as saying that if a man standing on the cowcatcher of an engine threw a ball forward, it would not make any difference with the velocity of the ball whether the train was running at full speed forward or backward or standing still. But the experiments of the American physicists, Michelson and Morley, who measured the speed of light and found it the same whether the earth was moving toward the source of the ray or away from it, or at right angles to its direction, confirm Einstein’s second assumption.

If we accept Einstein’s two primary postulates and his later “Principle of Equivalence” his theory clears up this ether-drift difficulty as well as various other riddles of the universe. It explains the shifting of the orbit of Mercury that Newton’s theory could never account for. It foretold the deflection of light by the sun’s gravitation that the observations on the eclipse of last May confirmed. A third test, the shifting of the lines of the solar spectrum toward the red end in a gravitational field, has not been met. Such technical points concern only physicists and astronomers, but Einstein’s relativity theory, which two out of the three experiments support, carries with it certain speculations as to time and space that are upsetting to current conceptions.

PARADOXES OF RELATIVITY

All three of Newton’s laws of motion are now questioned and the world is called upon to unlearn the lesson which Euclid taught it that parallel lines never meet. According to Einstein they may meet. According to Newton the action of gravitation is instantaneous throughout all space. According to Einstein no action can exceed the velocity of light. If the theory of relativity is right there can be no such thing as absolute time or way of finding whether clocks in different places are synchronous. Our yardsticks may vary according to how we hold them and the weight of a body may depend upon its velocity. The shortest distance between two points may not be a straight line. These are a few of the startling implications of Einstein’s theory of relativity. If he had put it forward as a mere metaphysical fancy, as a possible but unverifiable hypothesis, it would have aroused mere idle curiosity. But he deduced from it mathematical laws governing physical phenomena which could be put to the test of experiment. They have been tested in these two crucial cases and prove to be true.

In the preceding pages we have discussed the question of the relativity of motion and seen how impossible it is to tell, for instance, whether a train or a ship you are on is moving or not unless you can compare it with something that you are “sure” is stationary. But what are you sure is stationary? Nothing on earth surely, for the earth compared with the “fixed” stars is spinning around at the rate of about a thousand miles an hour and rushing around the sun at the rate of nearly 70,000 miles an hour. But are we sure the stars are fixed since we have nothing else to compare them with? You may remember Herbert Spencer’s illustration of the sea captain who was walking west on the deck of a ship sailing east at the same rate. Is he moving or not? If you are in the same boat, you say he is. If you are on shore when the ship is passing you say he is standing still and “marking time.” It all depends on the point of view.

Now you may readily admit that all motion is relative, not absolute, and yet you may balk at the idea that space and time are also relative, not absolute. But motion is merely simultaneous change of position in space and time, and why should we feel so certain about space and time when we have never seen either?

You may say, for instance, that you are sure your desk is _so_ long. But if I ask you _how_ long you have to say as long as something else. You may say it is a yard long. But how long is a yard? It is as long as some tape or stick marked “one yard,” and this in turn has been taken from some other yardstick, until you get back to the brass rod in London that is just as long as the distance from the tip of the nose of King Henry I to the end of his royal thumb. But such a standard of absolute measurement is unsatisfactory to everyone except an absolute monarchist. But apart from the difficulty of the present inaccessibility of King Henry’s nose and thumb, can we be confident that our yardstick keeps the same length while we are measuring with it? We must admit indeed that it is longer on a summer day than on a winter day, but can we be sure that it does not alter in length when we hold it upright or lay it horizontally? Or, rather, could we tell if it did change in length as it is changed in direction?

ARE YOU SURE OF YOUR SHAPE?

If you have ever been in any of those funny places at the amusement parks you will have noticed the convex mirrors there and how ridiculous they make other people look. If you cannot afford the nickel necessary for the study of optics in such an establishment you can contemplate your reflection in the side of a shiny tin cup or can. In a plane mirror you see a man who looks as you suppose yourself to be except that somehow you seem to have become left-handed. But when you look into a convex cylindrical mirror set upright you see a man thinner than you “really are.” Look into the same mirror set horizontal and you see a man shorter than you “really are.” You grin at the sight of such queer-looking creatures, but you notice that they are equally amused at your shape. Now how are you going to prove to the men in the curved glasses that they are mere caricatures and that you are not really built on the plan of either of these images? You naturally resort to measurement, as a scientist should. You cannot get into the mirror world to measure the tall man who pretends to represent you, but you can explain to him in the sign language what you want him to do and he instantly complies. You stand up a measuring rod at your side and show him that you are exactly 72 inches tall. He also sets up a rod and that also reads 72 inches. Never mind, let him use any kind of measure he likes, you will catch him when it comes to measurement of width with the same stick. You hold your rule across your shoulders and it reads 18 inches, that is, one-fourth your height. But he also measures his width with his rule and makes it just the same, 18 inches, although as you see him he looks at least six times as high as he is broad.

When the man in the middle looks at himself in a curved mirror he sees what he regards as a distorted image. The image on the right is thinner and seems taller because it is reflected from a cylindrical surface set upright. The image on the left is shorter and seems broader because it is reflected from a cylindrical surface set horizontally. But if the man and his image are measured by scales in the real world and the mirror world they come out the same. So, too, it would be impossible for us to find out if everything in the world were expanded or contracted in all directions. In other words, all measurements are relative. According to Einstein any body in movement is shortened in the direction of the line of motion while the transverse dimension remains the same. If, then, a man is being carried headlong through space with a velocity approaching the speed of light he would be shortened like the man on the left. If he were moving sideways he would be like the man on the right.

The man’s image in a plane mirror seems to him symmetrical but reversed. His right hand has somehow got over on his left side and vice versa. Such a transformation as the mirror seems to effect cannot be actually accomplished in ordinary space, but would conceivably be possible in a space of four dimensions.]

Now you are sure he is cheating--must have some sort of telescoping rod that contracts and expands according to the way he holds it. You point out to him that his measure is unreliable, but to your surprise his gestures seem intended to convince you that you instead are using the elastic rule. You shake your fist in his face--to which he responds with equal indignation--and then you turn to the squatty chap in the other mirror, hoping he will be amenable to reason. But he also measures himself as 72 inches high and 18 inches wide by his own rule. If you try the still queerer-looking fellow in the concavo-convex mirror who is distorted in all sorts of ways you will find that his rule lengthens and shortens and bends just enough to make him as symmetrical a man as yourself. And how can he be otherwise since he is the image of yourself?

You are therefore driven to doubt the invariableness of your own yardsticks. Suppose when you wake up tomorrow everything, including all means of measuring, is twice as big as it is today. Could you tell the difference? Would it make any difference? Would there be any difference? Is there any such thing as absolute distance? Are not all measurements relative?

Such questions had from the earliest times occupied the attention of speculative philosophers, but they passed from the realm of metaphysics to the realm of physics in 1886 when Michelson and Morley made their famous experiment on the speed of light in various directions. Their object was to find out if the ether, the hypothetical medium carrying the light waves, was stationary and drifted back through the earth as the earth moved onward. They devised an instrument of such delicacy that the stamp of a foot a hundred yards off would be noticeable. A ray of light was divided into two parts; one half was sent forward and back in the direction toward which that part of the earth where the experiment was made was moving at the time; the other half was sent back and forth across the line of this motion. But the two rays of light following different routes came back at the same instant and matched up exactly. In order to correct for any inequality in the instrument, Michelson and Morley turned it around so the arm that formerly pointed across the line of motion now pointed in the direction of that motion and the other arm pointed across, but that made no difference. The light traveled with the same velocity regardless of the motion of the earth.

This negative result was just as astonishing as if you should stand at a certain spot on the bank of a river half a mile wide and should send out two boats, one to go up the river half a mile against the current and then back with the current and the other boat to go across the river and back. If both boats should return at the same moment you would be puzzled to account for it. One way of accounting for it would be that your measurement of the half-mile course upstream had been a little short. This was the explanation of the Michelson-Morley experiment given by the Dutch physicist, Lorentz. He suggested that the arm of the instrument shortened a trifle as it was turned from across the line of the earth’s motion to the direction of that motion. The amount of shrinkage necessary to compensate for the ether drift would be exceedingly small. Besides how could you measure the change in the length of the arm if the rule you laid alongside of it altered in the same proportion? Lorentz’s explanation could not be disproved, yet it was so upsetting to our ordinary ideas of the stability of matter that it was hard to accept.

Einstein took Lorentz’s idea and made it one of the fundamental principles of his new theory of the universe and then deduced from this theory sundry very startling conclusions, some of which could be--and have been--confirmed by experiment. According to Einstein the size and shape of any body depends upon the rate and direction of its movement. For ordinary speeds the alteration is very slight, but it becomes considerable at rates approaching the speed of light, 186,000 miles a second. If, for instance, you could shoot an arrow from a bow with a velocity of 160,000 miles a second, it would shrink to about half its length, as measured by a man remaining still on earth. A man traveling along with the arrow could discover no change. No force could bring the arrow or even the smallest particle of matter to a motion greater than the speed of light, and the nearer it comes to this limit the greater the force required to move it faster. This means that the mass of a body, instead of being absolute and unalterable as we have supposed, increases with the speed of its movement. Newton’s laws of dynamics are therefore valid only for matter in motion at such moderate speeds as we have to deal with in our experiments on earth and in our observations of the heavenly bodies. When we come to consider velocities approximating that of light the ordinary laws of physics are subject to an increasing correction.

If a person calculates that he is attaining a speed faster than light he will seem to another observer to be moving the other way. That is, any motion above the speed of light is negative motion. Just as a tourist traveling more than 12,000 miles away from home in any direction will really be getting nearer home the farther he goes.

Such speculations would not have bothered anybody twenty years ago, for then the physicist did not have to handle any cases of such high speeds. But when radium was discovered it was found that this metal was continuously throwing off particles of negative electricity with approximately the speed of light. Now if these electrons are not matter they are at any rate the material of which matter is made. They can be detected and counted and tracked and deflected and speeded and weighed. They are very real things, perhaps the ultimate reality of all things, yet their extreme velocity carries them out of Newton’s world and into Einstein’s.

INTRODUCING THE FOURTH DIMENSION

Now Einstein’s world, as I said before, differs from the world in which we are accustomed to live in many particulars. It has four dimensions instead of three. One of these dimensions may be time. Time, too, must be relative, not absolute. This is even harder to imagine than the relativity of space.

As some schoolboy said: “If there were no matter in the universe the law of gravitation would fall to the ground.” Quite so.

WHAT IS MEANT BY DIMENSIONS

No dimensions:

A mathematical point. Has position but no size. Represented by a dot. Like this .

One dimension:

Has length but no breadth. Made by moving a point along straight in any direction. Represented by a line. Like this ----

Two dimensions:

A plane surface like this page.

Has length and breadth but no thickness.

Made by moving a line in a direction perpendicular to its length (that is, into the second dimension).

Represented by two straight lines of indefinite length perpendicular to each other.

The lines are called axes and are labeled _x_ and _y_.

The point where they meet, the origin, is marked O.

Like this

Three dimensions:

A solid like a cube.

Has length, breadth and thickness.

Made by moving a plane in a direction perpendicular to the other two (that is, into the third dimension).

Cannot be pictured on paper, but is indicated by three axes, _x_, _y_, and _z_, of which _x_ and _y_ are on the plane of the page and _z_ is supposed to be stuck up at right-angles to the other two. Stick a pin into the paper at the point O and you will have the third or _z_ axis.

Like this

Four dimensions:

Has length, breadth, thickness and extension into a fourth dimension, say time.

Made by moving a cube in a direction perpendicular to the other three (that is, into the fourth dimension).

Cannot be pictured on paper, but may be indicated by four axes, _x_, _y_, _z_ and _t_ (or _u_), each at right-angles to the other three.

Like this

More dimensions:

Any desired number of dimensions can be worked out mathematically but with increasing difficulty because of the impracticability of diagrammatical representation. We can generalize the idea by speaking of a “geometry of _n_ dimensions” where _n_ may stand for any number whatever from zero to infinity.

A line of a given length contains an infinite number of points.

A square of a given size contains an infinite number of lines.

A cube of a given size contains an infinite number of plane squares.

A tesseract (four-dimensional cuboid) of a given size contains an infinite number of solid cubes.

And what would there be left of space if you took everything out of it, and what would become of time if nothing ever happened? In other words are not space and time merely forms of thought, the framework of ideas, and if so cannot we fix them over to suit our need of new conceptions? As a matter of fact we do. We have constructed by the aid of Euclid and his successors a geometry of three dimensions that works perfectly for all ordinary requirements and if we need a fourth dimension to accommodate these new astronomical and physical phenomena we will build on the necessary addition to our conception of space. There was no use having a fourth dimension so long as we had nothing to put in it. For ordinary earth measurements (geometry) such as laying out a town lot we only use two dimensions, length and breadth. We speak of “flat ground” and “water-level” regardless of the fact that all our “straight” lines on the earth’s surface are really curves that come back to us after going 25,000 miles or less. It is only when measuring mile lengths that we have to correct for the curvature of the earth in the third dimension. So if, as seems probable, we shall have to make allowance in astronomical measurements for the curvature of the universe in a fourth dimension it will merely mean a little labor to the astronomers and it will relieve their minds of some of their perplexities. There is nothing more mystical or mysterious or “psychical” about a fourth dimension than about the other three. A dimension is simply a measurable direction and we can use five dimensions or _n_ dimensions if we need to.

It does not matter that we cannot “see” a figure in four dimensions even with our mind’s eye. Actually we cannot see any figure of more or less than two dimensions: we have to take the others on faith. Nobody can see the mathematician’s point because it has no dimensions, no size at all. The schoolboy says: “Let that be the point A,” and we let it be although what he is pointing at with his stick is not a point but a vast irregular splotch of white chalk on the blackboard. So, too, we cannot see a mathematical line because it has only one dimension, length and no breadth. But set four lines at right angles to one another and we get a square. This we can really see if the enclosed surface is of a different color such as a shadow or black print. Set six squares together at right angles and we get a cube. This we cannot see in its entirety at one time. All that we see when we look squarely at a cube is a square. If we look at it from an angle we see what looks like a square with a couple of lozenges on the sides. The retina of the eye is practically a plane surface, so all we can get is a two-dimensional projection of a solid.

The best way to get an idea of the construction of a cubical solid in four dimensions is to draw a diagram yourself and trace out in turn each of the eight cubes that inclose it. I am indebted to K. W. Lamson of Barnard College for the following sketch and directions:

Draw the four coördinate axes _OX_, _OY_, _OZ_, _OU_.

Lay off the unit _a_{1}a_{4}_ on the _X_ axis, _a_{1}a_{2}_ on the _Y_ axis, _a_{1}a_{7}_ on the _Z_ axis and _a_{1}b_{1}_ on the _U_ axis.

Draw the cube _a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}_ on the three axes _XYZ_.

Draw parallel to this the cube _b_{1}b_{2}b_{3}b_{4}b_{5}b_{6}b_{7}b_{8}_ on the _U_ axis.

Draw the cube _a_{1}a_{2}a_{3}a_{4}b_{1}b_{2}b_{3}b_{4}_ on the three axes _XYU_. This is partly drawn already.

Draw parallel to this the cube _a_{7}a_{8}a_{5}a_{6}b_{7}b_{8}b_{5}b_{6}_ on the three axes _XYU_.

This completes the figure.

There are four other cubes in the figure besides those described above:

The cube on the _XZU_ axes _a_{1}a_{4}b_{1}b_{4}a_{7}a_{8}b_{7}b_{8}_ and its opposite _a_{2}a_{3}b_{2}b_{3}a_{5}a_{6}b_{5}b_{6}_.

The cube on the _YZU_ axes _a_{5}a_{2}a_{7}a_{1}b_{5}b_{2}b_{7}b_{1}_ and its opposite _a_{6}a_{3}a_{8}a_{4}b_{6}b_{3}b_{8}b_{4}_.

The figure has: 16 corners, 32 edges, 24 bounding squares, 8 bounding cubes.

The heavy line _a_{1}b_{6}_ might be called the principal diagonal and makes an angle of 60 degrees with each of the four axes. It is foreshortened in the sketch, but its real length is twice that of one edge of the cube. Every line except this is on the outside of the four-dimensional figure.]

A four-dimensional cube-like solid if transparent and looked at with one eye would appear something like this. But it is obviously impossible to depict a four-dimensional figure on a two-dimensional surface like this page.--From “The Fourth Dimension Simply Explained,” Munn and Company, N. Y.]

Since our two eyes present us slightly different pictures of an object we infer from these its size, shape and distance, but this is guesswork.

Still we have a pretty clear idea of a cube although we have never seen it in its solidity. But the attempt to visualize the hypercube, the four-dimensional figure corresponding to the cube, strains our imagination to the breaking point. Some mathematicians endowed with constructive imaginations of high power claim to have got by long hard thinking some sort of a shadowy and fleeting perception of it, but their visions--if they are not imaginary--do not help out us ordinary folks. But if we cannot imagine--that is, image--the hypercube we know all about it, even its name. It is called the “tesseract,” and it is bounded by eight cubes just as the cube is bounded by six squares and the square by four lines. The tesseract has 24 square faces, 32 edges and 16 rightangular corners.

TIME AS THE FOURTH DIMENSION

Although we find it hard to conceive of a fourth dimension in space we have no such difficulty in case the fourth dimension is time. In fact, we use this idea all the while and could not get along without it. To fix the position of any event requires four dimensions. For instance, a man is shot. Where? At the corner of 7th Avenue and 42d Street, New York. This fixes the place by two coördinates crossing at right angles in a plane. But was it above or below this, on the twentieth floor of the Times Building or in the Subway? Knowing this fixes the third dimension, but we have still to fix its position in a fourth dimension, time. Was it today or last week and what hour? If then we find out all four we can distinguish this shooting from any that may have occurred in other places at the same time or at other times in the same place.

Or consider this simple illustration: Cut a strip of motion picture film into its separate scenes and pile them up in order till it is as high as it is broad. You have then a cubical event. Two dimensions of the cube are spatial; the third dimension is essentially temporal, although in a spatial form. If one of the films from the middle of the pack represents the present then the films below represent the past and those above the future. The people on the picture you picked out know only of the scene there depicted though they may have a fading memory of the past and a dim anticipation of the future. But to you who are outside of the film pack all the scenes are equally visible. They are all present to you. This is the way most Christians have conceived of God, as one to whom past and future form one eternal present, so he sees simultaneously all things that have been, are or will be.

If our pile of film were made up of snapshots taken one a day throughout a man’s life we should see at one glance his growth from babyhood to boyhood, to maturity and old age. We could turn the leaves of his life backward or forward as we will. Some day perhaps we shall have stereo-movies, scenes in three dimensions with time as the fourth.

This idea of time as a fourth dimension is not a new one. In 1754 d’Alembert, defining “dimension” in the Encyclopedia, wrote: “A brilliant man of my acquaintance believes that one may regard duration as a fourth dimension.” In 1903 Minkowski worked out the idea in mathematical form. H. G. Wells, always quick to catch up a new scientific theory to use as a plot for a story, wrote in 1895 of “The Time Machine,” a vehicle by which a man could travel back and forth in time as he can travel east and west in a motor car. In this he visits the future and finds mankind split into two species, a subterranean working class living on--literally--a pleasure-loving leisure class.

In “The Plattner Case” Wells tells of a chemical professor who was by an explosion knocked into--not the middle of next week as we commonly say--but into the fourth dimension of space. Ten days later he was knocked back again into our world but the only evidence of the truth of his story was that his heart beat on the right side and he was left handed and otherwise reversed in a way that would be impossible in a space of three dimensions.

We can turn a glove inside out in three dimensions and so make it just like its mate of the other hand, but we cannot turn a solid inside out except in four-dimensional space.

In another of his “Thirty Strange Stories” Wells tells “The Story of Davidson’s Eyes.” While Davidson was working in his London laboratory a lightning shock so affected his eyesight that he could not see the familiar objects about him which he could feel but looked instead at a South Sea island on the opposite side of the globe. This might be possible in a curved space of four dimensions although Wells professes to pooh-pooh such an absurd suggestion while he ingeniously insinuates it. George Macdonald in his fantastic romance “Lilith” also introduces the fourth dimension.

Points that are far apart if measured in three dimensions may be close together in the fourth. We can readily understand this if time is the fourth dimension, for events can happen at the same instant though thousands of miles apart. But it is not impossible to conceive of the fourth dimension as spatial instead of temporal if we approach the problem from a simpler standpoint. Let us think of ourselves as living in a “Flatland” of two dimensions with no thought of a third. There yet survive in enlightened America individuals who believe that “the sun do move” and who deny that the earth is “round like a ball.” That is, they do not recognize the curvature of the earth in the third dimension. But if such an individual were to travel in a “straight” line westward over the “level” land and water he would, much to his surprise, come back to his starting point which he had left 25,000 miles behind him.

A WORM’S-EYE VIEW OF THE WORLD

Suppose yourself a worm--the Bible says you are anyway--and crawling around on a sheet of paper. With your vermicular mind you doubtless would take a superficial view of the universe and find it as impossible to imagine a third dimension as man does a fourth. If in the course of your crawling you came across a triangle you might--if you were a measuring worm--pace it off and find that the distance from _A_ to _B_ was 8 inches, from _B_ to _C_ was 6 inches and from this data, if you knew the law of the hypothenuse, you might calculate that the distance from _A_ to _C_ was 10 inches. On measuring it you would find your prediction verified and so gain perfect confidence in your plane geometry. But unbeknownst to you, poor worm with your eyes fixed on the paper, some man may have picked up the sheet and crumpled it up or rolled it over so that _A_ and _C_ are only one inch apart--in the third dimension. The worm is right when he thinks the distance between these points is 10 inches: so is the man right when he says it is one inch. It depends on the point of view.

Now in Einstein’s view something of this sort happens to our three-dimensional space when matter gets into it. We know for instance that if you divide the circumference of any circle by the diameter the ratio figures out as 3.1415+. It has been calculated to 707 decimal places but we can dispense with the rest of them and call the whole thing Pi for short. Write it in Greek as π and it looks more learned. Now if you place a heavy particle, say a lead bullet, in the center of a circle the ratio of the diameter to the circumference, according to Einstein, becomes a little less than Pi, for the circle has been warped, so to speak, into the fourth dimension by the strain of gravitation. The difference in such a case is too small to be measurable by any known means, but it is supposed to be an actual, not an imaginary, deviation from the geometrical law.